OMAE HOW TO CARRY OUT METOCEAN STUDIES

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1 Proceedgs of the ASME 20 30th Iteratoal Coferece o Ocea, Offshore ad Arctc Egeerg OMAE20 Jue 9-24, 20, Rotterdam, The Netherlads OMAE HOW TO CARRY OUT METOCEAN STUDIES Judth va Os Hydraulc Egeerg Deltares Delft Hydraulcs P.O. Box 77, 2600 MH Delft, The Netherlads E-mal: Judth.vaOs@deltares.l Sofa Cares Hydraulc Egeerg Deltares Delft Hydraulcs P.O. Box 77, 2600 MH Delft, The Netherlads E-mal: Sofa.Cares@deltares.l ABSTRACT Metocea studes volve qute a lot of statstcal aalyses. The detal ad extet of the requred metocea codtos (waves, water level, currets ad wd) s study depedet. Nevertheless, most of the studes volve data valdato, determato of ormal ad extreme metocea codtos ad offshore to earshore trasformato of data. All of the above aspects of a metocea study ca be performed umerous dfferet ways, whch wll a way deped o the qualty ad amout of the avalable data. Data valdato, for example, ca be doe vsually ad umercally ad there are several methods to perform a extreme value aalyss. It s therefore eeded to have best practce gudeles to execute metocea studes a effcet ad stadardzed way so that accurate ad verfable results ca be obtaed. Deltares carres out a lot of metocea studes for the dustry ad s presetly workg o a method to stadardze the executo of metocea studes, by developg gudeles alog wth a MATLAB tool, called ORCA, whch tegrates the ma aspects of aalyzg metocea data. INTRODUCTION The Deltares gudeles am at stadardzg the executo of metocea studes, ad focus o the followg fve aspects: () Data valdato: Procedures ecessary for the determato of a "good dataset". (2) Normal codtos: Procedures ecessary for the determato of mea clmates. (3) Extreme codtos: Procedures for extreme value aalyss. (4) Sea State aalyss: Procedures for sea state aalyss. (5) Persstece statstcs: Procedures for the computato of persstece statstcs (ot addressed here). I ths paper, examples of challeges ecoutered the data aalyses ad our gudeles o how to approach them are gve ad llustrated by meas of worked out examples. Note that these gudeles are geerc ad focusg o statstcal aalyses. It s always ecessary whe carryg out such aalyses that the peculartes of the data questo are kow ad that the metocea processes mpactg the ste questo are uderstood. 2 DATA VALIDATION Each metocea study starts wth datasets of measured or modeled quattes. These datasets are the bass of the study. It s therefore very mportat to kow what s the qualty of the datasets. The most mportat propertes of a good data set are: () The dataset s homogeous. (2) The values the dataset are accurate. (3) The dataset covers a perod that s log eough to serve as a bass for the aalyss. (4) Each perod (moth, seaso) the data set s represeted equally well. (5) It cotas o urelable or faulty data such as urealstcally steep waves or urealstcally hgh wd speeds. The homogeety of a dataset should always be checked, for stace by checkg wth the data delverer f chages of some kd were mplemeted durg the measurg or modelg perod. The homogeety of a dataset ca be mproved by removg o-physcal treds or by restrctg the perod of used data to the perod whch the data are homogeeous. Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

2 The frst thg to do whe a ew dataset becomes avalable, s to perform a few basc qualty checks ad make adjustmets where ecessary, such as removal of treds, jumps ad outlers. Afterwards, the qualty checked dataset ca be compared to other datasets, error statstcs ca be determed ad the data ca be calbrated. I the ext paragraphs, three dfferet areas of data checks are descrbed: data qualty checks, data comparso ad error statstcs. 2. DATA QUALITY CHECKS The deal stuato would be that the qualty of the datasets avalable for a metocea study s suffcet, or that the qualty ca be mproved by calbratg wth aother data source of good qualty. Ths s ot always possble. However, eve wthout calbrato, a few basc checks ad adjustmets ca be performed to mprove the qualty of a dataset. Such checks ad adjustmets are descrbed ths secto. Checkg data coverage Ths check ams at determg the coverage of the orgal tmeseres. For every parameter, the umber of observatos per moth should be determed. For further aalyses, whch seasoal, yearly or mothly statstcs are to be computed, t s mportat to have a dataset whch has a equal coverage per perod: for example, f the moth of August has a average coverage of 95%, ad Jauary oly 77%, the aual clmate derved from the dataset much more weght wll be gve to the August data tha to the Jauary data. The moths that have the best coverage are most ft to be used the determato of mea statstcs. ORCA ca produce a table wth the data coverage per moth. Usg ths formato t ca be determed how may total years ca be created wth the data. Note that the coverage ca vary from parameter to parameter: wave heght data ca have a very hgh coverage whle the correspodg wave drecto may have a low coverage. removal of repeated records fllg of gaps