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2 Predcon of Wave Hegh Based on he Monorng of Surface Wnd 80 Tsukasa Hokmoo Graduae School of Mahemacal Scences, The Unversy of Tokyo Jaan 1. Inroducon The ocean wave s one of he hyscal facors whch cause serous sea dsasers, and s redcon rovdes he nformaon avalable for varous human acvy relaed o he sea. More han a half-cenury has assed snce orgnal heores for wave hndcasng echnques have been roosed n he oneerng aers such as Sverdru and Munk (1947) and Person, Neumann, and James (1960) and so on, he mehod and he echnque for wave redcon roblem have rogressed a grea deal, agans he background of recen rogresses n he echnologes of measuremen and comuaon. However, even a he resen, he redcon of he wave henomena s sll a dffcul roblem, and he echnology for wave redcon s gong on furher develomen. There are several reasons why he redcon of he henomena relaed o he sea sae s a dffcul roblem even now. One of he reasons s he comlexy of he hyscal mechansm on he wave develomen. When he sea s geng rough by wnd forcng, he sea surface movemen s affeced by he neracons among he meeorologcal facors, such as wnd moon and amosherc ressure, and he oograhcal nfluence whch vares by regon. I means ha he heorecal descron of he sea surface movemen, akng no accoun of he dynamc relaonsh among hese facors, s very comlcaed. And anoher reason s he dffculy of he feld measuremen a sea. I s ofen he case ha we can no carry ou consan monorng on he necessary meeorologcal facors, due o he lack of measuremen facles, sudden malfuncon of a measuremen nsrumen, and so on. In he radonal research on he wave redcon roblem, varous sascal mehods for he redcon of he sea sae daa have been roosed unl now. However, mos of such mehods have been consdered based on he measured daa obaned by buoys or shs. In Jaan, he Jaan Meeorologcal Agency has se u abou 1300 regonal saons for he ground-based meeorologcal monorng, whch s called Auomaed Meeorologcal Daa Acquson Sysem (AMeDAS), hroughou of hs counry, and over 80 sensors for ulrasonc wave hegh meers n he coasal areas. They rovde measured daa on wave hegh and varous meeorologcal facors consanly, whch are avalable va Inerne. I s hough ha he hyscal facors whch make nfluence on he sea condon, such as he wnd seed and wnd drecon, change wh saal and emoral correlaons. So, as an aroach o he above wave redcon roblem, we develo a sascal model for redcng he change of wave hegh from he change of surface wnd, obaned by consan ground-based observaon.

3 170 Oceanograhy 2 Wll-be-se-by-IN-TECH In hs chaer, we rovde wo ocs on he sascal modelng for he redcon of wave hegh. The frs oc s a modelng for redcng he change of wave hegh based on ocean wnd, by alyng he mehod roosed n our revous aer (Hokmoo and Shmzu (2008)). And he second oc s he develomen of a sascal model for redcng wave hegh, based on he change of surface wnd, obaned by ground-based observaon. Also, he effecveness n redcon usng he roosed models s examned by means of he numercal exermen. The secons below are organzed as follows. In he nex secon, we oulne radonal researches on he sascal models for he sea sae daa. In secon 3, we resen a mehod for he wave hegh redcon based on he measuremen of ocean wnd. In secon 4, we develo a model for redcng wave hegh based on he measuremen of surface wnd, obaned by ground-based observaon. And secon 5 rovdes a summary of he resul and dscusson of furher research on hs oc. 2. Tme seres models for he sea sae analyss I s well-known ha he wave moon under low wnd seed can be aroxmaed by Gaussan rocess. For he measured daa n hs asec, he lnear saonary me seres models roosed by Box and Jenkns (1976), such as auoregressve (AR) model or auoregressve movng average (ARMA) model, have been wdely used o consruc a redcor. In fac, varous alcaons o wave hegh daa (e.g., Cunha and Guedes (1999), Ym e. al. (2002)) and he wnd daa (e.g., Brown e. al. (1984), Danel and Chen (1991)) have been reored by many auhors. However, as for he wave moon durng he wave develomen rocess, he above saonary models do no gve reasonable redcons. There are also many models whch are alcable o he measured daa n he ransonal asec. One s sandard lnear nonsaonary me seres models. For examle, auoregressve negraed movng average (ARIMA) model (Box and Jenkns (1976)), he auoregressve model wh me varyng coeffcens (Kagawa and Gersch (1985)), and generalzed auoregressve condonal heeroskedascy (GARCH) model (Bollerslev (1986)) are used wdely o he nonsaonary me seres daa. Also, f he seed n changng sascal srucure can be regarded o be slow, we can aly a saonary AR model o he me seres daa n he local me nerval whch can be regarded o be saonary. For examle, a model for redcng he change of nonsaonary secral densy funcon of he sea surface movemen durng he wave develomen rocess was develoed based on hs conce (Hokmoo e. al. (2003)). There are also nonsaonary me seres models based on he decomoson of he rend and he oher comonens (e.g., Ahanassouls e. al. (1995), Sefanakos e. al. (2002), Walon and Borgman (1990)). One of neress when we rea he measured daa on he sea sae s how we rea he dreconal me seres daa on he wnd drecon. Ths roblem s serous when we consder a sascal model o he mulvarae me seres daa ncludng wave hegh, wnd seed and wnd drecon, because dreconal daa have a unque roery ha hey ake he values on he crcle. In he framework of dreconal sascs, varous mehodologes for sascal nferences on he dreconal daa have been roosed (for examle, Marda and Ju (2000)). Among hem, mulvarae regresson models, ncludng crcular and lnear varables, have been ofen roosed n envronmen sudes. Johnson and Wehrly (1978) consdered he heorecal background of he lnear aramerc regresson, whch has he lnear varable and he angular varable. And he exenson of her model has gven n Fsher and Lee (1992), SenGua (2004), SenGua and Ugwuowo (2006), and so forh. However, here have been

