Proceedngs of the 1 th Internatonal Sh Stablty Worksho 1 Onboard Sea State Estmaton Based on Measured Sh Motons ABSTRACT Ulrk D. Nelsen Techncal Unversty of Denmark, Kgs. Lyngby, Denmark Davd C. Stredulnsky Defence R&D Canada Atlantc, Halfax, NS, Canada It s ossble to obtan estmates of the sea state at the secfc oston of an advancng vessel by rocessng measurements of the vessel s wave-nduced resonses. The analogy to a wave rder buoy s clear, although the stuaton of an advancng sh s more comlex due to forward seed. The aer studes the wave buoy analogy, and a large set of full-scale moton measurements s consdered. It s shown that the wave buoy analogy gves farly accurate estmates of sea state arameters when comared to estmates from real wave rder buoys. KEYWORDS Sea state estmaton; moton measurements; advancng sh; decson suort systems. INTRODUCTION Sea state arameters, or the drectonal wave energy sectrum, around a sh are needed on a contnuous bass for navgatonal and oeratonal gudance to a sh's master. The lkelhood of arametrc roll, for examle, deends amongst others on the sea state n whch the sh oerates. Thus, f the sea state s contnuously estmated t s ossble to rase a warnng f, say, vessel seed or course s n a regon where arametrc roll can be trggered (Jensen, 11). The evaluaton of a vessel s erformance (Hansen and Lützen, 1) requres also nut of the sea state arameters, so onboard wave estmaton s hghly relevant for any tye of montorng and/or decson suort system on shs. In the lterature there are reorts (e.g. Isek and Ohtsu, ; Nelsen, 6, 8; Pascoal et al., 7; Tannur et al., 3) about the estmaton of sea state arameters usng measured sh resonses (e.g. moton data) where the sh, to make an analogy, acts as a wave rder buoy for whch reason the methodology s called the 'wave buoy analogy'. The fundamental nut to the wave buoy analogy s a set of resonse measurements where the ndvdual resonse bascally can be any one as long as a lnear (comlex-valued) transfer functon may be assocated to the resonse. The wave buoy analogy rovdes a robust alternatve to wave radars by utlsaton of onboard resonse measurements that are often carred out rresectvely on many of today s navy and commercal vessels. Consequently, the wave buoy analogy s also a relatvely nexensve estmaton concet, snce the system develoment s assocated to software only. Ths study consders sea state estmaton from full-scale moton measurements obtaned durng sea trals conducted by DRDC Atlantc on one of ther research vessels. In connecton wth the sea trals wave measurements were made usng tradtonal wave buoys and the study comares these measurements wth those estmated from the measured moton resonses of the sh by alcaton of the wave buoy analogy. Partcularly, the study nvestgates how the sea state estmates are nfluenced by the secfc
Proceedngs of the 1 th Internatonal Sh Stablty Worksho selecton of sh moton resonses consdered. The man objectve of the study s to further valdate the wave buoy analogy. CONCEPT AND THEORY PARAMETRIC MODELLING Man Prncle The concet of the wave buoy analogy bulds on a comarson between measurements of resonse sectra and calculated ones. On the bass of an error calculaton, acton s taken to mnmse the dscreancy between the measured and the calculated sectra and ths rocedure s reeated untl an accetable degree of convergence has been reached, see Fg. 1. Fg. 1: The fundamental rncle of the wave buoy analogy. The teraton roblem (Fg. 1) of the wave buoy analogy can be handled by dfferent aroaches. Two aroaches that have receved artcular nterest are formed by Bayesan modellng and arametrc modellng. Bayesan modellng reles on the fndng of the dscrete sectral comonents of a frequency-drectonal wave sectrum, whereas arametrc modellng assumes the drectonal wave sectrum to be formed by a set of arametersed wave sectra wth due account for drectonal sreadng. The two aroaches should not been seen as comettors but rather as comlementary, snce each method has ts own advantages and dsadvantages (e.g. Nelsen, 6, 7; Pascoal and Guedes Soares, 8). It s worth to menton that a thrd aroach buldng on Kalman flterng has been formulated by Pascoal and Guedes Soares (9) for resonse-based wave estmaton. However, as reorted by the authors, ths aroach s stll n a develong hase and needs further elaboraton. Indeendently of the estmaton aroach, the study of data from an advancng sh means that the so-called trle-valued functon roblem needs to be