wth dummy values. Oce a tmeseres s avalable wth equdstace tme data ad clearly detfed gaps, t s possble to remove outlers usg stadard test. Idetfyg ad removg outlers Idepedet of the varable uder cosderato, the exstece of outlers should be verfed by checkg: for devatos from the mea over a fxed perod, N x N x xcrt stdx ( ), where x, N std( x) x x 2 N, N s the umber of records ( the perod over whch x ad std( x ) are determed) ad crt the umber of stadard devatos a value ca be allowed to devate from the mea; for devatos magtude from oe tme step to the ext, x x crt2 std( x), where crt 2 s the umber of stadard devatos a value ca be allowed to devate from the prevous. Fgure shows a example of a dataset from whch outlers have bee removed. The outlers detfed usg devato from the mea are show red ad those detfed usg devatos magtude from oe tme step to the ext are show blue. The black le s the resultg dataset wthout outlers. Hs Oce a mmum requred coverage for a moth s defed, a dataset ca be assembled coverg a umber of years of data wth smlar data coverage per moth. Note that the years from whch the data s cosdered may dffer per moth. For stace whe complg a 4 year dataset wth at least 90% coverage per moth s ca be that data from Jauary 2000, 200, 2003 ad 2005 are cosdered, whereas data from February 200, 2002, 2004 ad 2005 are cosdered. Note that, whe usg the data to carry out extreme value aalyss, t may be ecessary to cosder also data from perods wth poor coverage, order ot to exclude storms from the aalyss. Furthermore, t should also be checked how a measurg devce has bee fuctog durg the measurg perod. If t has faled to regster for example the hghest peaks or the lowest wave heghts, the remag data wll be based. Creatg tmeseres wth equdstat tme step A umber of procedures may eed to be carred out to create equdstat tme step tmeseres, whch ca the be further qualty checked. The followg procedures should always be carred out: Sortg of the data tme Fgure - Red: orgal dataset, black: dataset wthout outlers The values of crt, crt 2, ad the perods over whch the mea ad stadard devatos are determed, cotrol the detfcato of outlers ad oe has to choose these crtera wsely, takg to accout the behavor of the varable uder cosderato. Sgfcat wave heghts (H s ), for example, ted to vary less quckly tha wd speeds (U 0 ). Also, the more tme there s betwee cosecutve records a dataset, the hgher the devatos magtude from oe tme step to the ext ca be. tme 2 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

3 The crtera have thus to be adjusted depedg o the type of data beg aalyzed ad the tme resoluto of the data. Note that the result of automatcally removg outlers should be checked carefully to avod the excluso of real evets. Other checks ad adjustmets Besdes the above procedures, ORCA also cludes the followg possbltes to mprove a dataset: checkg the data vsually graphs; detfcato of treds; chagg the tme terval of the data (to a tme terval larger tha the orgal oe); adjustg the wd speed to the 0m level, assumg a logarthmc profle (eutral atmospherc stablty), see []. Whe all the ecessary checks ad adjustmets have bee performed, the last step the data qualty assurace s to create the cosstet ad valdated, or good dataset. Also, for llustrato, some plots to vsualze the data ca be produced, see for example the wd roses Fgure 2. I the ORCA geerated scatter ad quatle plots, ftted treds les ca be preseted, cludg the parameters defg them. These ca be used to derve the correcto to be appled to the dataset that eeds calbrato. Note that there s a dfferece calbratg data based upo a ft to the whole data (as gve a desty scatter plot), or based upo a ft to the quatles (as gve the quatle plot). I the former case, the rage of values whch most data are preset, gets the most weght the ft. Sce most of the data usually fall the lower values, the foud relatoshps are ofte based to the errors the lower rages of the data. Whe estmatg relatos betwee percetles o extra weght s gve to the lower rages of the data ad the estmated relatos geerally better calbrate the hgher data values (for whch errors are geerally larger). Etre dataset Selected dataset Model data Fgure 3 - Example of a scatter plot cluded ftted tred les Fgure 2 - Example of wd roses. The bar legths dcate the occurrece percetages. Drectos are to the cetre of the roses. The umbers the cetre of the roses are the percetage of occurreces the lowest class. 2.2 DATA COMPARISON Calbrato ca be useful for stace cases where there s a lot of umercal model formato, ad oly few measuremets. I these cases, the avalable measuremets are ot suffcet to serve as a bass for data aalyss, but ca be used to calbrate the dataset produced by the umercal model. Useful ways of comparg two datasets are desty scatter plots ad quatle plots (see Fgures 3 ad 4). Desty scatter plots (Fgure 3) are advsed stead of stadard data plots sce the former provde formato o the amout of data each rego. If ecessary, pror to ths the data of the set to be calbrated ca be terpolated to the locato of the other data set. Model data Fgure 4 - Example of quatle plot cludg ftted tred les 3 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

