Applied Ocean
Research
ELSEVIER
Applied Ocean Research 25 (2003) 243-261www.elsevier.com/locate/apor
Estimating directional wave spectrum based on stationary
ship motion measurements
Eduardo A. Tannuri''''*, Joao V. Sparano'', Alexandre N. Simos'', José J. Da Cruz''
" Depailment of Naval Arch, and Ocean Engineermg, University of Sao Paulo, Sao Paulo, SP, Brazil^Department of Telecomnumicatlou and Control Engineering, Universiry of Sao Paulo, Sao Paulo, SP, Brazil
Received 10 July 2003; revised 12 December 2003; accepted 12 January 2004
Abstract
Useful information can be derived from on-board estimation of the directional wave spectrum of the sea, especially concerning feed
forward control of dynamically positioned systems. This work discusses the feasibility of using stationary ship motion measurements for
in-site directional wave spectrum estimation, focusing on the particular problems that may arise in the application of adapted estimation methods to this kind of system. Load variations, operational trim and other disturbances frequently occur, modifying the ship response to incident waves. Since the methodology depends on previous knowledge about the amplitude response operators, these errors may cause some degradation in the estimated spectrum. Furthermore, due to the large inertia of the ship, high frequency wave components are filtered, reducing the frequency range of the spectrum that can be estimated. These drawbacks are analyzed and their influence in final estimation is quantified, Roll motion of the ship presents non-linear and resonant behavior and is extremely sensitive to load variations. For this reason, sway motion was used instead of roll one, differently from common directional buoys algorithms that take into account the thi'ee motions in the vertical plane (heave, roll and pitch). A parametric estimation method, normally used for directional buoys measurements, was adapted to the case and tested. Exhaustive numerical trials were carried out using a tanker in two loading conditions (full and ballasted) and a pipe-laying barge, under typical wave spectra of Brazil's Campos Basin, either in unimodal or bimodal cases. Small-scale towing-tank results were also used as a first experimental validation concerning unimodal unidirectional sea states. Spectral estimations based on experimental results were also compared to those obtained by means of a Bayesian estimation method presented in literature, for the sake of comparison. Parametric Method presented best accuracy in all tested cases.
© 2004 Elsevier Ltd. A l l rights reserved.
Keywords: Directional wave spectrum; Estimation; FPSO
1. Introduction
A s offshore o i l production moves towards deeper waters, dynamic positioning (DP) systems become more and more important as an economical solution f o r the station-keeping o f f i o a t i n g production units. For D P operations under extreme conditions, feed f o r w a r d control may represent a significant improvement i n the efficiency o f the system, c o n c e r n i n g both, station-keeping behaviour and f u e l * Con-esponding author current address: Escola Poiitecnica da USP, Departamento de Engenharia Naval e Oceanica, A v . Prof Mello Moraes, 2231, CEP 05508-900, Cidade Universitaria, Sao Paulo Brazil. Tel.; -I- 55¬ 11-3091-5350; fax: + 55-11-3088-7989.
E-mail addresses: eduat@usp,br (E.A. Tannuri); joao.sparano@poli. usp.br (J.V. Sparano); alesimos@usp.br (A.N. Simos); jaime@lac.usp.br (J.J. Da Cruz).
consumption. The feed f o r w a r d control consists of p r o v i d -i n g -i n f o r m a t -i o n on the env-ironmental exc-itat-ion (waves, current and winds) to the system i n order to predict the D P response required f o r counteracting the estimated environ-mental forces.
W i n d feed f o r w a r d was used since first D P systems, because the measurement of w i n d velocity and direction can be done by anemometers w i t h acceptable accuracy. W a v e forces feed f o r w a r d control was firsdy introduced by Pinkster [ 1 ] , and he showed that i t can lead to a significant improvement i n D P performance. However, i t requires the esdmadon o f the Directional W a v e Spectrum ( D W S ) o f the sea acting on the production unit, what is the m a i n topic o f the present work.
The D W S is used to estimate the mean d r i f t forces acting on the vessel, by means of the w e l l - k n o w n d r i f t 0141-1187/$ - see front matter © 2004 Elsevier Ltd. A l l rights reserved,
244 E.A. Tanmiri el al. / Applied Ocean Research 25 (2003) 243-261
coefficients o f the hull. Such estimates are directly added to thrusters control forces (feed forwardloop). O f course, such estimates contain errors and also disregard the slow d r i f t forces (zero mean low-frequency wave forces), w h i c h are compensated by the conventional control feedback loop. So, a larger part of the wave mean force is compensated by the feed f o r w a r d loop, and the smaller part (which comes f r o m estimation errors) is counteracted by a feedback loop. Since the system dynamics strongly changes f o r different environmental conditions, a conventional single feedback controller presents performance degradation under an environmental condition that is different f r o m the one i t was used to tune a l l controller parameters, w h i c h consists i n a severe operational problem i n commercial DPS. The association o f the feedback and feed forwardloops eliminates such problem, because the feed forwardloop compensates the major part of the dynamics changes.
This paper discusses the feasibility o f estimating the D W S using on on-board monitoring of the first-order motions o f the unit, i n particular, a FPSO ( F l o a t i n g Producfion, Storage and Offloading) system based on a moored Very Large Crude Carrier ( V L C C ) .
I n the last 30 years, sea state measurements were c a n i e d out m a i n l y by moored directional buoys. Such devices provide good esdmates of wave spectrum, since they have w e l l Icnown dynamics and their motions can be accurately measured by accelerometers and tilt sensors [2]. I n some cases, buoys dynamics are even neglected i n the numerical processing of its data, considering the measured heave, pitch and r o l l equal to wave height and slopes. However, buoys are easily subjected to damage and loss, and present practical and economical drawbacks related to deep water m o o r i n g system installation.
Recently, wave-monitoring radar systems have been developed based on the analysis of temporal and spatial evolution of the radar backscatter i n f o r m a t i o n [3,4]. These systems may be instaUed on board, what eliininates the problems associated to inoored buoys. However, they require complex computational hardware and have a h i g h i n i t i a l cost.
The estimation o f the spectrum based on ship motions measurements may overcome such problems since i t requires simple instrumentation and computational hard-ware, and can be installed onboard. Some applications to running ships have already been described i n Ref. [ 5 - 7 ] f o r example.
This paper addresses the feasibility o f applying such wave m o n i t o r i n g system method i n stationary offshore systems such as large moored tankers converted to FPSO's systems or pipe-laying barges. O f course, several problems arise i n this particular case.
Concerning D P improvement by means o f feed f o r w a r d control, a VLCC-based FPSO certainly represents one o f the inost adverse applications f o r the methodology here proposed. I n view o f the ship large inertia, it does not
respond to smah waves. As a consequence, D W S w i t h l o w peak periods cannot be estimated accurately. Nevertheless, although high frequency wave components do not excite significantly the first order motions o f the ship, they m a y contribute significantly to the d r i f t forces, as the ship h u l l reflects them.
A l s o , the m e t h o d o l o g y applied depends on the previous knowledge about how the ship responds to wave incidence, w h i c h is modeled by the linear Response Ainplitude Operators ( R A O ' s ) . However, i n a FPSO several factors such as load variations, operational t r i m and other disturbances make the correct evaluation o f the R A O ' s a hard task.