4 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd only lmed aems o model mulvarae angular-lnear daa. In Hokmoo and Shmzu (2008), we develoed an angular-lnear me seres model o exress he dynamc srucure among wave hegh, wnd seed and wnd drecon, by exendng he mulle regresson model by Johnson and Wehrly (1978), and showed he effecveness on wave hegh redcon beween he change n ocean wnd and he change n wave hegh. Our neres here s ha he model whose srucure s smlar o he above model may be effecve for he descron of he dynamc relaonsh beween wave hegh and surface wnd. 3. Wave hegh redcon based on he change of ocean wnd In hs secon, we resen a sascal mehod for redcng he change of wave hegh from he moon of ocean wnd, based on Hokmoo and Shmzu (2008). The develomen of hs mehod was movaed by he measured daa obaned from ocean surveys usng a research sh, n Hunka-bay, Hokkado, Jaan. Fg. 1. A ma around he measurng on 3.1 In-su monorng on wave hegh and ocean wnd Fgure 1 shows a ma around he measurng on (42 17 N, E). We have measured he changes of relave sea surface level, wnd seed and wnd drecon n Hunka-bay. For he relave sea surface movemen, we measured relave dslacemen from he mean of he sea surface movemen over 10 mnues by usng an ulrasonc wave hegh meer of he research sh. Also, he changes n wnd seed and wnd drecon a abou 15 meers hegh from he sea surface were measured by usng an ulrasonc wnd meer. Afer he measuremen, we obaned he me seres daa on 1/3 sgnfcan wave hegh, mean wnd seed and mean wnd drecon for every 1 mnue, based on he measured records. Fgure 2 dslays he me seres daa obaned n he above, whch were measured on Dec. 2, From he o, he sgnfcan wave hegh (m), he wnd seed (m/s), and he wnd

5 172 Oceanograhy 4 Wll-be-se-by-IN-TECH drecon (rad.) are shown, where each samle sze s 90. I s noed ha he orgn of he wnd drecon daa s defned o be norh and he osve value means he clockwse drecon. Accordng o he weaher mas of he samlng day, as well as he days before and afer, he locaon of amosherc ressure formed ycal aern of wner n Jaan. In oher words, he hgh ressure area s exendng over he wes of Jaan Islands and he low ressure area s exendng over he eas. Under he background of hs locaon, he above observaon showed he endency ha he wnd drecon changed slowly from norh-wes o norh, and he wnd seed radly ncreased aroxmaely from 6m/s o 13m/s n mnues, and hen changed slowly n he range aroxmaely from 12m/s o 15m/s. On he oher hand, 1/3 wave hegh gradually grew u o abou 3.5 meers under he background ha he wnd seed ncreased and he wnd drecon changed slowly. The above daa can be regarded o be he measuremen of he wave develomen rocess, because wave hegh ncreases under he suaon ha he wnd seed becomes faser and he wnd drecon does no change so much. Wavehegh WH(m) mnues Wnd Seed WS(m/s) mnues Wnd Drecon WD(rad.) mnues Fg. 2. Measured daa on wave and wnd (from he o, 1/3 sgnfcan wave hegh (m), wnd seed (m/s), wnd drecon (rad.))

6 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd Some characerscs on he correlaon srucure In he followng, we make some relmnary analyses on he correlaon srucure of he measured daa, n order o nvesgae wha class of model s suable for exressng he change of he measured daa. In he followng, le{wh },{WS } and{wd } ( = 1,..., N)be ses of measuremens of sgnfcan wave hegh, wnd seed and wnd drecon, resecvely, where s he me on and N s he samle sze Crcular auocorrelaon of he wnd drecon daa Frs, we nvsgae he correlaon srucure of he dreconal me seres daa of wnd drecon. As a basc conce of exloraory crcular daa analyss, we refer o a book by Fsher (1993, Chaer 2) and use he followng wo ransformaons of WD x = cos(wd ), y = sn(wd ) (1) In order o exlore he ossbly of deecng changes of drecon, we use wo sascs; one s he cumulave sum (CUSUM) lo dslayed by he ons C = x, S = y (2) and he oher s he cumulave mean drecon lo {Θ c ; = 1,...}, such ha cos(θ c )=C / C 2 + S2, sn(θc )=S / C 2 + S2 (3) are sasfed smulaneously. CUSUM lo s dslayed n he lef of Fgure 3, where he horzonal axs denoes C and he vercal axs denoes S. Also, he cumulave mean dreconal lo s dslayed n he rgh of Fgure 3, where he horzonal axs denoes he me on and he vercal axs denoes Θ c. I s noed ha he change n sascal srucure of he dreconal me seres daa s admed, when he rend of CUSUM lo s clearly dfferen from he sragh lne whose sloe s one, and when he value of he cumulave mean dreconal lo s clearly dfferen from he consan value. The cumulave mean dreconal lo suggess he ossbly ha he dreconal me seres daa have a change on of sascal srucure a = 40 roughly, and n hs case he me seres exhbs nonsaonary. We also checked he sascal es of change n mean drecon by usng CrcSas (Chaer 11 of Jammalamadaka and SenGua (2001)). The resul showed ha here exss a change on a he me on = 42, whch suggesed ha he daa exhb nonsaonary. Now we are neresed n wheher here s clear dfference n he correlaon srucures, beween he case when we regard he wnd drecon daa o be crcular me seres daa and he case when we regard he daa o be lnear me seres daa. For he esmaon of correlaon, s necessary o subrac he rend of he daa. Therefore, we esmae he rend from he followng wo sandons. One s he esmaon by regardng he daa o be crcular me seres daa. In hs case, for esmang rend, we oban he smoohed seres of {x } and {y } by usng he locally weghed regresson (LOWESS). And hen, based on he smoohed seres, say {x } and {y }, we oban he smoohed rend of wnd drecon T, such ha x (x )2 +(y )2 = cos(t ), y (x )2 +(y )2 = sn(t ) (4)