4 2.3 ERROR ANALYSIS A partcularty of certa evrometal data (e.g. wave data) s that they ca be classfed to lear data (e.g. mea wave perod ad Hs) ad crcular data (e.g. MWD, ad drectoal spread), ad ths dstcto has to be take to cosderato whe carryg out error aalyss. Ideed, the statstcal techques for dealg wth these two types of data are dfferet: crcular (or drectoal) data requre a specal approach. Basc cocepts of statstcal aalyss of crcular data are gve [2] ad [3]. Dffereces betwee lear varables are ofte quatfed usg the followg stadard statstcs: the bas: y x ; 2 the root-mea-square error: RMSE ( yx) ; ; the scatter dex: 2 the correlato coeffcet: SI y y x x x 2 2 the symmetrc slope: r x y. x x y y x x y y 2 2 I all these formulae the x 's usually represet observatos (or the dataset whch s cosdered less ucerta or basele), the y 's represet the model results (or the dataset whch s cosdered more ucerta or wth a certa devato from the basele results) ad the umber of observatos. Note that the scatter dex advsed above s ot the ormalzed rootmea-square error; t cotas a bas removal term. Whe dealg wth crcular data, each observato s cosdered as ut vector, ad t s vector addto rather tha ordary (or scalar) addto that s used to compute the average of agles, the so-called mea drecto. Wrtg for a sample x { x,,..., }, C cos x ad sample mea drecto s S s x, the x TAN ( S C ), where ( C TAN ( S C) s the verse of the taget of S ) the rage [0, 2 ],.e., ta ( ), S C TAN ( S C ): ta ( S C ), S C ta ( ) 2, S 0, C 0 C 0 S 0, C 0. Eq. () ca be used to compute the bas betwee two crcular varables by substtutg x by y x. I a smlar way, the root-mea-square error ad scatter-dex betwee two crcular varables ca be computed. There are several crcular aalogues of the correlato coeffcet, but the most wdely used s the oe proposed [4], the so-called T-lear correlato coeffcet. Gve two sets ; () x { x,,..., }, y { y,,..., } of crcular data, the T- lear correlato coeffcet betwee x ad y s defed by s( x xj)s( y yj) j T, 2 2 s ( x x ) s ( y y ) j j j j ad satsfes T. I spte of the aaloges wth the lear case, t makes o sese to cosder a symmetrc slope for crcular data other tha oe. 3 NORMAL CONDITIONS 3. JOINT OCCURENCES AND SCENARIOS The determato of mea clmates play a mportat role evrometal mpact studes, shorele studes, etc. We descrbe here a umber of ways to determe mea clmates, o the bass of tmeseres of metocea varables, ad preset them. The two most commo ways of presetg a mea clmate metocea studes are bvarate: by mea of roses, whe both a magtude ad a drecto eed to be cosdered at oe or more locatos (see also Fgure 2); by meas of jot occurrece tables, especally whe the bvarate ad margal exact umber of occurreces eeds to be kow. Aother possblty that ORCA offers ad whch s advsable whe watg to sythesze spatal ad multvarate data s to represet a mea clmate as a (multvarate) umber of mea codtos, each represetg oe (mult-dmesoal) class. Such a set of mea codtos s called a scearo. I the b-dmesoal ad bvarate case a scearo could be the mea values of the Hs ad MWD data fallg the bdmesoal class 0.5m<H s <.0m ad 65 N<MWD<95 N, whch could be H s =0.9m ad MWD=83 N. Scearos are, however, geerally defed usg several varables at a umber of locatos. A example of a scearo s gve Fgure 5. Fgure 5 cotas for a gve scearo the followg formato: Mea values of the wave parameters ( blue), Mea values of the wd parameters ( red), These parameters are show at several locatos for the case where at the referece locato ( gree) the followg holds: the MWD s betwee 37.5 N ad 52.5 N, the H s s betwee ad.5 m ad the wd drecto s betwee 0 N ad 80 N. It should be checked whether a scearo s realstc,.e. whether the chose classfcato was adequate. Ths ca be doe by cotrollg the hstograms of the data used to compute each of the meas, order to check the spreadg of the data. For stace f a hstogram at a certa locato shows that the computed mea wd drecto for the cosdered scearo s 4 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

5 from the North but that all observatos fallg that class are ether from the East or from the West the the computed scearo s ot realstc ad the data should also be classfed terms of wd drecto. The tmeseres ca be trasformed to scearos. Each scearo represets a wave codto a certa class, for whch the occurrece probablty s coupled for a umber of locatos. I other words a scearo s a combato of waves, wd ad water level that could actually occur a certa area. These scearos ca therefore be used drectly as put for a wave propagato computato. Oce the wave propagato model has computed the earshore wave codtos assocated wth each scearo, the results ca be used to determe the earshore clmate. Ths ca be doe two dfferet ways: Fgure 5 - Example of a scearo The ma dfferece betwee defg a wave clmate as scearos or defg a wave clmate as roses or tables s the followg: The bass for a scearo s the classfcato ad the occurrece of the data at a user defed referece locato. The wave ad wd codtos at the other locatos are coupled to the referece locato o the bass of smultaeous occurrece. Therefore, each scearo, cosstg of multvarate mea codtos (mea wave perod, wd speed, etc) at a umber of locatos, has oe occurrece probablty. I a represetato of wave clmates roses or tables at a umber of locatos, each locato has ts ow dstrbuto of occurrece probabltes over the classes. There s o couplg of occurreces. 