I n such aspects, the V L C C represents a demanding test f o r the methodology proposed. For smaller offshore systems, such as d r i l l i n g ships and barges, the drawbacks mentioned above w i l l certainly be reduced.
Another relevant aspect to be analyzed regards the non-linear dynamic effects that also play an important role when oscillation amplitude increases. The analysis o f the R A O ' s o f a ship showed that the use o f sway m o t i o n is more appropriate than r o l l m o t i o n , since the f o r m e r does not present resonant behavior at the typical wave frequency range and is considerably less sensitive to changes i n loading configurations.
A Parametric estimation method was applied to the V L C C tanker subjected to typical waves o f Campos Basin, i n order to test the estimation principle.
Numerical trials emulated practical probleins i n the estimation o f R A O ' s , considering the f u l l and ballasted loading condition o f the FPSO. U n i and bimodal sea states were considered i n the numerical analysis. Errors i n the D W S and, consequently, i n the w a v e - d r i f t forces estiinations were quantified.
A first experimental v a l i d a t i o n was obtained by estimation o f the wave spectra generated at the I P T towing-tank using the motions measured on a small-scale V L C C model, subjected to such spectra. Due to tank limitations, however, only unimodal unidirectional seas could be tested.
Spectral estimations based on e x p e r i m e n t a l l y measured motions were also compared to those obtained by the application o f a non-parametric method presented i n literature, the Bayesian method. This method was chosen because i t is one o f the reference methods applied to directional buoys data processing [7] and has been successfully applied to running ships w i t h no uncertainty i n R A O ' s [ 8 ] . Furthermore, i t is a non-parametric method that, i n principle, is able to estimate any shape o f spectrum, even bimodal cases w i t h a swell sea associated w i t h a random local sea. The Parametric method used considers a 8-parameter model f o r the spectrum as proposed i n Ref. [9] and is also able to approximate bimodal spectra.
The potential of the methods is discussed and the probleins that were encountered are reported.
E.A. Tannuri el al./Applied Ocean Research 25 (2003) 243-261 245
2. P a r a m e t r i c estimation method
Assuming linearity between waves and ship response, the cross spectra o f ship motions time series <}>„,„ and the D W S are related by the R A O ' s through the f o l l o w i n g integral:
R A o , „ ( w , 0).RAo;;(w, d).sicü, e).de, ( i )
where R A O „ , ( « , % denotes the Response A m p l i t u d e Operator o f the m o t i o n in at frequency w and incidence direction 9 and S{w, 0) denotes the D W S .
The power spectrum w i d e l y used to describe wave energy frequency distribution, may be obtained f r o m directional spectrum by:
0)d0
A t this p o i n t i t is w o r t h noting that three ship motions w i l l be considered i n what f o l l o w s , namely, = 2 f o r sway,
m = 3 f o r heave a n d = 5 f o r pitch. Since 4>„m is real when m = n and complex otherwise, Eq. (1) w i l l give rise to 9
equalities.
B y d i v i d i n g the frequency range o f interest \_u)q, J i n J points, {oijYjl^, the discrete expression o f (1) is derived assuming the integrand to be constant on each interval A ö :
(/.„„,(«;•) = A ö £ R A O , „ , ( a ) , ) R A O ; ; , ( « , . ) 5 i . ( a ) , . ) , (2)
w i t h A e = 2'7T/K,Ski(Oj) = Sicoj, 9,,),
R A O „ , , ( « , . ) = RAO,„(cü,., 9k) and R A O ; ; , ( w ) = RAO;;(w, 0,),
The Parametric method is conceptually simple and is based on the f o l l o w i n g 10-parameter representation o f the b i m o d a l spectrum [ 9 ] : / 4 V H 4
V
Hs'r Xcos' ^ ' ( ^ ) e x p vAisi) 4 A : + l / f t ) „ ^4 i'i) where Ais) 2''~'r\s + 1) 7 r r ( 2 ^ + 1)is a normalization factor f o r the area under a cos^* curve, F is G a m m a f u n c t i o n , Sj represents the spreading o f the /th component, Hsj is the signihcant wave height. A,- is a shape parameter, 0„,; is the mean direction and cj„„- is the modal frequency.^ This f o r m u l a t i o n was proposed b y Ref. [ 9 ] and is a combination o f the 6-parameter model f o r the power
Tlie modal frequency h related with the peak period Tp, by the expression: Tp,- = 2ir/(u„„..
spectrum presented i n Ref. [10] (that is similar to w e l l -k n o w n J O N S W A P f o r m u l a t i o n ) w i t h the cos"^ m o d e l proposed i n Ref. [11]. Since i t considers t w o separated wave components (/ = 1 and 2), i t is capable o f representing a variety o f spectrum shapes including bimodal spectra.
Since A,- has weak influence on wave induced loads and ship motion, its value has been fixed as A,- = 1. The number of parameters to estimate has then been reduced. I n this case, the power spectrum related to (3) is reduced to a P i e r s o n -M o s k o w i t z spectrum. A s w i l l be shown i n this w o r k , even i n the presence o f a wave pattern i n w h i c h A,- 1, the relevant parameters are w e l l estimated w i t h this simplification. So, the final spectrum parameterization used i n the method is given by:
'^^ = Ï 2 . 4 ^ « , ; ^ A ( . , ) c o s - | - ^ j
4 V « / _ (4)
The Parametric method is based on the m i n i m i z a t i o n o f the quadratic error o f the motions predicted using the estimated spectrum and the measured ones. As already explained, the calculation is done assuming linearity between waves and ship response and uses the ship R A O ' s .
B y using E q . (2), f o r a given spectrum, the values o f
(knni(^j)^iO — j ^ J - 1) can be calculated. Denoting by 4'mni<^j)>iO < j < J - 1) the measured values o f the ship
motions spectrtim and considering the wave spectrum represented by the eight parameters o f the E q . ( 4 ) , namely:
X =[w,„^Hs^Sl 9,„i (ü„a Hsj sj 9,„2V
the quadratic error may be written as
Eix) =
- I 1/2
X Z ^4'<nni(^j) - 4>nmi(^j)f
Thus, the spectrum estimation problem is reduced to finding the m i n i m u m point o f the f u n c t i o n a l ^(.v). The f o r m o f Eix) is crucial f o r the convergence and the outcome f r o m the algorithm. The u n i f o r m weight over the whole co domain was used, since one has no prior knowledge about incident wave frequency range.
The Parametric approach is able to estimate u n i m o d a l or b i m o d a l spectra. I n b o t h cases the m e t h o d finds a 8-component v e c t o r x o f estimated parameters.
X = [(D,„i Hsi Sl 0,„i&)„,2 Hs2 S2 9„,2]^. For u n i m o d a l seas, one o f the estimated significant heights iHsi or HS2) is expected to be negligible.
A d i f f i c u l t y w i t h the Parametric M e t h o d is that i t leads to a non-linear p r o g r a m m i n g p r o b l e m , whose n u m e r i c a l solution requires a h i g h computational effort.