7 174 Oceanograhy 6 Wll-be-se-by-IN-TECH Fg. 3. Cumulave sum lo (lef) and Cumulave mean dreconal lo (rgh) Wnd Drecon (rad.) Lag(*1mn.) Fg. 4. Trend esmaon on wnd drecon based on {T curve) } (doed curve) and {T } (sold are sasfed smulaneously. Anoher s he rend esmaon based on lnear me seres daa, whch s gven n Kagawa and Gersch (1985). The rend of {WD } can be obaned by alyng he rend model, and WD = T + ζ, ζ N(0, σ 2 ζ ) (5) T T 1 = v, v N(0, σ 2 v) (6) where T s he random varable o exress he rend, σζ 2 and σ2 v are unknown varances of ζ and v, resecvely. Fgure 4 shows he rend esmaon based on T and T, where he doed curve means {T } and he sold curve means {T }. I looks ha here s no clear dfference beween {T } and {T }. So we esmae crcular auocorrelaon coeffcen based on he subraced seres, WD WD T. Based on crcular-crcular assocaon (Fsher

8 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd (1993, Chaer 6)), he samle crcular auocorrelaon coeffcen s gven by ˆρ (τ)= 4(A τ B τ C τ D τ ) [ (N 2 Eτ 2 Fτ)(N 2 2 Gτ 2 Hτ) 2 ], τ = 0, 1,... (7) 1/2 where τ s he me lag, and A τ = D τ = H τ = N τ cos WD cos WD N τ +τ, B τ = =1 N τ sn WD cos WD +τ, E τ = =1 N τ sn(2wd +τ) =1 =1 sn WD sn WD +τ, C τ = N τ cos(2wd ), F τ = =1 N τ cos WD sn WD +τ, =1 N τ sn(2wd N τ ), G τ = =1 =1 cos(2wd +τ), On he oher hand, he samle auocorrelaon funcon of he me seres daa WD s gven by ˆρ (τ)= N τ =1 (WD +τ WD )(WD WD ) =1 N (WD, WD ) 2 WD = 1 N N WD =1 (8) (9) Correlaon Lag(*1mn.) Fg. 5. Comaron beween { ˆρ (τ)} (bold lne) and { ˆρ (τ)} (doed lne) Fgure 5 dslays he esmaes of ˆρ (τ) and ˆρ (τ) (0 τ 30), where he vercal axs denoes he correlaon, he horzonal axs denoes τ n mnues, and he bold and doed lnes corresond o ˆρ (τ) and ˆρ (τ), resecvely. We observe ha hey change smlarly wh he same endency, alhough ˆρ (τ) akes slghly larger values han ˆρ (τ) when τ s small. I s evaluaed from hs resul ha he samle crcular auocorrelaon coeffcen can be aroxmaed by he lnear correlaon o some exen. Also, suggess he ossbly ha s suffcen o exress he dynamc srucure of he measured daa by usng he lnear me seres model.

9 176 Oceanograhy 8 Wll-be-se-by-IN-TECH Lag () () Lag () () Table 1. Cross correlaon funcons ( () {WD } and {WH }, () {sn(wd )} and {WH } ) Cross correlaon among wnd seed, wnd drecon and wave hegh Nex, we focus on he cross correlaons among{ws },{WD } and{wh }. We esmaed he cross correlaon funcon beween {WH } and he varables {WS }, {WD } and {sn(wd )}, by usng he me seres daa afer subracon of her rends esmaed by LOWESS mehod. Fgure 6 shows an esmaed resul of he cross correlaon funcon beween {WS } and {WH } by usng he samle cross correlaon funcon γ(τ) (τ = 0,±1,...), γ(τ)= N τ =1 (WH WH)(WS +τ WS) N=1 (WH WH) 2 N =1 (WS WS) 2, WH = 1 N N =1 WH, WS = 1 N N WS (10) =1 where he horzonal means he me lag n mnues and wo arallel lnes denoe Barle s bounds (.e.,±1.96n 1/2 ). I suggess he ossbly ha he change n wnd seed affecs he one of wave hegh afer mnues Correlaon Lag (*1mn.) Fg. 6. Cross correlaon funcon beween {WS } and {WH } wh Barle s bounds For esmaon of he correlaon beween {WD } and {WH }, s of neres how we rea he dreconal varable WD. Table 1 gves esmaed values of he cross correlaon funcons n he wo cases, () {WD } and {WH } and () {sn(wd )} and {WH }. I s observed ha he absolue value of () ends o ake larger values han he one of (). Ths resul suggess he ossbly ha s execed o mrove he redcon accuracy by adong he varables sn(wd ) (and cos(wd )) as he exlanaory varables, nsead of usng WD.

