THE INFLUENCE OF HINDCAST MODELING UNCERTAINTY ON THE PREDICTION OF HIGH RETURN PERIOD WAVE CONDITIONS

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1 Prodngs of OMAE4 3rd Intrntonl Confrn on Offshor Mhns nd Art Engnrng Jun -5, 4, Vnouvr, Brtsh Colub, Cnd OM AE4-56 THE INFLUENCE OF HINDCAST MODELING UNCERTAINTY ON THE PREDICTION OF HIGH RETURN PERIOD WAVE CONDITIONS Dnl C. Brookr Shool of Ol nd Gs Engnrng Unvrsty of Wstrn Austrl Ndlnds, W.A. 697 E-l: Goffry K. Col Shool of Ol nd Gs Engnrng Unvrsty of Wstrn Austrl Ndlnds, W.A. 697 E-l: Json D. MConoh Woodsd Enrgy Ltd Adld Trr Prth, W.A. 6 E-l: json.onoh@woodsd.o.u Kywords: Extr Vlu Anlyss, Dsgn Wv, Hndstng, Unrtnty, Dt Anlyss, Mont-Crlo, Bootstrp, Probblty ABSTRACT Extr vlu nlyss for th prdton of long rturn prod t-on ondtons s oftn bsd upon hndst studs of wnd nd wv ondtons. Th rndo rrors ssotd wth hndst odlng r not usully norportd whn fttng n xtr vlu dstrbuton to hndst dt. In ths ppr, odfd probblty dstrbuton funton s drvd so tht odlng unrtnts n b xpltly nludd n xtr vlu nlyss. Mxu lklhood stton s thn usd to norport hndst unrtnty nto rturn vlu stts nd onfdn ntrvls. Th thod prsntd hr s oprd gnst sulton thnqus for ountng for hndst rrors. Th nflun of rndo rrors wthn odld dtsts on prdtd yr rturn wv stts s dsussd. INTRODUCTION Th stndrd pproh to stton of hgh rturn prod wv hghts s to ft n xtr vlu dstrbuton (.g. Wbull, Gubl, Gnrlsd Prto) to xstng wv dt nd prdt th wv hght for gvn rturn prod usng th fttd dstrbuton. Whn hndst dt s usd nstd of tul surd dt, th s produr for dstrbuton fttng nd rturn vlu prdton s usully usd. Wv hndstng nvolvs so dgr of rror du to splftons/ssuptons n th nlytl odl usd nd surnt rrors n th nput dt. Whlst oprson gnst surd dt s oftn usd for lbrton of hndst odl, th fft of hndst odlng rrors on rturn vlu stts s not lwys onsdrd. An portnt pont to onsdr s tht hndst odl n b lbrtd to gv unbsd stts of wv hght, but stll rsult n bsd rturn vlus du to rndo vrton. Gvn tht hndstng unrtnts r known nd n b quntfd, thr s quston s to whthr thr nglt durng xtr vlu nlyss s pproprt. In ths ppr, thod by whh odlng unrtnty n b xpltly nludd n xtr vlu nlyss s prsntd nd oprd wth rsults fro Mont-Crlo nd bootstrp studs. Ths nw thod s bsd upon th us of odfd probblty dstrbuton funton whh norports both nvronntl nd hndst odlng unrtnty. Mxu lklhood stton of ths odfd dstrbuton s usd to norport hndst unrtnty nto rturn vlu stts nd onfdn ntrvls. Th thod prsntd hr s usd to hk th robustnss of yr rturn wv stts for so typl dtsts subjt to known hndst rror. Copyrght 4 by ASME

2 BACKGROUND Hndstng of t-on dt usng torologl hrts s oonly usd s n ltrntv for drt surnt of wnd nd wv vlus durng stors. Suh hndstng studs typlly show so dgr of bs nd rndo vrton whn oprd gnst surd vlus. Bs s usully lntd by lbrtng hndst odl rsults usng ny vlbl surnts n th r (.g. v rgrsson thods). Th rndo vrton howvr rns; typlly n vrton btwn surnts nd odl prdtons of round. s obsrvd for sgnfnt wv hght vlus [,]. A sttr n of -5% s rprsnttv of odrn hndsts, whr th sttr n (S.I.) s dfnd s: σ SI.. = µ s whr σ s th stndrd dvton of th dffrn btwn surd nd odld vlus nd µ s s th n of th surd wv hghts usd for oprson. Drt surnt of on dt (.g. v wv buoys) lso nvolvs rndo rrors, prtulrly th surnt of wv hghts. Ths rrors r, howvr, gnrlly uh