Empirical Estimation of Probability Distribution of Extreme Responses of Turret Moored FPSOs

Transcription

1 Empirical Estimatio of Probability Distributio of Extreme Resposes of Turret Moored FPSOs Amir H. Izadparast Research ad Developmet Group, SOFEC. Housto, Texas, USA Aru S. Duggal Research ad Developmet Group, SOFEC. Housto, Texas, USA ABSTRACT I this article, the probability distributio of dyamic resposes of turret moored FPSOs is closely studied ad the effects of o-liearity o the respose distributio ad the extreme statistics are evaluated. For this purpose, sample data sets obtaied from two experimetal model tests studyig the respose of typical exteral turret moorig systems desiged for deepwater ad shallow-water coditios are utilized. The focus here is o the extreme statistics of the moorig lie tesio ad vessel horizotal offset. The probability distributio of measured data are estimated usig commoly used distributio models of liear ad o-liear radom variables. Additioally, the applicatio of three-parameter Rayleigh distributio model i estimatig the probability distributio of liear ad o-liear resposes of turretmoored FPSOs is studied. The performace of these distributio models i represetig the sample distributio ad predictig extreme statistics is evaluated. KEY WORDS: o-liear respose, Turret moored system, Extreme statistics, Probability distributio estimatio, three-parameter Rayleigh ITRODUCTIO Some of the dyamic resposes of turret moored Floatig Productio, Storage, ad Offloadig (FPSO) uits i extreme eviromets are kow to be o-liear radom variables. Amog those are the horizotal offset of the FPSO ad the tesio i the moorig legs. The o-liearity i these variables depeds o may factors icludig the moorig system desig, evirometal coditios, the vessel characteristics, etc. I aalysis of these turret moored FPSOs, complex aalytical, umerical, ad experimetal methods are utilized to model the o-liear resposes. Additioally, statistical models are applied to predict the probability distributio of the radom resposes. The probability distributios are evetually used to predict the extreme statistics, e.g. expected maximum ad most probable maximum, ad to obtai the desig values. It is commo i the field of offshore egieerig to assume that the amplitudes of a radom process follow the Rayleigh distributio law. The desig guidelies ad recommedatios usually use the Rayleigh model to estimate the extreme statistics of moorig lie tesio ad vessel offset (see e.g. API, ABS, ad DV ). The Rayleigh distributio model assumes that the radom process ca be approximated as a liear radom variable. This assumptio may result i sigificat uderestimatio of extreme statistics whe the respose is ot liear. Previous studies o the slow-drift respose of floatig structures i irregular seas idicate that these radom variables could be highly o-liear (e.g. aess 98, Stasberg 99, Stasberg 99, Stasberg ). Liu ad Bergdahl (998) show that the probability distributio of extreme moorig lie tesios caused by wavefrequecy excitatios may also deviate from the Rayleigh distributio of the liear radom variables. I this study, the probability distributio of moorig lie tesio ad vessel horizotal offset are closely studied. For this purpose, experimetal data sets obtaied from two model tests studyig the respose of exteral turret FPSOs i extreme evirometal coditios are used. The cases studied here represet typical deepwater ad shallow-water moorig system desigs. The probability distributio of low-frequecy, wave frequecy, ad total moorig lie tesio as well as the low-frequecy horizotal offset is estimated usig commoly used probability distributio models. As a alterative, the threeparameter Rayleigh distributio model is applied to estimate the probability distributio of turret moored reposes. The three-parameter Rayleigh distributio model was origially developed by Izadparast ad iedzwecki (9, ) for secod-order Stokes type variables. Here, the performace of this model i capturig the probability distributio of low-frequecy ad wave-frequecy resposes is evaluated. PROBABILITY DISTRIBUTIO OF O-LIEAR AMPLITUDES The amplitudes of a radom variable are defied as the maximum observatio betwee each two cosecutive zero-upcrossigs. The ormalized form of amplitudes is obtaied from a () where is the mea, is the stadard deviatio of, ad a is the amplitude. Assumig that is a liear arrow-baded radom variable, it ca be show that the amplitudes follow a Rayleigh distributio law with cumulative distributio fuctio (CDF) of