10 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd A sascal modelng on he change of wave hegh by wnd forcng Suose ha we redc he fuure values of wave hegh {WH N+l ; l = 1,..., L}, based on he hsorcal daa {WH, WS, WD } ( = 1,..., N). We sar our consderaon by assumng ha he me seres {WH }, {WS } and {WD } are saonary, afer alyng a roer ransformaon (he deal s descrbed laer n hs secon). We wre he change of {WH } as WH =m L + + β (1) WH + K k=1 β (3),k cos(k WD )+ K k=1 β (4),k sn(k WD ) β (2) WS + ε (1), ε (1) WN(0, σ 2 WH ) (11) where and K are orders, m L s he unknown mean, β s are unknown weghs, and ε (1) s he random varable whch follows a whe nose rocess wh E(ε (1) )=0 and V(ε (1) )=σwh 2. Smlarly, we wre WS =m S + + γ (1) WH + K γ (3) k=1,k cos(k WD )+ K k=1 γ (4),k sn(k WD ) γ (2) WS + ε (2), ε (2) WN(0, σ 2 WN ) (12) and sn(h WD ) and cos(h WD ) (h = 1,...,K) as sn(h WD )=m h + + δ (1) WH + K δ (3) k=1,k cos(k WD )+ K k=1 δ (4),k sn(k WD ) δ (2) WS + δ (h), δ (h) WN(0, σh 2) (13) and so forh, where m s, m h, γ s and δ s are unknown weghs. Pu he sae vecor a me by y (K) (WH, WS, cos(wd ), sn(wd ),..., cos(k WD ), sn(k WD )) (14) Then we can wre y (K) =m (K) + A (K) 1 y (K) A(K) y (K) + δ(k), δ(k) WN(0, Σ (K) ) (15) where m (K) s he unknown mean vecor, A (K) ( = 1,..., ) s he unknown coeffcen marx, and δ (K) follows he mulvarae whe nose rocess wh mean 0 and he dserson marx Σ (K). Ths s a mulvarae vecor auoregressve model of he h order, and herefore, he esmaes for elemens of unknown marces A (K) can be obaned by usng he leas squares mehod (e.g., Brockwell and Davs (1996)). Thus, we can consruc an l-se (l = 1,..., L) ahead redcor based on (15) by ŷ (K) N+l = ˆm(K) + Â (K) 1 z (K) N+l 1 + Â(K) 2 z (K) N+l Â(K) z (K) N+l (16)

11 178 Oceanograhy 10 Wll-be-se-by-IN-TECH and z (K) N+l m = y(k) N+l (l ), z(k) N+l m = ŷ(k) N+l (l > ), where  s he leas squares esmaor of A, The redced values of WH N+l (l = 1,...,L) can be obaned from he redcon of ŷ (K) N+l. However, he model (15) wh he sae vecor (14) has a drawback n comuaonal asec. I s robable ha he accuracy of he esmaes of arameer becomes worse when boh K and become large, because (15) has (2+2K)+(2+2K) 2 unknown arameers o be esmaed. For mrovng he redcon accuraces, he dmenson of he sae vecor ŷ (K) should be small. In order o akng accoun of he mulle dreconal nformaon wh he small numbers of varables, we focus on he followng lnear sum WD (K) ω 1 cos(wd )+ω 2 sn(wd )+ + ω 2K 1 cos(k WD )+ω 2K sn(k WD ) (17) where ω ( = 1,..., 2K) are unknown weghs. And we roose o use he model (15) wh he sae vecor ỹ (K) (WH, WS, WD (K) ) (18) Here, s necessary o deermne he omum order K and he value of ω. For deermnng ω, we nroduce he conce of rncal comonen analyss. WD (K) can be wren as WD (K) = Ω K D(K) (19) where Ω K =(ω 1,...,ω 2K ) and D (K) =(cos(wd ), sn(wd ),..., cos(k WD ), sn(k WD )). We selec he values of Ω K so ha V( WD (K) )=Ω K Σ(K) Ω K (20) s maxmzed under he consrans Ω K Ω K = 1, where Σ (K) s he dserson marx of D (K). Ω K can be obaned as he egenvecor b (K) of he egen equaon, Σ (K) b (K) = λb (K) (21) Le λ 1 λ 2K be 2K egenvalues of he egen equaon. We choose he egenvecor whch (K) corresonds o λ 1 wh un norm, say b M, wh K fxed. We esmae WD (K) by WD (K) = b (K) M D(K) (22) As for he selecon of he order K, we choose he value of K such ha he squared sum of he redcon errors, S l (K)= N l 1 N l N + 1 (WH +l ŴH (K) +l) 2 (23) =N s mnmzed for every l, where ŴH (K) +l s he redced value by (16) and N s a refxed value. For selecon of n (15), we use AIC (Akake Informaon Creron), under he value of K s fxed.

