Probabilistic Model for Wind Speed Variability Encountered by a Vessel

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1 Natral Rsorcs, 014, 5, Pblshd Onln Octobr 014 n ScRs. Probablc Modl for Wnd Spd Varablty ncontrd by a Vssl Igor Rychlk 1, Wngang Mao 1 Mathmatcal Scncs, Chalmrs Unvrsty of chnology, Götborg, Swdn Dpartmnt of Shppng and Marn chnology, Chalmrs Unvrsty of chnology, Götborg, Swdn mal: rychlk@chalmrs.s, wngang.mao@chalmrs.s Rcvd 6 Ag 014; rvsd 1 Sptmbr 014; accptd Octobr 014 Copyrght 014 by athors and Scntfc Rsarch Pblshng Inc. hs work s lcnsd ndr th Cratv Commons Attrbton Intrnatonal Lcns (CC BY). Abract As a rslt of socal awarnss of ar msson d to th s of fossl fls, th tlzaton of th natral wnd powr rsorcs bcoms an mportant opton to avod th dpndnc on fossl rsorcs n ndral actvts. For xampl, th martm ndry, whch s rsponsbl for mor than 90% of th world trad transport, has alrady artd to look for soltons to s wnd powr as axlary proplson for shps. h practcal nallaton of th wnd faclts oftn rqrs larg amont of nvmnt, whl ncrtants for th corrspondng nrgy gans ar larg. hrfor a rlabl modl to dscrb th varablty of wnd spds s ndd to mat th xpctd avalabl wnd powr, coffcnt of th varaton of th powr and othr atcs of ntr,.g. xpctd lngth of th wnd condtons favorabl for th wnd-nrgy harvng. In ths papr, wnd spds ar modld by mans of a spato-tmporal transformd Gassan fld. Its dpndnc rctr s localzd by ntrodcton of tm and spac dpndnt paramtrs n th fld. h modl has th advantag of havng a rlatvly small nmbr of paramtrs. hs paramtrs hav natral physcal ntrprtaton and ar atcally fttd to rprsnt varablty of obsrvd wnd spds n RA Intrm ranalyss data. Kywords Wnd Spds, Wnd-nrgy, Spato-mporal Modl, Gassan Flds 1. Introdcton In th ltratr typcally cmlatv drbton fncton (CDF) of wnd spd W, say, s ndrood as th long-trm CDF of th wnd spds at som locaton or rgon. h drbton can b ntrprtd as varablty of W at a randomly takn tm drng a yar. Wbll drbton gvs oftn a good ft. Lmtng tm span to, How to ct ths papr: Rychlk, I. and Mao, W. (014) Probablc Modl for Wnd Spd Varablty ncontrd by a Vssl. Natral Rsorcs, 5,

2 I. Rychlk, W. Mao for xampl, Janary month affcts th W CDF smply bcas, as t s th cas for many gophyscal qantts, th varablty of W dpnds on sasons. o avod ambgty whn dscssng th drbton of W, tm span and rgon ovr whch th obsrvatons of W ar gathrd nd to b clarly spcfd. By shrnkng th tm span to a sngl momnt t and gographcal rgon to a locaton p, on obtans (n th lmt) th drbton of W( p, t). hs s sd as th drbton of W n ths papr. Obvosly th long-trm CDF can b rtrvd from W p, t drbtons by mans of an avrag of th local drbtons, vz for a fxd locaton p th local 1 s+ S ( W w) = ( W(, t) w) d, t S p (1) s whr S can b a month, a sason or a yar. Smlarly th long-trm CDF ovr a rgon A, say, s proportonal to s+ S ( W( p, t) w) dd t p. A s In ordr to dntfy th drbtons at all postons p and tms t, va amont of data ar ndd. Hr th rconrcton of W from nmrcal ocan-atmosphr modls basd on larg-scal mtorologcal data, calld also ranalyss, s tlzd to ft a modl. h ranalyss dos not rprsnt actal masrmnts of qantts bt xtrapolatons to th grd locatons basd on smlatons from complx dynamcal modls. It s dfnd on rglar grds n tm and spac, and hnc convnnt to s. In ths papr, th RA Intrm data [1] prodcd by ropan Cntr for Mdm-Rang Wathr Forcas s sd to ft th modl. Howvr th modl can also b fttd to othr data s,.g. to satllts wnd masrmnts whch has also good spatal covrag, s []. Modlng spatal and tmporal dpndnc of wnd spd s a vry complx problm. Modls proposd n th ltratr ar rvwd n [3]. Hr w propos to s th transformd Gassan modl, whch assms that thr xs a dtrmnc fncton G( w ), possbly dpndnt on locaton p, sch that X( p, t) = GW ( ( p, t) ) s Gassan. h X ( p, t) fld s dfnd by th man m( p, t) and covaranc rctr ov( X ( p1, t1), X ( p, t) ). Obvosly for a gvn transformaton G and many yars of hnd-ca, on cold mat th covaranc for any par ( p 1,t1), ( p,t), s,.g. [4] [5]. Howvr sch an approach s lmtd to rlatvly small grds n spac. mployng th mprcal covarancs n tm and spac wold rslt n hg matrcs, whch lmt th applcablty of sch mprcal approach. Consqntly, a smpl paramtrc modl that catchs only som aspcts of th wnd spd varablty, mportant for a partclar applcaton, s of practcal ntr. h mnmal rqrmnts on th modl ar that t shold provd a corrct mats of long-trm drbtons of th wnd spds, accrat prdctons of avrag dratons of th xtrm wnds condtons and rlabl mats of CDFs of top spds drng orms ncontrd by a vssl. In ordr to dmonrat th capablty of th proposd modl for sch mnmal rqrmnts, ths papr s organzd as follows: In Scton, a gnral conrcton of non-atonary modl for wnd spd varablty n tm and spac s prsntd. Scton 3 prsnts probablc modl for th vlocty of orms movmnts. hn atcal proprts of som orms charactrcs ar dscrbd n Scton 4. h physcal ntrprtatons of th ntrodcd paramtrs ar also gvn n ths scton and n Appndx 1. In Scton 5, on board masrd wnd spds ar sd to valdat th proposd modl, whr th long trm CDFs of ncontrd wnd spds and prsnc atcs ar sd. otal forty rots ar sd, s Fgr 1. h tm whn rots wr sald ar wll sprad ovr a yar. Fnally n Scton 6, mans to smlat th ncontrd wnd spds ar brfly rvwd. Papr closs wth thr appndxs contanng somwhat mor tchncal mattrs.. ransformd Gassan Modl and Long-rm CDFs In ths scton w shall ntrodc th transformd Gassan modl for th varablty of wnd spds. In partclar th transformaton G makng th transformd wnd spd data X= GW normally drbtd wll b prsntd. Sasonal modl for th man and varanc of X s gvn and assmd normalty of X valdatd. h wnd spd W( p, t) s th tn mnts avrag of th wnd spd masrd at poston p, dfnd n dgrs of longtd and lattd, whl t s th tm of th yar. W wll s th transformaton G( w) = w, a whr a s a fxd conant that dpnds on th locaton p, vz a (, ) (, ) ( p = ) X p t W p t () h paramtr a s nonngatv wth convnton that th cas a = 0 corrsponds to th logarthm. W as- X p, t s normally drbtd. X, t m, t σ p,t, rspctvly, dpnd both on poston and sm that Man and varanc of ( p ), dnotd by ( p ), 838

