Modelling of multipeaked directional wave spectra

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Applied Ocean Research 31 (2009) 1 3 2 - 1 4 1

ELSEVIER

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Applied Ocean Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p o r

O C E A N R E S E A R C H

Modelling of multipeaked directional wave spectra

A.V. Boukhanovsky, C. Guedes Soares*

Centre/or Marine Teclmoiogy and Engineering. Technical Umversity of Lisbon. Instituto Superior Técnico. Av. Rovisco Pais. W49-001. Lisbon. Portugal

A R T I C L E I N F O A B S T R A C T

Article histo,y A n a p p r o a c h f o r m o d e i i i n g o f m u l t i p e a k e d d i r e c t i o n a l w a v e s p e c t r a is p r o p o s e d For m o d e l i d e n t i f i c a t i o n Received 17 December 2008 a n u m e r i c a l o p t i m i z a t i o n t e c h n i q u e t h a t uses t h e r a n d o m l i n e a r s e a r c h ^ g o n t h m ,s a p p h e d . T h Received in revised f o r m t e c h n i q u e a l l o w s t h e fitting o f s p e c t r a l m o d e l s t o m e a s u r e d o r h m d c a s t d a t a . T h e HIPOCAS h m d c a s t d a t a 20 May 2009 f o r N o r t h A t l a n t i c are u s e d f o r a n a p p l i c a t i o n S t u d y . . , ^ » „

Accepted 14June 2009 © 2 0 0 9 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d . Available online 7 July 2009

Keywords:

Multipeaked spectra Double peak spectra Wave spectra Direcdonal specta

1. Introduction

Design o f m a r i n e s t r u c t u r e s is based o n e s t i m a t e s o f t h e w a v e i n d u c e d loads g e n e r a l l y c a l c u l a t e d i n t h e f r e q u e n c y (ƒ) a n d d i r e c t i o n ( 0 ) d o m a i n s , w h i c h r e q u i r e i n f o r m a t i o n a b o u t t h e d i r e c t i o n a l w a v e s p e c t r u m S ( f , 9). The f o r m o f t h e d i r e c t i o n a l w a v e s p e c t r u m reflects t h e p h y s i c a l processes t h a t g o v e r n t h e e n e r g y balance b e t w e e n t h e local w i n d stress o f sea surface, i n t e r a c t i o n b e t w e e n w i n d w a v e s ( W ) a n d t h e s w e l l s (S) c o m i n g f r o m o t h e r areas, g e n e r a t e d at earlier t i m e s a n d b y o t h e r w i n d fields. Hence, t h e t r a d i t i o n a l a p p r o a c h f o r s p e c t r a l m o d e l l i n g as S ( f , 0 ) = S ( J ) ( l ( f , 6 ) , w h e r e t h e f r e q u e n c y s p e c t r u m

S ( f ) is a p p r o x i m a t e d b y t h e o n e - p e a k e d P i e r s o n - M o s k o w i t z s p e c t r u m f o r d e v e l o p e d sea states a n d b y a JONSWAP s p e c t r u m f o r d e v e l o p i n g seas (see [ 1 ] ) , a n d t h e d i r e c t i o n a l s p r e a d i n g d i s t r i b u t i o n (l(J, 9) is s i n g l e - p e a k e d also (see [ 2 ] ) .

M o s t studies o f w a v e s p e c t r a l m o d e l l i n g t h a t have b e e n d o n e i n t h e past have addressed o n l y single peaked spectra. O c h i a n d H u b b l e [ 3 ] a n d Guedes Soares [ 4 ] have p r o p o s e d s p e c t r a l m o d e l s t o describe t h e c o m b i n e d s i t u a t i o n o f s w e l l a n d w i n d sea. Guedes Soares [ 5 ] , a m o n g o t h e r s , has s h o w n t h a t t h e t w o peaked spectra s t i l l occur w i t h a r e l a t i v e l y h i g h f r e q u e n c y , w h i c h m a k e s i t i m p o r t a n t t o use t h e m i n t h e d e s i g n a p p r o a c h . Guedes Soares a n d Nolasco [ 6 ] have p r o p o s e d r e p r e s e n t a t i v e values f o r t h e 4 p a r a m e t e r s o f t h e d o u b l e peak m o d e l o f Guedes Soares [ 4 ] r e l e v a n t f o r t h e sea states o f t h e Portuguese coastal c l i m a t e .

* Corresponding author. TeL: +351 218417957.

E-mail address: guedessOmar.ist.utl.pt (C. Guedes Soares).

T o r s e t h a u g e n [ 7 ] has f u r t h e r g e n e r a l i z e d t h e m o d e l o f Guedes Soares [ 4 ] , also a d o p t i n g t h e JONSWAP m o d e l s f o r each o f t h e w a v e systems, b u t i n s t e a d o f u s i n g a c o n s t a n t (average) peak e n h a n c e m e n t f a c t o r i n each m o d e l , he a l l o w e d t h i s t o v a r y , l e a d i n g t o a m o d e l w i t h seven p a r a m e t e r s , w h o s e r e p r e s e n t a t i v e values he d e t e r m i n e d f o r t h e c o n d i t i o n s o f t h e N o r w e g i a n coast.

It has t a k e n m a n y years b e f o r e t h e ship a n d o f f s h o r e i n d u s t r y r e a l i z e d t h e i m p o r t a n c e o f a c c o u n t i n g f o r this c o m b i n e d seas s i t u a t i o n i n t h e d e s i g n a p p r o a c h , b u t r e c e n t years have seen a n i n c r e a s i n g i n t e r e s t i n t h e a d o p t i o n o f these cases i n i n d u s t r i a l a p p l i c a t i o n s , as i n d i c a t e d i n [ 8 - 1 1 ]. This i n t e r e s t has m o t i v a t e d t h e p r e s e n t paper, w h i c h deals w i t h a m e t h o d to i d e n t i f y t h e e x i s t i n g w a v e systems i n a g i v e n w a v e s p e c t r u m a n d t o f i t p a r a m e t r i c m o d e l s to each o f t h e i n d i v i d u a l systems. I t w i l l be s h o w n t h a t t h e r e s u l t i n g m o d e l o f t h e w a v e s p e c t r u m is capable o f d e s c r i b i n g c o m p l e x sea c o n d i t i o n s i n t h e f r e q u e n c y a n d d i r e c t i o n d o m a i n s t h a t r e p r e s e n t s e x i s t i n g w i n d w a v e s a n d s w e l l c o m b i n a t i o n s .