12 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd As observed n Fgure 2, he me seres daa of WH, WS and WD durng he wave develomen rocess exhb nonsaonary. We follow he mehod of ARIMA model by Box and Jenkns (1976) and focus on he dfferenced me seres. In oher words, we regard he dfferenced seres o be saonary and hen f and x (K) = B (K) 1 x (K) B(K) x (K) + ǫ(k), ǫ (K) WN(0, Σ ǫ (K)) (24) x (K) ( WH, WS, WD (K) ) (25) where s he back-shf oeraor such ha WH = WH WH The effec of angular-lnear srucure on he redcon of wave hegh In he followng, we examne he avalably of he roosed mehod hrough he evaluaon of he redcon accuracy on wave hegh. For hs urose, we carred ou he numercal exermens on redcon accuracy by usng he measured daa shown n Fgure 2. The rocedure of he redcon exermen s as follows. Frs, we f he model (24) o he mulvarae me seres daa {WH, WS, WD ; = 1...,50} and hen oban he redcon values of WH u o 5 ses ahead (1 se corresonds o 1 mnue). Nex, we f he model o he me seres daa from =2 o =51 and oban he redced values n he same way. Afer reeang hs rocedure, he redcon accuracy s evaluaed based on he redced values and realzaons. As crera for evaluaon, we defne he mean absolue error (MAE) and he correlaon coeffcen (COR) by MAE(l) 1 M COR(l) M WH () N+l ŴH() N+l (26) M (WH() N+l WH() (l))(ŵh () N+l ŴH () (l)), M (WH() N+l WH() (l)) 2 M (ŴH() N+l ŴH () (l)) 2 WH(l)= 1 M M WH () N+l, ŴH(l)= 1 M M ŴH () N+l (27) where l s he redcon se (l = 1,..., 5), WH () s he realzaon of WH a he h exermen, ŴH () s he redced value of WH a he h exermen, and M s he number of reeons of he exermen. MAE gves beer evaluaon as he redced value ges closer o he observaon. COR s defned as he samle correlaon beween he observaons and redced values, n order o evaluae he degree of accordance o her rends. We frs nvesgae wheher he angular-lnear srucure of he roosed model gve he osve effec on he redcon accuracy of wave hegh. For hs urose, we analyze wheher s ossble o mrove he redcon accuracy by akng no accoun he varables {sn(k WD ), cos(k WD )} (k = 1,..., K), nsead of usng he varable WD drecly. In hs exermen, we comare he redcon accuracy by usng he model (24), under assumng he followng hree sae vecors. The frs s he vecor conssed from he dfference of WH, WS and WD, x ( WH, WS, WD ) (28)

13 180 Oceanograhy 12 Wll-be-se-by-IN-TECH MAE COR L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L= Table 2. Predcon accuracy by usng he model (24) wh he sae vecor (28) (M=35) MAE COR L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L= Table 3. Predcon accuracy by usng he model (24) wh he sae vecor (29) (M=35) MAE COR L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L= Table 4. Predcon accuracy by usng he model (24) wh he sae vecor (25) (M=35, K=25) The second s x ( WH, WS, cos(wd ), sn(wd )) (29) And he hrd s (25), he roosed mehod. Tables 2, 3 and 4 show MAE s and COR s n he above hree cases, resecvely. I s noed ha each exermen was carred ou under he condon ha he order was fxed n he range from 1 o 5. Overall, he resul by usng (29) ends o gve smaller MAE s han he one by usng (28). I suggess he ossbly ha akng no accoun he angular-lnear srucure s effecve for mrovng he redcon accuraces by he redcor based on (28). The resul of COR also shows he smlar endency. I s noed ha, as he order and he redcon se L are larger, he redcon accuracy based on (29) becomes worse o ake negave correlaons. The redcon based on he model wh he sae vecor (25) ends o gve he bes redcon accuracy among he hree models. Ths suggess ha he rncal comonen srucure of (25) worked effecvely, whch conrbued o he mrovemen of he redcon accuracy.

14 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcng he change of wave hegh from surface wnd In he revous secon, we develoed a sascal model for exlanng he dynamc relaonsh beween ocean wnd and wave hegh. Now we consder he redcon roblem on wave hegh based on he moon of he surface wnd, observed by ground-based observaon. For hs urose, we develo a new mehod by alyng he model resened n he revous secon. Also, n order o evaluae he avalably of he develoed model, we comare he redcon accuraces beween he roosed model and radonal me seres models. Hokkado Okushr Esash Mor Hakodae Ohma Masumae Wave recorder Pacfc ocean Fg. 7. Locaons of he sensor for ulrasonc wave hegh meer (nvered rangle) and major AMeDAS saons around he sensor (black crcles) 4.1 Ground-based observaon on surface wnd and measuremen of wave hegh In Jaan, as descrbed n Inroducon, many saons of AMeDAS and he sensors for ulrasonc wave hegh meers have been locaed n varous regons and coasal areas of hs counry by he Jaan Meeorologcal Agency. In he followng, we consder a case sudy on redcon of he wave hegh n Masumae-ok, he sea area n he souhwes of Hokkado. Fgure 7 shows a ma of he locaons of he sensor of a wave hegh meer n Masumae-ok ( N, E) and major AMeDAS saons locaed around he sensor. The monorng of he changes of wnd seed and wnd drecon, and he measuremen of wave hegh are carryng ou consanly, and he measured daa are avalable va Inerne. For he followng analyss, we obaned he daase of wave hegh measured n Masumae-ok and he daases of wnd seed and wnd drecon monored a he AMeDAS saon n Masumae-cho, whch s locaed roughly 5 km away from he measurng on of wave hegh. Fgure 8 dslays he changes n he sgnfcan wave hegh (m) measured n Masumae-ok, and wnd seed (m/s) and wnd drecon (rad.) monored n Masumae-cho, whch were measured every hour on he hour. They are he records for every four seasons n he erod from Arl 2010 o February As he daases for four seasons, we obaned he measured daa n he erod from Ar. 1 o May 31 for srng, Jul. 1 o Aug. 31 for summer, Oc. 1 o Nov. 31 for auumn and Jan. 1 o Feb. 28 of 2011 for wner. The measured daa on wnd seed and wnd drecon are rovded as he mean value over he as 10 mnues, and he measured daa on 1/3 sgnfcan wave hegh s calculaed based he sea surface daa over