3 I. Rychlk, W. Mao Fgr 1. h consdrd rots n th valdaton procss. tm. h tmporal varablty of man and varanc s approxmatd by sasonal componnts wth trnds dfnd as follows m p, t = m p + m p cos πt + m p sn π t + m p t, (3) ( σ ) = ln p, t b p b p cos πt b p sn π t b p t. (4) Hr t has nts yars. hs typ of modl has bn sd n th ltratr, s.g. classcal papr [6]. Rmark 1 For a fxd poston p, th paramtrs a and m n qaton (3) ar fttd smltanosly n sch a way that th danc btwn yarly long-trm mprcal CDF of W( p, t) a m( p, t) and a Gassan drbton s mnmzd. Mor prcsly for a wnd data at fxd poston p and paramtr a ( 0,1) w valat xt = wt a and ft rgrsson qaton (3). hn th rsdal = xt mt s valatd and ts mprcal CDF matd. Nxt a danc btwn th mprcal CDF and th Gassan CDF (fttd to ) s valatd. Fnally th paramtr a * that mnmzs th danc s th mat of a. A tabl of a and m vals as fncton of th locaton p s cratd. As addtonal paramtrs of th modl wll b matd nw colmns wth paramtrs mats wll b addd to th tabl. For xampl, havng matd a and m th varanc σ ( p,t), dfnd n qaton (4), s fttd sng an addtonal assmpton that proprts of th wnd spd changs slowly n tm. hn th paramtrs b ar savd n th tabl. Mor dtals of modl matons ar gvn n Appndx 3. Valdaton of Gassanty of X( p, t) n yars of data W(, t) p wr sd to mat paramtrs a, m and b n qaton (3), (4) n North Atlantc on a grd of 0.75 dgr. Fgr prsnts th mats of paramtr a. At offshor locatons a vals vary arond 0.8 whl clos to shor or nlands locatons a can b mch smallr. Not that small vals of a ndcat largr dpartrs of th obsrvd wnd spds drbton from th Gassan on. Usflnss of th proposd modl rls on th accracy of th approxmaton of X (, t) drbton. h Gassanty of th procss X X (, t) p CDF by Gassan = p has bn valdatd for th Northrn Atlantc. An xampl of condctd valdatons s shown n Fgr 3. In th fgr th lft plot shows tn yars of W procss lmtd to two wks n th mddl of Fbrary, at locatons ( 0,60), ( 10,40), ( 40,50), ( 0, 45), plottd on th normal probablty papr. (It s assmd that th wnds ar atonary for sch short prod of tm.) In th rght plot of th fgr th transformd data X= GW s plottd on normal probablty papr. On can s that X CDFs ar wll approxmatd by th Gassan drbtons. In th rght plots of Fgr 4, th andard dvaton σ ( p,t), dfnd n qaton (3), s prsntd for Fbrary and Ag, rspctvly. On can s that th andard dvaton changs consdrably wth th gographcal locaton bt s lss dpndnt on sason. W trn nxt to prsntaton of varablty of th paramtr 839

4 I. Rychlk, W. Mao Fgr. Vals of paramtr a n th transformaton qaton (). Fgr 3. Lft: n yars of wnd spds W (t) wth t lmtd to Fbrary at th for locatons. ( 0, 60), ( 10, 40), ( 40, 50), ( 0, 45) plottd on normal probablty papr. Rght: ransformd wnd spds X (t) lmtd to Fbrary at th for locatons plottd on normal probablty papr. h vals of paramtr a n transformaton qaton () ar a = 0.850, 0.675, 0.875, 0.875, rspctvly. ( p, t),.. th man of X (, t) m p dfnd n qaton (). Howvr snc nts of m ar not physcal w choos to show th varablty of th mdan spd (, t) m(, t) 1 a( p) p p (5) µ = nad. h vals of th mdan for Fbrary and Ag ar prsntd n two lft plots of Fgr 4. As xpctd, wnd spds ar hghr n wntr than n smmr. Fnally w chck whthr th rgrssons qatons (3) and (4) sd to modl sasonal varablty of m and σ lads to accrat mats of th long-trm CDF of W at poston p. mployng Gassanty assmpton of X CDF th thortcal long-trm CDF of wnd spds at a fxd poston p, dfnd n qaton (1), s gvn by whr ( x) wnd spds ( W > w) a( p) ( p, ) ( p, t) 1 s+ S w m t ( W w) = d, t S Φ s σ Φ s th CDF of a andard Gassan (normal) varabl. In Fgr 5, th yarly probablts for comptd sng qaton (6) at for locatons n North Atlantc ar compard wth th mprcal mats. (h locatons ar markd by crosss n Fgr ) On can s that th agrmnt btwn th mats s xcllnt. (6) 840