2 (Loguet-Higgis 9) exp F x x () Rayleigh distributio has bee widely used for ocea egieerig applicatios to predict the extreme statistics. However, it is well kow that the distributio of o-liear radom variables deviates from Rayleigh distributio. This issue is more sesible o the tail of the distributio where the o-liearity has a larger cotributio. A simplified represetatio of the secod-order radom variables ca be obtaied by assumig that the o-liear term is closely related to the squared of the liear variable, specifically () where is the amplitude of the o-liear process. Usig this trasformatio ad the distributio of liear amplitudes Eq. (), the CDF of becomes F xexp x () which is the well-kow Expoetial distributio. The Expoetial probability distributio is commoly used to describe the tail distributio of secod-order radom variables. A more geeral form of the Expoetial distributio was itroduced by Stasberg (99) for estimatio of extreme values of o-liear slow-drift resposes. The model was used by Fyllig ad Stasberg (99) ad Stasberg (99) for extreme offsets ad achor lie loads of turret moored systems ad reasoable agreemet betwee the model predictios ad experimetal data was observed. Later o, Stasberg () updated his Expoetial model to improve its performace for systems with very high ad very low low-frequecy dampig. I this model the o-liear amplitudes are defied as A B () ad cosequetly the CDF of the Expoetial distributio is chaged ito x F xexp B () A The three-parameter Weibull distributio is aother distributio model that has bee widely used to estimate the probability distributio of o-liear amplitudes. The structural form of the Weibull distributio is defied by three parameters, i.e. scale, shape, ad locatio parameters, specifically x F xexp () The Weibull distributio is a more geeral form of the Rayleigh distributio ad assumes the followig relatio betwee the liear ad o-liear radom amplitudes. (8) A Weibull distributio with,, ad reduces to the Rayleigh distributio of liear amplitudes. The three-parameter Weibull distributio model is usually used as a powerful data aalysis tool but the model parameters do ot have clear physical iterpretatios. Izadparast ad iedzwecki (9, ) itroduced the three-parameter Rayleigh distributio model for the amplitudes of secod-order Stokes type radom variables. I this model, the o-liear radom variable is defied usig the quadratic trasformatio (9) where is the amplificatio of the liear term, is the amplificatio of the quadratic term, ad is the shiftig betwee liear ad oliear variables. The three-parameter Rayleigh distributio essetially combies the cotributio of Rayleigh ad Expoetial distributios. As show i previous studies (Stasberg 99, 99, ad ad Fyllig ad Stasberg 99) the probability distributio of slow-drift respose is usually betwee the bouds defied by Rayleigh distributio ad Expoetial distributio ad therefore the threeparameter Rayleigh distributio should be able to model those behaviors. The three-parameter Rayleigh model assumes that the liear ad o-liear terms are phased-locked ad their peaks happe at the same time; therefore, the three-parameter Rayleigh model is appropriate for represetig the probability distributio of large amplitudes. Applyig the radom variable trasformatio rule o Eq. (9), the CDF of the three-parameter Rayleigh model for is obtaied as F xexp () 8 where, x () I the case of, the CDF becomes F x exp H x exp () 8 8 where H x is the step fuctio ad has a value of uity for x ad is zero for x. EXTREME STATISTICS The CDF of the maxima max i idepedet ad idetically distributed evets ca be obtaied from the ordered value statistics theory (Leadbetter, Lidgre, ad Rootze 98), specifically F x F x max () From that, the expected maximum E max ca be estimated from the itegratio E max xdf x () max It ca be show that for large, all the distributio models itroduced i the previous sectio belog to the Gumbel maximal domai of attractio with asymptotic distributio of F xexpexpx a max b () where a ad b are the Gumbel distributio parameters. The Gumbel distributio parameters are related to the parameters of the threeparameter Rayleigh distributio model as a ll () b l The relatio betwee the Gumbel distributio parameters ad the parameters of the Weibull distributio is obtaied i the form of a l () b l l The estimates of a ad b for the other distributios ca be obtaied by makig the followig substitutios i Eq. () For Rayleigh distributio Eq. ():,, ad For Expoetial distributio Eq. ():,, ad