15 182 Oceanograhy 14 Wll-be-se-by-IN-TECH Wavehegh Wavehegh WH(m) Wnd Seed WS(m/s) Wnd Drecon WD(rad.) WH(m) Wnd Seed WS(m/s) Wnd Drecon WD(rad.) (a) Srng (Ar. 1- May 31) (b) Summer (Jul. 1- Aug. 31) Wavehegh Wavehegh WH(m) Wnd Seed WS(m/s) Wnd Drecon WD(rad.) WH(m) Wnd Seed WS(m/s) Wnd Drecon WD(rad.) (c) Auumn (Oc. 1- Nov. 31) (d) Wner (Jan. 1- Feb. 28) Fg. 8. The changes of 1/3 sgnfcan wave hegh (m) (o), wnd seed (m/s) (mddle) and wnd drecon (rad.) (boom) for four seasons (Ar Feb. 2011)

16 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd he as 25 mnues. I s noed ha he orgn of he wnd drecon daa s defned o be norh and he osve value means he clockwse drecon. D(WS()) (a) D(WS) (m) CCF lag D(cos(WD())) (b) D(cos(WD)) CCF lag D(WS()cos(WD())) (c) D(WScos(WD)) CCF lag Fg. 9. Tme seres los of { WS }, { cos(wd )}, { (WS cos(wd ))} (lef column) and he cross correlaon funcons n he cases (a)-(c) (rgh column) 4.2 Cross correlaon among he measured daa Frs, we consder how we rea he measured daa of he wnd drecon. In he followng, le {WS }and{wd } be he measured me seres daa on wnd seed and wnd drecon of he surface wnd. Fgure 9 shows he me seres los of { WS }, { cos(wd )} and { (WS cos(wd ))}, and he cross correlaon funcons n he 3 cases, (a) { WS } and { WH }, (b) { cos(wd )} and { WH }, and (c) { (WS cos(wd ))} and { WH }, whch were esmaed by (10), where he doed lnes mean he Barle s bounds. We observe ha he case (c) gves larger cross correlaon han he cases of (a) and (b). 4.3 Modelng he change of wave hegh by akng no accoun he change of surface wnd We consder a sascal model o exress he change n wave hegh based on he change n surface wnd, monored a an AMeDAS saon. Followng he resul n he revous secon, we buld a nonsaonary me seres model focusng on he change of (WS cos(wd )). We wre WH as

17 184 Oceanograhy 16 Wll-be-se-by-IN-TECH WH = α WH + K k=1 β,k (WS cos(kwd )) + K k=1 γ,k (WS sn(kwd ))+ ε 1,, ε 1, WN(0, σ 2 WH ) (30) where and K are orders, (α, β, γ) are unknown coeffcens and ε 1, s he random varable whch follows a whe nose rocess wh E(ε 1, )=0 and V(ε 1, )=σwh 2. Smlarly, we wre (WS sn(hwd )) = α (h) + WH + K k=1 K k=1 β (h),k (WS cos(kwd )) γ (h),k (WS sn(kwd ))+ ε(h) 2, (31) (WS cos(hwd )) = α (h) + WH + K k=1 K k=1 β (h),k (WS cos(kwd )) γ (h),k (WS sn(kwd ))+ ε(h) 3, (32) for h = 1,..., K, where ε (h) 2, WN(0, σ2,h 2 ) and ε(h) 3, WN(0, σ3,h 2 ). Pu he sae vecor a me on by he (2K+1) dmensonal vecor y (K) ( WH, WC 1, WS 1,..., WC K, WS K ) (33) where WC h = WS cos(hwd ) and WS h = WS sn(hwd ) (h = 1,...,K). Then he above models can be rewren by a mulvarae AR model, y (K) = A (K) 1 y (K) A(K) y (K) + δ(k), δ(k) WN(0, Σ (K) ) (34) where A (K) ( = 1,...,) are unknown coeffcen marces and δ (K) follows he mulvarae whe nose rocess wh mean 0 and he dserson marx Σ (K).Anl-se ahead redcor can be consruced by ŷ (K) N+l =  (K) 1 z (K) N+l 1 + Â(K) 2 z (K) N+l Â(K) z (K) N+l, z (K) N+l m = y(k) N+l (l ), ŷ(k) N+l (l > ) (35) where  (K) s he leas squares esmaor of A (K). Thus he l-se ahead redced values, WH N+l (l = 1,..., L), can be obaned by he redcor ŷ (K) N+l.