5 I. Rychlk, W. Mao Fgr 4. Lft top Mdan wnd spd µ [m/s], dfnd n qaton (5), n Fbrary; Lft bottom Mdan wnd spd n Ag; Rght top Standard dvaton of X, comptd by mans of qaton (4) n Fbrary; Rght bottom h andard dvaton n Ag. Fgr 5. Comparsons of mats of th long-trm probablty ( W > w) for yarly wnd spds varablty qaton (1) at for locatons dfnd n dgrs of longtd and lattd: ( 0, 60), ( 10, 40), ( 40, 50) and ( 0, 45). h sold ln s th probablty comptd sng qaton (6) wth S = 1 yar. Somwhat mor rrglar lns ar th matd probablts basd on tn yars of hnd-ca data. 841

6 I. Rychlk, W. Mao 3. Vlocty of a Wnd Storm W t,.g. = 15 m/s. h bordr of a orm s a -lvl { p p }. h bordr changs as orms mov, grow or fall. In a classcal papr [7] Longt- Hggns has ntrodcd vlocts to dy movmnts of random srfacs. hr ar svral dfntons of vlocts proposd n th ltratr, s [8]. Hr w wll s vlocty n a fxd drcton, say. h drcton wll b calld th man azmth of a orm. It wll b dfnd n Rmark, s also xampl 1. As comary w s th convnton that th drcton soth to north has azmth = 0 whl azmth α = 90 for th drcton w to a. Followng [8] th vlocts n th drcton and 90 ar gvn by A orm occrrng at tm t s a rgon whr ( p, ) contor : W(, t) = W W,, t t V = V 90 = W W 90 whr W t s th tm drvatv of th wnd spd, W and W ar th drctonal drvatvs havng azmths, 90, rspctvly. hs ar valatd at a poston p on th -lvl contor. Not that tm t s 90 fxd. h gnral assmpton of ths papr s that paramtr a dos not dpnd on tm and changs mch slowr n spac than th wnd spd W vars, s Fgr. Hnc th gradnt W = ( Wx, Wy, Wt) can b approxmatd as follows, whr 1 a 1 W 1 a X X, (8) X s th gradnt of X-fld. Hnc vlocts dfnd n qaton (7) can b approxmatd by Xt Xt V =, V =. 90 X X 90 For a homognos Gassan fld th vlocts hav mdan vals qal to v ( ( p, ), ( p, )) ar ( X ) ( ( p, ), t ( p, )) ov X t X ov X t X t t t = v = 90,, 90 ar ( X 90 ) s [8] for proof. h spds n drctons and 90 wll b dnotd by v, v, rspctvly. h azmth s chosn n sch a way that th drctonal drvatvs X 90, X ar ncorrlatd, s Rmark for 90 som dscssons abot th choc of. Rmark h angl dpnds on proprts of th covaranc matrx Σ of th gradnt X ( p, t). Mans to mat th matrx Σ ar gvn n Appndx 3. For svral rasons, s [9] for dtald dscsson, t s convnnt to rotat th coordnat sym so that th partal drvatvs X x and X y bcom ncorrlatd. Lt A b th rotaton by angl arond th t-axs matrx makng covaranc btwn X x and X y zro. hn lt dnot by Σ th covaranc matrx of th X n th rotatd coordnats vz. (7) (9) (10) Σ = A Σ A, (11) whr ar X s th th lmnt havng ndx 11 n th matrx Σ whl ov( X ( p, t), X t ( p, t) ) has ndx 13. Usng th lmnts th mdan vlocty v n qaton (10) can b comptd onc th matrx Σ has bn valatd. xampl 1 Lt consdr th followng fld t x t X( xyt,, ) = σ1r1cos π π + φ1 + σrcos π + φ (1) L whr R 1, R and φ 1, φ ar ndpndnt varabls havng Raylgh, nform CDF, rspctvly, and hnc X s a sm of two ndpndnt Gassan flds. h fr componnt s a harmonc wav movng along th x-axs wth vlocty L/ whl th scond trm can b ntrprtd as colord nos. Obvosly X x s ndpndnt of X y and hnc = 90, s Rmark. Frthr A s th transpos of A. Now th ( ( p ) t ( p )) σ ( p ) t ( p ) ov X, t, X, t = π L, ov X, t, X, t =

7 I. Rychlk, W. Mao ar X = π σ L and hnc th mdan vlocts qaton (10) ar gvn by whl 1 v = L,0, v = 0,0. In ths smpl xampl th mdan vlocts agr wth th vlocty of th harmonc wav movng along th x-axs. In Fgr 6, varablty of th mdan vlocts v and v n qaton (10) s compard. In th top 90 plots sasonal varablty of v s llratd by showng dffrncs btwn th vlocts n Fbrary and Ag. h maxmal man spd n th top plots s abot 45 km/h whl mnmal s zro. Smlar comparson for th vlocty v s gvn n th bottom plots, whr th maxmal spd s abot 19 km/h. Gnrally on 90 can say that th orms mov far n wntr than n smmr, and th angl also changs btwn th sasons. For xampl, n th North Atlantc th orms mov bascally n avrag from w to a whl n th smmr months th drcton s oppost n lattd of arond 0 dgrs. 4. Statcs of ncontrd Wnd Spds Man sbjct of th papr s dvlopmnt of a smpl modl dscrbng varablty of wnd spds tm srs ncontrd by a vssl or at a fxd locaton. In ths scton w wll dfn th modl and gv mans to mat th long-trm CDF of ncontrd wnds; xpctd draton and rngth of an ncontrd orm. A shp rot s a sqnc of postons p, say, a shp ntnds to follow. W assm that a shp wll follow raght lns btwn th postons havng azmth α, say. A voyag arts at tm s and wll la for S days. sh Intal poston p ( s), azmths α and shp spds v ( t ), t [ ss, + S], dfn ts poston p at any tm t drng a voyag. hn th ncontrd wnd spds ar gvn by 90 ( ) A shp salng along a rot ( p, t) = ( x, y, t), t [ ss, S] W t = W p t, t, s t s+ S. (13) +, has vlocty Fgr 6. op mats of th mdan vlocts, km/h, th wndy fld movs n drcton n Fbrary and Ag. h color corrsponds to spd. h hgh spd (orang) s 45.1 km/h whl th low (bl) s 0 km/h. Bottom Comparsons of th mdan vlocts v 90 n Fbrary and Ag. h hgh spd s 18.6 km/h. 843