246 E.A. Tannuri et al./Applied Ocean Research 25 (2003) 243-261
Table 1
VLCC characteiistics
Properties Full loaded - 9 0 % loaded - 8 0 % loaded - 4 0 % loaded - 3 0 % loaded
Mass (M) 321900 ton 271650 ton 257500 ton 126100 ton 96680 ton
Moment of inertia (Z^) 2.06 X lO' t o n n r 1.78 X 10' t o n n r 1.65 X l O ' t o n n r 0.71 X 10' t o n m - 0.62 X l O ' ton n r
Length (L) 320 m 320 m 320 m 320 ni 320 m
Draft (T) 21.47 m 19.00 m 17 m 9.00 m 7 m
Breadth (S) 54.5 m 54.5 m 54.5 m 54.5 ra 54.5 m
Wetted Suif.(S) 27342 25770 n r 24198 m- 17910 m- 16998 m^
3. Sensitivity analysis
Tlie estimation methods are strongly dependent on the R A O ' s used, since they contain i n f o r m a t i o n about sliip response to incident waves.
R A O ' s can be obtained experimentally, either i n f u l l -scale or model--scale tests, but i t is usual to apply a wave-body interaction software to p e i f o r m their evaluation. Such evaluation may present some inaccuracies due to the f o l l o w i n g factors:
• Non-linearities—the definition o f the R A O ' s is based on
the hypothesis that the relation between wave excitation and ship response is linear. O f course, this siniphhcation is vahd f o r small oscillations around the e q u i h b r i u m position; however, i n critical situations, the ship response amplitudes may reach h i g h values, and non-linear effects become relevant. This problem is more critical f o r r o l l motion, since it is strongly dependent on viscous forces, w h i c h are non-linear i n nature. Furthermore, the large number o f risers and m o o r i n g lines increase the non-linear viscous damping o f the system.
® Loading conditions—the ship response depends on its
load distribution. For the case analyzed i n the present w o r k this problem is crucial, since i n a FPSO o i l stored i n the tanks may represent 6 0 % o f the total weight o f the system. Moreover, o i l free surface effects may influence ship response, although this is generally not taken into account during R A O ' s calculation.
I n the present w o r k , R A O ' s inaccuracies were emulated b y uncertainties i n the d r a f t value, w h i c h are related to the second factor explained above. The method was applied to the tanker V i d a l de Negreiros, a V L C C w i t h the characteristics listed i n Table 1.
R A O ' s were evaluated f o r the 5 loading conditions presented i n Table 1 using a wave-body interaction software ( W A M I T ) . For example. Figs. 1-4 show the amplitude o f R A O ' s f o r a 135° wave heading incidence.
W h e n wave spectrum estimation methods are applied to directional buoys, they normally use heave, p i t c h and r o l l motions, measured by accelerometers and i n c l i n -ometers [ 2 ] .
Obviously, the wave response o f a buoy subjected to waves can be obtained either numerically or experimentally, since it is a small body w i t h I m o w n parameters and dynamics. So, i t can be said that its R A O ' s are generally I m o w n w i t h high level o f accuracy.
O n the other hand, the dynamical behavior o f a large body such as a V L C C tanker may be extremely d i f f i c u l t to be predicted and it may present h i g h sensitivity to non-modeled effects, as previously exposed.
Indeed, f o r the V L C C under consideration. F i g . 4(a) confirms that r o l l response is strongly dependent on loading conditions, m a i n l y due to its resonant behavior. I n this case, the use o f r o l l m o t i o n i n the estimation methods w o u l d lead to non-robust results. Also, non-linear effects have much more influence i n r o l l than i n any other first-order ship motion. O n the other hand sway m o t i o n presents a l o w
so 90 too 0.2 O.-l 0.6 O.i Frequency (rad/s) 1 1.2 (a) . (b) Fig, 1. (a) Sway and (b) Heave RAO amplitude for 100, 90 and 80% loading conditions.
E.A. Tannuri et al./Applied Ocean Research 25 (2003) 243-261 247
0.014
Fig. 2. (a) Roll and (b) Pitch RAO amplitude for 100, 90 and 80% loading conditions.
Fig. 3. (a) Sway and (b) Heave RAO amplitude for 40 and 30% loading conditions.
sensitivity to loading conditions as illustrated by Figs. 1 and 5.
R o l l R A O is an odd f u n c t i o n o f the incidence angle, and i t takes into account i f waves come f r o m port or starboard. T h i s i n f o r m a t i o n is not contained i n p i t c h and heave motions, since the coiTesponding R A O ' s are even functions of incidence angle. Since sway R A O is also an odd f u n c t i o n
and taking into account the considerations above, i t can advantageously replace r o l l i n the estimation procedure.
The estimation method was then applied to sway, heave and p i t c h motions. However, even i n this case, some sensitivity to loading conditions is expected, since R A O ' s phase plots present a significant variation specially f o r heave and p i t c h motions, as can be seen i n Figs. 6 and 7.
0J)1 •a om oas S < asm O ^ om a oms S •3 OXDl ' I om .1 otm g O j B O I z Frequency (rad/s) (a) Frequency (rad/s) (b) Fig. 4. (a) Roll and (b) Pitch RAO amplitude for 40 and 30% loading conditions
248 EA. Tannuri et ai./Applied Ocean Researdi 25 (2003) 243-261
500
1 1 , , , . . 1
0 0.2 0.4 0.6 0.8 1 1.2 Frequency (rad/s)
Fig. 5. Pliase of sway RAO for 80, 90 and 100% loading conditions.
The influence o f such variations i n the final estimated spectrum is analysed i n the next section.
4. Numerical tests
A s a first test o f the methodology, accuracy and robustness, a series o f n u m e r i c a l s i m u l a t i o n s was p e i f o r m e d , using as example the V L C C V i d a l de Negreiros i n a variety o f environmental conditions.
The first set o f trials was carried out i n non-real conditions, supposing that the R A O ' s are perfectly k n o w n . T h i s was done i n order to evaluate the n u m e r i c a l performance and mathematical behavior o f the methods. The ship is considered f u l l y loaded; so, 100% R A O ' s are used both to generate ship motions ("real" model) (by means o f E q . (1)) and i n the algorithm ( ' i d e a l ' model).
A s i t should be expected, the Parametric M e t h o d generated good estimates and converged to the real spectrum w i t h good accuracy.
I n order to evaluate the sensitivity o f the methods to some degree of uncertainty i n R A O ' s , the second set o f trials used different loading conditions i n the generation o f ship motions ("real" model) and i n the estimation algorithm ("ideal" model), emulating an uncertainty i n the R A O ' s .
lOOh
1 1 , , . , , 1
0 0.2 0.4 0.6 0.8 t 1.2 Fi'cqucncy (rad/s)
Fig. 6. Phase of heave RAO for 80, 90 and 100% loading conditions.
500 Ö 200
1
100 O 2 0 w a: -100 -200 Frequency (rad/s)Fig. 7. Phase of pitch RAO for 80, 90 and 100% loading conditions.
T w o l o a d i n g conditions were considered i n these analyses:
• Fully loaded condition: the 9 0 % loaded ship R A O ' s were used i n the generation o f motions ('real' model) and the 100% loaded ship R A O ' s were used i n the estimation method ("ideal" model).