This m e t h o d generalizes t h e o n e o f Guedes Soares [ 4 ] , w h i c h d e a l t o n l y w i t h f r e q u e n c y spectra, by e x t e n d i n g t h e search o f w a v e s y s t e m s t o t h e d i r e c t i o n d o m a i n . T h i s g e n e r a l i z a t i o n is n o w possible because, p r e s e n t l y , m o s t o f t h e m e a s u r e d w a v e data is m t h e f o r m o f d i r e c t i o n a l spectra, w h i l e i n t h e 1980's o n l y f r e q u e n c y spectra w a s g e n e r a l l y available. Generally, t h e s p e c t r a l m o d e l o f c o m b i n e d seas is p r e s e n t e d as t h e s u m o f w i n d a n d s w e l l s p e c t r a l c o m p o n e n t s Sw,Ss ( f o r single o r several s w e l l s y s t e m s ) . I n t h e f r e q u e n c y d o m a i n , t h e i n d i v i d u a l s p e c t r a l c o m p o n e n t s are d e s c r i b e d u s i n g Sw, Ss as a t r a d i t i o n a l a p -p r o x i m a t i o n t o t h e s i n g l e - -p e a k e d s -p e c t r u m . The s w e l l c o m -p o n e n t o f t h e m u l t i p e a k e d s p e c t r u m w a s d e s c r i b e d b y a Gaussian f u n c -t i o n i n [ 1 2 ] w h i l e O c h i a n d H u b b l e [ 3 ] used a G a m m a -t y p e spec-t r u m a n d Guedes Soares [ 4 ] a n d laspec-ter T o r s e spec-t h a u g e n [ 7 ] a d o p spec-t e d spec-t h e

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A.V. Boukhanovsky, C. Guedes Soares/Apphed Ocean Research 31 (2009) 132-141 133

JONSWAP m o d e l . As s w e l l o f t e n has a n a r r o w s p e c t r u m , t h e JON-SWAP s p e c t r u m w i t h a p p r o p r i a t e peak e n h a n c e m e n t f a c t o r has b e e n s h o w n b y Goda [ 1 3 ] to be able t o r e p r e s e n t s w e l l . F u r t h e r -m o r e , as s w e l l results f r o -m a w i n d sea t h a t propagates a w a y f r o -m t h e g e n e r a t i o n area, i t is n a t u r a l t h a t a y o u n g s w e l l is d e s c r i b e d b y t h e same spectral m o d e l as w i n d sea (JONSWAP).

O c h i [ 1 4 ] a n a l y z e d 8 0 0 spectra o f w i n d w a v e s at n i n e Ocean W e a t h e r Stations i n t h e N o r t h A t l a n t i c , a n d s h o w e d t h a t t h e m o s t g e n e r a l r e p r e s e n t a t i o n f o r s u c h spectra w o u l d be c o m p o s i t i o n o f t h e set o f o n e - p e a k e d spectra d e f i n e d b y a s i m p l e set o f p a r a m e t e r s .

Guedes Soares [ 4 ] has separated t h e w a v e c o m p o n e n t s f r o m t h e analysis o f scalar spectra, w h i c h w a s t h e i n f o r m a t i o n available t h e n . This g e n e r a l a p p r o a c h w a s f u r t h e r d e t a i l e d i n [ 6 ] , w h i l e Ro-d r i g u e z a n Ro-d GueRo-des Soares [ 1 5 ] Ro-d e v i s e Ro-d a m e t h o Ro-d t h a t w o u l Ro-d l e n Ro-d i t s e l f to a n easier i m p l e m e n t a t i o n i n a n a u t o m a t e d i d e n t i f i c a t i o n process. Basically these m e t h o d s go t h r o u g h a l l s p e c t r a l o r d i n a t e s a n d d e t e r m i n e t h e i r local m a x i m a ( s p e c t r a l peaks) as w e l l as t h e m i n i m a b e t w e e n peaks. The s e p a r a t i o n is d o n e at t h e m i n i m a , a l -t h o u g h a f -t e r f i -t -t i n g s p e c -t r a l m o d e l s fixed a-t -t h e peak f r e q u e n c i e s , t h e i r tails w o u l d c o n t r i b u t e t o t h e spectral e n e r g y at h i g h e r f r e -quencies t h a n t h e s e p a r a t i o n f r e q u e n c y . W a n g a n d H w a n g [ 1 6 ] p r o p o s e d a n i n t e r e s t i n g a l t e r n a t i v e t h a t w a s based o n t h e c o n s i d e r a t i o n s t h a t t h e s p e c t r a l steepness above a g i v e n f r e q u e n c y w o u l d be m a i n l y d e p e n d e n t o n t h e w i n d sea. T h e r e f o r e , c a l c u l a t i n g this p a r a m e t e r as a f u n c t i o n o f f r e q u e n c y a n d d e t e r m i n i n g its m a x i m u m w o u l d i d e n t i f y t h e s e p a r a t i o n p o i n t b e t w e e n t h e t w o w a v e systems. I n t h e i r e x a m p l e s , t h e y have s h o w n t h a t t h e i r s e p a r a t i o n p o i n t w o u l d c o i n c i d e w i t h t h e one t h a t w o u l d be d e t e r m i n e d b y t h e a l g o r i t h m o f Guedes Soares [ 4 ] . They a r g u e d t h a t t h e i r a p p r o a c h w a s b e t t e r because t h e y w o u l d a v o i d g o i n g t h r o u g h all s p e c t r a l o r d i n a t e s to d e t e r m i n e local m a x i m a a n d m i n i m a . H o w e v e r , t h e y have t o use all s p e c t r a l o r d i n a t e s t o c a l c u -late t h e i r f r e q u e n c y d e p e n d e n t steepness f u n c t i o n , a n d a f t e r w a r d s t h e y have t o d e t e r m i n e its m a x i m a . T h e r e f o r e , a l t h o u g h b e i n g a n e l e g a n t a p p r o a c h , i t does n o t p r o v i d e a n y d i f f e r e n t r e s u l t t h a n w h a t Guedes Soares [ 4 ] w a s o b t a i n i n g .

The i m p r o v e m e n t i n t h e t e c h n o l o g y o f m e a s u r i n g w a v e s , m a d e d i r e c t i o n a l w a v e spectra m o r e w i d e l y available at a l a t e r stage, a n d o t h e r approaches w e r e d e v i s e d t o take advantage o f t h e d i r e c t i o n a l i n f o r m a t i o n also. T h i s is t h e case o f t h e a p p r o a c h o f G e r l i n g [ 1 7 ] , f u r t h e r e x t e n d e d b y H a n s o n a n d Philips [ 1 8 ] . T h e m a i n a p p r o a c h is s i m i l a r , as i t starts b y l o o k i n g f o r local m a x i m a n o w i n t h e c o m p l e t e space o f f r e q u e n c y a n d d i r e c t i o n . A f t e r w a r d s , s t a r t i n g f r o m t h e m a x i m a , t h e s p e c t r a l o r d i n a t e s associated w i t h i t are i d e n t i f i e d . Some h e u r i s t i c rules are d e v e l o p e d f o r t h e m i n i m u m d i s t a n c e b e t w e e n peaks a n d m i n i m u m e n e r g y i n c o m p o n e n t s i n a w a y s i m i l a r t o w h a t is d e s c r i b e d i n [ 6 ] f o r t h e case o f scalar spectra. The a p p r o a c h o f H a n s o n a n d Philips [ 1 8 ] a l l o w s t h e t r a c k i n g o f t h e source o f s t o r m s b u t i t r e q u i r e s i n f o r m a t i o n a b o u t t h e w i n d a n d r e q u i r e s a g r i d o f data p o i n t s . I t is w e l l a d a p t e d t o t r e a t data p r o d u c e d b y w a v e m o d e l s b u t i t is less a d e q u a t e t o analyze data c o l l e c t e d b y a single d i r e c t i o n a l b u o y .

Ewans et a l . [ 1 9 ] have c o m p a r e d b o t h m e t h o d s a p p l i e d t o m e a -s u r e d w a v e -spectra a n d have c o n c l u d e d t h a t t h e y give c o m p a r a b l e results, a l t h o u g h i n s o m e cases t h e m e t h o d o f Guedes Soares [ 4 ] w a s m o r e accurate t h a n t h e o n e o f H a n s o n a n d P h i l i p s [ 1 8 ] .