3 For Stasberg s Expoetial distributio Eq. (): A, AB Assumig that max follows the Gumbel probability distributio fuctio, the expected maximum ca be estimated from E max a b EM (8) where EM. is the Euler-Mascheroi costat. As examples, the ormalized expected maximum of the Rayleigh ad Expoetial distributios i E.8 ad E max.9, respectively. evets are Aother issue to be discussed is the umber of idepedet cycles i a storm with a certai duratio (e.g. hr storm). The desig guidelies (e.g. API, ABS, ad DV ) recommed estimatig the umber of cycles as Tstorm Tz (9) where Tstorm is the storm duratio ad T z is the average meaupcrossig period of the process. The mea-upcrossig period of a wave-frequecy process is i order of -sec, which results i 8- cycles i a hr storm. The period of a low-frequecy process is commoly i order of -sec ad the umber of cycles i a hr storm is about 8-. For a true arrow-baded process T z ca be cosidered as the correlatio time of the process ad the correlatio betwee the cosecutive amplitudes is isigificat. I a actual process with a spectrum of fiite width, the cosecutive amplitudes are correlated ad the umber of idepedet cycles i a sigal is differet from the umber of cycles estimated by Eq. 9. It is commo to assume that the wave-frequecy respose is arrow-baded ad use the total umber of cycles for extreme estimatio. This approximatio usually results i slight overestimatio of extreme statistics. Applicatio of Eq. 9 for low-frequecy resposes remais questioable as these processes are usually ot arrow-baded. To address this issue, aess (989) estimated the umber of statistically idepedet cycles i slow-drift resposes as a fuctio of the lowfrequecy dampig i the system. Stasberg () estimated the umber of idepedet cycles i a low-frequecy respose by substitutig Tz i Eq. 9 with the correlatio time of the sigal defied as () where is the badwidth of the spectrum calculated from S f df S f df () S f is the oe-sided spectrum, ad f is the frequecy. This approach is followed here to estimate the umber of idepedet cycles i lowfrequecy sigals. Estimatig the umber of idepedet cycles i a sigal of combied wave-frequecy ad low-frequecy compoets is eve more challegig. Here, the umber of observed cycles are used for which is ot theoretically justified. Defiig a better estimate for the umber of cycles of combied low ad wave frequecy compoets requires further studies ad is ot i the scope of this article. MODEL PARAMETERS I order to use the Rayleigh distributio of the liear amplitudes Eq. () ad the Expoetial distributio Eq. (), oe eeds the estimates of the mea ad stadard deviatio of the radom process. This makes these two models very attractive whe oly limited iformatio about the radom variable is available. Stasberg s expoetial model requires some iformatio about the characteristics of the spectrum of max the o-liear respose ad the iput waves. It is worth metioig that the parameters of the Rayleigh distributio model, Expoetial distributio, ad Stasberg s Expoetial model ca be estimated from frequecy domai aalysis of the system while i order to estimate the parameters of the three-parameter Weibull distributio model ad the three-parameter Rayleigh distributio model, a sample timeseries is required. Estimates of Stasberg s model parameters, i.e. A ad B, ca be obtaied from (Stasberg ) A/ A sm y () B D M A where, M is the ratio of the badwidth of the wave group spectrum ad the badwidth of the respose spectrum, M wave group respose () The spectrum badwidth is estimated from Eq. ad the other parameters used i Eq. are estimated from s M M y M A A M M D D s M y y A y D s M M A.. () The parameters of the three-parameter Weibull ad the three-parameter Rayleigh distributio are estimated usig a sample set of amplitudes. There are umerous parameter estimatio methods that ca be used to estimate the parameters of a probability distributio, e.g. method of maximum likelihood, method of least squares, method of momets, etc. Izadparast ad iedzwecki () used two momet based parameter estimatio methods, i.e. covetioal method of