3.2 TRANSFORMATION OF TIMESERIES I may metocea studes, the trasformato of wave data from a offshore locato to earshore s ecessary order to compute earshore mea wave clmates. For that, a wave propagato model, such as SWAN, whch computes wave codtos o a umercal grd, usg wave boudary codtos ad wd felds, s used. The wave boudary codtos ad wd felds ca be derved from the avalable offshore datasets. The trasformato of wave data from a offshore locato to earshore ca be doe a few ways: I prcple, each record from log-term tmeseres (cotag data o wave, wd ad water level) could be used as put for a wave propagato computato to derve the assocated earshore codto. I practce however, ths s ofte ot feasble gve the large amout of computatos that would have to be executed. A mult-dmesoal matrx ca be set up, coverg the etre rage of values for all relevat parameters the tmeseres. Each class the matrx has a certa occurrece probablty. The dffculty here s that the occurrece probablty dffers per locato, whch complcates the dervato of relable boudary codtos.. Gve that each scearo has a assocated durato, the computed earshore codtos ad assocated duratos of all the scearos ca be used to compute earshore wave roses ad jot occurrece tables. 2. The scearos ad assocated earshore codtos ca also be used to determe relatoshps betwee the offshore ad earshore codtos. These relatoshps ca tur be used to trasform offshore log-term tmeseres to earshore. Subsequetly, the earshore tmeseres ca be used to compute earshore wave roses ad jot occurrece tables. Roses ad tables computed usg the secod procedure wll prcple show more spreadg ad are deemed more realstc. The procedure volves three separate steps:. The relato betwee offshore ad earshore parameters s determed for each scearo. For lear varables, the rato s determed, e.g. H s-earshore /H s-offshore. For crcular varables, the dfferece (agular offset) s determed, e.g. MWD earshore MWD offshore, accoutg for the crcularty of the data (cf. Secto 2.3). 2. If the tmeseres cotaed oly records wth data cocdg exactly wth the codtos the scearos, the trasformato would be complete after the frst step. I realty however, ths s ot the case ad therefore some terpolato s eeded. For the terpolato betwee the earshore-offshore relatos determed Step, two addtoal parameters are cosdered. These are the parameters alog whch the bvarate terpolato s performed. They are therefore the parameters o whch the relato betwee the earshore ad the offshore values most lkely depeds. I most of the cases, these parameters are Hs ad MWD because geeral they determe the earshore-offshore relato: prcple, the hgher the offshore wave heght, the smaller the rato betwee earshore ad offshore wave heght (hgher waves break earshore). Furthermore, the drecto from whch waves approach the coastle s a mportat factor for the earshore wave heght (e.g. because of shelterg behd a slad) ad earshore drecto. However, other parameters may be deemed more relevat, lke the wd speed ad drecto. 3. The last step the trasformato s a qute smple oe. Nearshore tmeseres are costructed by multplyg the values the offshore tmeseres wth the values the 5 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

6 trasformato matrces (or addg them to the values, case of drectos). Ths o-parametrc method to trasform data from offshore to earshore ca also be used for other purposes such as predcto, calbrato ad exteso of tmeseres. 4 EXTREME CONDITIONS Oe of the ma applcatos of extreme value theory s the estmato of the oce per m year (m-yr) retur value, the value whch s exceeded o average oce every m years, based o some lmted amout of data, usg extreme value aalyss (EVA). Extreme value theory provdes aalogues of the cetral lmt theorem for the extreme values a sample. Accordg to the cetral lmt theorem, the mea of a large umber of radom varables, rrespectve of the dstrbuto of each varable, s dstrbuted approxmately accordg to a Gaussa dstrbuto. For example, the sea surface elevato s ofte modeled as a sum of several dvdual radom waves ad therefore ts dstrbuto ca ofte be assumed to be Gaussa. Accordg to extreme value theory, the extreme values a large sample also have a approxmate dstrbuto that s depedet of the dstrbuto of each varable. A very good troducto to EVA s gve [5]. I order to expla the basc deas of extreme value theory, let us wrte M max X,, X, where X, X2 s a sequece of depedet ad detcally dstrbuted (..d.) radom varables wth dstrbuto fucto F. I ts smplest form, the extremal types theorem states the followg: If there