This p a p e r presents t h e g e n e r a l i z a t i o n o f t h e m o d e l l i n g o f Guedes Soares [ 4 ] f o r d i r e c t i o n a l spectra o f c o m p l e x seas, w h i c h does n o t r e q u i r e m o r e i n f o r m a t i o n t h a n t h e s p e c t r a l o r d i n a t e s . I t is d e m o n s t r a t e d h o w one can fit t h e m o d e l t o m e a s u r e d or h i n d c a s t data, a n d t h e d i f f e r e n c e b e t w e e n t h e m o d e l l i n g t e c h n i q u e s f o r d i r e c d o n a l spectra a n d f o r t h e f r e q u e n c y d o m a i n o n l y are s h o w n . The HIPOCAS h i n d c a s t data [ 2 0 , 2 1 ] are used as a n e x a m p l e o f a p p l i c a t i o n o f t h e m e t h o d .

2. Formulation of multipeaked directional wave spectra

A general sea state consists o f a l o c a l l y g e n e r a t e d w i n d sea a n d one or m o r e s w e l l s y s t e m s . I f t h e n o n - l i n e a r i n t e r a c t i o n b e t w e e n t h e w i n d waves a n d t h e s w e l l are n e g l i g i b l e , t h e n t h e t o t a l d i r e c t i o n a l s p e c t r u m is expressed as: S ( f , 0 ) = £ s , ( / , 0 ) _{( 1 )} 1=0 w h e r e t h e i n d e x value i = 0 is associated w i t h w i n d w a v e s , a n d N is t h e s w e l l systems n u m b e r . The f u n c t i o n s S i ( f , 6) a n d S j ( f , 9) at

i ^ j a l l o w f o r o v e r l a p p i n g over s o m e values o f (ƒ, 0 ) . Each s p e c t r a l c o m p o n e n t S, is c h a r a c t e r i z e d b y t h e set o f t h e i r m o m e n t s mu,

w h i c h are c o r r e s p o n d i n g to characteristics o f sea state, e.g. w a v e h e i g h t h (as t h e m e a n w a v e h e i g h t h o r s i g n i f i c a n t w a v e h e i g h t hs), z e r o - c r o s s i n g w a v e p e r i o d T a n d m e a n w a v e d i r e c t i o n 6 [ 2 2 ] . The t o t a l w a v e characteristics o f Eq. ( 1 ) are expressed f r o m t h e characteristics o f t h e s p e c t r a l c o m p o n e n t s , e.g. f o r d o u b l e p e a k e d s p e c t r u m o f w i n d w a v e s ( I V ) a n d t h e s w e l l ( S ) : ( 2 ) ( 3 ) H e r e , t h e w e i g h t c o e f f i c i e n t s 7]w, o f t h e w i n d w a v e s a n d s w e l l c o m p o n e n t s are: 'Is hi

The m e a n w a v e d i r e c t i o n o f a c o m p l e x sea is expressed as: ( 4 ) ( 5 ) ( 6 ) e = a r g ( c o s 0 - F i s i n e ) , w h e r e s i n e = r]w sin9w + m s i n 0 s ,

cos

9 = rivj

cos 0w -F

r]s

cos

9$.

The Eqs. ( 2 ) ( 6 ) a l l o w one to o b t a i n t h e d i r e c t i o n a l c h a r a c t e r i s -tics o f c o m p l e x seas b y m e a n s o f o p e r a t i o n s w i t h characteris-tics o f t h e s p e c t r a l c o m p o n e n t s , o n t h e a s s u m p t i o n t h a t t h e y have b e e n c o r r e c t l y s e p a r a t e d .

U n f o r t u n a t e l y , t h e clear s e p a r a t i o n o f t h e c o m p o n e n t s i n Eq. ( 1 ) is s o m e t i m e s a d i f f i c u l t p r o b l e m . I n t h i s case Eqs. ( 2 ) - ( 6 ) are c o n s i d e r e d as a n u n d e r d e t e r m i n e d n o n l i n e a r e q u a t i o n s y s t e m , as t h e n u m b e r o f u n k n o w n variables i n t h e right side o f Eqs. ( 2 ) - ( 6 ) is g r e a t e r t h a n t h e n u m b e r o f c o m p l e x sea c h a r a c t e r i s t i c s . T h e a d d i t i o n a l d e s c r i p t i o n o f t h e s p e c t r a l shape, u s i n g t h e a n a l y t i c a l m o d e l s o f sea w a v e s p e c t r u m , is r e q u i r e d f o r s o l v i n g t h a t p r o b l e m . T h e m o d e l o f single s p e c t r u m S ( / , 0 ) i n Eq. ( 1 ) is r e p r e s e n t e d as a p r o d u c t [ 2 3 ]

SiS, 9)= Sana, 9),

( 7 ) o f t h e f r e q u e n c y s p e c t r u m S(J) a n d t h e d i r e c t i o n a l s p r e a d i n g d i s -t r i b u -t i o n (l(J, 9), w h i c h s i m p l i f i e s t h e p a r a m e t e r i z a t i o n p r o c e -d u r e . T r a -d i t i o n a l l y , t h e m o s t c o m m o n is t h e S(J) r e p r e s e n t a t i o n b y m e a n s b y t h e g e n e r a l i z e d JONSWAP s p e c t r u m :

Si!) = Af-"

e x p

[-Bf-"] Y^^^

( 8 )

w h e r e A a n d B are t h e scale a n d t h e s h i f t c o e f f i c i e n t s , w h i c h d e -p e n d o n w a v e h e i g h t a n d -p e r i o d ; -p o s i t i v e values (I<, n) are t h e shape c o e f f i c i e n t s , y is t h e peakedness c o e f f i c i e n t a n d S ( f ) is t h e peak i n t e n s i t y f u n c t i o n . Eq. ( 8 ) is t h e g e n e r a l i z a t i o n o f a l o t o f w e l l k n o w n s p e c t r a l a p p r o x i m a t i o n s . I f y = 1 , t h e Eq. ( 8 ) is t h e t r a d i -tional W a l l o p s s p e c t r u m [ 2 4 ] f o r d i f f e r e n t c o n d i t i o n s . For t h e case

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134 AV. Boukhanovsky. C. Cuedes Soares / AppUed Ocean Research 31 (2009) 132-141

k = n + 1, t h e f r e q u e n c y s p e c t r u m is a p p r o x i m a t e d w i t h t h e s o -c a l l e d G a m m a - s p e -c t r u m :

S r ( f . mojp, n) = mo exp . ( 9 )

N o t e t h a t S r ( f j p , n ) d f = mo, a n d t h a t t h e spectral peak occurs f o r ƒ = fp. The h i g h - f r e q u e n c y t a i l o f t h e s p e c t r u m is p r o p o r t i o n a l to ƒ " " . The peakedness a n d w i d t h o f t h e s p e c t r u m are g o v e r n e d b y the v a l u e o f n. W e l l - k n o w n w i n d w a v e s p e c t r u m o f t h i s f o r m is P i e r s o n - M o s k o w i t z s p e c t r u m f o r n = 5 [ 2 5 ] a n d t h e D a v i d a n - M a s s e l s w e l l s p e c t r u m [ 2 6 ] f o r n = 6. I f y > 1, t h e s p e c t r a l shape depends o f t h e S ( f ) f u n c t i o n . For n = 5 Eq. ( 8 ) b e c o m e s t h e w e l l k n o w n JONSWAP s p e c t r u m [ 2 7 ] . For t h e a p p r o x i m a t i o n o f Q ( f , Ö) i n Eq. ( 7 ) t h e f o l l o w i n g e x p r e s s i o n is u s e d : a ( f , 0 , 0 , s ) = | Q c o s 2 ^ ^ ' ( 0 - 0 ( / ) ) , | e - 0 ( f ) | < 7 r / 2 , [ 0 , \9-ë(f)\>7v/2, w h e r e Cs is a n o r m a l i z i n g p a r a m e t e r such t h a t :