• Ballasted c o n d i t i o n : the 3 0 % loaded ship R A O ' s were used i n the generation o f motions ("real" model) and the 40% loaded ship R A O ' s were used i n the estiriiation method ( ' i d e a l ' model).
Three types o f numerical tests were carried out. I n section A , unimodal sea states w i t h peak pseriod between 7 and 20 s were considered. This analysis was used to estimate the cut-o f f frequency cut-of the methcut-od bcut-oth f cut-o r the f u l l and ballasted conditions o f the V L C C tanker. Section B and C presents the application o f real Campos Basin D W S , considering 1 and 100-year unimodal i n section B and bimodal cases i n section C.
4.1. Analysis of unimodal spectrum witli peaic periods bePi veen 7 and 20 s
I n i t i a l l y , a u n i m o d a l sea-state w i t h significant wave height Hs=\m and peak period Tp between 7 and 20 s was considered. Three possible wave-ship headings were assumed f o r m o t i o n generation, namely 90, 135 and 180°, as shown b y F i g . 8. A large value f o r spreading coefficient 5 i = 60 was used i n order to emulate an unidirectional wave pattern. The incident wave spectrum has A; = 1.
Fig. 9 presents the results o f the Parametric M e t h o d f o r the f u l l y loaded condition. The m a x i m u m estimation eiTors obtained f o r the three heading angles are shown as f u n c t i o n o f peak period. I t can be seen that f o r periods higher than 11 s, the m a x i m u m errors f o r height, period and direction are smaller than 7.5%, 2.5% and 1.5°, respectively.
The loss o f accuracy f o r smaller wave periods can be explained by the fact that the ship does not respond to such waves, as already said. Indeed, considering f o r instance a beam-sea incidence o f a 2 m height wave. Table 2 shows
EA. Tannim et al./Applied Ocean Research 25 (2003) 243-261 249
Fig. 8. Wave-ship headings considered in the analysis.
the m a x i m u m amphtude o f ship motions f o r Tp = 8.5 s and > Tp = 1 7 s. I t can be seen that ship m o t i o n is extremely reduced f o r wave periods below 6 s.
The analysis above was performed to dehne a ' c u t - o f f ' frequency o f the method when applied to the f u l l y loaded V L C C . As a consequence, i t was decided that estimated spectra w i t h estimated peak period bellow l i s should be discarded, since they may present large relative errors. For periods higher than 11 s, the estimated spreading coefficient varied between 57 and 74.
The results f o r the ballasted condition are presented i n Fig. 10. For peak periods above 10 s, the m a x i m u m errors are bounded by 9, 3.5% and 1.3°, respectively. I n this case, the en'ors are smaller than i n the f u l l loaded condition, and the estimates remain satisfactory f o r a slightly broader range o f wave periods (Tp > 10 s). This fact was expected since the ballasted ship inertia is smaller than f o r the f u l l loaded condition, and i t responds more intensely to waves. I n the ballasted case, the estimated spreading coefficient varied between 43 and 63.
As already said, accurate estimation o f mean d r i f t forces and moment plays an important role i n modern feed f o r w a r d D P Systems. Such forces ai'e evaluated by a simple spectral
crossing, given by:
/•CO r2TT
0 J o
M D s{cü, e).Dj{cü, e).do.dcü (5)
where / = 1 f o r surge force, / = 2 f o r sway force, / = 6 f o r y a w moment and D is the d r i f t coefficient, obtained by potential wave theory and numerical integration along ship h u l l .
For the sake o f illustration, the mean d r i f t forces were evaluated using both the real and the estimated spectra. The relative errors are presented i n F i g . 11 ( f u l l y loaded c o n d i d o n ) and F i g . 12 (ballasted c o n d i t i o n ) . A 180° incidence f o r surge force, 90° incidence f o r sway force and
135° incidence f o r y a w moment were considered. The eiTors can be considered satisfactory (smaller than 20%) f o r wave
Table 2
Maximum amplitude of ship motions
Tp =8.5 s r p = 1 7 s
Sway 0.3 m 1.1 m
Heave 0.5 m 1.7 m
250 E.A. Tannuri et al. /Applied Ocean Reseaich 25 (2003) 243-261
B a l l a s t e d C o n d i t i o n - E s t i m a t i o n E r r o r s
13 15
P e a k P e r i o d (s)
Fig. 10. Maximum estimation errors for unimodal waves with Hs - 1 m and T„ = 7 - 2 0 s—ballasted condition.
F u l l C o n d i t o n - IWean d r i f t l o a d s e s t i m a t i o n e r r o r s
30%
11 13 15
P e a k P e r i o d ( s )
Fig. 11. Estimation errors for mean drift forces, with Hs = 1 m and T^ = 5 - 2 0 s—Fully loaded condition.
B a l l a s t e d C o n d i t i o n - M e a n drift l o a d s e s t i m a t i o n e r r o r s
30%
11 13 15
P e a k P e r i o d ( s )
E.A. Tannuri et al. /Applied Ocean Research 25 (2003) 243-261 251 B a l l a s t e d C o n d i t i o n - E s t i m a t i o n E r r o r s 20% 18% 16% v> : 14% liJ 12% XS O 10% 'C! <D Q. 8% •s c ro 6% 4% '& 2% 0% -Significant Height - Peak Period - Direction 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0 , 1
ro
@ 13 11 13 15 P e a k P e r i o d ( s )Fig. 13, Maximum estimation errors for unimodal waves with Hs •
19
periods higher than the cut-off period. This level o f accuracy is adequate f o r control purposes, since modern controllers are robust i n the sense that they satisfy performance require-ments under bounded modeling and estimation errors. A n example o f integration between the proposed Parametric Estimation M e t h o d and a robust controller is presented i n Ref. [ 1 2 ] .
Notice that the eirors are larger i n the forces than i n the estimated spectrum parameters. This can be easily under-stood i f we observe that the significant wave height had the largest estimation errors i n almost all tests and the forces are proportional to the square o f it (see Eqs. (4) and (5)).
Extreme wave conditions are normally characterized b y larger periods, as w i l l be exposed i n section B . So, even w i t h the loss o f accuracy o f the present method f o r short period waves, i t w i l l be able to be used i n extreme wave f e e d f o r w a d control.
The method was also applied to B G L - 1 , a pipe-laying barge that operates i n Brazilian waters. The barge length is
121.9 m and its mass is 17177 ton. A 10% error i n draft is also assumed f o r the application o f the method. F i g . 13 presents estimation eirors f o r 135° incidence angle. Indeed, errors are sinaller than those obtained f o r the FPSO, c o n f i r m i n g the fact that the method is better f o r smaller ships since the ship motions have larger amplitudes. I n this case, the m i n i m u m peak period that can be estimated is smaller than 7 s.