8 I. Rychlk, W. Mao sh whr v sh = ( x y ) = v ( α α ) sh v, sn,cos, (14) t s th shp spd at tm t. (Rcall that th x axs has azmth 90 whl th y-axs has azmth 0.) In th followng w wll s th transformd Gassan fld qaton () to modl th ncontrd wnd W t, vz spds h procss 1 ( ) ( ) a p t 1 at X t s Gassan wth man mt = m( p, t) and varanc σ σ (, t) = p, rspctvly. h long-trm CDF of ncontrd wnd spds s dfnd by W t = X p t, t = X t. (15) 1 s+ S ( W w) = ( W w) d t, w 0. S (16) s h CDF gvn n qaton (16) cold b b matd by fttng an approprat drbton to avalabl data. (Wbll drbton s oftn sd.) Altrnatv approach s to compt th thortcal CDF, vz Drbtons of Storm Charactrcs at mt 1 s+ S w ( W w) = d. t S Φ (17) s σ Smlarly as n Scton 3 w wll say that a shp ncontrs ormy condtons at tm t f wnd spd W ( t ) xcds som fxd lvl. (In th xampls w wll s = 15 m/s.) Smlarly, t ncontrs wndy wathr condtons at tm t f wnd spd s abov th mdan,.. W > µ ( p, t). In Fgr 7 wnd spds ncontrd along a rot n Octobr ar prsntd. h ppr ntrvals mark tms n orms whl th lowr ntrvals show th prods of wndy wathr ncontrd by a vssl. h thn ln, shown n th lowr plot, llrats varablty of th ncontrd mdan wnd spd. h rgon of ormy condtons conss of tm ntrvals whn th wnd spd s conantly abov thrshold. h ntrvals wll b calld orms. hn lt N dnot th nmbr of ncontrd orms. For xampl, Fgr 7. Illraton of th dfnton of ormy, wndy wathr rgons. op A rot takn n Octobr; Bottom Sold thck ln shows th onboard masrd wnd spds. h thn sold ln prsnts varablty of th mdan wnd spd along th rot. h ntrvals plottd at lvl 15 m/s rprsnt tms whn shp ncontrs th ormy wathr whl ntrvals plottd at lvl zro marks th ncontrd wndy wathr rgons. 844

9 I. Rychlk, W. Mao N = 3 n Fgr 7. h dratons of orms ar dnotd by, whl th hgh wnd spd drng a orm by A, = 0,, N. h probablty drbtons of th charactrcs wll b dfnd nxt. In ordr to ffcntly wrt down th formlas for th CDFs w nd som addtonal notaton ntrodcd nxt. Lt th nmbr of ncontrd orms for whch vnt (atmnt) A s tr b dnotd by N ( A ). For xampl, N ( A > w) s th nmbr of ncontrd orms for whch wnd spds xcd a thrshold w, whl N ( > t) s th nmbr of orms that la longr than t. Obvosly N = N( > 0), snc all > 0, s th nmbr of pcrossngs of lvl by th ncontrd wnd spds. h mprcal probablty that a orm la longr than t hors can now b wrttn as follows mp ( t) ( > ) N t > = N Nxt th thortcal, basd on modl, probablty of vnt A,.g. ( A) [ N ] A= > t, wll b dfnd by ( > ) N N t A =,.g. ( > t ) =. [ N ] h proposd modl qaton (15) wll b valdatd by comparng th mprcal drbton of orms rngth A and th avrag dratons of orms wth thortcally comptd ( A > w) and. Mor complx orms atcs cold also b sd to valdat th modl bt t wold rqr a ddcatd nmrcal softwar, s.g. [10] and rfrncs thrn, to valat N ( A). Hnc t wll not b sd hr. In th followng only a smpl bond ntrodcd n [11], and th xpctatons [ Nw ] [ N ],, ( A > w) w [ N ] W > W cl = S, = S, [ N ] wll b sd for valdaton prposs. In qaton (0), cl dnots tm prod whn wnd spd s nntrrptdly blow th thrshold,.. a tm prod btwn orms. h qaton (0) wll b provd n Appndx. In ordr to valat qaton (19) and qaton (0), th formla for [ N w ] s ndd. h xpctd nmbr of pcrossngs of lvl w by W can b comptd sng th gnralzd Rc s formla [1], vz. s+ S + [ ] = w s 0 W t, W t s also [13]. Hr W s th tm drvatv of (18) (19) (0) N zf z, w dd, z t (1) W t. Rmark 3 Consdr a atonary Gassan procss X wth man m and varanc of pcrossngs of lvl w by X n tm ntrval of lngth S thn classcal rslt of Rc [1] gvs σ. Lt N w b th nmbr S [ Nw ] = ar ( X ( 0) ) xp ( w m) σ. () πσ Consqntly th avrag danc btwn pcrossng of th man lvl m by X s 4.. valaton of [ N w ] [ ] = π σ (3) ar X ( ( 0) ) From dfnton of th ncontrd wnd spd procss W t follows that th nmbr of pcrossngs of th lvl at w by W ( t ) n th ntrval [ ss, + S] s qal to th nmbr of pcrossngs of th lvl w by X ( t ). Snc w ar prmarly ntrd n modlng wnd flds n offshor locatons w assm that th fld s homognos n a rgon wth rads of abot 100 km and atonary for a prod of copl of wks. (h as- 845