£

Q ( 0 , ö p , s ) d e = 1 . ( 1 0 ) (11)

Eq. ( 1 0 ) is close to t h e w e l l - r e c o g n i z e d cos-2s d i r e c t i o n a l d i s t r i b u t i o n w h i c h is c o n s i d e r e d by L o n g u e t - H i g g i n s e t al. [ 2 2 ] a n d M i t s u y a s u et al. [ 2 8 ] . H o w e v e r , i n d e p e n d e n t l y o f t h e values o f s t h e Eq. ( 1 0 ) does n o t a l l o w s p r e a d i n g m o r e t h a n 180 degrees, w h i c h is essential f o r the s e p a r a t i o n o f i n d e p e n d e n t w a v e systems i n c o m p l e x sea spectra (see Section 3).

The p a r a m e t e r s reflects t h e range o f d i r e c t i o n a l s p r e a d i n g . The v a l u e o f s is increasing w i t h t h e o v e r a l l i n t e n s i t y o f sea w a v e s . The d e p e n d e n c y o f 6 ( f ) and s ( f ) o n t h e f r e q u e n c y are expressed as

ê ( f ) = 9 p + A , ( f - f p } a n d s ( f ) f ( 1 2 )

w h e r e 6p, Sp are t h e values o f p a r a m e t e r s a t t h e peak o f a w a v e s y s t e m . Here /S > 0 f o r ƒ < fp, a n d ^ < 0 i n t h e o p p o s i t e case. It is a p p r o p r i a t e t o p o i n t o u t t h a t t h e r e are several a l t e r n a t i v e r e p r e s e n t a t i o n s o f s p r e a d i n g f u n c t i o n s , e.g. b y means o f w r a p p e d n o r m a l d i s t r i b u t i o n , S e c h - 2 s - d i s t r i b u t i o n , Poisson a n d V o n Mises d i s t r i b u t i o n s [ 2 ] . But t h e s m a l l d i f f e r e n c e b e t w e e n t h e m [ 2 9 ] a l l o w s u s i n g Eq. ( 1 0 ) as t h e s i m p l e a n d t h e f a v o r i t e d i r e c t i o n a l d i s t r i b u t i o n w i t h a l o t o f a p p l i c a t i o n s i n e n g i n e e r i n g practice. Thus, t h e c o n s t r u c t i o n o f t h e p a r a m e t r i c m o d e l b y Eq. ( 1 ) r e q u i r e s t h e k n o w l e d g e o f t h e t o t a l n u m b e r o f w a v e systems N. M o r e o v e r , f o r each s y s t e m i = 0, N t h e set o f p a r a m e t e r s Si = ( m ^ ' \ 9l,'\ n,, s|,'*, Af,l3i) is r e q u i r e d f o r t h e G a m m a -based m o d e l ( o n Eq. ( 9 ) ) , a n d S , = (mf,/p"', , y,-, s<'*, A f , A ) -f o r t h e JONSWAP-based m o d e l ( o n Eq. ( 8 ) ) .

3. Model identification through fitting

There are d i f f e r e n t approaches f o r t h e e s t i m a t i o n o f the spectral p a r a m e t e r s i n Eq. ( 1 ) , u s i n g t h e t e c h n i q u e of f i t t i n g to data (see e.g. [ 3 0 ] ) . For t h e r e d u c t i o n o f d i m e n s i o n a l i t y , consider t h e s i m p l i f i e d p r o b l e m , w h e r e t h e d i r e c t i o n a l p a r a m e t e r s s, 9 are n o t d e p e n d e n t o n t h e f r e q u e n c y , e.g. s = Sp,ö = öp a n d Ap = P = 0. T r a d i t i o n a l l y , t h e clearest a p p r o a c h is t h e explicit o n e , w h o s e idea is t h e fitting o f t h e s p e c t r a l c o m p o n e n t s , w h e r e t h e p a r a m e t e r s o f peaks ( m o , / p , ö p ) are o b t a i n e d d i r e c t l y f r o m t h e spectra, a n d t h e shape p a r a m e t e r s ( n , s ) or ( y , s ) are fixed i n accordance t o t r a d i t i o n a l r e c o m m e n d a t i o n s , e.g. n = 5 , s = 2 a n d y = 3.3 f o r w i n d w a v e s . T o d a y this m e t h o d is p o p u l a r t o

a p p r o x i m a t e m u l t i p e a k e d spectra, because i t does n o t r e q u i r e t h e c o m p l i c a t e c o m p u t a t i o n a l p r o c e d u r e s o f G e r l i n g [ 1 7 ] , o r H a n s o n a n d Philips [ 1 8 ] .

The first step i n t h e p r e s e n t a p p r o a c h ( a n d i n t h e p r e v i o u s ones) is t o locate t h e peaks ( / p " , 0 p ' ) i n t h e t w o - d i m e n s i o n a l s p e c t r u m . It is possible to use l i n e a r search f o r i d e n t i f y i n g local m a x i m a o n t h e g i v e n s p e c t r u m . A local m a x i m u m is d e f i n e d i f t h e v a l u e o f t h e m a t r i x e l e m e n t is h i g h e r t h a n t h e v a l u e o f a l l its n e i g h b o r s . H o w e v e r , s o m e t i m e s l i t t l e fluctuations are d e f i n e d as local m a x i m u m . So i t is necessary to define i f t h e r e is a s i g n i f i c a n t m a x i m u m o r n o t . This task becomes easier after c o m p l e t i o n o f second s t e p — i s o l a t i n g peaks.

I s o l a t i n g peaks a l l o w s t h e d e f i n i t i o n o f t h e area f o r each m a x -i m u m -i n w h -i c h every p o -i n t belongs to a c o r r e s p o n d -i n g peak. T o d i v i d e t h e w h o l e s p e c t r u m i n t o s u c h areas, i t is possible t o use t h e f o l l o w i n g a l g o r i t h m . T h e s p e c t r u m is p r e s e n t e d as a s p e c t r a l m a -t r i x w h i c h c o n -t a i n s -t h e values o f spec-tral d e n s i -t y i n -t h e g r i d p o i n -t s for t h e d i f f e r e n t frequencies a n d d i r e c t i o n s ( i n r o w s a n d c o l u m n s , c o r r e s p o n d i n g l y ) . To decide t o w h a t m a x i m u m each e l e m e n t o f t h e s p e c t r u m m a t r i x belongs to, i t is c o m p a r e d w i t h all its e i g h t n e i g h -bors i n t h e g r i d t o d e t e r m i n e w h i c h one is higher. I f all o f t h e m are l o w e r , t h e n t h e s t a r t i n g e l e m e n t is a local m a x i m u m , or else i t b e -longs to t h e m a x i m u m t o w h i c h be-longs its highest n e i g h b o r , a n d t h i s step s h o u l d be r e p e a t e d f o r it. So f o r each p o s i t i o n , a p a t h t o o n e local m a x i m u m can be d e f i n e d . A f t e r t h e second step, each peak's area, as t h e set o f the p o i n t s , w h i c h are associated w i t h t h e same local m a x i m u m , is d e f i n e d . A f t e r w a r d s , t h e values o f t h e s p e c t r a l m o m e n t s m ^ are c o m p u t e d b y i n t e g r a t i o n o v e r each area.