exst sequeces of costats 0 ad such that P M z Gz ( ) as, where G s a odegeerate dstrbuto fucto, the G must be a Geeralzed Extreme Value (GEV) dstrbuto, whch s gve by z exp, for 0 Gz ( ) z exp exp, for 0, The parameters, ad are called the locato, scale ad shape parameters, ad they satsfy, 0 ad. For 0 the GEV s the Gumbel dstrbuto, s sad to have a type I tal ad z ; for 0 the GEV s the Fréchet dstrbuto, s sad to have type II tal ad z (the doma of z has a lower boud); for 0 the GEV s the (reverse) Webull dstrbuto, sad to have a type III tal ad z (the doma of z has a upper boud). The m-yr retur value (for m>) based o the AM method/gev dstrbuto, z, s gve by m (2) z m - l, for 0 m l l, for 0. m The extremal types theorem gves rse to the aual maxma (AM) method of modelg extremes, whch the GEV dstrbuto s ftted to a sample of block maxma (e.g. to aual maxma, though baual, mothly or eve daly maxma ca of course be used as well). Sce the tmeseres avalable typcally cover oly a few decades, the sample szes of aual maxma data are usually small. Cosequetly, the estmates of the parameters of the GEV dstrbuto, ad hece the estmates of the retur values, have large varaces (ucertates). Ths has motvated the developmet of a more sophstcated method based o exceedaces of a threshold. The approach based o the exceedaces of a hgh threshold, hereafter referred to as the Peaks-over-Threshold (POT) method, cossts of fttg the geeralzed Pareto dstrbuto (GPD) to the peaks of clustered excesses over a threshold, the excesses beg the observatos a cluster mus the threshold, ad calculatg retur values by takg to accout the rate of occurrece of clusters. Uder very geeral codtos ths procedure esures that the data ca have oly three possble, albet asymptotc, dstrbutos (the three forms of the GPD gve below) ad, moreover, that observatos belogg to dfferet peak clusters are (approxmately) depedet. I the POT method, the peak excesses over a hgh threshold u of a tmeseres are assumed to occur tme accordg to a Posso process wth rate u ad to be depedetly dstrbuted wth a GPD, whose dstrbuto fucto s gve by y, for 0 Fu( y) y exp, for 0, where, y>0, 0 ad ( ( y )) 0. The two parameters of the GPD are called the scale ( ) ad shape () parameters. Whe 0 the GPD s sad to have a type I tal ad amouts to the expoetal dstrbuto wth mea ; whe 0 t has a type II tal ad t s the Pareto dstrbuto; ad whe 0 t has a type III tal ad t s a specal case of the beta dstrbuto. If 0, just as wth the GEV dstrbuto, the support of the GPD has a upper boud. The m-yr retur value based o a POT/GPD aalyss, gve by (3) (4) z, s m 6 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

7 z m u {( um) }, for 0 u l( um), for 0. The choce of the threshold the POT method (whch s aalogous to the choce of block sze the block maxma approach) represets a trade-off betwee bas ad varace: too low a threshold s lkely to volate the asymptotc bass of the model, leadg to bas; too hgh a threshold wll geerate fewer excesses wth whch to estmate the model, leadg to hgh varace. A mportat property of the POT/GPD approach s the threshold stablty property: f a GPD s a reasoable model for excesses over a threshold u 0, the for a hgher threshold u a GPD should also apply; the two GPD s have the same shape parameter ad ther scale parameters are related by u u uu, whch ca be reparameterzed as * u 0 0. Cosequetly, f 0 s a vald threshold for excesses to follow the GPD the estmates of both * ad should rema early costat above 0. Ths property of the GPD ca also be used to fd the mmum threshold at whch a GPD model apples to the data. We advse the use of the threshold stablty property to choose the threshold o the bass of whch the sample of peak excesses s selected ad used to estmate the GPD. More precsely: look for threshold values aroud whch the estmate of the shape parameter shows the least varato. We have automatzed such a choce of the threshold ORCA usg the followg procedure for a dataset wth N yrs of data:. POT samples wth at least 0 ad at most 0*N yrs peaks are collected by systematcally decreasg the threshold, ad for each of these samples GPD fts are obtaed. Note that f there s a POT sample wth, say, 20 peaks, the t does ot follow that there s also a POT sample wth 9 peaks, sce dfferet peaks may have the same value ad eve a small crease of the threshold ca elmate more tha oe of the peaks collected at a lower threshold. 2. For each sample sze, a set of parameter estmates based o sample szes ragg from -l to +l peaks, where l s some fxed value (see below), are obtaed, ad the stadard devato (v) ad the lear slope (b) of such a set of estmates s computed. I the case of the shape parameter, for example, ths procedure yelds oe stadard devato for each value of, ad each stadard devato quatfes the varablty of the parameter estmates aroud a wdow of 2l+ sample szes (2l+=(+l)-(-l)+). 