18 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd MAE COR Model L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L=5 () () () (v) (v) Table 5. MAE s and COR s based on srng daa 4.4 Evaluaon of he redcon accuracy In he followng, we evaluae he effecveness of he roosed mehod by means of he redcon exermen whch s smlar o he one gven n he subsecon 3.4. The rocedure for he exermen s as follows. We selec he me on o sar redcon randomly n he range of he daase. And hen f he roosed model o he measured me seres daa for 100 (.e., samle sze s 100), and oban he redced values u o 5 ses ahead (1 se corresonds o 1 hour). Afer reeang he rocedures, we evaluae he redcon accuracy by MAE and COR. For evaluaon of he redcon accuracy, we also oban he redced values when we used radonal me seres models. The models nroduced for comarson are defned as follows; () WH = α WH + δ 1,, δ 1, WN(0, σ1 2) () WH = β WH + δ 2,, δ 2, WN(0, σ 2 2 ) () y = A 1 y A y + δ 3,, δ WN(0, Σ 3, ), y =( WH, WS ) (v) y = B 1 y B y + δ 4,, δ WN(0, Σ 4, ), y =( WH, WS cos(wd )) (v) y = C 1 y C y + δ 5,, δ WN(0, Σ 5, ), y =( WH, (WS cos(wd ))) where {α, β, A, B, C } are unknown arameers. () and () are unvarae me seres models based on wave hegh. The former s a saonary AR() model and he laer s a nonsaonary ARIMA(,1,0) model. () s a mulvarae AR model akng no accoun he wnd seed as a covarae, and (v) and (v) are mulvarae AR models akng no accoun wnd seed and wnd drecon as covaraes. I s noed ha f he changes of wnd seed and wnd drecon are deenden, he redcon accuracy of (v) becomes beer han ha of (v). Table 5 shows MAE s and COR s of he above fve models, based on he measured daa n srng. The number of reeons s 130. I s noed ha we seleced he order of he model by Akake Informaon Creron (AIC). By he comarson beween () and (), we confrm ha he nonsaonary ARIMA model gves beer redcon erformance han he saonary AR model. Also, he comarsons beween () and (), () and (v), and () and (v) show he endency ha he model akng no accoun he change of wnd moon as covarae mroves he redcon accuracy when we used he unvarae me seres model on wave hegh. Furhermore, he comarson beween (v) and (v) shows he endency ha he redcon accuracy by usng (v) becomes beer, whch suggess ha here exss he deendency beween wnd seed and wnd drecon.

19 186 Oceanograhy 18 Wll-be-se-by-IN-TECH (A) Srng MAE COR Model L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L=5 () () () (v) (v) (B) Summer MAE COR Model L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L=5 () () () (v) (v) (C) Auumn MAE COR Model L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L=5 () () () (v) (v) (D) Wner MAE COR Model L=1 L=2 L=3 L=4 L=5 L=1 L=2 L=3 L=4 L=5 () () () (v) (v) Table 6. Comarsons of MAE and COR for every season (Ar Feb. 2011) 4.5 Robusness on he redcably of wave hegh for every season I s also of neres wheher or no he effecveness n redcon usng he develoed model s robus hroughou a year. In Jaan, here exss unque characerscs on he ressure aern for every season. Therefore, s necessary o nvesgae wheher he model has he ably o mrove he redcon accuraces by radonal models, for all seasons of a year. Table 6 shows MAE s and COR s obaned by usng he measured me seres daa for four seasons n he erod from Arl 2010 o February Overall, he resul has he endency ha he roosed model (v) has he ably o gve he bes redcon accuracy among he fve models, alhough he degree on mrovemen of he accuracy are dfferen for every season. 5. Concluson Our goal n hs chaer s he develomen of a sascal aroach for redcng he change of wave hegh, based on he measured daa of he surface wnd obaned by ground-based

20 Predcon of Wave Hegh Based on he Monorng of Surface Wnd Predcon of Wave Hegh Based on he Monorng of Surface Wnd observaon. In secon 3, we resened a mehod for redcng he change of wave hegh based on ocean wnd, whch was roosed by Hokmoo and Shmzu (2008). And n secon 4, we develoed a model for redcng he wave hegh from he change of surface wnd, by alyng he model gven n he revous secon. The evaluaon on he redcon accuracy suggesed he ossbly ha he mehod roosed n secon 4 mroves he redcon accuraces by usng he redcors based on radonal me seres models. As descrbed a he begnnng, he hyscal facors whch macs on he change n he sea sae wll change wh correlaons on sace and me. A he resen, he models resened n hs chaer do no have saal srucure. For examle, he develomen of he model, akng no accoun he dreconal change of he wnd drecon observed a mulle AMeDAS saons, wll be avalable for deeer undersandngs on he dynamc neracon beween he moons of wnd and wave. 6. References Ahanassouls, G.A., Sefanakos, C.N. (1995). A nonsaonary sochasc model for long-erm me seres of sgnfcan wave hegh, Journal of Geohyscal Research, 100(C8), Bollerslev, T.(1986). Generalzed Auoregressve Condonal Heeroskedascy, Journal of Economercs, 31, Box, G.E.P., Jenkns, G.M. (1976). Tme Seres Analyss, Forecasng and Conrol (revsed edon), Holden-Day, San Francsco. Brockwell, P.J., Davs, R.A. (1996). Inroducon o Tme Seres and Forecasng, Srnger-Verlag, New York. Brown, B.G., Kaz, R.W., Murhy A.H. (1984). Tme seres models o smulae and forecas wnd seed and wnd ower, Journal of Clmae and Aled Meeorology, 23, Cunha C, Guedes S.C. (1999). On he choce of daa ransformaon for modellng me seres of sgnfcan wave hegh, Ocean Eng, 26, Danel, A.R., Chen, A.A. (1991). Sochasc smulaon and forecasng of hourly average wnd seed sequences n Jamaca, Sol Energy, 46(1), Fsher, N.I. (1993). Sascal Analyss of Crcular Daa, Cambrdge Unversy Press, Cambrdge. Fsher, N.I., Lee, A. J. (1992). Regresson models for an angular resonse, Bomercs, 48, Johnson, R.A., Wehrly, T.E. (1978). Some Angular-Lnear Dsrbuons and Relaed Regresson Models, Journal of he Amercan Sascal Assocaon, 73, Hokmoo, T., Kmura, N., Iwamor, T., Amaga, K., Huz, M. (2003). The effecs of wnd forcng on he dynamc secrum n wave develomen: A sascal aroach usng a aramerc model, Journal of Geohyscal Research, 108(C10), Hokmoo, T., Shmzu, K. (2008). An angular-lnear me seres model for wave hegh redcon, Ann Ins Sa Mah, 60, Kagawa, G., Gersch, W. (1985). A Smoohness Prors Tme-Varyng AR Coeffcen Modelng of Nonsaonary Covarance Tme Seres, IEEE Transacons on Auomac Conrol, 30, Marda, K.V., Ju, P.E. (2000). Dreconal Sascs, John Wley, New York. O Carroll (1984). Weaher modellng for offshore oeraons, The Sascan, 33,