In fact, t h e values o f t h e s p e c t r a l shape p a r a m e t e r s ( n , s, y ) v a r y s t r o n g l y a m o n g t h e w a v e systems and t h e g e o g r a p h i c a l c o n -d i t i o n s . Hence, t h e q u a l i t y o f fitting m a y be i m p r o v e -d b y m e a n s o f an a d d i t i o n a l p r o c e d u r e f o r n u m e r i c a l o p t i m i z a t i o n o f these p a -r a m e t e -r s , w h e n t h e w a v e s y s t e m n u m b e -r N a n d t h e p a -r a m e t e -r s

( m f , f p ' \ ö p ' ) are o b t a i n e d d i r e c t l y f r o m t h e s p e c t r u m . The m e a -sure o f fit w i t h respect t o Eq. ( 1 ) is t h e f u n c t i o n a l t o be m i n i m i z e d :

r ( 3 )

n

2 [ s * ( j , e ) - s ( f , i S ) ] ' d / d ö ^ m i n . N.S

( 1 3 ) Here S * ( » ) is t h e t a r g e t s p e c t r u m , a n d S ( f , 9,S)is t h e p a r a m e t r i c m o d e l o f s p e c t r u m b y Eq. ( 1 ) . N u m e r i c a l l y , S*(/i(, 9i) = S^, are t h e values o f t h e target s p e c t r u m at r e g u l a r g r i d p o i n t s . This t e c h n i q u e is k n o w n as semi-explicit a p p r o a c h .

In s o m e cases, i t is possible to o b t a i n t h e p a r a m e t e r s

m f , fp'^, ö p ' d i r e c t l y f r o m the s p e c t r u m w i t h r a t h e r h i g h p r e c i s i o n , b u t f o r m u l t i - p e a k e d spectra w i t h some s m o o t h e d peaks, t h e results o f t h e p a r t i t i o n i n g and peak i s o l a t i o n by an e x p l i c i t m e t h o d , can be u n r e a l i s t i c a n d even i n c o r r e c t (see [ 3 1 ] ) . For t h i s p u r p o s e , t h e f u l l y implicit t e c h n i q u e is used, w h e r e all t h e p a r a m e t e r s o f Eq. ( 1 ) , i n c l u d i n g N w a v e s y s t e m s , are o b t a i n e d b y m e a n s o f n u m e r i c a l o p t i m i z a t i o n o f Eq. (13). H o w e v e r , i n t h i s case, a single s o l u t i o n o f p r o b l e m by Eq. ( 1 3 ) is o n l y achievable w h e n t h e f o l l o w i n g a u x i l i a r y c o n s t r a i n t s are c o n s i d e r e d :

? ^ : | / p " ' - / p ^ ' ' | > 5 ; v | ö < " - ö , ^ ' ' | > 5 « ,

i , j = Ö r N , i ^ J . ( 1 4 ) Physically Eq. ( 1 4 ) means t h a t t h e peaks i,j are separated e i t h e r i n

f r e q u e n c i e s o r i n d i r e c t i o n s b y an offset l a r g e r t h a n a p r e d e f i n e d l i m i t S. I f t h e s p e c t r u m S* is d e f i n e d o n a r e g u l a r g r i d , t h e values Sf, SH are i n t e r p r e t e d as t h e m a x i m u m g r i d steps o v e r t h e f r e q u e n c y a n d t h e d i r e c t i o n , respectively.

Eq. ( 1 3 ) includes b o t h discrete ( N ) a n d c o n t i n u o u s ( S )

p a r a m e t e r s . I t assumes a s e q u e n t i a l o p t i m i z a t i o n p r o c e d u r e o f c o n t i n u o u s p a r a m e t e r s S f o r t h e values N = 0, 1 , 2, . . . . There

(4)

A.V. Boukhanovsky, C. Guedes Soares/ AppUed Ocean Research 3J (2009) 132-141 ₁₃₅

are m a n y o p t i m i z a t i o n s t e c l i n i q u e s available [ 3 2 ] , b u t ttie f a m i l y o f g r a d i e n t m e t h o d s is t h e m o s t o b v i o u s classical choice. H o w e v e r , f o r t h e p r o b l e m o f Eq. ( 1 3 ) w i t h w a v e spectra data, t h e r a n d o m search m e t h o d s are m o r e c o n v e n i e n t due t o t h e i r l i n e a r ( n o t p o w e r ) s c a l a b i l i t y w i t h r e s p e c t t o t h e n u m b e r o f p a r a m e t e r s . A l s o , t h e r a n d o m search a l g o r i t h m s also take i n t o a c c o u n t t h e r e s t r i c t i o n s o f the Eq. ( 1 4 ) r a t h e r s i m p l y . H e r e t h e a l g o r i t h m o f c o o r d i n a t e - w i s e l i n e a r r a n d o m search is u s e d : Sk+i =Sk--^\ [ f o r y N ( S k + , ) <Jn(Sk), i n case o f " s u c c e s s f u l " k + 1 - i t e r a t i o n , a n d o t h e r w i s e : '^k+X randici) i=Tjn f o r ; ( S t + i ) > ; ( S ' k ) V Sfc+, 6 m. ( 1 5 ) ( 1 6 ) H e r e is t h e v a r i a b l e step o f t h e i t e r a t i o n s , is t h e r a n d o m d i r e c t i o n ( m e a n i n g i n c r e a s i n g (-1-1) o r d e c r e a s i n g ( - 1 ) o f one o f the c o m p o n e n t s o f t h e v e c t o r S). The o p e r a t o r rand(») is t h e r a n d o m choice p r o c e d u r e s e l e c t i n g s o m e o t h e r d i r e c t i o n i n t h e m -d i m e n s i o n a l space o f p a r a m e t e r s , a n -d 3^ is t h e c o n s t r a i n t s area. The p a r a m e t e r s •^|/^, xj/2 d e f i n e t h e r e l a x a t i o n degree i n t h e n e i g h b o r h o o d o f t h e l o c a l e x t r e m e . T h e o r e t i c a l l y , t h e r e l a t i o n ilr^^f^^"''^ = 1, w i t h p = 0 . 2 7 , leads to t h e " b e s t " r e l a x a t i o n f o r a n y f o r m u l a t i o n d e f i n e d b y t h e Eq. ( 1 3 ) [ 3 3 ] . I n t h e c o m p u t a t i o n s b e l o w , t h e v a l u e s o f i/f] = 1.50 a n d 1/^2 = 0.86 are u s e d . The v a l u e o f t h e i n i t i a l s t e p e ' " ' is e q u a l t o 10% o f t h e i n i t i a l v a l u e o f t h e c o r r e s p o n d i n g c o m p o n e n t o f S. I t e r a t i o n s i n Eqs. ( 1 5 ) a n d ( 1 6 ) are p e r f o r m e d u n t i l t h e v a l u e o f J is m i n i m i z e d , h a v i n g achievecl t h e l o c a l m i n i m u m w i t h t h e r e q u i r e d t o l e r a n c e , w h i c h is 0 . 1 % i n t h e p r e s e n t c o m p u t a t i o n s . The a b o v e m e n t i o n e d p r o c e d u r e p r o v i d e s o p t i m a l v a l u e s o f t h e p a r a m e t e r s f o r S ( f , 6, a ) at a n y fixed n u m b e r N o f w a v e s y s t e m s . I n fact, t h e r e are n o clear f o r m a l a l g o r i t h m s f o r i d e n t i f y i n g t h e e x a c t n u m b e r N o f w a v e systems i n a g i v e n s p e c t r u m . O n l y t h e m a i n s p e c t r a l p e a k 6^^^ can be d e f i n e d w i t h c e r t a i n t y f o r a n y S(J, 9). Hence, t h e fitting p r o c e d u r e is c o n s t r u c t e d u s i n g t h e s e q u e n t i a l i n c r e m e n t o f t h e n u m b e r o f w a v e s y s t e m s i n t h e i n i t i a l s p e c t r u m . O n t h e f i r s t stage, t h e p r o b l e m b y Eq. ( 1 3 ) is s o l v e d f o r o n e - p e a k e d spectra ( N = 0 ) a n d t h e c o u p l e {Sp°\ 0 p ° ' ) is c o n s i d e r e d as a n i n i t i a l a p p r o x i m a t i o n o f t h i s p a r t o f t h e p a r a m e t e r s v e c t o r S. For t h e shape p a r a m e t e r s n, s ( f o r t h e G a m m a - b a s e d m o d e l ) t h e i n i t i a l values a d o p t e d are n'°^ = 5, s^"^ = 4 w h i c h c o r r e s p o n d t o the P i e r s o n - M o s k o w i t z s p e c t r u m f o r s t o r m w a v e s , o r ( f o r JONSWAP-based m o d e l ) = 3.3, s'") = 4 . Physical c o n s i s t e n c y o f t h e i t e r a t i o n s c h e m e i n Eqs. ( 1 5 ) a n d ( 1 6 ) r e q u i r e s s o m e a d d i t i o n a l c o n s t r a i n t s : e.g. n > 2 , s > 1 a n d y > 1 t o assure r e a l i s t i c m o m e n t s o f a n y s p e c t r u m h a v i n g t h e s t r u c t u r e d e f i n e d b y Eqs. ( 7 ) - ( 9 ) . M o r e o v e r , a n u p p e r l i m i t can be set f o r t h e shape p a r a m e t e r s n, s, y f r o m p r a c t i c a l reasons. A t t h e n e x t i t e r a t i o n , t h e significance o f t h e (N F 1 ) -t h -t e r m i n -t h e Eq. ( 1 ) is c h e c k e d b y a p p l y i n g -t h e c o n d i -t i o n / ' ^ ' ( i 5 ' ) / J ^ ^ + " ( S ' ) < a. N u m e r i c a l e x p e r i m e n t s s h o w e d t h a t t h e best v a l u e is a = 1.5, w h i c h means t h a t i f t h e d i s c r e p a n c y i n Eq. ( 1 3 ) f o r N -F 1 t e r m s is less t h e n 1.5 t i m e s f o r N t e r m s , t h e n N t e r m s are e n o u g h f o r t h e a p p r o x i m a t i o n . The i n i t i a l values