3. The threshold, or sample sze, that s the used for makg fereces s the oe yeldg the smallest (v+b) value computed bullet 2. (5) I prevous aalyses, several tests were carred out to determe the best choce of the wdow sze o whch the stadard devatos of the secod bullet are computed. It tured out that l=0 was a good choce: usg about 2 (2l+ for l=0) estmates, the automatcally determed threshold cocdes ofte wth the oe that we would have chose by vsual specto of plots. I most cases the results are sestve to chages l for l betwee 0 ad 5. Wth larger values of l the threshold chose s ofte too low. Metocea data wave, water level, wd speed, etc. are typcally avalable every hour to every 6 hours, depedg o whether they cosst of measuremets or of model results. These data have to be regarded as statstcally depedet, the sese that the value of, say, the Hs at a gve tme s to a certa extet predctable from ts value 6 hours earler. However, order to carry out a extreme value aalyss t s requred that the data used to ft the dstrbuto be approxmately depedet. Cosequetly, the samples used the AM ad POT methods have to be extracted from the orgal tmeseres such a way as to make ths assumpto realstc. I the case of the AM method, ths s acheved by esurg that f the aual maxmum of a gve year belogs to a storm that straddles that year ad the ext, the the aual maxmum of the followg year s ot to be pcked wth that storm. I the case of the POT method t should be doe by a process of declusterg whch oly the peak (hghest) observatos clusters of successve exceedaces of a specfed threshold are retaed ad, of these, oly those whch some sese are suffcetly apart (so that they belog to more or less depedet storms ) are cluded the POT sample. Based o storm durato studes, t has bee cocluded that the hypothess of depedece of the POT data s teable by treatg cluster maxma at a dstace of less tha 2-4 days apart as belogg to the same cluster (storm). There are several umercal methods avalable for the estmato of the parameters of extreme value dstrbutos, such as the method of probablty weghted momets (PWM) ad the maxmum lkelhood (ML) method. Based o smulato study results [6], where t was cocluded that for datasets wth less tha 00 years of data: the POT/GPD estmates are more accurate tha those of the AM/GEV ad the method of PWM should, o the bass of ts error characterstcs ad robustess, be preferred to the ML method. Aother parameter that eeds to be estmated the POT/GPD aalyss s the yearly cluster rate. It ca geerally be estmated by the average umber of clusters/peak excesses per year. However, for yearly seres wth dfferet umbers of observatos (gaps), the estmato of u should accout for the gaps the data. I order to accout for gaps, u s estmated by dvdg the total umber of peaks by the actual umber of years of data (the total perod of vald data). It s ofte the case that gaps the summer moths should ot be accouted for. 7 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

8 Based o what s sad above, we recommed the use of the POT/GPD method for EVA ad that the parameters of the GPD be estmated usg the method of PWM. Nevertheless, whe possble, we also recommed that the POT/GPD ad AM/GEV estmates be compared. Furthermore, whe estmatg dstrbutos t s ot oly mportat to obta (pot) estmates, but also to provde a dea about ther ucertaty. There are several methods for computg cofdece tervals (whch reflect the ucertaty of the estmates) for the parameters (see [7]). We recommed that adjusted bootstrap cofdece tervals be used. Recappg, the ma steps volved a EVA are as follows:. The selecto of a sample of observed extreme values. Gve a tmeseres of some physcal quatty the Peaks over Threshold (POT) ad Aual Maxma (AM) are the advsed data selecto methods. 2. The selecto of a extreme value dstrbuto as a statstcal model for the data produced the frst step. Accordg to theory the geeralzed Pareto dstrbuto (GPD) should be selected whe the sample s collected by meas of the POT method, whle the case of the AM method a geeralzed extreme value (GEV) dstrbuto should be chose. 3. The selecto of a method to estmate the ukow parameters of the dstrbuto chose Step 2. We advse that the PWM method be used. 4. The selecto of a method to quatfy the ucertaty the estmates of the parameters of the dstrbutos ad the assocated retur values. We recommed that adjusted bootstrap cofdece tervals be used. A example s gve usg a 24 year Hs data set, fxg l=0 ad a cluster dstace to 2 days the aalyss. The chose POT sample Fgure 6, the result of the procedure used to choose the threshold s show Fgure 7 ad the retur value plot Fgure 8. Fgure 6 Selected POT data. Blue: all values. Gree: values hgher tha the chose threshold (3.8 m). Red: the maxmum values of depedet evets (more tha 48 hours apart). Fgure 7 Threshold plot wth the threshold suggested by the automatc procedure (3.8 m) dcated by vertcal les. Fgure 8 - Retur value plot. 5 SEA STATE ANALYSIS I metocea studes the kowledge of the dstrbuto of a dvdual wave heght H a sea state s ofte eeded order to estmate hgh quatles (e.g. H 0.