momets ad method of liear momets (L-momets) to estimate the parameters of threeparameter Rayleigh distributio. It was show i their study that the performace of the two parameter estimatio methods are similar for large samples while method of L-momets is more robust for small sample sizes. Here, the method of L-momets is applied to estimate the model parameters of the three-parameter Weibull ad the threeparameter Rayleigh distributios. I method of L-momets, the estimates of the parameters are obtaied by equatig the distributio momets with their correspodig ubiased sample L-momets. This will give a system of equatios to be solved for the ukow parameters. It ca be show that the relatios betwee the first three sample L- momets, i.e. l, l, ad l ad the parameters of three-parameter Rayleigh model are,.9 l l l l () where is the well-kow Gamma fuctio. Similarly, the relatios betwee the sample L-momets ad the three parameters of the Weibull model are derived as l l t l l ()

4 The formulatio for estimatig the sample L-momets from a sample ca be foud i Hoskig ad Wallis (99). SAMPLE DATA The data sets used i this study are obtaied from two large-scale model tests performed o exteral turret-moored FPSOs. The first experimet represets a typical deepwater moorig system (water depth of more tha m) ad the secod experimet represets a typical moorig system desiged for shallow-water coditios (water depth of less tha m). A brief descriptio of these moorig systems is provided below. Due to cofidetiality cosideratios, more details of these projects caot be reported. Deepwater Moorig System: the moorig system cosists of taut moorig lies grouped i three budles of four moorig lies. The lies are desiged i a chai-polyester-chai cofiguratio. Shallow-water Moorig System: the moorig system cosists of cateary moorig lies grouped i four budles of three moorig lies. The lies are desiged i a chai-heavy chai-chai cofiguratio. The samples used here are measured durig site-specific -year retur period seastates with colliear wid, wave, ad curret. Each test was ru for a duratio equivalet to 9hr full-scale. The focus here is o the moorig lie top tesio ad the vessel low-frequecy offset. For this purpose, the tesio measuremets i the most loaded lie (widward) ad the least loaded lie (leeward) as well as the vessel surge offset at the turret locatio are selected. Fig. shows the power spectrums of the top moorig lie tesio ad the surge offset for the deepwater ad shallow-water examples. Similarly, i Fig., the power spectrums for the shallow-water example are preseted. ote that i both figures, the spectrums are ormalized by the sample variace. As expected ad also show i both Fig.s ~, the moorig lie lowfrequecy tesio is respodig to the slowly varyig surge offset. The wave-frequecy tesio is maily caused by the vertical motio at the locatio of the moorig lie origiated by vessel heave ad pitch motios. The spectrum of the widward lie tesio of the deepwater moorig system show i Fig. idicates that the low-frequecy tesio has sigificatly larger cotributio to the total tesio tha the wave-frequecy tesio. For the leeward lie of the deepwater moorig system, the cotributio of low-frequecy ad wave-frequecy compoets is comparable. I the case of shallow-water moorig system, see Fig. (), the total tesio of both widward ad leeward lies is domiated by the low-frequecy tesio compoet. S T ( f ) / T (/sec) Lie T (Leeward) Lie T (Widward) X Motio f ( Hz ) Fig.. The power spectrums of lie tesio ad vessel motio of the deepwater example. S X ( f ) / X (/sec) S T ( f ) / T (/sec) Lie T (Leeward) Lie T (Widward) X Motio f ( Hz ) Fig.. The power spectrums of lie tesio ad vessel motio of shallow-water example. SAMPLE PROBABILITY DISTRIBUTIOS The characteristics of the quatile distributio of the samples itroduced i the previous sectio are studied i this sectio. The quatile distributio (also kow as iverse of CDF) defies the relatio betwee the value of the radom variable ad the probability of exceedace P defied as u where u x F x. I frequecy domai aalysis, the low-frequecy ad wave-frequecy calculatios are performed idepedetly ad the the extreme statistics of these two compoets are combied with the mea of the process to estimate the extreme statistics of the total process. The challege i this approach is to model the correlatio betwee the wave-frequecy ad