( f p ' ^ \ fip*^') f o r N > 1 are o b t a i n e d d i r e c t l y f r o m t h e s p e c t r u m , i f s e c o n d a r y peaks are c l e a r i y v i s i b l e . I n o t h e r cases, these i n i t i a l v a l -ues are r e c a l c u l a t e d as / p ' ^ ' =f^''-'^ + Sf, 9^ = + So w i t h respect t o c o n s t r a i n t s b y Eq. ( 1 4 ) .

A l l steps o f t h i s p r o c e d u r e are s h o w n i n t h e Fig. 1. The t a r g e t d i r e c t i o n a l s p e c t r u m (Fig. 1(a)) has t h r e e c l e a r peaks, b u t t h e

f r e q u e n c y s p e c t r u m S { f ) a n d i n t e g r a t e d d i r e c t i o n a l s p r e a d i n g f u n c t i o n ( 2 ( 0 ) h a v e t w o peaks each o n l y . I n t h e first step ( N = 0) t h e i m p l i c i t a p p r o a c h p r o d u c e d the u n r e a l i s t i c a p p r o x i m a t i o n (Fig. 1(b)), w h e r e the s i n g l e - p e a k e d s p e c t r u m is s m o o t h e d a m o n g t h e data peaks. N e v e r t h e l e s s , i n t h e s e c o n d step ( N = 1) t h e p o s i t i o n s o f the t w o m a i n w a v e systems are l o c a l i z e d i n t h e d i r e c t i o n a l d o m a i n (Fig. 1(c)). I n the t h i r d s t e p ( N = 2 ) all t h r e e peaks are a p p r o x i m a t e d a d e q u a t e l y (Fig. 1(d)). A d d i t i o n a l l y i n Fig. 1 t h e values o f the n o r m a l i z e d d i m e n s i o n l e s s a p p r o x i m a t i o n e r r o r

E r r o r : ( 1 7 )

are s h o w n . It is seen t h a t , i n t h e first step, the i n i t i a l e r r o r is d e c r e a s i n g m o r e t h a n 3.7 times, a n d i n s e c o n d s t e p i t is decreasing 3 t i m e s ( w h e n t h e s e n s i t i v i t y p a r a m e t e r a i n t h e c o m p u t a t i o n a l p r o c e d u r e is e q u a l t o 1.5).

W i t h respect t o t r a d i t i o n a l d e t e r m i n i s t i c o p t i m i z a t i o n m e t h o d s (e.g. g r a d i e n t t e c h n i q u e s ) the r a n d o m search m e t h o d does n o t n e e d g r a d i e n t c a l c u l a t i o n because i t uses r a n d o m steps for m o v i n g the c u r r e n t p o s i t i o n i n t h e space o f p a r a m e t e r s . Since i t uses a r a n d o m s t e p , i t is p o s s i b l e t o d e f i n e a g l o b a l m a x i m u m . M o r e o v e r , t h i s m e t h o d r e q u i r e s o n l y one t a r g e t f u n c t i o n (see Eq. ( 1 3 ) ) c a l c u l a t i o n p e r step. W h i l e this c a l c u l a t i o n takes time, i t is a n i m p o r t a n t b e n e f i t t o the a l g o r i t h m .

4. Comparative analysis with other multipeaked spectral

models

Guedes Soares [ 4 ] has p r o p o s e d a f r e q u e n c y s p e c t r a l m o d e l t o describe t h e c o m b i n e d s i t u a t i o n o f s w e l l a n d w i n d sea o n t h e basis o f Eq. ( 1 ) , w h e r e t h e c o m p o n e n t s are t h e JONSWAP spectra. T h e l e a d i n g c o n t r o l v a r i a b l e f o r t h i s m o d e l is t h e r a t i o b e t w e e n s p e c t r a l peaks o f s w e l l (S) a n d w i n d w a v e s ( W ) c o m p o n e n t s , i.e.:

Sr = ( 1 8 )

I n t h e f r a m e o f m o d e l i d e n t i f i c a t i o n , t h e v a l u e o f Sn a n d t h e f r e -quencies fp^'^^ are e s t i m a t e d d i r e c t l y f r o m t h e m e a s u r e d spec-t r u m , a n d hR = (hs/hw)^ is r e c a l c u l a t e d t h r o u g h Eq. ( 1 8 ) . T h i s p a r a m e t e r i z a t i o n is e n o u g h for t h e case w h e n t h e f r e q u e n c y spec-t r u m has spec-t w o clear peaks.