10 I. Rychlk, W. Mao smptons ar lkly to fal n clos to coa or nland locatons.) Undr th assmpton mt = 0 and X ( t ) and X ar ndpndnt. Consqntly th ntgral n qaton (1) can b wrttn n a mor xplct way, vz. [ N ] ( w at mt ) s S1 ar X + σ ( ) w = d. t (4) s π σ In th followng w shall s an addtonal paramtr σ τ = π ar X and wrt qaton (4) n an altrnatv form [ N ] τ dfnd by ( w at mt ) s+ S 1 σ s Not that f X s atonary, thn [ ] wndy condtons la for. Proprts and mans to valat dscssd n Appndx Valdaton of th Modl (5) w = d. t (6) τ τ =, sn qaton (3). Hnc τ s th avrag tm prod that τ sng physcally ntrprtabl paramtrs ar h proposd modl s valdatd by nvgatng th accracy of th thortcally comptd drbtons wth th mprcal drbtons matd from data. Frly at fxd postons p th thortcal atcs of th orm charactrcs A, and cl wll b compard wth mats of th atcs drvd sng tn yars of hndca data. Scondly, th long-trm wnd spd drbtons ncontrd by vssls ar compard wth th thortcally comptd drbtons sng th modl and th mats drvd from th hnd-ca. h xpctd nmbr of ncontrd pcrossng wll also b sd n th valdatons. Howvr atcs of ncontrd orm charactrcs wll not b sd n th valdaton procss. hs s bcas th wnd spds masrd on-board shps ar basd by captans dcsons to avod salng n havy orms, rportd also n [14]. Som valdatons of th modl at nland locatons was prsntd n [15] Drbtons of Storm Charactrcs A, and cl at a Fxd Poston Consdr a boy at poston p thn X = X ( p, t). h paramtr τ, s qaton (5), s thn gvn by σ τ = π. (7) ar X p, t ( t ) In Fgr 8 vals of th paramtr τ valatd sng qaton (7) for Fbrary and Ag ar prsntd. In offshor locatons τ s lss than two days, whch s mch shortr than th atonarty prod assmd to b abot 3 wks. Hnc th paramtr τ s th xpctd tm prod th wnd spds xcds th mdan and that τ s approxmatly conant for abot a month. h vals τ, prsntd n Fgr 8, wll b fr sd to valdat th approxmaton of probablty that a orm obsrvd at poston p wll hav wnd spds xcdng a lvl w > = 15 m/s,.. formla qaton (19). Nxt by combnng formlas qaton (0) wth qaton (16) and qaton (4) th xpctd draton of a orm wll b comptd and thn compard wth th obsrvd avrag dratons. h xpctd draton of calmr wathr,.. tm ntrvals whn wnds spds ar conantly blow th thrshold, wll b comptd n a smlar way and thn sd n th valdaton h probablts ( A > w) and xpctatons, cl ar comptd for a prod S = 1 yar and postons p markd by crosss n Fgr. h rslts prsntd n Fgr 9 and abl 1 show vry good agrmnt btwn th obsrvd orm charactrcs at th for locatons and th thortcally comptd charactrcs. 846

11 I. Rychlk, W. Mao Fgr 8. Comparson of spatal varablty of τ ( s) (5) for a boy. (op) Fbrary, (Bottom) Ag., dfnd n qaton Fgr 9. Probablts ( A > w), = 15 [m/s], that wnd n a orm xcds lvl w drng on yar at for locatons havng longtds and lattds; ( 0, 60), ( 10, 40), ( 40, 50) and ( 0, 45). h sold lns ar τ t probablts comptd sng qaton (19) and qaton (5) wth a, and σ matd at th locatons. h rrglar lns ar th matd probablts sng tn yars of hnd-ca data. abl 1. Long-trm (on yar) xpctd orm/calm dratons n days. Poston = 15 m/s cl cl = 18 m/s cl cl ( 0, 60) ( 10, 40) ( 40, 50) ( 0, 45)

12 I. Rychlk, W. Mao 5.. Valdaton-Wnd Spds ncontrd by Vssls Masrmnts of th wnd spd ovr grond,.. tn mnts avrags, rcordd ach tn mnts on-board som shps, ar sd to valdat th proposd modl. Snc th data ar rcordd mch dnsr than th hnd-ca w hav rmovd hgh frqncs from th sgnals (prods abov 1.5 hor wr rmovd sng FF). h data sd n ths dy s lmtd to th North Atlantc and wrn rgon of Mdtrranan Sa. h accracy of th thortcally comptd long-trm drbton of ncontrd wnd spd wll b nvgatd. Fr a sngl voyag opratd n lat Ag, shown n th top lft plot of Fgr 10, s consdrd. In rght top plot of th fgr, th masrd wnd spds, shown as sold ln, ar compard wth th matd wnd spds sng hnd-ca, dashd dottd ln. On can s that th two sgnals ar rasonably clos. In th lft bottom plot of Fgr 10, tn thn lns show th mprcal long trm probablts ( W > w) comptd for hnd-casd basd mats of wnd-spds that wold b ncontrd f th shp wr salng th sam rot vry yar. On of th mats s not vsbl snc t s vry clos to th ( W > w) matd sng th on-board masrd wnd spd, th thck sold ln. h tn mats show larg varablty btwn yars. h rglar sold ln s th thortcally comptd ( W > w). It s clos to th avrag of th tn mats drvd from th hnd-ca (not shown n th fgr). W concld that for th consdrd rot th thortcal long-trm drbton of wnd dscrbs wll long-trm varablty of wnds along th rot. Smlar conclsons can b drawn from Fgr 11 lft plot wr th combnd long-trm drbtons for all 40 voyags ar Fgr 10. op lft A rot sald from rop to Amrca n lat Ag. op rght Wnd spds masrd on-board a vssl (sold rrglar ln) compard wth thr mats drvd from th hnd-ca data (dashd dottd ln); Bottom lft Comparsons of mats W > w, plottd on th logarthmc scal, for th voyag. of th long-trm probablty h thck smooth ln s th probablty comptd by sng qaton (6). h lss smooth thck ln s th probablty matd sng th on-board masrd wnd spds. h thn rrglar lns ar th probablts matd from th hnd-ca data for tn dffrnt yars;, plottd on th logarthmc scal, for Bottom rght Comparsons of th mats of [ N w ] th voyag. h thck smooth ln s [ N w ] comptd by sng qaton (1). h thck N w rrglar ln s th N w valatd from th on-board masrd wnd spds. h dashd dot- sng hnd-ca drvd wnd spds for th rot sald n td ln s th mat of [ ] tn yars. 848