sllr thn thos ssotd wth hndstng nd wll not b onsdrd drtly n ths ppr. An xpl plot oprng surd to odld sgnfnt wv hghts s shown n Fgur. Ths dt hs bn obtnd on Austrl s North Wst Shlf; h pont orrspondng to th xu wv hght obsrvd/prdtd durng pssg of tropl ylon. Vlus hv bn norlzd by th prdtd yr rturn prod sgnfnt wv hght, H. Th vlus shown n Fgur r thos ftr lbrton of th hndst odl v lnr rgrsson... th ndvdul odld vlus r orrtd usng: odl H = ( H b) / odl, orr s = H + b whr th onstnts nd b hv bn dtrnd v lnr rgrsson of hndst prdtons gnst surnts : H odl Aftr orrton, th n rror s qul to zro, hn ny systt bs n th hndst vlus s lntd... n s odl ε = ( H H )/ n = = It would usully b pproprt to st b= to nsur tht th hndst odl n b snsbly xtrpoltd byond th hghst vlbl surnt. Ths hs bn don n ths s. Ths typ of lbrton sh s ost pproprt whn th vlbl surnt dtst s ndqut for dstrbuton fttng nd hndstng s bng usd to sgnfntly xtnd th surnt dtst. For ths dtst, th dstrbuton of th rsduls s odl, orr ε = ( H H ) n b wll pproxtd by th norl (Gussn) dstrbuton nd s tkn to b ndpndnt of wv hght (.. n bsolut rror). By nspton of Fgur, th ssupton of ndpndn pprs rsonbl for ths st of dt, wth th rndo vrton ssotd wth th lrgr stors slr to tht sn for th sllr ons. Slr oprsons usng th s hndst odl n dffrnt lotons n Austrln wtrs dsply low orrlton btwn ε nd wv hght, s dos th publshd dt n rf. [] nd []. It should b notd howvr tht th ssupton of ndpndn of rndo rror y not b tru for ll hndst odls n urrnt us; t y sots b or pproprt to dsrb th rror s funton of wv hght. Modld Wv Hght Msurd Wv Hght Fgur - Coprson of Msurd Sgnfnt Wv Hghts (Norlzd by H ) Agnst Corrtd Hndst Prdtons - Austrln North Wst Shlf Tkng ndpndnt, norlly dstrbutd rrors, w n xprss th hndst unrtnty v th pdf: fε ( ε ) = whr σ s th stndrd dvton of ε ι. ε σ Clbrton of hndst odl to lnt bs dos not nssrly rsult n unbsd xtr vlu prdtons. Th Copyrght 4 by ASME

3 ddton of ny rndo rror to dtst rsults n or hghly skwd prl dstrbuton provdd th rror s sytr (Fgur ). Hn, th fft of usng spl of hndst dt s to nrs th skwnss of th spl nd so ovr-prdt rturn vlus. Ths ppr s prrly onrnd wth thods for sttng th dgr of ovr-stton tht s lkly gvn tht th gntud of th hndst rror n b rsonbly ssssd usng th σ lultd durng lbrton. Tru Wv Populton 3) A rndoly dtrnd rror s ddd to h spl pont, tkn fro th dstrbuton of hndst rror vlus (.. drwn fro norl dstrbuton of zro n nd stndrd dvton σ ). 4) For h spl, nw xtr vlu ft s ondutd, nd rturn prod stts r d. 5) Th n of th rsultnt rturn vlus s thn found An stt of th ount of bs du to hndstng rrors s obtnd by tkng th dffrn btwn ths vlu nd th rturn vlu stt obtnd fro th orgnl dt. Rndo Error Skwd Populton Wv Hght Wv Hght Wv Hght A brodly slr pproh for ssssng unrtnts s dsrbd n [3]. Usd n ths wy, th ont-rlo pproh ould b qully wll dsrbd s prtr bootstrp thod. Anothr bootstrp thod tht ks or drt us of vlbl dt s th rsdul bootstrp thod dsrbd blow. Th Rsdul Bootstrppng Approh Th rsdul bootstrp thod s slr, but ks us of splng fro th tul dt rthr thn fro fttd dstrbutons: ) For h dt pont n th st of lbrtd hndst vlus, dd rndoly drwn rsdul (rror) fro th st of rsduls, ε ι (spld wth rplnt). ) Rpt ths lrg nubr of ts (.g. 5), nd ondut n xtr vlu ft on h dtst to obtn rturn prod stts. 3) Fnd th n of th st of rturn vlus. Fgur - Efft of Hndst Unrtnty on