low-frequecy compoets correctly. Several studies have bee doe o methods of combiig these compoets (see e.g. aess 989b ad Liu ad Bergdahl 999). It is a commo idustry practice to either use the API formulatio (API ) or coservatively estimate the extreme statistics of the total process from a simple summatio of the extreme wave-frequecy ad extreme lowfrequecy compoets with the mea of the process. I time domai aalysis, the statistics of the total process ca be estimated directly from the sample results. Here, to better study the characteristics of the probability distributios, the probability distributios of wavefrequecy tesio, low-frequecy tesio, ad the total tesio are idividually studied. I the case of vessel offset, oly the probability distributio of the low-frequecy surge offset is importat. Here, the ormalized samples are obtaied from a a wave low a wave, low, () wave low where, is the measured timeseries, stadard deviatio of, is the mea of, S X ( f ) / X (/sec) is the a are the amplitudes of defied as the maximum observatio betwee each two cosecutive mea-crossigs, ad the terms wave ad low refers to the estimates of the wavefrequecy ad low-frequecy timeseries. I Fig. the quatile distributio of the ormalized wave-frequecy tesio amplitudes of the deepwater moorig system are preseted. Additioally, the quatile distributios of the Rayleigh model of liear amplitudes (Eq. ), the Expoetial model (Eq. ), the three-parameter Weibull model (Eq. ), ad the three-parameter Rayleigh model (Eq. ~) are preseted. Similarly, the quatile distributios of the ormalized low-frequecy tesio amplitudes ad ormalized total tesio amplitudes are preseted i Fig.s ~, respectively. For the low-frequecy tesio, the quatile distributio of the Stasberg s Expoetial model (Eq. ) is also preseted.

5 -wave PAR-WEIBULL PAR- PAR-WEIBULL PAR- - - P = ( - u ) a. Widward lie P = ( - u ) a. Widward lie - - -wave PAR-WEIBULL PAR- PAR-WEIBULL PAR- - - P = ( - u ) b. Leeward lie Fig.. Probability distributio of the amplitudes of wave-frequecy tesio of deepwater moorig system P = ( - u ) b. Leeward lie Fig.. Probability distributio of the amplitudes of total tesio of deepwater moorig system low EXP-STASBERG PAR-WEIBULL PAR- -low EXP-STASBERG PAR-WEIBULL PAR- -low - SAMPLE EXP -ST ASBERG PAR-WEIBULL PAR- - - P = ( - u ) a. Widward lie - P = ( - u ) b. Leeward lie Fig.. Probability distributio of the amplitudes of low-frequecy tesio of deepwater moorig system P = ( - u ) Fig.. Probability distributio of the amplitudes of slow drift motio of deepwater moorig system. As show i Fig., except for the last three observatios, the wavefrequecy tesio amplitudes of widward ad leeward lies closely follow the Rayleigh distributio of liear amplitudes. For these samples, Expoetial distributio sigificatly overestimates the amplitudes. It has bee observed that the estimates of the threeparameter Weibull ad the three-parameter Rayleigh model are i a close agreemet ad they are both successful i capturig the sample probability distributio. The low-frequecy tesio amplitudes of the widward lie (Fig. ) shows some level of o-liearity ad the sample distributio deviates from the Rayleigh distributio. The low-frequecy tesio amplitudes of the leeward lie, however, behave more liearly ad follow the Rayleigh distributio more closely. For both widward ad leeward examples, the Expoetial distributio teds to overestimate the large - -

6 amplitudes with small probability of exceedace. It has bee observed that the Stasberg s Expoetial model cosiderably performs better tha the origial Expoetial model. As expected, the samples of lowfrequecy tesio amplitudes cotai limited umber of observatios which could cause some cocers about the performace of the threeparameter Weibull ad the three-parameter Rayleigh distributios. However, as ca be see i Fig. both models are successful i capturig the o-liearity i the low-frequecy tesio amplitudes of the widward lie ad capturig the distributio of the low-frequecy tesio amplitudes i leeward lie. The quatile distributio of the total tesio of the widward lie shows some deviatio from the Rayleigh distributio which is a idicatio of weak o-liearity i this sample. The total tesio of the leeward lie ca be very well approximated by the Rayleigh distributio model. For both total tesio samples, Expoetial model sigificatly overestimates