13 I. Rychlk, W. Mao Fgr 11. Lft Comparsons of th mats of th long-trm probablty ( W > w) for dashd-dotd ln s th mat of th forty voyags. h sold smooth ln s th probablty comptd sng qaton (6). h W > w sng tn yars of hnd-ca whl th rrglar ln s th mat of th wnd spds ncontrd by vssls. Rght Comparsons com- of th mats of [ N w ] for th forty voyags. h sold smooth ln s th [ N w ] ptd sng qaton (1). h dashd dottd ln s an mat of [ N w ] sng tn yars hnd-ca whl th rrglar ln s th on-board obsrvd N w. shown. Basd on th rslts prsntd n Fgr 10 and Fgr 11, w concld that th thortcal long trm drbton of wnd spds ncontrd by a salng vssls agrs wll wth th drbton drvd sng hndca; and scondly that th rotng syms sd n plannng a rot s sccssfl n slctng rots wth calmr wnd condtons than avrag on. basd on hnd-ca (dashd In Fgr 10, bottom rght plot, and n Fgr 11, rght plot, mats of [ N w ] dottd ln) and th obsrvd N w (th sold rrglar ln) ar compard wth th thortcal [ ] N w comptd sng qaton (4) for rots shown n Fgr 1 and Fgr 10 top lft plot. On can s that th lns ar clos xcpt for th hgh wnd spds. h obsrvd crossngs of hgh wnd spds (sold rrglar ln) ar fwr than thortcally prdctd. hs w attrbt to s of rotng programs that sccssflly choos calmr roots than th avrag on. hs clam s also spportd by ds of th mat of [ ] N w drvd from 10 yars of hnd-ca, shown as th dashd ln. On can s that ths mats ar hghr than on-board obsrvd N w for wnd spds abov 1 m/s. 6. Smlaton of th ncontrd Wnd Spds Common xprnc says that wnd spds vary n dffrnt tm scals,.g. drnal pattn d to dffrnt tmpratrs at day and nght; frqncy of dprssons and ant-cyclons whch sally occr wth prods of abot 4 days and annal pattrn. o follow th clam th transformd obsrvd wnd spd fld x( p, t) s dcomposd nto for parts whch contan prods abov 40 days, btwn 40 and 5 days, btwn 5 and 1 day and th nos. For ach fld varancs σ and th covaranc matrcs of th gradnt vctor Σ wr matd,.. th transformd Gassan modl fttd. Now for any voyag on can compt paramtrs τ, thn ndpndntly smlat th ncontrd for X t and fnally transformaton componnts along a shp rot. Addng th componnts gvs smlaton of 1 W = X at gvs smlatd wnd spds that cold b ncontrd along a rot. Mor prcsly, for a shp rot ( p, t), s t s+ S, on fnds paramtrs a( t ), mt, τ, = 1,,4, thn X ( t ) s smlatd by σ and 849

14 I. Rychlk, W. Mao Hr 4 X t mt t f = 1 t s B s = + + σ ( ) d. τ t (8) B ar ndpndnt Brownan motons whl th krnls f τ 14 1 = π xp π. τ f τ ar gvn by t τ h procss X ( t ) s Gassan wth man mt and th covaranc fncton gvn by τ( s) + τ ( s) = 1 ( ) 4 τ π ( t s) τ ( s) + τ ov( X, X ( s) ) = σ σ( s). (9) τ Obvosly th ntgrals n qaton (8) hav to b comptd nmrcally. hs s carrd ot sng th followng approxmaton + ( ) t τ ( ) f t s d B, s f t s j dsz τ j (30) whr Z j, = 1,, 4, ar ndpndnt zro man varanc on Gassan random varabls, whl ds = s j+ 1 s j. Hr s j forms an qdant grd covrng th doman of th krnl f τ. (In th cas whn wnd spds ar smlatd on vry dns grd thn t s rcommndd to slghtly smooth th paramtrs a( t ), σ and τ.) h proposd modl gvs mans for ffcnt smlaton of wnd spds along any shp rots. h paramtrs a( t ), mt, σ and τ ar spcfd by mans of qaton (8) and qaton (30). Altrnatvly on can smlat X ( t ) sng covarancs dfnd n qaton (9) and som of many mthods to s- mlat Gassan vctors. h algorthm basd on qaton (30) s prfrabl whn dnsly sampld wnd spds along a long shp rot ar ndd. For xampl, for a rot dfnd n Fgr 1 top plot that was sald for 400 hors gvng 400 rcordd wnd spds, t took lss than 100 sconds on laptop to smlat 100 wnd W t ar prsntd as thn sold lns n spd profls along th rot. Fv of th smlatd profls of j Fgr 1. op A rot sald n Northrn Atlantc n Aprl; Mddl h τ t ; Bottom Wnd xpctd lngth of ncontrd wndy wathr prod spds masrd on-board a vssl (sold thck ln) hnd-ca prdcton (dashd dottd ln) and fv smlatons of th wnd spds by mans of qaton (15) and qaton (8) (thn sold lns). 850