21 188 Oceanograhy 20 Wll-be-se-by-IN-TECH Person, W.J., Neumann, G, and James, R.W. (1960). Praccal Mehods for Observng and Forecasng Ocean Waves By Means of Wave Secra and Sascs, U.S. Navy Hydrograhc Offce, Rern edon. SenGua, A. (2004). On he consrucons of robably dsrbuons for dreconal daa, Bullen of he Calcua Mahemacal Socey, 96(2), SenGua, A., Ugwuowo, F.I. (2006). Asymmerc crcular-lnear mulvarae regresson models wh alcaon o envronmenal daa, Envronmenal and Ecologcal Sascs, 13, Sefanakos, C.N., Ahanassouls, G.A., Barsow, S.F. (2002). Muvarae me seres modellng of sgnfcan wave hegh, Proceedngs of Inernaonal Socey of Offshore and Polar Engneers Conference, III, Sverdru, H.U., Munk, W.H. (1947). Wnd sea and swell: Theory of relaon for forecasng, U.S. Navy Hydrograhc Offce, Washngon, D.C., No.601. Walon, T. L., Borgman, L.E. (1990). Smulaon of non-saonary, non-gaussan waer levels on he grea lakes, Journal of he ASCE, Waerway Por, Coasal, and Ocean Engneerng Dvson, 116(6), Ym, J.Z., Chou, C., Ho, P. (2002). A sudy on smulang he me seres of sgnfcan wave hegh near he keelung harbor, Proceedngs of Inernaonal Socey of Offshore and Polar Engneers Conference, III,

22 Oceanograhy Eded by Prof. Marco Marcell ISBN Hard cover, 348 ages Publsher InTech Publshed onlne 23, March, 2012 Publshed n rn edon March, 2012 How narorae o call hs lane Earh when s que clearly Ocean (Arhur C. Clarke). Lfe has been orgnaed n he oceans, human healh and acves deend from he oceans and he world lfe s modulaed by marne and oceanc rocesses. From he mcro-scale, lke coasal rocesses, o macro-scale, he oceans, he seas and he marne lfe, lay he man role o manan he earh equlbrum, boh from a hyscal and a chemcal on of vew. Snce ancen mes, he world's oceans dscovery has brough o humany develomen and wealh of knowledge, he meahors of Ulysses and Jason, reresen he culural growh ganed hrough he exloraons and dscoveres. The modern oceanograhc research reresens one of he las froner of he knowledge of our lane, deends on he oceans exloraon and so s srcly conneced o he develomen of new echnologes. Furhermore, oher scenfc and socal dsclnes can rovde many fundamenal nus o comlee he descron of he enre ocean ecosysem. Such muldsclnary aroach wll lead us o undersand he beer way o reserve our "Blue Plane": he Earh. How o reference In order o correcly reference hs scholarly work, feel free o coy and ase he followng: Tsukasa Hokmoo (2012). Predcon of Wave Hegh Based on he Monorng of Surface Wnd, Oceanograhy, Prof. Marco Marcell (Ed.), ISBN: , InTech, Avalable from: h:///books/oceanograhy/redcon-of-wave-hegh-based-on-he-monorng-ofsurface-wnd InTech Euroe Unversy Camus STeP R Slavka Krauzeka 83/A Rjeka, Croaa Phone: +385 (51) Fax: +385 (51) InTech Chna Un 405, Offce Block, Hoel Equaoral Shangha No.65, Yan An Road (Wes), Shangha, , Chna Phone: Fax:

23 2012 The Auhor(s). Lcensee InechOen. Ths s an oen access arcle dsrbued under he erms of he Creave Commons Arbuon 3.0 Lcense, whch erms unresrced use, dsrbuon, and reroducon n any medum, rovded he orgnal work s roerly ced.