I n Fig. 2 t h e r e are e x a m p l e s o f f r e q u e n c y s p e c t r a fitting b y Eqs. ( 1 ) , ( 8 ) a n d ( 1 8 ) i n c o m p a r i s o n w i t h t h e above d e s c r i b e d d i r e c t i o n a l m o d e l o n t h e base o f Eqs. ( 1 3 ) - ( 1 6 ) . T h e h i n d c a s t w a v e d a t a f r o m HIPOCAS dataset [ 3 4 ] near Canaries are u s e d for case s t u d i e s . T h e r e w e r e t w o sorts o f d i r e c t i o n a l m o d e l c o m p o n e n t s i n Eq. ( 1 ) : as G a m m a - s p e c t r u m (Eq. ( 9 ) w i t h t h e v a r i a b l e p a r a m e t e r n), a n d as JONSWAP s p e c t r u m (Eq. ( 8 ) w i t h t h e fixed p a r a m e t e r n = 5 a n d v a r i a b l e p a r a m e t e r y). For q u a n t i t a t i v e c o m p a r i s o n , t h e D e v i a t i o n I n d e x (D.I.), p r o p o s e d b y L i u [ 3 5 ] a n d a d o p t e d b y Guedes Soares a n d H e n r i q u e s [ 3 6 ] is e s t i m a t e d as: 100 S*(fi) S*(fi)(fi+i - f i ) m o ( 1 9 )

I n t h e Table 1 t h e values o f D.I., associated w i t h t h e Fig. 2 ( a ) - ( c ) are p r e s e n t e d .

I n Fig. 2(a) t h e s i m p l e t w o - p e a k e d s p e c t r u m w i t h clear s e p a r a t i o n o f w i n d w a v e s a n d s w e l l c o m p o n e n t s is s h o w n . F r o m the Table 1 i t is c l e a r l y seen t h a t t h e D.I. f o r all t h e cases is l o w e r t h a n 16. U s u a l l y , i f t h e v a l u e s o f D.I. are l o w e r t h a n 2 5 - 3 0 t h e m o d e l fitting is c o n s i d e r e d s a t i s f a c t o r y [ 3 5 ] . T h u s , all t h e t h r e e m o d e l fittings are r a t h e r close w i t h each o t h e r a n d w i t h t h e i n i t i a l d a t a .

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136 A.V. Boukhanovsky, C. Guedes Soares/ Applied Ocean Research 31 (2009) 132-141

b ,

ƒ

O 90 180 2 7 0 3 6 0 0 0.2 0.4 O 9 0 180 2 7 0 3600 0.1 0.2

180 270 3 6 0 0 0.2 0.4

Fig. 1. Example of multi-peaked directional wave spectrum fitted by means of the i m p l i c i t technique by Eqs. ( 1 3 ) - ( 1 6 ) . (a) Initial example ( N o r t h Atlantic coast of Portugal, HIPOCAS data), ( b ) , ( c ) , ( d ) Sequential steps of method ( N = 0, 1, 2).

Table 1

Deviation Index for comparison of multipeaked spectral models.

Case study Models

Directional model by Eqs. (1) and ( 1 3 ) - ( 1 6 ) Model by Guedes Soares 14|

Gamma-spectrum (9) J0NSW?AP(8) 01.02.1999 0:00 (Fig. 2(a)) 09.02.1999 9:00 (Fig. 2(b)) 11.02.1999 0 : 0 0 ( F i g . 2(c)) 15.9 14.6 11.3 11.3 21.6 21.1 13.1 33.8 66.0

I n Fig. 2 ( b ) , (a) a m o r e c o m p l i c a t e d case o f t w o - p e a k e d w a v e s p e c t r u m w i t h d o m i n a t e d s w e l l a n d w e a k w i n d w a v e s is s h o w n . The second peak is l a t e n t - o n l y the l o n g s p e c t r a l tail is observable. For this case S r = 0 by Eq. ( 1 8 ) , a n d o n l y a s i n g l e - p e a k m o d e l is a d o p t e d . N e v e r t h e l e s s , u s i n g Eqs. ( 1 3 ) - ( 1 6 ) a l l o w s one t o o b t a i n m o r e r e l i a b l e e s t i m a t i o n s w h i c h c o n t a i n b o t h t h e w i n d sea a n d t h e s w e l l c o m p o n e n t s . The f i t t i n g s b y Eqs. ( 1 3 ) - ( 16), w h e r e D.1. = 14.6, are closer to t h e i n i t i a l data t h a n t h e m o d e l by Eqs. ( 1 ) , ( 8 ) a n d ( 1 8 ) , w h e r e D.I. = 33.8.

I n Fig. 2 ( c ) t h e c o m p l e x sea case w i t h t w o s w e l l s a n d the w i n d w a v e s is s h o w n . O n l y a single s p e c t r a l peak is v i s i b l e i n t h e s p e c t r u m . The second s w e l l c o m p o n e n t a n d t h e w i n d w a v e s c o m p o n e n t f o r m t h e l o n g s m o o t h e d s h e l f o n t h e t a i l o f the s p e c t r u m . For this case, the m o d e l by Eqs. ( 1 ) , ( 8 ) a n d ( 1 8 ) leads to u n s a t i s f a c t o r y f i t t i n g ( D . I . = 6 6 . 0 ) , b u t a d o p t i n g t h e p r o p o s e d m o d e l o n t h e basis o f Eqs. ( 1 3 ) - ( 1 6 ) a l l o w s one t o select a l l the

peaks a n d to o b t a i n a r a t h e r g o o d r e p r e s e n t a t i o n o f t h e s p e c t r a l shape ( D . I . = 11.3).

W h i l e this c o m p a r i s o n is u s e f u l to s h o w t h e f l e x i b i l i t y o f t h e p r e s e n t l y p r o p o s e d m e t h o d , it m u s t be n o t e d t h a t the c o m p a r i s o n w i t h t h e m o d e l o f Guedes Soares [4] is s t r e t c h i n g i t b e y o n d its scope, w h i c h is t o m o d e l d o u b l e peaked spectra. I n t h e o r i g i n a l p a p e r , i t is suggested t h a t a s i m i l a r a p p r o a c h c o u l d be a d o p t e d for spectra w i t h m o r e c o m p o n e n t s b u t , i n t h a t case, a d d i t i o n a l r a t i o s S r (Eq. ( 1 8 ) ) w o u l d n e e d t o be c o n s i d e r e d .

Fig. 2 s h o w s t h a t t h e p r i n c i p a l advantage o f t h e t e c h n i q u e based o n Eqs. ( 1 3 ) - ( 1 6 ) is i n v o l v i n g t h e d i r e c t i o n a l i n f o r m a t i o n for f i t t i n g o f c o m p l e x - s h a p e m u l t i p e a k e d spectra. For a d e t a i l e d e x p l a n a t i o n o f this f e a t u r e . Fig. 3 s h o w s a c o m p a r i s o n o f t h r e e d i f f e r e n t approaches ( e x p l i c i t , s e m i - e x p l i c i t a n d i m p l i c i t , w h i c h are d e s c r i b e d i n Section 3) for d i f f e r e n t types o f d i r e c t i o n a l m u l t i p e a k e d spectra. I n the c o l u m n Fig. 3 ( a ) ( l - 4 ) the e x a m p l e o f

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A V . Boukhanovsky, C. Guedes Soares / Applied Ocean Research 31 (2009) 132-141 137 Table 2

Characteristics of wave spectra occurrence for different regions of North A t l a n t i c

/?s, m Ps,% P2P,% Pm.%

North of the Britain (64N, 6 W )