A Bayesian M e t h o d was also implemented, based on Ref. [7,8]. Extensive numerical simulations were earned out to adjust all parameters i n v o l v e d i n such methodology i n order to obtain the best possible results. F i g . 14 presents the results f o r the f u l l y loaded condition. I t can be seen that such method is strongly sensitive to R A O ' s uncertainties, leading to estimation errors much higher than those obtained w i t h the Parametric M e t h o d . The results obtained f o r the ballasted condition are slightly better, but they still present unacceptable errors. A l t h o u g h the Bayesian M e t h o d is much faster than the Parametric one, since i t requires only Full C o n d i t o n - E s t i m a t i o n E r r o r s
7,00 9,00 11,00 13,00 15,00 17.00 19.00
P e a k P e r i o d ( s )
252 EA. Tanmiri et al./Applied Ocean Researcli 25 (2003) 243-261
Fig. 15. Ship heading considered in the 1-yeai- and 100-year wave analysis.
a quadratic programming algorithm instead o f a generic non-linear m i n i m i z a t i o n one, the results are poor.
A l l other numerical tests performed confirmed that the accuracy of the Parametric M e t h o d is higher than the one presented by the Bayesian M e t h o d . So, only the results o f the Parametric Method are presented i n the f o l l o w i n g .
4.2. B. Application to extreme 1-year and 100-year unimodal waves in Campos Basin
The previous analysis was used to estimate the cut-off frequency o f the method both f o r the f u l l and ballasted conditions o f the V L C C tanker.
The method was then applied to 1-year and 100-year waves i n Campos Basin. The ship heading considered i n the present analysis is 157.5° w i t h respect to N o r t h direction, as shown i n F i g . 15.
The results obtained f o r 1-year waves are shown i n Table 3. I t can be seen that f o r the f u h load condition, the eiTor is smaller than 7% f o r wave height, 1.2% f o r peak period and 1.9° f o r mean direction. Even f o r W and N W waves, w h i c h have peak periods smaller than the estimated c u t - o f f period o f 11 s, the errors are acceptable.
For ballasted condition, the estimation eiTors are smaller than 7, 2.3% and 1.8° f o r height, period and mean direction, respectively.
The SE 1-year spectrum is represented i n F i g . 16. I n this wave spectrum contour plot, the circles conespond to
the indicated frequencies o f 0.4, 0.8 and 1.2 rad/s. I t can be clearly seen that the overall shape and directional spreading of the estimated spectra are quite similar to the real one, both f o r the f u l l y loaded and ballasted conditions.
As the method presented good results when applied to 1-year waves, even better results were expected i n the case o f 100-yeai- waves, since the significant height and peak period are higher, g i v i n g rise to higher ship motions. Indeed, the results f o r 100-year waves are presented i n Table 4, where i t can be seen that the m a x i m u m eirors f o r the f u l l loaded condition are o f 4 , 1 % and 1.6° f o r height, period and direction, respectively. As expected, these eirors are smaller than those obtained f o r the L y e a r wave estimation. Furthermore, f o r the ballasted condition the errors are smaller than 2 % , 0.5% and 0.7° f o r height, period and direction, respectively.
I n the analyses presented i n Tables 3 and 4, s^ = 60 and Aj = 1 were used again. The estimated spreading coefficient varied between 40 and 70.
I t must be emphasized that i n a l l the cases exposed above, the Parametric M e t h o d returned the f u l l vector o f parameters
X = [o)„a HsiSie„aCJ,„2Hs2S20,„2V. Since the real specttum
in these cases is unimodal, one o f the estimated significant heights was indeed negligible, and the corresponding peak was disregarded by a post-processor algorithm.
4.3. Bimodal spectra
Finally, the method was applied to typical b i m o d a l spectra observed i n Campos Basin. A n exhaustive analysis was carried out, considenng the highest probability sea-states i n the region, using the data recorded during 5 years by a directional buoy operated by Petrobras [ 1 3 ] .
Since b i m o d a l sea states presented i n this section are not extreme, the present analysis w o u l d not be applied i n a D P feedforwad wave compensation technique. The only objec-tive here is to c l a r i f y and e x e m p l i f y the estimation method. Table 5 presents the results f o r t w o different cases, summarizing a l l the analyses done. The ship is headed to Southeast direction.
Table 3
Annual waves in Campos Basin
Real spectrum parameters Estimated parameters (loaded condition) Estimated parameters (ballasted condition) Hs(m) r p ( s ) eini") Hs (m) Tp (s) dm {°} Hs(xn^ T ^ dm (") N 4.2 12.4 0.0 4.1 NE 3.9 12.0 45.0 3.7 E 3.7 11.7 90.0 3.7 SE 4.5 12.7 135.0 4.2 S 5.1 13.4 180.0 5.0 SW 5.7 14.1 225.0 5.3 W 3.0 10.7 270.0 3.1 NW 3.0 10.7 315.0 3.1 12.5 0.7 4.1 12.4 -07 12.1 45.0 3.9 12.1 46.8 11.7 88.1 3.6 11.7 91.0 12.8 136.2 4,4 12.7 134.8 13.4 179.6 5.0 13.4 178.9 14.2 225.0 5.7 14.1 225,0 10.7 269.1 2.8 10.7 271.2 10.4 313.6 2.9 10.7 315.1
EA. Tannuri et al./Applied Ocean Researcli 25 (2003) 243-261 253
_ 180°.S
-Real (Hs=4.5m;Tp= 12.7s)
Estim. Full (Hs=4.2m Tp=12,7s; A9=1.2°) Estim. Ballasted (Hs=4.4m; Tp=12.7s; Ae=-0.2°)
Fig. 16. SE 1-year spectrum.
Table 4
Centenary waves in Campos Basin
Real spectram parameters Estimated parameters (loaded condition) Estimated parameters (ballasted condition) Hs (m) TpCs) em (°) lis (m) 7'p(s) Hs (m) T,(s) em (") N 6.3 14.6 0.0 6.2 14,6 - 1 . 2 6,2 14.6 - 0 . 6 NE 5.4 13.8 45.0 5.3 13.8 46.4 5,3 13.8 45,7 E 4.7 12.9 90.0 4.6 13.0 91.3 4.7 12,9 90.0 SE 6.7 15.1 135.0 6.6 15.2 135.8 6,7 15.1 134.7 S 7.0 15.3 180.0 6.9 15.3 180.0 7,0 15,3 179.5 SW 7.8 16.2 225.0 7.8 16,2 224.2 7,7 16.2 225.4 W 4.9 13.2 270.0 4.9 13,2 269.5 4,8 13.2 270.7 N W 4.9 13.2 315.0 4.7 13,4 313.4 4,8 13.2 315.6 Table 5
Bimodal spectra estimation
Peak 1 Peak 2 Cond Hs\ T 1 Al .51 ei Hsj A2 s2 92 A Real 1.76 12.05 1.5 17.10 182.20 1.08 5.93 1,2 23.70 98.30 Full 2.02 11.50 1.0(*) 19.30 180.30
-
- - - -Ballasted 1.67 12.09 1.0(*) 17.20 180.80-
-
-
-
-B Real 1.52 6,12 1.5 19.90 186.30 0.64 11.64 1.2 23.50 140.50 Full _ --
--
0.75 10,94 1.0(*) 13.89 141.72 Ballasted-
-
- - 0.69 10,99 1.0(*) 8,56 141.16254 E.A. Tamniri et al. /Applied Ocean Research 25 (2003) 243-261
(a) (b)
Fig. 17. Bimodal spectrum in condition A. (a) Real spectrum (b) Estimated spectrum in full loaded condition.
Fig. 18. Maximum ship motions for both peaks (Cond, A — f u l l loaded ship).
Table 6
Bimodal spectra estimation
Cond Peak 1 Peak 2
Hs^ Al i-l 01 Hs2 7-p2 A2 s2 02
C Real 1,76 12,05 1,5 17.10 182.2 1.08 12.05 1.2 23.70 98.3
Full 2,18 11,45 1,0(*) 13.60 175,2 0.96 12,16 1,0(*) 13.90 97.8
' Ballasted 2,02 11,68 1,0(*) 14.32 180.3 1.02 12.08 1.0(*) 20.25 97.3
(*) this parameter was not estimated. It is supposed Aj = 1, as already explained.
Tlie first contiition i n Table 5 (Cond.A) corresponds to the case where the first peak (the peak w i t h the highest significant height) has a peak period o f 12.05 s and the waves come f r o m Southern direction. The second peak has a higher frequency, w i t h 5.93 s peak period. As expected, the second peak could not be recovered neither in loaded nor i n baUasted condition, since its peak period is sinaller than the cut-ofl" period previously determined (10 s f o r ballasted ship and l i s f o r loaded one). The first. peak, however, was estimated w i t h an error bellow 14% f o r height, 5% period and 2° f o r direction. I t can be seen that spreading estimation presents higher errors w h i c h is
Table 7
VLCC characteristics in 3 loading conditions considered in experiments Properties Full Intermediate Ballasted
loaded
Mass (M) 302028 ton 198944 ton 115838 ton
Draft (T) 21.0 m 14,7 m 9,0 m Roll radius 16,83 m 16.92 m 23.22 m of gyration Pitch radius 86,31 m 82.43 m 90.54 m of gyration Yaw radius 80.00 m 80.00 m 80.00 m of gyration
E.A. Tanmiri el al./Applied Ocean Researdi 25 (2003) 243-261 255
Fig. 19. Wave incidence directions of experiments (angles with respect to longitudinal axis of the ship).
expected since the ship motions are less dependent on i t . Once again, f o r all the spectrum parameters estimation results f o r the ballasted case are better than f o r the f u l l loaded ship. The method evaluated the f u l l estimated p a r a m e t e r v e c t o r JV = {u>mi H^i s\ 0,„i co,„2 Hs2 ö , „ 2 ] ^ Since the second estimated frequency w„,2 was greater than the c u t - o f f frequency previously obtained, the post-processor algorithm disregarded the corresponding peak.
Furthermore, even i n the presence o f an incident spectrum w i t h A,- 5^ 1, the estimation accuracy o f Hsj, (i),„j and 6„,j is good.
This type o f spectrum, i n which the peak w i t h highest significant height has the peak period higher than 10 s, corresponds to 36% o f all the registered spectra i n the analysis carried out i n Campos Basin [13].
The second class o f spectra is represented by Cond.B i n Table 5, i n w h i c h the second peak (namely, the peak w i t h the lowest significant height) presents the peak period greater than 10 s (in this case, 11.64 s). I t can be seen that
256 E.A. Tannuri et al. /Applied Ocean Research 25 (2003) 243-261
the second peak was recovered w i t h errors smaller than 17% f o r height, 6% f o r period and 1.2° f o r direction. The first peak could not be estimated, since its peak period is smaUer than the cut-off period both i n ballasted and f u l l loaded condihons. This class o f spectrum corresponds to 16% o f the registered cases i n Ref. [13].
As expected, the 48% remaiinng registered spectrum presented both peaks w i t h periods smaller than 10 s. Consequendy any peak o f this class o f spectra could not be recovered by the method neither i n f u l l loaded nor i n ballasted condition. Fortunately these sea states do not induce h a r m f u l motions i n the system.
T h e first s p e c t r u m ( C o n d . A ) is represented i n Fig. 17(a), w i t h the ship headed to Southeast direction. The first peak comes f r o m South (182.2°) and the second peak comes f r o m East (98.3°). The estimated spectrum i n the full-loaded condition is shown i n F i g . 17(b). I t can be seen that the first peak has been recovered w i t h a slight deformation i n the shape mainly f o r high-frequency components. As already discussed, the second peak has not been recovered.
In order to e x e m p l i f y the relative importance o f the peaks, the m a x i m u m amplitude o f each m o t i o n is evaluated f o r the previous case, assuming i n d i v i d u a l incidence o f the peaks. The results are presented i n Fig. 18 where i t can be seen that the second peak induces small motions i n the ship indeed.
A n artificial case (Condition C) was also analysed, a i m i n g to detennine the a b i l i t y o f the method i n estimating bimodal spectra i n w h i c h both peaks have average zero-crossing periods greater than 8 s. Indeed, Table 6 shows that the both peaks are estimated w i t h errors smaller than 24% f o r wave height, 5% f o r period and 7° f o r direction. A g a i n , it can be seen that the spreading parameter estimation presents higher errors, w h i c h is expected since the motions o f the ship are less dependent on them.
5. E x p e r i m e n t a l results
The Parametric M e t h o d was apphed to model-scale experiments carried out i n I P T t o w i n g tank. A 1:90 reduced model o f V L C C V i d a l de Negreiros was used. Three loading conditions were considered whose m a i n characteristics transposed to real scale are given i n Table 7.
The ship was moored by 8 linear springs, representing a spread m o o r i n g system w i t h d i f f e r e n t i a l c o m p l i a n c e ( D I C A S ) . The i n i t i a l heading o f the model was 2 0 2 , 5 ° w i t h respect to the Northern direction.
Experiments were conducted considering only unimodal and unidirectional sea states. F i g . 19 shows the f o u r wave incidence directions considered.
A n optical reference system was used to measure a l l ship motions. Wave height were measured by a capacitive wave probe installed close to the model (Fig. 20).
A l l trials lasted approximately 47 s (equivalent to 7.5 m i n i n real scale), and the measurements were performed w i t h a samphng rate o f 51 H z (5,4 H z i n real scale). Cross spectra f u n c t i o n s o f ship motions were evaluated by W e l c h M e t h o d (Welch, 1967), w i t h 4096-sample F F T , 256-4096-sample Hanning w i n d o w i n g w i t h 128-sample overlap.
I n the present experiment, ship yaw was almost constant, w i t h variations smaller than 2° due to the stiff springs used
Table 8
1, 10 and 100-year extreme wave conditions
Incidence 1 -year 10-year 100-year
Incidence Hs T Hs T Hs T -"p .\ 5.7 m 137 s 6,9 m 14,6 s 7,8 m 15.3 s B 5.1 m 13,2 s 6,1 m 14.0 s 7.0 m 14,7 s C 4.5 m 10,3 s 5,5 m 10.8 s 6.7 m 11,3 s D 3,9 m 8,5 s 4,7 ni 9.0 s 5.4 m 9,4 s
E.A. Tannuri et al. /Applied Ocean Research 25 (2003) 243-261 257
Table 9
Parametric method results—estimation errors in parenthesis
Cond. bicidence Trial Hs real (m) Tp real (s) Hs est (m) Tp est (s) Dir. est
Full loaded A (180°) 1 6.4 14.6 7.3 (15%) 13.0(- 10%) 180,2° (0°) 2 7.0 14.8 7.2 (3%) 13.9(-6%) 180,2° (0°) 3 7.8 13.3 8.5 (9%) 12.6(-5%) 180.2° (0°) 4 9.0 14.5 9.8 (8%) 13,9(-4%) 180.2° (0°) B (225°) 1 4.9 15.2 6.0 (22%) 14.8 ( - 2 % ) 217.4° ( - 8 ° ) 2 6.1 12.9 7.7 (26%) 13,2 (3%) 225,4° (0°) 3 9.1 127 9.0 ( - 1%) 13.6 (7%) 2 1 4 , 4 ° ( - 11°) 4 7.3 14.3 8.8 (21%) 14,1 ( - 1%) 2 1 4 , 7 ° ( - 10°) C (270°) 1 5.3 10.8 4.3 ( - 18%) 11.0 (1%) 269,1° (1°) 2 5.3 10.9 4.2 ( - 19%) 11,3 (4,8%) 269,1° (1°) 3 6.8 10.9 5.2 ( - 2 4 % ) 11,5 (5%) 269,3° (1°) 4 7.8 11.0 5.8 ( - 2 5 % ) 11,5 (5%) 269.6° (0°) D (315°) 1 4.3 10.0 5.4 (26%) 9.8 ( - 2 % ) 337.0° (22°) 2 4.9 9.5 6.7 ( - 5 % ) 9,7 (2%) 339.0° (24°) 3 5.7 10.1 7.5 (31%) 9,4 ( - 7 % ) 358.1° (43°) 4 6.0 10.2 5.0 ( - 18%) 10.2 (0%) 351.2° (36°) Intermediate A (180°) 1 6.6 14.9 6.4 ( - 4 % ) 13,4(- 10%) 180,0° (0°) 2 6.8 15.2 6.8 (0%) 13.7(-10%) 180.0° (0°) 3 8.2 13.6 9.0 (10%) 12.7 ( - 6 % ) 180.0° (0°) 4 9.4 14.6 9.6 (2%) 13,5 ( - 7 % ) 180.0° (0°) B (225°) 1 5.0 15.2 5.8 (15%) 14.6 ( - 3 % ) 221,7° ( - 3 ° ) 2 6.3 13.3 6.6 (4%) 13.2 ( - 1 % ) 217,8° ( - 7 ° ) 3 8.6 13.1 8.4 ( - 2 % ) 13,5 (3%) 222,5° ( - 3 ° ) 4 7.7 14.0 8.3 (8%) 14,0 (0%) 222,4° ( - 3 ° ) C (270°) 1 4.8 10.8 5.2 (8%) 11,3 (4%) 268.8° ( - 1°) 2 6.2 10.9 6.0 ( - 4 % ) 12,6 (16%) 269.4° ( - 1°) 3 7.8 10.3 6.5 ( - 16%) 11,8 (15%) 269.3° ( - 1°) 4 6.6 10.7 6.5 ( - 1%) 11.3 (6%) 269.1° ( - 1°) D (315°) 1 4.3 9.8 5.9 (36%) 9.8 (0%) 353.8° (39°) 2 5.3 9.6 6.2 (18%) 9.4 ( - 2 % ) 344.5° (30°) 3 5.1 lO.I 8 7 (70%) 9.6 ( - 5 % ) 354,7° (40°) 4 5.9 10.2 6.7 (14%) 9,9 ( - 3 % ) 354.9° (40°) Ballasted A (180°) 1 6.7 14.6 6.7 (0%) 13,5 ( - 8 % ) 180.0° (0°) 2 7.1 14.3 6.9 ( - 4 % ) 14,1 ( - 2 % ) 180.0° (0°) 3 8.4 14.1 8.4 (0%) 13,2 ( - 6 % ) 180.0° (0°) 4 9.3 14.9 10.2 (9%) 13,4(-10%) 180.0° (0°) B (225°) 1 4.9 15.2 5.9 (20%) 14.3 ( - 6 % ) 224,6° (0°) 2 6.1 13.3 7.0 (15%) 12.9 ( - 3 % ) 2 2 0 . 0 ° ( - 5 ° ) 3 8.4 13.6 8.7 (4%) 13.3 ( - 2 % ) 224.0 ° ( - 1 ° ) 4 7.5 14.3 8.3 (11%) 13.9 ( - 3 % ) 225,7 ° (1°) C (270°) 1 4.1 10.8 5.1 (24%) 11.4 (5%) 269.5° ( - 1 ° ) 2 6.2 11.2 6.2 (0%) 12.5 (12%) 269.7° (0°) 3 6.2 10.3 7.5 (20%) 11,3 (11%) 269,7° (0°) 4 6.1 10.7 7.0 (15%) 11,4 (7%) 269.6° (0°) D (315°) 1 4.4 10.1 7.7 (74%) 9,5 ( - 6 % ) 351.0° (36°) 2 4.8 9.1 8.7 (80%) 9,2 ( - 5 % ) 337,7° (23°) 3 5.3 10.0 10.1 (0%) 9,5 ( - 5 % ) 358.2° (43°) 4 5.8 10.1 8.3 (42%) 9.5 ( - 6 % ) 350.8° (36°)
i n tiie m o o r i n g system. However, real applications can present h i g h e r y a w v a r i a t i o n s , w h i c h changes the incident wave direction w i t h respect to the ship and influences the estimate. Such problem i s n ' t addressed i n the present work, but i t was treated by Ref. [ 1 4 ] , i n which the authors performed the average o f spectral blocks of the W e l c h method f o r each heading separately.
The real incident wave spectrum was obtained by means o f wave-height time series measured by wave-probe. I t was used f o r c o n f r o n t a t i o n w i t h spectrum estimated b y Parametnc M e t h o d using only ship motions measurements. For example, F i g . 21(a) shows wave-height time series f o r an incidence B experiment, and F i g . 21(b) contains the respective wave power spectrum.
E.A. Tannuri et al./Applied Ocean Research 25 (2003) 243-261
E.A. Tanmni el al./Applied Ocean Reseaich 25 (2003) 243-261 259
Frequency(iad/s)
Fig. 24. Estimated and measured wave power spectra. Full loaded condition, incidence A, trial 3.
For unimodal waves, the significant wave height is related to wave power spectrum by:
Hs = 4yJ S{co)dcd
The peak period Tp is the period w i t h m a x i m u m power spectrum, and it can be shown to correspond to the modal frequency previously defined (Tp = ITT/CO,,,). These par-ameters (Hs and Tp) were evaluated f o r each experiment, using the wave power spectrum obtained by wave-height time series. Such parameters were used f o r comparison w i t h the estimates obtained by the Parametric M e t h o d . 1, 10 and 100-year extreme wave conditions were used f o r each incidence usually observed i n the Campos Basin. The significant wave height and peak period f o r these cases are given i n Table 8.