% ) or the average umber of waves a partcular class. These dstrbutos are usually descrbed terms of a cumulatve dstrbuto fucto, defed by F( h) P( H h) for h 0, where PE ( ) deotes the probablty of the evet E ad H h s the evet that a wave heght H s smaller tha or equal to the umercal value h. The probablty that a wave heght H falls wth a certa rage ca be computed terms of F as P h H h P H h P H h F( h ) F( h ) for h h2. Quatles of the dstrbuto of wave heghts are also computed terms of F. By defto, the quatle of probablty / N of F s the value of wave heght that s exceeded o average oce every N waves. If we deote ths value by H (or by H, f probabltes are expressed / N 00 % N percetages rather tha proportos), we the have, by defto, that NPH ( H/ N) N[ FH ( / N)], whch s the expected umber of exceedaces of H amog the N waves, equals. I other words, H / N s the soluto to the equato F( H/ N ) N. More geerally, the quatle of probablty p (0 p ) s the soluto H p to the / N 8 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

9 equato F( H p) p ; the terpretato terms of the average umber of exceedaces amog N waves refers to the case where p / N. Ofte oe s terested computg H 0.% (take N,000 above). Also of terest s the value of the wave heght that s expected to be exceeded oce a sea state comprsg about N waves; ths case, oe ca compute N ss as N DT ss m0,2 ss, (6) where D s the sea state wth durato ( secods), ofte assumed to be of 3 hours. Note that whe Tm 0,2 0 s, the N ss s about,000. Idvdual wave heght dstrbutos have bee wdely studed lterature. I [8] the Raylegh dstrbuto s proposed as a good model for the dstrbuto of dvdual wave heghts of waves wth a arrow bad eergy spectrum deep waters (see e.g. [9]). Its dstrbuto fucto s wrtte here as 2 h Fh ( ) exp( ), h 0 (7) Hrms where Hrms 8m0. Sce the work of Loguet-Hggs [8], several studes have show that the Raylegh dstrbuto descrbes the dstrbuto of dvdual wave heght deeps water waves reasoably well ad s therefore wdely used. For shallow waters varous alteratves to the Raylegh dstrbuto have bee proposed whch accout for depth-lmted breakg, thus mposg restrctos o the dstrbuto of hgher dvdual wave heght a shallow water sea state. Assumg the valdty of the Raylegh dstrbuto deep waters, the shallow water alteratves should prcple coverge to the Raylegh dstrbuto deep waters. Oe of the most recet ad more wdely used shallow water dstrbuto s the Composte Webull dstrbuto [0], hereafter also referred to as the Battjes ad Groeedjk dstrbuto. I ORCA a adjusted verso of the Battjes ad Groeedjk dstrbuto has bee mplemeted. The adjustemets, whch are descrbed below, were doe to assure that the Battjes ad Groeedjk dstrbuto coverges to the Raylegh dstrbuto deep waters. I [0], Battjes ad Groeedjk argue that the dstrbuto of wave heghts shallow waters s better descrbed by a combato of two Webull dstrbutos, whch they refer to as a Composte Webull wave heght dstrbuto, k h exp h Htr H Fh ( ) (8) k2 h exp h Htr. H2 The dstrbuto ca be wrtte terms of ormalzed wave heghts, h hh, as rms k h exp h H H Fh ( ) k2 h exp h H H 2 Equato (9) has fve ukows: tr tr (9) H tr, H, H 2, ad 2. H, H 2 ca be computed from the followg two costrats: I order to have cotuty of the dstrbuto fucto at H Htr H2 Htr H tr, 2, ad to assure cosstecy wth the Raylegh dstrbuto, the secod momet of the pdf s made equal to. Furthermore, [0] the expoets ad 2 are fxed as 2 ad o the bass of measuremet fts. Estmates ca thus be obtaed from (9) varyg H tr, as doe [0], Table 2. For H tr 2.75 the estmates of h x coverge to those of the (ormalzed) Raylegh dstrbuto. The trastoal wave heght s a fucto of the foreshore slope ad the water depth: Htr ta d. (0) Cosstetly wth the kow spatal lag the process of breakg compared to depth chages, a steeper bottom leads to a hgher H tr mplyg less depth-depedet breakg ad thus fewer waves devatg the sea state from the Raylegh characterstcs. I [0], Battjes ad Groeedjk express the root-measquare wave heght as a fucto of the water depth ad m 0, H rms m m d. () 0 As d, Hrms m0 2.69, whch [0] followg the aalyss of feld data of [] s argued to be the vald rato for sea waves deep waters wth a broad baded frequecy spectrum. Recall that the Hrms m 0 rato equals 8 whe a arrow frequecy spectrum s assumed. Fgure 9 shows for a gve example that as the depth creases, the Battjes ad Groeedjk estmates approach the Raylegh dstrbuto estmates, actually overshootg them, ad the the Battjes ad Groeedjk estmates decrease, covergg to a value well below the Raylegh lmtg value (sce () wll coverge to 2.69 stead of 8 as fxed (7)). The results Battjes ad Groeedjk (2000) were computed usg a faulty mplemetato of (6) whch the frst argumet of (a,x) was take as a=2/3.5+ stead of a=2/ Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