consdered. Ths roblem stems from the transformaton of encounter frequences nto true wave frequences whch s made dffcult due to the Doler effect. In the lterature, some studes are restrcted to FPSO vessels, (e.g. Tannur et al., 3; Smos et al., 1; and Pascoal et al., 7) whch means that the trle-valued functon roblem s avoded. Isek and Ohtsu () showed how to nclude resonse measurements from an advancng sh, and the resent study deals wth sea state estmaton from a sh wth forward seed. In the remanng art of ths secton, the fundamental theory of the wave buoy analogy wll be brefly outlned, when arametrc modellng (e.g. Nelsen, 6; Tannur et al., 3; Pascoal et al., 7) s aled. The detals can be found n the mentoned lterature. Governng Equatons The governng equaton system of the wave buoy analogy orgns from the electrc flter analogy (St. Dens and Person, 1953), whch assumes lnearty between the resonse sectrum S j (ω e ) of the th and jth resonses and the drectonal wave sectrum E(ω e,θ) S j ( e ) ( e, ) j ( e, ) E( e, ) d (1) where ω e and θ are the encounter wave frequency and the relatve wave headng, resectvely. The left-hand sde s the measured cross (resonse) sectrum whereas the rght-hand sde s the calculated cross sectrum usng the estmated wave sectrum E(ω e,θ) and the resonse amltude oerators (RAOs) n terms of comlexvalued transfer functons Φ(ω e,θ). The bar denotes the comlex conjugate. The measured cross sectra,.e. the left-hand sde of Eq. (1), are obtaned by cross-sectral analyss, and n the resent model multvarate autoregressve
Proceedngs of the 1 th Internatonal Sh Stablty Worksho 3 modellng s aled (Neumaer and Schneder, 1; Nelsen, 6). In ths study, the on-ste wave sectrum s obtaned by arametrc modellng whch means that the comlete frequency-drectonal sectrum s estmated through an otmsaton roblem derved from Eq. (1). The soluton to the roblem s a set of (otmsed) wave arameters that, together wth a arametersed wave sectrum, characterse the sea state. The arametersed drectonal wave sectrum s chosen to be a ffteen-arameter tr-modal sectrum that allows for mxed sea such as wnd and swell, snce the sectrum s a summaton of three base sectra. Bascally, the sectrum s smlar to the tenarameter sectrum suggested by Hogben and Cobb (1986), although they consder a summaton of only two arametersed fve-arameter sectra. The aled arametrc exresson reads 1 3, 1 H s, E(, ) A( s ) 1 1 ( ), 1 s, mean cos ex () wth H s beng the sgnfcant wave heght, λ s the shae arameter of the sectrum, θ mean s the mean relatve wave drecton, ω s the angular eak frequency, and s reresents the sreadng arameter. The constant A(s) s ntroduced to normalse the area under the cos s curve. The constant s gven by s1 ( s 1) A( s) (3) (s 1) where Γ denotes the Gamma functon. It should be noted that the sreadng arameter(s) s s not ncluded n the otmsaton, whch means that a total of twelve arameters are to be otmsed. The sreadng arameter s modelled as a functon of wave frequency and as functon of the rncal arameter s max, cf. Goda () 5 cel ( / ) s s.5 cel( / ) s max max,, () where s max = 5 n ths analyss, as ths value characterses wnd waves and/or swells wth a short decay dstance, cf. Goda (), whch s assumed to be alcable for the gven envronmental condtons and geograhcal area. In Eq. (), cel rounds towards lus nfnty and s a numercal technque ntroduced to stablse the otmsaton. The actual otmsaton roblem s establshed by mnmsng the dfference between the left- and the rght-hand sde of Eq. (1), wth the wave sectrum E(ω,θ) gven by Eq. (). The mlementaton of the roblem s, however, not straght forward due to the trle-valued functon roblem that must be taken nto account for advancng shs. Moreover, the resent model consders an energy conservaton requrng the th order sectral moment of the left- and rght-hand sde to be dentcal. Fnally, constrants are set for the relatonsh between sgnfcant wave heght and zero-crossng erod, and t s also taken nto account that wave estmatons between two consecutve tme nstants cannot vary too much, leavng out detals and further argumentaton. FULL-SCALE MOTION MEASUREMENTS Sea Trals In the followng, sea states wll be estmated on the bass of moton measurements obtaned durng sea trals. The sea trals have been conducted by DRDC Atlantc n Nov./Dec. 8 usng the Canadan Navy research sh CFAV Quest (Fg. ). The rncle artculars of Quest are shown n Table 1. A comlete descrton of the sea trals s gven n Stredulnsky (1) but t s noteworthy that n ths analyss consderaton s gven to 16 sets of trals comrsng a total of 96 runs each of aroxmately 5 mnutes duraton. Durng all runs, moton measurements were recorded together wth nformaton about the on-ste sea state obtaned by three wave rder buoys. The moton measurements were taken aroxmately
Proceedngs of the 1 th Internatonal Sh Stablty Worksho Table 1: Prncle artculars of CFAV Quest. Dslacement Length between erendculars Beam Draft Block coeffcent Metacentrc heght 35 t 71.6 m 1.8 m.8 m.59.31 m 5 km southeast of Halfax, cf. Fg. 3, n close vcnty to the MEDS (Marne Envronmental Data Servce) moored buoy at Staton C137; the other two wave buoys were a TRIAXYS TM and a Traxys MINI TM deloyed n a drftng mode. The moton measurements were obtaned by the DRDC Sh Moton ackage (Brunt, 3) whch was nstalled at the sh center-of-gravty (COG). The sh moton sgnals were dgtsed usng a lato comuter wth a LabVew data acquston card samlng at Hz. Tme hstores recorded from the moton ackage ncluded the roll angle, roll rate, tch angle, tch rate, yaw rate and longtudnal, lateral, and vertcal acceleratons and data was stored to fle every mnutes. Fg. 3: Voyage ma of sea trals. Fg. : Run attern of tral no. 1. Fg. : The Canadan Navy research sh CFAV Quest. For each set of trals, sx runs were conducted so that sx dfferent relatve wave headngs were exerenced. The run attern of tral no. 1 s seen n Fg., where run 1 corresonds to head sea, run to followng sea, run 3 to starboard bow, run to ort quarter, run 5 to ort beam, and run 6 corresonds to starboard beam. All other trals had ( relatve ) run atterns smlar to that of tral no. 1, although the absolute headng would not be the same. Transfer Functons of the Sh The comlex-valued transfer functons have been calculated usng the n-house code SHIPMO7 (McTaggart, 1997) at DRDC Atlantc. SHIPMO7 s to a large extent a classcal str theory code (Salvesen et al., 197), but the code ncludes aendage and vscous forces and t uses an teratve rocedure to obtan the roll amltude and effectve lnearsed damng for the rescrbed sea condtons. Other detals of the code, ncludng addtonal references to
Proceedngs of the 1 th Internatonal Sh Stablty Worksho 5 verfcaton studes of SHIPMO7, can be found n Stredulnsky and Thornhll (11). Selectng Resonses for Sea State Estmaton Tycally, t s consdered as a good comromse to nclude three resonses at a tme (Nelsen, 6), when sea state estmaton s carred out on the bass of Eq. (1). If fewer resonses are used there may be a lack of nformaton, whereas more than three resonses ncreases the comutatonal tme sgnfcantly. The secfc selecton of whch three resonses to nclude n the estmaton analyss forms a general roblem. In ths analyss 8 combnatons, or sets, of three resonses have been formed from the moton measurements recorded. The consdered sets are: Set 1: {roll angle, tch angle, vertcal acc.} Set : {roll angle, tch angle, horzontal acc.} Set 3: {roll angle, tch rate, vertcal acc.} Set : {roll angle, tch rate, horzontal acc.} Set 5: {roll rate, tch angle, vertcal acc.} Set 6: {roll rate, tch angle, horzontal acc.} Set 7: {roll rate, tch rate, vertcal acc.} Set 8: {roll rate, tch rate, horzontal acc.} SEA STATE ESTIMATION Results The analyss of data has been conducted as a ostvoyage rocess and the results of the wave buoy analogy (WBA) are gven n Fgs. 5-7 that show the sgnfcant wave heght, H s, the zero-crossng erod, T z, and the absolute mean wave drecton, ν, resectvely. For the ndvdual wave arameter, the true value, obtaned as the mean value of the three wave rder buoys (MEDS, TRIAXYS, Traxys MINI), s ncluded n the fgures wth legend DRDC. The estmated sgnfcant wave heghts of the three wave rder buoys shows n some cases more than a lus/mnus 15% scatter comared to the mean value. Smlarly, a dfference of more than lus/mnus 1 second s observed for the crossng erod, whereas the mean wave drecton between the buoys dffers more than lus/mnus deg. n some cases. The corresondng scatter bands have been ncluded n Fgs. 5-7. Two outcomes of the wave buoy analogy are resented n Fgs. 5-7: The results corresondng to the wave estmatons obtaned by use of Set of the moton measurements, and the results acheved when all the estmatons obtaned n the ndvdual runs by Sets 1 to 8 are averaged. H s [m] 5 3 1 WBA (Set ) WBA (mean) DRDC (+/ 15%) 1 3 5 6 7 8 1 11 1 13 1 15 16 17 Tral (runs) Fg. 5: Estmates of sgnfcant wave heght by wave buoy analogy (WBA) and wave rder buoys (DRDC). T z [s] 1 8 6 WBA (Set ) WBA (mean) DRDC (+/ 1 sec.) 1 3 5 6 7 8 1 11 1 13 1 15 16 17 Tral (runs) Fg. 6: Estmates of zero-crossng erod by wave buoy analogy (WBA) and wave rder buoys (DRDC). ν [deg.] 3 1 WBA (Set ) WBA (mean) DRDC (+/ deg.) 1 3 5 6 7 8 1 11 1 13 1 15 16 17 Tral (runs) Fg. 7: Estmates of the mean (absolute) wave drecton by wave buoy analogy (WBA) and wave rder buoys (DRDC). Dscusson Based on Fgs. 5-7 t s seen that the results of the wave buoy analogy comare reasonably well wth the mean estmates by the wave rder buoys for nearly all runs. Snce the true wave arameters are known, as a result of the deloyment of real wave rder buoys n the sea tral area, t s easy to select the set of moton measurements that leads to the most accurate sea state arameters, when the wave buoy analogy s aled for wave
Proceedngs of the 1 th Internatonal Sh Stablty Worksho 6 estmaton. In ths way, Set of the moton measurements has been selected. The selecton has been based on a combned root-mean-squarre dfference calculaton consderng all wave arameters (H s, T z, ν). It s clear that such a selecton s osble only because the true wave arameters are known. Under normal oeratonal condtons, consderng any tye of vessel at an arbtrary oston n the ocean, the wave arameters are unknown and the selecton of the best set of resonses must be based on other means. Untl recently, no research has studed the develoment of aroaches that can be used to select automatcally the most sensble set of resonses for artcular oeratonal condtons, when the wave buoy analogy s used for wave estmaton. However, Lajc (1) and Lajc et al. (1) have develoed deas that can be useful for the secfc urose. The deas have been derved wthn the feld of control theory, and relmnary results ndcate that an automatcal selecton of a set of resonses s ossble. In the future t wll be of nterest to study ths toc further and to aly and elaborate on the aroaches (Lajc, 1; Lajc et al. 1) usng, e.g. the set of full-scale data studed n ths aer. As an alternatve to select one secfc set of moton measurements t s nterestng to note (Fgs. 5-7) that the mean value, formed by the average of the 8 sets of resonses, rovdes also a far estmate of the ndvdual wave arameter for all runs. The standard devaton on the sgnfcant wave heght of the 8 sets are seen n Fg. 8. The sreadng s normalsed wth the mean value of the wave rder buoys and s seen to be less than 3 ercent n all runs. Thus, t s a reasonable oton to consder the 8 sets all together. However, the dsadvantage of ths aroach s the ncreased comutatonal tme, whch would be an ssue for real-tme onboard sea state estmaton. On the other hand, for the secfc mlementaton of the wave buoy analogy, usng arametrc modellng, t takes aroxmately 1 mnutes to obtan the sea state estmate of one run usng all 8 sets of moton measurements (ntel CORE TM 5,. GHz). Consequently, t would be feasble to udate the sea state estmate at a reasonable tme nterval. [%] 3 1 1 3 5 6 7 8 1 11 1 13 1 15 16 17 Tral (runs) Fg. 8: The relatve sreadng on sgnfcant wave heght usng the 8 dfferent sets of measurements. Although mxed-sea condtons can be estmated by the wave buoy analogy, the resent analyss lacks a detaled study of the agreement of the actual dstrbuton of wave energy wth frequency and drecton. Ths choce has been made due to sace lmtatons. However, not only ntegrated wave arameters but the comlete dstrbuton of wave energy must be correct for decson suort systems to gve relable gudance wth resect to crtcal wave-nduced events. A more comrehensve study of the consdered data should therefore be made. CONCLUSIONS It has been shown that the wave buoy analogy can be used to obtan sea state arameters at the oston of an advancng sh. In the aer, fullscale moton measurements from sea trals conducted by DRDC Atlantc were studed. The agreement between estmates of wave arameters by the wave buoy analogy and and by real wave rder buoys was good. More detaled comarsons of the frequency-drectonal dstrbuton of energy lacks and should be carred out also n the future. REFERENCES Brunt, E. (3). Moton Sensor Acquston System, DRDC Atlantc TM 3-139, Defence R&D Canada Atlantc. Goda, Y. (). Random Seas and Desgn of Martme Structures, Advanced Seres on Ocean Engneerng, Vol. 15, World Scentfc. Isek, T. and Ohtsu, K. (). Bayesan estmaton of drectonal wave sectra based on sh motons, Control Engneerng Practce, Vol. 1,. 5-3.
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