the large amplitudes. The three-parameter Weibull ad the three-parameter Rayleigh model foud to be reasoably accurate i capturig the probability distributio of oliear widward tesio amplitudes ad the liear leeward tesio amplitudes. Similar to Fig., the quatile distributios of the ormalized lowfrequecy vessel surge amplitudes are show i Fig.. As show here, the quatile distributio of the ormalized surge amplitudes is similar to that of the low-frequecy tesio amplitudes of the widward lie. The quatile distributios of the ormalized wave-frequecy tesio amplitudes, low-frequecy tesio amplitudes, ad total tesio amplitudes of the shallow-water moorig system are preseted i Fig.s ~ 9. Comparig the distributios of tesio amplitudes of the shallow-water moorig system to those of the deepwater moorig system, it ca be cocluded that the respose of shallow-water moorig system is cosiderably more o-liear. The tesio amplitudes, especially the low-frequecy tesio amplitudes, are almost expoetially distributed ad the Rayleigh distributio sigificatly uderestimates the amplitudes. The wave-frequecy ad low-frequecy tesio amplitudes ad cosequetly the total tesio amplitudes of the widward moorig lie seems to be slightly more o-liear tha the tesio amplitudes measured i the leeward lie. I this example, the Stasberg s Expoetial model foud to be a reasoable approximatio of the tail distributio of the low-frequecy tesio amplitudes. I all studied cases, the three-parameter Weibull model ad the threeparameter Rayleigh closely follow the sample distributios. As compared to the distributios estimated for the deepwater example, the differece betwee the tail of the three-parameter Weibull distributio ad the tail of the three-parameter Rayleigh model is more sesible i the shallow-water examples. The three-parameter Rayleigh distributio cosistetly has heavier tail ad predicts larger amplitudes with small probability of exceedace. Similar to what was observed i the deepwater example, the quatile distributio of the ormalized low-frequecy surge amplitudes show i Fig. is similar to that of the low-frequecy tesio amplitudes of the widward lie show i Fig. 8. -wave -wave PAR-WEIBULL PAR- - PAR-WEIBULL PAR- - - P = ( - u ) a. Widward lie - P = ( - u ) b. Leeward lie Fig.. Probability distributio of the amplitudes of wave-frequecy tesio of shallow-water moorig system. -low -low EXP -ST ASBERG PAR-WEIBULL PAR- - EXP -ST ASBERG PAR-WEIBULL PAR- - P = ( - u ) a. Widward lie P = ( - u ) b. Leeward lie Fig. 8. Probability distributio of the amplitudes of low-frequecy tesio of shallow-water moorig system. - -

7 PAR-WEIBULL PAR- - PAR-WEIBULL PAR- - - P = ( - u ) a. Widward lie - P = ( - u ) b. Leeward lie Fig. 9. Probability distributio of the amplitudes of total tesio of shallow-water moorig system. -low EXP -ST ASBERG PAR-WEIBULL PAR- - - P = ( - u ) Fig.. Probability distributio of the amplitudes of slow drift motio of shallow-water moorig system. The estimates of the expected maximum of the ormalized moorig lie tesio i a hr storm for the deepwater ad shallow-water moorig systems are provided respectively i Table ~. The estimates of the ormalized expected maximum of the low-frequecy surge offset are similar to those of the low-frequecy tesio of the widward lie. I these tables, the sample estimate of the expected maximum is obtaied by dividig the total 9hr timeseries ito three hr samples ad averagig the maximum of the three samples. The values show i the parethesis represets the rage of the maximum amplitudes observed i the three hr samples. As show here, the variability i the sample maximums is sigificat. I additio to the differeces betwee the probability distributio of wave-frequecy ad low-frequecy resposes discussed before, the cosiderable differece i the umber of cycles of these sigals plays a importat role i the differece betwee the ormalized expected maximum of the two processes. I Table ~, the predictios of differet distributio models are compared. The results show i these tables cofirm the observatios made earlier comparig the quatile distributios. I geeral, the extreme estimates of Rayleigh ad Expoetial distributios respectively defie the lower ad upper bouds of the estimates. Rayleigh distributio teds to uderestimates the extreme statistics, which could result i sigificatly uder predicted extremes i case of o-liear