15 I. Rychlk, W. Mao Fgr 1 bottom plot. h masrd wnd spds ar prsntd as th sold thck ln whl dashd dottd ln s th hnd-ca basd mat of th spds. Not that paramtrs σ and τ ar smply comptabl from th paramtrs σ, τ alon. Hnc th thortcal long-trm drbtons and atcs of orm charactrcs can b comptd by mans of mthods dscssd n prvos sctons. 7. Conclson A atcal modl for th wnd spd fld varablty n tm and ovr larg gographcal rgon has bn proposd. h modl was fttd to RA Intrm ranalyzd data. Valdaton ts show vry good match btwn th drbtons matd from th data and th thortcal comptd on from th modl. h modl was also sd to mat rsk of ncontrng xtrm wnds and th thortcal mats agr wll wth th mprcal on. Ralc wnd profls can b smlatd sng th modl. Acknowldgmnts Spport of Chalmrs nrgy Ara of Advanc s acknowldgd. Rsarch was also spportd by Swdsh Rsarch Concl Grant and by Knt and Alc Wallnbrg Stftls. h athors also wold lk to thank Wallns Lns AB for provdng s wth onboard wnd masrmnt data. Rfrncs [1] D, D.P., Uppala, S.M., Smmons, A.J., Brrsford, P., Pol, P., Kobayash, S., Andra, U., Balmasda, M.A., Balsamo, G., Bar, P., Bchtold, P., Bljaars, A.C.M., van d Brg, L., Bdlot, J., Bormann, N., Dlsol, C., Dragan, R., Fnts, M., Gr, A.J., Hambrgr, L., Haly, S.B., Hrsbach, H., Hólm,.V., Isaksn, L., Kållbrg, P., Köhlr, M., Matrcard, M., McNally, A.P., Mong-Sanz, B.M., Morcrtt, J.J., Park, B.K., Pby, C., d Rosnay, P., avolato, C., hépat, J.N. and Vtart, F. (011) h RA-INRIM Ranalyss: Confgraton and Prformanc of th Data Assmlaton Sym. Qartrly Jornal of th Royal Mtorologcal Socty, 137, [] Baxvan, A., Cars, S. and Rychlk, I. (008) Spato-mporal Statcal Modllng of Sgnfcant Wav Hght. nvronmtrcs, 0, [3] Monbt, V., Allot, P. and Prvoo, M. (007) Srvy of Stochac Modls for Wnd and Sa Stat m Srs. Probablc ngnrng Mchancs,, [4] Carals, G., Rados, K. and Zrvos, A. (010) h ffct of Spatal Dsprson of Wnd Powr Plants on th Crtalmnt of Wnd Powr n th Grk Powr Spply Sym. Wnd nrgy, 13, [5] Kss, P. and János, I.M. (008) Lmtatons of Wnd Powr Avalablty ovr UROP: A Concptal Stdy. Nonlnar Procsss n Gophyscs, 15, [6] Brown, B.G., Katz, R.W. amd Mrphy, A.H. (1984) m Srs Modls to Smlat and Forca Wnd Spd and Wnd Powr. Jornal of Clmat and Appld Mtorology, 3, [7] Longt-Hggns, M.S. (1957) h Statcal Analyss of a Random, Movng Srfac. Phlosophcal ransactons of th Royal Socty A, 49, [8] Baxvan, A., Podgórsk, K. and Rychlk, I. (003) Vlocts for Movng Random Srfacs. Probablc ngnrng Mchancs, 18, [9] Baxvan, A. and Rychlk, I. (006) Maxma for Gassan Sas. Ocan ngnrng, 33, [10] Podgórsk, K., Rychlk, I. and Machado, U..B. (000) xact Drbtons for Apparnt Wavs n Irrglar Sas. Ocan ngnrng, 7, [11] Rychlk, I. and Ladbttr, M.R. (000) Analyss of Ocan Wavs by Crossng and Oscllaton Intnsts. Intrnatonal Jornal of Offshor and Polar ngnrng, 10, [1] Rc, S.O. (1944) h Mathmatcal Analyss of Random Nos Part I and II. Bll Sym chncal Jornal, 3, [13] Rychlk, I. (000) On Som Rlablty Applcatons of Rc Formla for Intnsty of Lvl Crossngs. xtrms, 3, [14] Mao, W., Rngsbrg, J.W., Rychlk, I. and Storhag, G. (010) Dvlopmnt of a Fatg Modl Usfl n Shp 851

16 I. Rychlk, W. Mao Rotng Dsgn. Jornal of Shp Rsarch, 54, [15] Rychlk, I. and Mdanagc, A. (013) A Spatal-mporal Modl for Wnd Spds Varablty. Dpartmnt of Mathmatcal Scncs, Dvson of Mathmatcal Statcs, Chalmrs Unvrsty of chnology and Unvrsty of Gothnborg, Gothnbrg,