10 Caveat: The Battjes ad Groeedjk dstrbuto was developed the cotext of a shallow water foreshore,.e., whe waves propagate over a getly slopg shallow rego towards the coast. The model has ot bee developed for regos wth horzotal bottoms, as s ofte the stuato wd parks ad North Sea offshore platform regos. Fgure 9 - Comparsos betwee H0.% Hs estmates of the Raylegh, Battjes ad Groeedjk ad adjusted Battjes ad Groeedjk dstrbutos. H s =4m, slope =:250. Fgure 0 shows H 0.% ad H rms as computed usg the Battjes ad Groeedjk approach versus the water depth. Cotrary to what seems to be the case for H 0.%, the estmates of H 0.% behave rather well. They crease wth the water depth up to a depth from whch they rema costat. Thus, for suffcetly deep waters, h s always below H tr ad the ormalzed dstrbuto fucto (9) equals the ormalzed Raylegh dstrbuto. O the other had, H rms coverges as the depth creases rather slowly to ts asymptotc value of However, the formulato of [0] order to obta h x from h, h should be multpled by ther defto of H rms, (). Ths deormalzato has as cosequece that the H0.% estmates do ot show the ce asymptotc behavor of H 0.%. Ths problem s llustrated usg H 0.%, but the problem s obvously the same f other quatles are used. The adjusted Battjes ad Groeedjk dstrbuto ca thus be used to estmate a gve hgh quatle wave heght, whch ofte wll be close to the maxmum wave heght. I fact, t s stadard practce to defe the maxmum wave heght a sea state, H max, as the H 0.% (the wave heght that s exceeded oce every 000 waves). The crest heght, H c, assocated wth such a hgh wave, ofte also eeds to be estmated. Gve that such extreme waves are ofte o-lear, the assocated crest heght should ot be estmated usg lear wave theory,.e. as half the wave heght. Our advce s that t should be computed usg the olear potetal theory of Reecker-Feto, see [2]. Ther expressos are mplemeted ORCA ad eed as put the water depth, the estmated extreme wave heght ad the correspodg wave perod. Based o a aalyss of a large umbers of measuremets, Goda has show [3], that the most lkely wave perod assocated wth the hghest waves a sea state s closely related to the peak wave perod T p. Accordg to Goda, ths wave perod s 0.9 to.0 T p. Our stadard practce s to take the wave perod assocated wth the maxmum wave heght a sea state (or H 0.% ) T Hmax equal to the peak wave perod; T Hmax =T p. Let us suppose that for a gve locato the 00-yr Hs s 0m wth a assocated peak wave perod of 0s, that the local depth s 25m ad the bottom slope :00. The the assocated extreme wave heght estmates are: H 0.% =5.2m, H c =.43m, H 0.% /d=0.6, H 0.% /H s =.52 ad H c / H 0.% =0.75. Note that f the locato was deep waters the we would have had H 0.% /H s =.86 ad H c / H 0.% =0.5. Fgure 0 Varato wth depth of 0.% usg (8). H ad Hrms / 0 m computed I order to obta well behaved Battjes ad Groeedjk estmates, ORCA Battjes ad Groeedjk dstrbuto results are adjusted as follows: The estmates from the Battjes ad Groeedjk dstrbuto are gve whe the Battjes ad Groeedjk estmates do ot exceed the Ralegh dstrbuto estmates ad H tr Otherwse, the Raylegh dstrbuto estmates are provded. The adjustmet descrbed above assures that the Battjes ad Groeedjk estmates coverge to the Raylegh estmates deep waters as ca be see the les marked as adjusted Battjes ad Groeedjk estmates Fgure 9. Fgure Hs ad MWD jot occurreces table. The adjusted Battjes ad Goeedjk dstrbuto s also used to compute jot occurreces of dvdual wave heght 0 Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use:

11 ad say wave drecto. Supposg that aga for a locato wth a local depth of 25m ad a bottom slope :00, the jot occurrece table of the Hs ad the MWD s as gve Fgure, the the correspodg jot occurrece table of the dvdual wave heght ad MWD s as gve Fgure 2. [] Goda, Y., 979: A revew o statstcal terpretato of wave data. I: Report of the Port ad Harbour Research Isttute, Japa 8 pp [2] Reecker, M.M. ad J.D. Feto, 98: A Fourer approxmato method for steady water waves. J. Flud Mechacs, 04, [3] Goda, Y., 978: The observed jot dstrbuto of perods ad heghts of sea waves. Proc. 6th It. Cof. o Coastal Egeerg, Hamburg. ASCE, New York, Fgure 2 Idvdual wave heght ad MWD jot occurreces table. ACKNOWLEDGMENTS The authors are thakful to Rob Morelsse ad Carle Bos for ther substatal cotrbutos to the developmet of ORCA. REFERENCES [] Kome, G. J., L. Cavaler, M. Doela, K. Hasselma, S. Hasselma ad P. A. E. M. Jasse, 994: Dyamcs ad Modellg of Ocea Waves. [2] Marda, K.V., 972. Statstcs of drectoal data. Academc Press, (Lodo ad New York). [3] Fsher, N.I., 993. Statstcal aalyss of crcular data. Cambrdge Uv. Press, 277 pp. [4] Fsher, N.I. ad A.J. Lee, 983. A correlato coeffcet for crcular data. Bometrka, 70, pp [5] Coles, S., 200: A Itroducto to the Statstcal Modellg of Extreme Values. Sprger Texts Statstcs, Sprger- Verlag: Lodo. [6] Cares, S., 2009: A comparatve smulato study of the aual maxma ad the peaks-over-threshold methods. SBW-Belastge subproject 'Statstcs'. Deltares Report [7] Cares. S., 2007: Extreme wave statstcs. Cofdece tervals. WL Delft Hydraulcs Report H [8] Loguet-Hggs, M.S. 952: O the statstcal dstrbuto of the heghts of sea waves. J. Mar. Research,, [9] Holthujse, L.H., 2007: Waves oceac ad coastal waters. Cambrdge Uversty Press. [0] Battjes, J.A. ad H.W. Groeedjk, 2000: Wave heght dstrbutos o shallow foreshores, Coastal Egeerg. 40, Copyrght 20 by ASME Dowloaded From: o /25/204 Terms of Use: