1 A Study on Parametrc Wave Estmaton Based on Measured Shp Motons Ulrk Dam NIELSEN * and Tosho ISEKI ** Abstract The paper studes parametrc wave estmaton based on the wave buoy analogy, and data and results obtaned from the tranng shp Shoj-maru are compared wth estmates of the sea states obtaned from other measurements and observatons. Furthermore, the estmatng characterstcs of the parametrc model are dscussed by consderng the results of a smlar estmaton concept based on Bayesan modellng. The purpose of the latter comparson s not to favour the one estmaton approach to the other but rather to hghlght some of the advantages and dsadvantages of the two approaches. Keywords: Seakeepng, onboard wave estmaton, full-scale shp moton measurements, parametersed wave spectra, nonlnear optmsaton 1. Introducton Sea state parameters, or the drectonal wave power spectrum, around a shp are needed on a contnuous bass for navgatonal and operatonal gudance to a shp's master. The lkelhood of large wave-nduced acceleraton, for example, depends amongst others on the sea state n whch the shp operates. Thus, f the sea state s estmated contnuously t s possble to rase a warnng f vessel speed and course s n a regon where the wave-nduced acceleraton may become crtcal to, say, stowed contaners on a contaner vessel. The evaluaton of a vessel s performance (1) requres also nput of the sea state parameters, so onboard wave estmaton s hghly relevant for any type of montorng and/or decson support system on shps. In the lterature there are studes (,3,4,5,6) concernng the estmaton of sea state parameters usng measured shp responses (e.g. moton data) where the shp, to make an analogy, acts as a wave rder buoy and for ths reason the methodology s called the 'wave buoy analogy'. The fundamental nput to the wave buoy analogy s a set of response measurements where the ndvdual response bascally can be any one as long as a lnear (complex-valued) transfer functon may be assocated wth the response. The wave buoy analogy provdes a robust alternatve to wave radar by utlsaton of onboard response measurements that are often carred out rrespectvely on many of today s navy and commercal vessels. Consequently, the wave buoy analogy s also a relatvely nexpensve estmaton concept, snce the system development s assocated wth software only. Ths study consders sea state estmaton from fullscale moton measurements obtaned durng experments wth the tranng shp Shoj-maru. In connecton wth the experments wave measurements were made also usng a wave radar montorng system and, moreover, vsual observatons were carred out.. Theory.1 Parametrc model of wave spectrum The parameterzed drectonal wave spectrum E(ω, θ) s chosen to be a ffteen-parameter tr-modal spectrum (1) that allows for mxed sea such as wnd and swell. E, E ( ) G (, ) 1 4 1 4 E ( ) p, 4 4 4 4 1, p exp 4 s G (, ) A( s ) cos s1 ( s 1) A( s) (s 1) 3 1 H s, 4 ( ) mean, 1 (1) * ** Member: Techncal Unversty of Denmark, (Kgs. Lyngby, Denmark) udn@mek.dtu.dk Member: Tokyo Unversty of Marne Scence and Technology (Tokyo, Japan) sek@kayoda.ac.jp
where ω and θ are the wave frequency and the relatve wave headng, respectvely. E (ω) s a one dmensonal wave spectrum wth ω p, H s, and λ beng the peak frequency, the sgnfcant wave heght and the shape parameter, respectvely, of the spectrum. G (ω, θ) s the drectonal dstrbuton functon, where θ mean and A(s) are the mean relatve wave drecton and a constant to secure normalsaton. A(s) s evaluated usng Gamma functon of the spreadng parameter s. Table 1 Prncpal partculars of T.S. Shoj-maru. Length, L pp 46.00 m Breadth, B mld 10.00 m Depth, D mld 6.10 m Draught, T mld.65 m Dsplacement 659.4 t. Governng equaton system The governng equaton system of the wave buoy analogy orgnates from a lnear assumpton relatng the response spectrum S j (ω e ) of the th and jth responses to the drectonal wave spectrum E(ω e,θ) S j ( e ) E d ( e, ) j ( e, ) ( e, ) () S j (ω e ) s the measured cross spectrum whereas the rght-hand sde s the calculated cross spectrum usng the estmated E(ω, θ) and the response ampltude operators Φ(ω e,θ) of the consdered shp responses. The bar denotes the complex conjugate. An optmsaton problem can be establshed by mnmsng the dfference between the left- and the rght-hand sde of Eq. (), wth the wave spectrum E(ω,θ) gven by Eq. (1). Thus, the least squares method s appled and the nonlnear optmsaton can be solved drectly usng, e.g., MATLAB by nvokng fmncon, whch s a bult-n functon based on sequental quadratc programmng. The mplementaton of the problem s, however, not straght forward due to the trple-valued functon problem that must be taken nto account for advancng shps (). Moreover, the present model consders an energy conservaton requrng the 0 th order spectral moment of the left- and rght-hand sde to be dentcal, leavng out detals and further argumentaton for reasons of space lmtaton. 3. Full-scale Moton Measurements Shp moton tme hstores were measured usng the tranng shp Shoj-maru of Tokyo Unversty of Marne Scence and Technology. A photo and prncpal partculars of the shp are shown n Fgure 1 and Table 1, respectvely. The experments were carred out on July 5th 007, January 3rd 008 and March 11th 009. The locaton of the expermental area was off Nojma Cape n Chba Prefecture, Japan. Fg. 1 Photography of T.S. Shoj-maru. The wave condtons of the expermental area were partally observed at Nojma Cape lghthouse usng the wave montorng radar system operated by the Thrd Regonal Japan Coast Guard Headquarters. Fgure shows the cover area whch s a fan-shaped regon between 100 and 1.7 degrees and dstance between 0.5 and mles from Nojma lghthouse. The trajectores of the T.S. Shoj-maru are also ndcated n the fgure. To measure changes n shp motons wth respect to the encounter angle of waves, the shp motons were recorded durng 90-mnute manoeuvres nvolvng straght sectons and changes n course. Table shows the operatonal condtons whch nclude courses and mean speeds of the shp for each straght runnng, the drectons and speeds of true wnd and wave condtons measured by vsual observatons. These data ndcate that the drecton of Fg. Expermental area wth shp trajectores.
3 Run ID Vessel course [deg.] Table Operatonal condtons. Vessel Wnd Wnd Wave speed dr. speed dr. [kts] [deg.] [m/s] [deg.] Wave heght [m] (007/07/5) A1 90 1 110 3 110 1.0 A 70 1 110 4 110 1.0 A3 90 8 110 4 110 1.0 A4 70 8 110 4 110 1.0 (008/01/3) B1 90 10 350 11 100 1.9 B 10 10 0 9 100 1.9 B3 330 10 340 10 50 1.6 B4 90 8 340 11 50 1.6 B5 70 8 0 9 50 1.6 (009/03/11) C1 90 10 350 5 70 1. C 180 10 330 4 70 1. C3 315 10 0 4 90 1.0 C4 5 10 350 3 90 0.9 C5 45 10 350 190 0.8 C5 70 10 350 10 0.7 waves around Nojma Cape dstrbuted from 100 to 10 degrees whle waves came from North-East drecton at offshore. The wave condtons observed by the wave montorng radar system are summarsed n Table 3. The table shows also the wave data reported by the wave analyss of Japan Meteorologcal Agency (JMA). The dscrepancy between the observatons s probably a result of the land shape as shown n Fgure. The sea area around Nojma Cape s n a shadow zone wth respect to the NE waves. For ths wave drecton, t s, n prncple, possble to compare the results of the parametrc estmaton to observed data of only the wave montorng radar system n the cover area. When the T.S. Shoj-maru was navgatng far off Nojma Cape on 3rd January 008, however, the wave drecton was NE as reported by JMA. Shp motons and the poston were measured usng a fber optc gyro and ts bult-n GPS system (JCS7401GA, Japan Avaton Electroncs Industry, Lmted). Just for reference, the relatve wave heght was measured by a supersonc wave sensor (UH- 401,KENEK) nstalled at the bow. These data were sampled every 0.1 s and recorded n the hard dsk of a notebook PC through the RS-3C port. Fgure 3 shows one of the measured set of tme hstores of ptch angle, roll angle and vertcal acceleraton. The data was measured on 3rd January 008 and the total tme span was 90 mnutes. As shown n Fgure, combnaton manoeuvres nvolvng Table 3 Measured wave parameters by wave radar montorng system (Radar) and by Japan Meteorologcal Agency (JMA). Date 007 07/5 008 01/3 009 03/11 Wave dr. [deg.] Wave perod [sec.] Wave heght [m] Radar JMA Radar JMA Radar JMA 137 ESE 11 10 1.0 0.6 10 NE 10 10 1.6.0 18 E 19 11 1.3.3 Fg. 3 Measured tme hstory data of ptch angle, roll angle and vertcal acceleraton (Run ID B1-B5). 1-mnute straght sectons changed the encounter angle to waves and caused the change of ampltude of tme hstores. The begnnngs of large ampltude ptchng motons can be seen at 10 and 60 mnutes, where the measurements lkely relate to head sea condtons. Ths agrees well wth the wave drecton observed by the wave montorng radar system at Nojma lghthouse. On the other hand, the begnnng of a large ampltude rollng moton can be seen at 47 mnutes n the measured tme hstory. Ths was measured durng offshore runnng wth course of 330 degrees and can be consdered as a result of drect encounter wth the NE waves. Fgure 4 shows the results of the cross spectrum analyss for the data B1. The actual tme hstores correspond to the frst 10 mnutes of Fgure 3. In ths analyss, the MAR (Multvarate Auto Regressve) modellng procedure was used and the optmum order of the model was chosen by MAICE (Maxmum AIC Estmaton) method. The boldfaced sold lnes and lght-faced sold lnes n the fgure represent the real and magnary part of the cross spectra, respectvely. As mentoned above, the course of 90 degrees s n the
4 Fg. 5 Measurements of sgnfcant wave heght. Fg. 4 Response spectra obtaned by MAR modellng (Run ID B1). head seas condton. Therefore, t can be seen that the spectra of ptchng and vertcal acceleraton have rather wde feet. 4. Parametrc Wave Estmaton 4.1 Results The analyss of data has been conducted as a postvoyage process and the results of the parametrc wave estmaton are gven n Fgures 5-7 that show the sgnfcant wave heght, the wave perod and the absolute wave drecton, respectvely. The wave parameters were estmated usng the data of ptch angle, roll angle and vertcal acceleraton. For the ndvdual wave parameter, the measurements obtaned from the vsual observatons, from the wave radar montorng system and from the Japan Meteorologcal Agency (JMA) are also ncluded. It should be noted that t s dffcult to estmate the wave perod from vsual observaton for whch reason Vsual observaton s not ncluded n Fgure 6. Moreover, t s not clear from the measurements by the wave radar montorng system whch characterstc wave perod s consdered. However, for the wave perods of JMA and the parametrc model, the results are based on the mean perod T = m 0 /m 1, where m 0 and m 1 are the 0 th and 1 st order spectral moment, respectvely. Fnally, t s noteworthy that only one set of measurements have been obtaned from the wave radar montorng system and from JMA for each of the three expermental sets, (A1,...,A4), (B1,...,B5) Fg. 6 Measurements of wave perod. Fg. 7 Measurements of absolute wave drecton. and (C1,...,C6), consdered. Ths fact explans why the measurements behave as constants wthn the ndvdual expermental sets for the wave radar montorng system and for JMA. 4. Dscusson Fgure 5 reveals that the estmates of sgnfcant wave heght by the parametrc model compare reasonably well wth the other sets of measurements.
5 However, for the expermental sets B1,...,B5 there s a tendency for the parametrc model to gve wave heghts on the smaller sde. It s dffcult to explan exactly why ths s so but the observaton could relate to flterng, whch s an nherent problem of the wave buoy analogy. Thus, t s n general possbly only to estmate those waves whch are felt by the shp when the wave buoy analogy s appled for wave estmaton. Ths phenomenon s dscussed n rather detals n the lterature (5,7,8), but t should be noted that, typcally, the phenomenon s most pronounced for larger szed vessels. Another nterestng observaton that can be made from Fgure 5 s that the vsual observatons confrm the parametrc model n the reducng trend of the wave heght experenced n expermental sets C1,...,C6. The agreement between the measurements of the wave perod and the correspondng estmates by the parametrc model s consdered far. It s not known why the wave radar montorng system gves a wave perod whch s far from the results by JMA and the parametrc model for data sets C1,...,C6. Smlar to the other two wave parameters, Fgure 7 shows that the wave drecton obtaned by the parametrc model, on average, agrees reasonably well wth the other three sets of measurements. However, there are two dstnct outlers of the parametrc model when focus s turned to sets A and B5. The reason for these two outlers s that the parametrc model estmates an extra second peak n the wave spectrum n these two cases. The consequence s a bmodal wave spectrum, where only the one peak should be consdered as a part of the true spectrum. As a drect extenson of ths dscusson t should be noted that although mxed-sea condtons can be estmated by the parametrc model, the present analyss lacks a detaled study of the agreement of the actual dstrbuton of wave energy wth frequency and drecton. Ths choce has been made partly due to space lmtatons and partly because the nformaton s not avalable for the other three sets of measurements. However, not only ntegrated wave parameters but the complete dstrbuton of wave energy must be correct for decson support systems to gve relable gudance wth respect to crtcal wave-nduced events. In the future, t could therefore be nterestng to conduct a more comprehensve study of the consdered data wth respect to the dstrbuton of wave spectral energy. 4.3 Comparson wth Bayesan estmaton The wave buoy analogy bulds typcally on ether parametrc wave estmaton, as descrbed n ths paper, or Bayesan wave estmaton (,3,5,9,10). The man dfference between the two concepts les n the approach used to handle the mnmsaton related to Eq. (), leavng out further detals. The outcome of the parametrc and Bayesan wave estmaton for the studed data s summarsed n Table 4. As the table shows, the two estmaton concepts gve not dentcal results when ntegrated wave parameters (wave drecton, wave perod, wave heght) are compared. The comparson s vsualsed graphcally by the plot n Fgure 8. The plot shows normalsed wave parameters obtaned by the parametrc approach on the x-axs and correspondng parameters obtaned by the Bayesan approach on the y-axs. Each wave parameter (wave drecton, wave perod, wave heght) has been normalsed wth the maxmum value of the specfc parameter obtaned from ether of the approaches. From the plot n Fgure 8 t appears evdently that the Bayesan estmaton provdes the largest wave heght n all cases. The stuaton s almost opposte when the wave perod s consdered, snce the parametrc approach gves hgher values n nearly all cases. The reason for ths to be so s because the Bayesan approach solves the equaton system, Eq. (), dscretely (frequency-wse and drectonal-wse) Table 4 Wave parameters by parametrc and Bayesan wave estmaton. Run Wave dr. [deg.] Wave perod [sec.] Wave heght [m] ID Param Bay Param Bay Param Bay (007/07/5) A1 90 110 8.4 10 0.8 0.9 A 65 85 13 3.3 1.0 1.5 A3 95 110 6.0 10 0.6 1.0 A4 130 100 11 3.8 1.0 1.3 (008/01/3) B1 85 115 7.6 5.9 0.9 1. B 110 60 7.4 4.7 0.8 1.9 B3 45 115 5.5 4.7 0.9. B4 90 55 6. 4.1 0.8.6 B5 90 10 11 5.6 1..0 (009/03/11) C1 90 15 10 7.0 1.3 1.8 C 15 95 8.0 7.9 1.0.6 C3 65 155 7.9 6.0 0.9 1.6 C4 170 80 6.5 4.8 0.8.7 C5 45 170 8.5 6.0 0.9 1.8 C6 70 106 8.5 4.5 1.0.3
6 Fg. 8 Normalsed wave parameters obtaned by parametrc versus Bayesan wave estmaton. Normalsaton made wth maxmum value of the specfc parameter. whch means that t s easer to fulfl the requrement wth respect to conservaton of energy compared to what s the stuaton for the parametrc estmaton, where the shape of the spectrum s gven by defnton. On the other hand, the dscrete soluton by the Bayesan estmaton leads sometmes to shapes of wave spectra whch have lttle resemblance wth (true) physcally observed ones. The consequence s that the frequency-wse dstrbuton of energy n the Bayesan estmaton can be shfted, and ths probably explans why the Bayesan approach gves wave perods on the lower sde as can be seen from Fgure 8. There can be made no general notes on the estmaton of wave drecton by the two estmaton concepts. Thus, t s seen by the plot n Fgure 8 that the ponts correspondng to estmates of the wave drecton spread arbtrarly. Ths latter observaton agrees well wth the fact that the frequency-wse dstrbuton of energy, dscussed above, does not have an nfluence on the wave drecton. 5. Conclusons In the paper, parametrc wave estmaton was dscussed as a means to obtan sea state parameters from an advancng shp by usng the wave buoy analogy. As part of the analyss of full-scale moton measurements from the tranng shp Shoj-maru, comparsons wth ntegrated wave parameters were made wth other wave measurements based on vsual observatons, a wave radar montorng system, and analyses carred out by Japan Meteorologcal Agency. The paper contaned also results derved from Bayesan wave estmaton, whch s a smlar estmaton concept that can be appled n the wave buoy analogy. All together the followng conclusons can be drawn: (1) The agreement between estmates of wave parameters by the parametrc approach and by other measurements s reasonable. () Detaled comparsons of the frequencydrectonal dstrbuton of energy have not been shown; for operator gudance the complete energy dstrbuton s often necessary. Ths type of comparson could therefore be nterestng to perform n the future. (3) Parametrc and Bayesan wave estmaton gave not dentcal results. The explanaton for ths s due to the handlng of the mnmzaton problem, whch s establshed as the governng equaton system n the wave buoy analogy. Acknowledgements The measurement of the wave montorng radar system at Nojma lghthouse was fully supported by the Thrd Regonal Japan Coast Guard Headquarters and Japan Rado Co.,Ltd. The authors express sncere grattude to the above organzatons. References (1) Hansen, S.V. and Lützen, M.: Performance Montorng and Shp Modellng by the Bond Graph Method, Proc. 11 th PRADS, Ro de Janero, Brazl, 010. () Isek, T. and Ohtsu, K.: Bayesan estmaton of drectonal wave spectra based on shp motons, Control Engneerng Practce, Vol. 1, pp. 5-30, 000. (3) Nelsen, U.D.: Estmatons of on-ste drectonal wave spectra from measured shp responses, Marne Structures, Vol. 19, pp. 33-69, 006. (4) Nelsen, U.D.: Introducng two hyperparameters n Bayesan estmaton of wave spectra, Probablstc Engneerng Mechancs, Vol. 3, pp. 84-94, 008. (5) Pascoal, R., Guedes Soares, C. and Sørensen, A.J. Ocean wave spectral estmaton usng vessel wave frequency motons, Journal of Offshore Mechancs and Arctc Engneerng, Vol. 19, pp. 90-96, 007. (6) Tannur, E.A., Sparano, J.V., Smos, A.N. and Da Cruz, J.J.: Estmatng drectonal wave spectrum
7 based on statonary shp moton measurements, Appled Ocean Research, Vol. 5, pp. 43-61, 003. (7) Nelsen, U.D.: Response-based estmaton of sea state parameters nfluence of flterng, Ocean Engneerng, Vol. 34, pp. 1797-1810, 007. (8) Smos, A.N., Tannur, E.A., Sparano, J.V., and Matos, V.L.F.: Estmatng wave spectra from the motons of moored vessels: Expermental valdaton, Appled Ocean Research, Vol. 3, pp. 191-08, 010. (9) Isek, T. and Terada, D.: Bayesan Estmaton of Drectonal Wave Spectra for Shp Gudance Systems, Internatonal Journal of Offshore and Polar Engneerng, Vol 1, pp 5-30, 00. (10)Nelsen, U.D. and Isek, T.: Estmaton of Sea State Parameters from Measured Shp Responses - The Bayesan Approach wth Fxed Hyperparameters, Proc. OMAE010, Shangha, Chna, 010. Questons and answers Hrosh Ishda (Kobe Unversty): It s assumed that there s a lnear relatonshp between shp s moton and ocean waves. But t seems the relatonshp depends on the condtons of ocean waves and the characterstcs of shp s moton. That s, t seems both have a non-lnear relatonshp under some wave condtons. The observatons were made on the relatvely low seas, where the wave heght was from one to two meters. It seems the wave heght s lower than that whch shp s moton and the wave condton have a lnear relatonshp. Is t possble to apply your method to estmate the wave condton n such a low wave condton? Ulrk Dam NIELSEN: It s the authors opnon that the wave heghts are not small but good enough for the lnear relatonshp. Theoretcal calculatons of transfer functons of T.S. Shoj-maru have been compared wth experments wth good agreement (Fgure 9). In the experments, an 1/17 model shp was used and the wave heghts were set from 1.6 cm to.8 cm to keep the wave slope constant. The wave heghts correspond to about 1 m n the full scale. The graphs show that the lnear assumpton s qute reasonable. The authors beleve that the methods can be appled to the wave condtons descrbed n the paper. Fg. 9 Comparsons of transfer functons of heavng, ptchng and rollng. The lnes and markers denote the results of calculatons and experments. (Encounter angle =150 deg, Froude No.=0.)