Spl Dstrbutons NUMERICAL SIMULATION METHODS FOR ACCOUNTING FOR MODELING ERROR Two wll stblshd sulton bsd thods for dlng wth probl of ths typ r Mont-Crlo thods nd bootstrppng. Th Mont-Crlo Approh An xpl of th Mont-Crlo pproh to ount for hndst rror s nurl sulton of th followng stps: ) An xtr vlu dstrbuton s fttd to lbrtd hndst dtst of sz N. ) A lrg nubr of spls (.g. ) of sz N r rndoly drwn fro ths fttd dstrbuton. On gn, n stt of th ount of bs du to hndstng rrors s obtnd by tkng th dffrn btwn ths vlu nd th rturn vlu found fro th orgnl dt. Not tht ths thod bootstrps th hndst rrors only, t s lso possbl to bootstrp th orgnl st of wv hghts s wll to ount for unrtnty du to spl sz. Both ths nurl thods r spl to pply, but suffr fro so potntlly srous drwbks whn dlng wth odlng rror: ) In both ss, w r ddng rror to dt tht lrdy ontns rror. W r ssung tht th bs du to ddtonl rror s quvlnt to th nhrnt bs du to usng ontntd hndst dt. Ths y b vld n lot of sttstl ppltons but s dbtbl for us n xtr wv nlyss du to tl snstvty, splly for lrg rrors. 3 Copyrght 4 by ASME

4 ) Extr wv nlyss usully uss th pksovr-thrshold pproh, whh rqurs dt bov gvn thrshold vlu only. Ethr th xdns ovr th thrshold r odld s Prto dstrbutd or truntd for of trdtonl xtrl dstrbuton (.g. Wbull) s usd. Both Mont-Crlo nd bootstrppng sulton thnqus los thr vldty whn onsdrng suh truntd dt - th ddton of ngtv rror to ponts nr th thrshold rsults n loss of dt, rtflly rdung th populton dnsty nr th thrshold pont. A thod by whh ths probls n b ddrssd s by onvolvng th hndst rror nto th xtr vlu dstrbuton usd to ondut th sttstl nlyss. In ths wy th ffts of hndst rror n b xpltly nludd, nd vld stts of bs for both truntd nd untruntd hndst dtsts n b obtnd. Ths pproh s rfrrd to hr s th onvoluton ntgrl pproh. CONVOLUTION INTEGRAL APPROACH If w ssu tht th hndst dt s rndo vrbl Z oprsd of th su of two rndo vrbls X (dstrbuton of th undrlyng wv populton) nd E (hndst odlng unrtnty), th probblty dnsty funton of Z (f Z ) s gvn n trs of th pdfs of X nd E by th onvoluton ntgrl: Z = X + E f () z = f () x f ( zx) Z X E If X nd E r both Gussn, ths ntgrl hs losdfor soluton, othrws ths ntgrl nnot b sply solvd prt fro so spl ss (s for xpl [4]). Th ntgrl s nonthlss nbl to nurl soluton. Mxu lklhood stton of ths onvolutd dstrbuton to st of hndst dt wll norport th hndst unrtnty nto ll rturn vlu stts nd onfdn ntrvls. If thr s vdn tht hndst rror s proportonl to wv hght n ltrntv forulton ould b usd: Z = XE fz() z = f X() x fe( z/ x) In prnpl, ny ssud dstrbuton for th undrlyng wv populton nd ny dsrpton of hndst unrtnty n b usd wth ths thod. In prt, th xu lklhood thod n gv poor stts for so dstrbutons (.g. th Gnrlsd Prto), n ths s th onvoluton thod y not b fsbl. To llustrt th pplton of th onvoluton ntgrl pproh, w wll ssu tht th wv populton dstrbuton s two-prtr Wbull n for nd th hndst odlng rror s ndpndnt nd norlly dstrbutd. by: As dsussd bfor, th hndst rror n b dsrbd f E ( ε ) = ε σ Th undrlyng populton dstrbuton s gvn by th stndrd Wbull forul wth sl prtr nd shp prtr : x x fx ( x) = Aountng for thrshold vlu, t, blow whh dt s not usd, th onvoluton ntgrl bos: f (; z,,) t = Z = ( z x) x x σ ( z x) x x σ t x ( z x) x σ x ( z x) x σ t dz dz Ths n b solvd to rsonbl lvl of ury usng ny nurl ntgrton thod (.g. dptv Spson's rul) ovr so fnt bounds (.g. -). Applton of th xu lklhood stton thod thn nvolvs xzng th loglklhood funton for prtrs nd for th st of dt z..z n = Loglklhood fn. = log fz( z;,, t) n 4 Copyrght 4 by ASME