resposes. The Expoetial distributio teds to overestimate the extreme statistics, which could results i too coservative estimates for liear ad weakly o-liear resposes. It has bee see that the Stasberg s Expoetial model is robust i predictig the extreme statistics of low-frequecy resposes ad its estimates of extreme statistics match the sample estimates reasoably well. I the case of deep-water moorig system, the estimates of the three-parameter Weibull model ad the three-parameter Rayleigh model are reasoably close ad agree with the sample estimates. The predictios of the three-parameter Rayleigh distributio model for the highly o-liear amplitudes of the shallow-water moorig system are cosistetly larger tha those of the three-parameter Weibull distributio. I the studied examples, the three-parameter Rayleigh distributio model seems to be performig better or as well as the threeparameter Weibull distributio. Table. ormalized expected maximum of the moorig lie tesio of the deepwater moorig system. Model Sample Wave. (. -.) Widward Low. (. -.8) Total. (. -.) Wave. (. -.) Leeward Low. (. -.) Total.9 (.9 -. ) Rayleigh Expoetial Par. Rayleigh Par. Weibull Stasberg Expoetial Table. ormalized expected maximum of the moorig lie tesio of the shallow-water moorig system. Model Sample Wave.8 (. -.9) Widward Low. (. -.) Total.8 (.9 -.) Wave. (.8 -.) Leeward Low. (. -.) Total. (. -.) Rayleigh Expoetial Par. Rayleigh Par. Weibull Stasberg Expoetial

8 COCLUSIOS The mai goal of this study was to study the characteristics of the probability distributio of moorig lie tesio ad low-frequecy vessel horizotal offset of turret moored FPSO. For this purpose, the sample data sets obtaied from two experimetal model tests of exteral turret moored systems i extreme evirometal coditios are used. The examples studied here represet typical deepwater ad shallow-water moorig systems. I the case of moorig lie tesio, the behavior of the wave-frequecy, low-frequecy, ad total tesio is idividually studied. The statistics of both most loaded lie (widward) ad least loaded lie (leeward) are preseted. It has bee observed that the respose of the shallow-water moorig system is cosiderably more o-liear tha the respose of deepwater moorig system. I both moorig systems, the o-liearity is more sesible i the widward lie tha the leeward lie. I the case of deepwater moorig system the o-liearity i the moorig lie tesio is maily sourced from the low-frequecy tesio, while i the shallow-water example both wavefrequecy ad low-frequecy tesio compoets are highly oliear. I both shallow-water ad deepwater desigs the characteristics of the distributio of low-frequecy surge motio is similar to those of low-frequecy tesio of the widward lie. I order to estimate the probability distributio of o-liear radom variables four distributio models, i.e. Expoetial, Stasberg s Expoetial, three-parameter Weibull, ad three-parameter Rayleigh, are utilized. The models are used to estimate the sample distributio ad the statistics are compared to those of Rayleigh distributio of liear amplitudes. The Rayleigh distributio is commoly used i offshore idustry to estimate the extreme statistics. Usig the measured sample data, it is cofirmed that the Rayleigh distributio sigificatly uderestimates the extreme statistics of o-liear radom variable. The Expoetial probability distributio defied the upper limit for the studied examples ad overestimated the extreme statistics whe the respose is ot highly o-liear. Stasberg s modificatio to Expoetial distributio foud to improve the performace of the origial model. The three-parameter Rayleigh distributio model was cosistetly successful i estimatig the probability distributio of data. The model has the flexibility to capture the distributio of liear ad o-liear variables. I most studied cases, the estimates of the threeparameter Rayleigh distributio models were reasoably close to those of the widely used three-parameter Weibull distributio. The differece betwee the tails of the two models was sesible for highly o-liear resposes of the shallow-water system. For those samples, the threeparameter Rayleigh distributio model estimated larger expected maximum, which were closer to the sample estimates. distributios usig the method of L-momets, J of Applied Ocea Research, Vol, pp -. 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