The results o f the Parametric M e t h o d are presented i n Table 9. The experiments w i t h estimated peak period smaller than U s ( f u l l loaded) and 1 0 s (bahasted and intermediate loading) are shaded (all experiments w i t h incidence D ) . These results must be discarded since the numerical analyses presented i n section 4 showed that these w a v e conditions cannot be estimated b y the Parametric Method.^ Indeed, large estimation eirors were o b t a i n e d f o r these e x p e r i m e n t s , r e a c h i n g 8 0 % f o r significant height and 7 0 ° f o r mean direction. However, peak period is accurately estimated, w i t h errors smaller than 7%. Hence a comparison o f the estimated peak period w i t h its theoretically predicted range permits the analyst to decide whether the estimates o f the remaining parameters
^ The numerical analysis was not carried out for intermediate loading condition. The minimum peak period obtained for ballasted ship (10s) was also used for this condition.
are trustable or not. For the f u l l loaded case i n wave incidence C, the real peak period is close to the theoretical estimation l i m i t o f 11 s and the estimates lie i n the range between 11.0 and 11.5 s, just above the l i m i t . I n these cases, the results may not be discarded, but an error up to 25% i n wave height estimation is observed. This fact occurs because o f the p r o x i m i t y o f the theoretical estimation l i m i t . So, a period margin must be established i n order to avoid these cases.
As already said, the Parametric M e t h o d returned the f u l l vector o f parameters x = [M,„I HsiSi6,„iCi)„,2Hs2S20,„2V. H o w e v e r , f o r a l l experiments, one o f the estimated significant height was indeed n e g l i g i b l e , and a post-processor algorithm discarded the corresponding peak. This fact confirms the method ability to i d e n t i f y unimodal sea-states.
The spreading coefficient 5 is theoretically i n f i n i t e f o r unidirectional sea states, l i k e those generated i n the t o w i n g tanlc. For all experiments conducted w i t h wave incidence C, the estimated value was ^ = 85, w h i c h is h i g h enough and the waves can be classified as unidirectional. For incidences A and B incidence the estimated spreading coefficient varied i n the range 1 2 - 3 0 , indicating some energy d i r e c t i o n a l spreading. F i g . 22 contains the estimated wave spectrum contour plots f o r each incidence ( A , B and C ) , c o n f i i m i n g thus the spread computed f o r incidences A and B .
The wave reflection i n the tank walls may explain this fact. Indeed, f o r incidence C, waves diffracted and reflected by the model propagate parallel to the tank borders. I n A and B incidences, however, some diffracted and reflected wave components propagate perpendicularly to the walls, being reflected and reaching the ship m o d e l again. I n these cases, a higher d i r e c t i o n a l spreading is expected.
A n analysis o f a l l the results obtained shows that the Parametric M e t h o d produced estimates w i t h errors smaller than 2 5 % f o r wave significant height, 15% f o r peak period and 11° f o r mean direction. Figs. 23(a) and (b) show the results, showing a smaller dispersion i n period w h e n compared to height. F i g . 23(c) shows estimated directions f o r a l l valid experiments. Errors up to 11° i n incidence B were observed.
Fig. 24 shows the estimated and measured wave power spectra f o r the experiment w i t h f u l l loaded ship, incidence A , test 3. It must be emphasized that the method produces good results even i n the presence o f a wave spectrum w i t h a general shape different f r o m the P i e r s o n - M o s k o w i t z spectrum. Indeed, i t can be seen that the measured spectrum presents higher energy concen-tration than the estimated one. I t resembles a J O N S W A P s p e c t r u m ( w i t h A, > 1), w i d e l y used to describe undeveloped sea states and swell waves. I n this case the wave parameters were accurately estimated, w i t h a 9% error i n significant height and a — 5 % error i n peak period estimates.
260 E.A. Tannim et al. /Applied Ocean Research 25 (2003) 243-261
Real peak p e r i o d (s)
The experiments were also analysed using the non-parametric Bayesian IVIethod previously menhoned. As it can be seen i n F i g . 25, the estimadon eiTors w i t h the Bayesian M e t h o d are higher than those obtained w i t h the Parametric Method, reaching 35% for significant height, 3 1 % f o r peak period and 30° f o r direcdon.
These results suggest the superiority o f Parametric M e t h o d i n the present case.
6. Conclusions
The feasibility of wave spectrum estimation based on stationary ship m o t i o n measurements was analyzed i n the present paper. Attention was given to the identification o f critical problems that may arise when such methods are applied to large tanker ships. Such algorithms play an important role i n f e e d f o r w a r d controllers used i n D P o f ships.
E.A. Tannuri et al./Applied Ocean Research 25 (2003) 243-261 261
The estimation methods depend on tiie previous k n o w l -edge o f the response o f the ship when subjected to waves expressed i n the f o r m o f R A O ' s . Practical limitarions mainly due to non-linear effects and variations i n loading conditions appear i n the application o f the methodology to FPSO's.
This problem is partially addressed here by considering the variations o f R A O ' s due to the uncertainty i n siup loading—non-linear effects have not been taken into account i n the paper. It was verified that r o l l motion is extremely sensitive to these variations, due to its resonant behavior. Then, unlike the usual applications o f directional buoys, the use o f sway instead o f r o l l was proposed. This change can be made since, like r o l l , sway R A O is also an odd f u n c t i o n o f the incidence angle, preserving the directional information contained i n the r o l l motion.
A Parametric Estimation M e t h o d was applied to a moored V L C C . The methodology was tested under typical wave conditions o f the Campos Basin i n numerical and towing-tank experiments.
First-order m o t i o n measurements are the basis f o r the estimation procedure proposed. So, the method w i l l not w o r k properly f o r high-frequency wave spectra, since they do not induce s i g n i f i c a n t (first order) ship motions. Spectrum components w i t h peak period smaller than a p r e - c a l c u l a t e d c u t - o f f p e r i o d cannot be accurately estimated. Numerical experiments con-esponding to several peak periods were useful to determine the cut-off period f o r FPSO both i n f u l l loaded and ballasted conditions.
Other numerical experiments emulated critical 1 and 100 year unimodal Campos Basin sea condition, as w e l l as typical bimodal states. For a l l cases, when the peak period was larger than the cut-off period, accurate estimates were obtained. T o w i n g - t a n k experiments confirmed the previous numerical analysis.
Bayesian non-parametric method was also applied. I t has s h o w n to be s t r o n g l y sensitive to uncertainties i n R A O values, w i t h worse results f o r both numerical and towing-tank experiments.
T o close the paper, it has been shown that a wave spectrum estimation algorithm based on FPSO motions measurements is feasible, provided that some care is taken. Since the ship is being used as a wave sensor, its dynamics must be w e l l k n o w n i n order to p e r f o r m good estimates o f wave spectra.
Acknowledgements
This project was sponsored by P e t r ó l e o Brasileiro S/A (Petrobras). Authors are grateful to Eng. Msc. A n d r é J.P.
Leite, f r o m Petrobras and to the technical assistance o f Professor J o s é A . P. Aranha, f r o m the Naval A r c h , and Ocean Engineering Department o f University o f Sao Paulo. T h i r d author acknowledges the financial support o f Petroleum National Agency ( A N P ) and the f o u r t h author thanks both to FAPESP (Proc. N o . 97/04668-1) and to CNPq (Proc. N o . 304071/85-4).
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