17 Appndx 1: Comptaton of ar X ( t ) on nds to ntrodc a tm dpndnt gradnt vctor X ( Xx, X y, Xt) ( p, t) h paramtr τ was dfnd n qaton (5), vz. τ = πσ ar X ar ( X ) vctor of drvatvs ( xt yt ) sh Obvosly v = ( xt, yt ) and X = v X = v Σ v I. Rychlk, W. Mao. In ordr to valat = and th v =,,1. (31), whr Σ s th covaranc matrx of th gradnt vctor X, whr s th scalar prodct. Hnc ar X t t t t (3). h matrx Σ has to b matd n th rgon of ntr. In Appndx 3 a sktch of th maton procdr s gvn. In th followng w shall gv an altrnatv formla for ar ( X ) whch mploys a physcally ntrprtabl paramtrs whch cold b sfl for comparson of stablty of a trad for s of wnd sals or othr mans to harv wnd nrgy. Paramtr ( t ) Rgons h varanc ar X τ as a Fncton of Wnd, Shp Vlocts and Gomtrcal Szs of Wndy s ndpndnt of th choc of coordnat sym. Hr w wll s th rotatd coordnat sym by azmth calld n Scton 3 man azmth of a orm. h matrx Σ n th rotatd Σ t and has th followng dagonal lmnts coordnat sym wll b dnotd by ( ) t σ = ar X p, t, σ = ar X p, t, σ = ar X p, t, (33) and followng off-dagonal lmnts ( X ( p t) Xt( p t) ) X ( p t) Xt( p t) σ = 0, σ =,,,, σ =,,,. (34) sh sh h shps vlocty = v ( sn α,cosα ) sh sh v = v ( α ) ( α ) Obvosly v s n th rotatd coordnats gvn by sh sh = v Σ v sn,cos. ar X t t,1 t t,1 and aftr som algbra ( ) σ00 σ00 ar X t = σ00 v + σ00 v + σ v v, 90 (35) σ00 σ00 whr th ncontrd vlocty,.g. th dffrnc btwn th shp vlocty and th wnd fld vlocty s, n th rotatd coordnats, gvn by sh sp ( v v ) = ( v v ( α ( t t )) v v ( α ( t ))), sn, cos. (36) In ordr to ntrprt componnts n qaton (35), w nd to ntrodc som addtonal paramtrs that dscrb avrag sz of wndy wathr rgons and som rrglarty factors. Rcall that wndy wathr condtons rgon at tm t s th rgon conss all p whr wnd spds xcds th mdan µ ( p,t). Now w shall ntrodc paramtrs rlatd to avrag sz of wndy rgon n drctons and 90. h paramtrs wll b dnotd by L and L, rspctvly. h thrd paramtr s th avrag prod th wndy wathr la las at a fxd poston p. h paramtrs ar dfnd 90 by σ σ σ L = π, L = π, = π, (37) σ σ σ s qaton () and Rmark 3. Obvosly th vals of paramtrs ar slowly changng fnctons of poston X p, t wr homog- and tm and that why w call thm local szs of wndy rgons. Howvr f th fld 853

18 I. Rychlk, W. Mao nos and atonary thn th paramtrs wold b qal to th avrag lngth btwn pcrossngs and downcrossng of th mdan by wnd spd W n drcton, + 90 and n tm, s [9] for dtals. Now by mltplyng both sds of th qaton (35) by ( π ) and dvdng by σ, w obtan that whr 1 ( τ ) v ( L v L α α ) = , α σ =, α = 90 σ00σ00 σ00σ00 σ (38) ar sfl rrglarty factors. Roghly, smallr vals of th factors hghr rsks of xtrm orms, s [9] for mor dtals. Frthr, f α + = 1 thn th srfac X drfts, vz α 90 If p has rotatd coordnats thn τ = dr X ( p, t) = X ( p v t,0 ). dr v n rotatd coordnats s qal to ( v, v ) 1. Fnally v ( 1 ) ( 1 L + v L + α α ) For a homognos wnd fld v L s th rcprocal of th avrag tm spnd n wndy wathr rgon whn salng wth azmth, smlar ntrprtaton can b gvn to v L whl s th avrag tm th wndy wathr s obsrvd by a shp at r or a boy. hs paramtrs can b matd from th on-board masrd sgnals or gvn sbjctv vals basd on xprnc. Usflnss of qaton (39) and qaton (4) ls n possblty of prdctng rsks for ncontrng xtrm orms sng asly avalabl paramtrs whch hav clar physcal manng. Appndx : Proof of qaton (0) Lt assm that W s s a smooth procss. Usng Fbnn s thorm [ N ] ( > ) + N d N d 0 > t t 0 t t = = + s + N > t d t = 1 d { W t } t, whr ( x ) A Snc 0 s f x A and zro othrws. Agan by Fbn s thorm and hnc + [ N ] 90.. (39) 1 s th ndcator fncton of th A takng val 1 ( ) s+ s+ 1 dt PW t dt s { W = } s > = Appndx 3: maton of Paramtrs PW ( > ) [ N ] h paramtrs of th modl hav bn fttd for th North Atlantc. Hr th RA Intrm data has bn sd, althogh n ftr work w plan to also s data from satllt basd snsors. A momnt s mthod and rgrsson ft wr mployd to mat th paramtrs. In ths scton w gv a short dscrpton of th appld maton procdr. In th followng th masrd wnd spds at a locaton wll b dnotd by wt. Stp 1: For a fxd gographcal locaton and 0< a < 1 th transformd wnd spd wt a s comptd and th man qaton (3) fttd sng LS rgrsson. mprcal cmlatv drbton fncton (CDF) and Gas-. 854

19 . Paramtr a * mnmzng th danc btwn th two ds- mt. Frthr th a san (CDF) ar fttd to th rsdal wt mt * trbtons s slctd as an mat of a. h corrspondng man m ( t ) s an mat of a * rsdal xt = wt m s valatd and thn sd to mat paramtrs Stp : maton of sgnals x ( t ), 1,, 4 from of x1 ( t ). h sgnal x x x1. h sgnal x3 x x1 x ( t ). Fnally, x4 = x x1 x x3. Stp 3: For a sgnal x ( t ) th paramtrs smng atonarty of t j hn I. Rychlk, W. Mao σ n th followng ps. =. h sgnal x 1 s matd as follows; fr on fltrs ot x t (s Stp 1) th harmoncs wth prods shortr than 40 days. h rsltng sgnal s an obsrvaton t s drvd by fltrng ot harmoncs wth prods blow 5 days from th sgnal t s drvd by fltrng ot harmoncs wth prods blow 1 day from th sgnal σ ar matd as follows. For a sqnc of tms t j, as- x s for s n a nghborhood of 10 days arond t j, mats of σ ar fond. σ ar matd by fttng sasonal componnts, smlar to qaton (3), to sqncs of obsrva- ( t, t ) j j σ. tons Stp 4: maton of Σ ( p, t),.. th covaranc matrx of th gradnt vctor valatd at ( p,t). h covaranc matrx s dfnd by sx covarancs btwn th partal drvatvs of X. h fnctons ar changng slowly wth sason bt spatal varablty can b hgh, partclarly at coaal and nland locatons. Consqntly w ft sx sasonal componnts to th covarancs for ach of postons p on a grd wth msh 0.75 dgr. h componnts ar matd n a smlar way as dscssd n Stp

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