5 W n us ultvrt nurl xzton sh suh s th Qus-Nwton ln srh thod to solv for th prtrs. Ths produr y b slow, but hs th dvntg tht th Hssn trx of th lklhood funton s pproxtd s prt of th xzton lgorth. Du to th syptot proprts of xu lklhood sttors th (pproxt) Hssn trx n b usd drtly to provd stts of th vrn of th sl nd shp prtrs. Durng th ln srh produr th Hssn s pproxtd s: LogL LogL A LogL LogL Th ovrn trx of th prtrs s thn gvn by: Vr() ˆ Cov(,) ˆ ˆ = Cov(,) ˆˆ Vr () ˆ [ A] Th 'tru' rturn vlu (y) for gvn rturn prod (RP) nd nnul rt of ourrn of wv hghts xdng th thrshold (λ) s found by solvng: y = FX λrp / ˆ ˆ t = ˆ log + λrp ˆ whr F X s th uultv dnsty funton of x. Vrn n th rturn vlu s thn obtnd usng th dlt thod: Vr( ˆ) Cov( ˆ, ˆ) y y y y y y Vr( y) = Cov( ˆ, ˆ) Vr( ˆ) λ λ Vr( ˆ λ) Ths thod s n pproxt on nd should b usd wth dgr of uton, s lulton of th Hssn s pron to nurl rror. An ltrntv pproh s to us th profl lklhood thod to drv onfdn ntrvls. Ths thod s pplbl whn lrg nubr of oprsons btwn th hndst odl nd surnts hv bn d nd th stt of σ n b rsonbly xptd to b urt. If, howvr, thr s srty of Mtrx of sond prtl drvts T nforton vlbl for lbrton of th hndst odl, w n nstd us th vlbl rsdul rrors, ε, drtly n th fttng produr... w xs th lklhood of obtnng th two spls z nd ε j : n Loglklhood fn. = log[ f ( z ;,, t, σ )] = Z ε j + log[ πσ ] j= σ In ths forulton, th hndst stndrd dvton, σ, bos on of th prtrs to b lultd long wth nd. In ths wy, w not only ount for th unrtnty n th odld wv hghts but lso th unrtnty n th ount of rror nvolvd wth hndstng. COMPARISON STUDY USING KODIAK WAVE DATA A publly vlbl dtst of stor sgnfnt wv hghts, surd ovr prod of 9 yrs t th Kodk st off th Alskn ost, hs bn dstrbutd by th Intrntonl Assoton for Hydrul Rsrh (IAHR) [6]. W shll rtflly ntrodu rndo rror to ths dtst n ordr to tst th ury of th proposd onvoluton ntgrl pproh for ssssng bs, nd to opr t gnst nurl sulton thods. Th Kodk dtst s shown n Fgur 3, long wth truntd -prtr Wbull ft of th dt. A thrshold lvl of 6.85 s usd hr, orrspondng wth totl of 49 wv hght surnts. A stor ourrn frquny of 49/9=.579 stors/yr s usd to obtn rturn prod stts. Bsd on ths dt, th nonl yr rturn prod wv hght s % onfdn ntrvls du to splng vrblty r nludd on th plot (syptot norl bounds). In ordr to ssss th dgr of bs n th yr rturn vlus du to rndo rror n th dt, odfd dtsts wr rtd. Norlly dstrbutd rndo rrors of zro n nd.8 stndrd dvton wr ddd to th Kodk dtst to provd odfd dtsts 3. Rfttng of th truntd Wbull dstrbuton to h odfd dtst to obtn yr rturn vlus produd th rsults shown n Fgur 4. As xptd, th gnrl trnd s postv bs n th prdtd yr wv vlu, wth th ount of bs rngng fro -.6 to Th n bs du to th ddton of rndo rror ws In othr words, for ths st of dt,.8 stndrd dvton n th ury of h surnt 3 Ths rrors r usd for llustrtv purposs only nd do not nssrly rflt th hndst unrtnty of th Kodk dtst. 5 Copyrght 4 by ASME

6 wll, on vrg, rsult n.388 ovrstt of th yr wv vlu (n ovrstton of round 3.3%). Wv Hght () Rturn Prod (yrs) 5 5 Prdtd Quntl Fgur 3 - Truntd Wbull Plot for Kodk Dt Rltv Frquny Bs n th yr Wv du to th Inluson of Modlng Error () Fgur 4 - Hstogr of Clultd Bs n H du to th Inluson of Modlng Error (Rsults fro Dtsts, Consstng of Norlly Dstrbutd Error of Zro Mn nd.8 Stndrd Dvton ddd to th Kodk Dt) A tst spl wth bs nr ws sltd for us n oprson study of th Mont-Crlo, rsdul bootstrp, nd onvoluton ntgrl thods for ssssng bs. Th rsults r prsntd n Tbl. trtons wr usd for both th Mont-Crlo nd th rsdul bootstrp nlyss. Th rror rsduls wr usd drtly n th onvoluton ntgrl nlyss. Tbl - Coprson of Dffrnt Mthods of Esttng Bs n yr Rturn Prod Wvhghts for Known Hndstng Error (σ =.8) Atul Kodk Dt Avrg Efft of Error Mont- Crlo Rsdul Bootstrp Convoluton Intgrl Prdtd H Bs % Error n Bs Estt % -49 % -4.6 % It n b sn tht th onvoluton ntgrl pproh pprs to ssss odlng bs rsonbly wll. Th Mont- Crlo nd Rsdul Bootstrp thods ppr lss urt. Whlst ths rsults donstrt tht hndst unrtnty n hv n fft on th prdton of dsgn ondtons, th vrton du to ths unrtnty rns qut sll n oprson to th splng unrtnty (s for xpl th onfdn ntrvl n Fgur 3). For yr ondtons, th ddd oplxty of usng th onvoluton ntgrl pproh dos not ppr wrrntd gvn tht th dgr of bs s lkly to b lss thn.5 for typl lvls of hndst unrtnty. Howvr, n ss whr thr s lrg dgr of unrtnty n th hndst odl, prhps du to thr bng only fw rlbl surnts of stor ondtons vlbl n th rgon, or thr s nd to xn vry long rturn prod vlus, suh s n rlblty study, th fft of hndst unrtnty s uh or portnt. Ths pont s llustrtd by Tbl, whh shows rsults for th yr ondton for hndst stndrd dvton of.. Not tht f hndst unrtnty s ngltd, th yr wv hght s ovrsttd by ovr tr. Tbl - Coprson of Dffrnt Mthods of Esttng Bs n yr Rturn Prod Wvhghts for Known Hndstng Error (σ =.) Atul Kodk Dt Avrg Efft of Error Mont- Crlo Rsdul Bootstrp Convoluton Intgrl Prdtd H Bs % Error n Bs Estt % -6.6 % +.3 % 6 Copyrght 4 by ASME

7 CONCLUSIONS It hs bn shown hr tht th us of hndst dt to ondut xtr vlu nlyss rsults n sll but sgnfnt postv bs n th lulton of rturn vlus of t-on prtrs suh s sgnfnt wv hght. Gvn tht hndst unrtnty n b wll sttd v lbrton studs, thr s no rson why th fft of hndst unrtnty nnot b norportd nto n xtrl nlyss. Ths ppr prsnts thod to oplsh ths, nd donstrts tht ths nw thod prfors bttr thn xstng nurl sulton thods n prdtng bs, prtulrly whn onsdrng truntd dtsts. Tsts usng stndrd on wv dtst ndt tht th fft of hndst unrtnty on rturn vlus s of prtulr portn for rlblty studs, nd n rgons whr th surnt dtbs vlbl for odl lbrton s sll. ACKNOWLEDGMENTS Th uthors wsh to grtfully knowldg th support nd ssstn provdd by Woodsd Enrgy Ltd. nd by Prof. In Js of Murdoh Unvrsty. Fnnl support fro th Cooprtv Rsrh Cntr for Wldd Struturs s lso knowldgd. REFERENCES [] R A.M. nd Crdon V.J. 98, Tst of Wv Hndst Modl Rsults Agnst Msurnts Durng Four Dffrnt Mtorologl Systs, Offshor Thnology Confrn, Houston, USA, OTC 433 [] Hvr S., 994, Unrtnts Rltd to Hndst Wv Dt, Pro. Int. Conf. Offshor Mhns nd Art Eng., Vol. II, p57-65 [3] Erl M.D. nd Br, L. 98, Effts of Unrtnts on Extr Wv Hghts, J. Wtrw. Port Costl On Dv. Pro. ASCE, 8, WW4, p [4] Vn Gldr P.H.A.J.M. 999, Sttstl Mthods for th Rsk-Bsd Dsgn of Cvl Struturs, Co. on Hydrul nd Gothnl Eng. [6] vn Vlddr G., God Y., Hwks P., Mnsrd E., Jsus Mrtn M., Mthsn M., Pltr E. nd Thopson E. 993, Cs Studs of Extr Wv Anlyss: A Coprtv Anlyss, Pro. WAVES 93 Conf., 6-8 July, Nw Orlns, USA p Copyrght 4 by ASME