0 - 2 2.2 0.5 7.9 3.9 9.1 1.10 2 - 4 12.5 4.6 22.3 10.5 24.4 1.01 4 - 7 14.6 1.3 10.4 1.5 8.5 1.05 > 7 5.0 - 2.2 0.7 2.3 0.88 34.3 6.4 42.8 16.6 44.3 1.01 Azores (40N, 26W) 0 - 2 0.9 3.8 9.8 12.5 16.8 0.91 2 - 4 14.5 6.4 17.7 15.2 26.9 1.08 4 - 7 11.4 2.4 3.5 0.2 3.1 1.46 > 7 1.7 > 7 1.7 Ep 28.5 12.6 31.0 27.8 46.8 1.15 Sines (38N, 9W) 0 - 2 0.1 37.2 13.9 4.0 6.6 1.14 2 - 4 1.2 29.9 7.3 2.2 3.3 1.17 4 - 7 0.8 2.1 0.5

_-

1.23 >7 0.5

_-

0.1

_-

1.22 Ef 2.7 69.2 21.8 6.2 9.9 1.19 Canaries (29.25N, 15.25W) 0 - 2 0.2 4 3 17.2 2.7 6.9 1.34 2 - 4 12.1 13.6 3 4 9 5.2 12.8 1.18 4 - 7 2.4 3.5 3.0 0.4 0.4 0.96 >7 0.4

_-

-

_-

-

-Ep 15.1 21.5 55.1 8.3 20.1 1.16 w e l l - s e p a r a t e d d o u b l e - p e a k e d s p e c t r u m is p r e s e n t e d . It is clearly seen t h a t all t h r e e approaches a l l o w one t o o b t a i n r a t h e r r e l i a b l e s e p a r a t i o n o f c o m p o n e n t s . B u t the f i t t i n g by t h e e x p l i c i t a p p r o a c h seems r o u g h e r due t o t h e f i x i n g o f shape p a r a m e t e r s i n Eq. ( 1 ) .

I n t h e c o l u m n Fig. 3 ( b ) ( l - 4 ) t h e m o r e c o m p l i c a t e d case o f a t h r e e - p e a k e d s p e c t r u m is c o n s i d e r e d . H e r e , the q u a l i t y o f t h e a p p r o x i m a t i o n s by b o t h e x p l i c i t a n d s e m i - e x p l i c i t approaches is l o w e r , t h a n t h e i m p l i c i t one (Fig. 3 ( b ) ( 4 ) ) . I n t h e c o l u m n Fig. 3(c)( 1 - 4 ) t h e e x a m p l e o f " s m o o t h e d " d o u b l e - p e a k e d s p e c t r u m is s h o w n . This s p e c t r u m consists o f t w o w a v e c o m p o n e n t s , b u t o n l y t h e single s w e l l peak is c l e a r l y seen i n t h e g r a p h . As a result, o n l y t h e f u l l y i m p l i c i t a p p r o a c h by Eqs. ( 1 3 ) - ( 1 6 ) a l l o w s t h e f i t t i n g o f t h e m o d e l a d e q u a t e l y to data. T h u s , s u c h analysis a l l o w s one to a d o p t t h e i m p l i c i t a p p r o a c h b y Eqs. ( 1 3 ) - ( 1 6 ) as t h e m o s t r e l i a b l e t e c h n i q u e o f a u t o m a t i c s e p a r a t i o n a n d m o d e l i d e n t i f i c a t i o n f o r m u l t i p e a k e d d i r e c t i o n a l w a v e spectra.

5. Model study on the HIPOCAS data

For s t u d y i n g t h e p e r f o r m a n c e o f t h e f i t t i n g a p p r o a c h , t h e HIPOCAS 4 4 years h i n d c a s t data o v e r t h e N o r t h A t l a n t i c w e r e used [ 3 4 ] . Four d i f f e r e n t p o i n t s w i t h s i g n i f i c a n t l y d i f f e r e n t w a v e c o n d i t i o n s are c o n s i d e r e d . For each p o i n t , the dataset o f 2 9 2 0 d i r e c t i o n a l w a v e spectra ( f r o m 1 9 9 9 ) o n a d i s c r i m i n a t i o n o f 25 f r e q u e n c i e s a n d 2 4 d i r e c t i o n s w e r e f i t t e d to t h e m o d e l by Eq. ( 1 ) b y t h e f u l l y i m p l i c i t a p p r o a c h by Eqs. ( 1 3 ) - ( 1 6 ) . Such a m o u n t o f d a t a is n o t e n o u g h f o r q u a n t i t a t i v e c l i m a t i c c h a r a c t e r i z a t i o n , b u t i t is possible to o b t a i n s o m e q u a l i t a t i v e k n o w l e d g e a b o u t s p e c t r a l s t r u c t u r e o f c o m p l e x sea states i n d i f f e r e n t regions o f t h e N o r t h A t l a n t i c , w h i c h is u s e f u l f o r t h e assessment o f the m o d e l adequacy. I n Table 2 t h e e s t i m a t i o n o f several characteristics o f t h e dataset are p r e s e n t e d w i t h respect t o levels o f s i g n i f i c a n t w a v e h e i g h t hs

a n d t o t a l l y over all sea states (X'p). The values Pw a n d Ps d e n o t e t h e f r e q u e n c i e s o f o c c u r r e n c e (%) o f s i n g l e - p e a k e d s p e c t r u m o f w i n d w a v e s a n d t h e s w e l l , r e s p e c t i v e l y . It is seen t h a t these values v a r y a l o n g t h e ocean. For e x a m p l e , t h e v a l u e o f Pw varies f r o m 2.7% at t h e coast o f P o r t u g a l (Sines) t o 34.3% at t h e N o r t h e r n coast o f

1.2 0.8 0.6 0.4 0.2 1 j 1 1 1 * * * 1 2 _ 3 / r \v\ If, \\ \ ll \\\

A r \

-I l l l l 0.05 0.1 0.15 0.2 0.25 0.3 frequency, H z 0.35 1 I l l l l

• •• /

2 li i\l Ï l\\

• \ \

1 1 1 * -0.05 0.1 0.15 0.2 0.25 frequency, H z 0.3 0.35 0.15 0.2 0.25 frequency, H z

Fig. 2. Comparative analysis o f models for multipeaked frequency spectrum. 1 - l n i t i a l data (see Table 1), 2 - E q s . ( 1 3 ) - ( 1 6 ) w i t h Gamma-spectrum components, 3 - E q s . (13)-( 16) w i t h JONSWAP spectrum components, 4 - E q s . (1), ( 8 ) and (19).

B r i t a i n . For Ps v a l u e t h e r e is t h e o p p o s i t e b e h a v i o r — f r o m 69.2% at t h e coast o f P o r t u g a l (Sines) d o w n to 6.4% at t h e N o r t h e r n coast o f B r i t a i n . The t o t a l o c c u r r e n c e o f s i n g l e - p e a k e d spectra is 36.6%-71.9%.

The values P2P a n d P^p d e n o t e the f r e q u e n c i e s o f o c c u r r e n c e (%) o f d o u b l e - p e a k e d a n d m u l t i - p e a k e d ( t h r e e o r m o r e peaks) spectra, r e s p e c t i v e l y , w h i c h consist o f w i n d w a v e s a n d s w e l l , o r several s w e l l s y s t e m s . The P2P v a l u e varies f r o m 21.8% a t t h e coast o f P o r t u g a l (Sines) t o 5 5 . 1 % a r o u n d t h e Canaries. A n a l o g o u s l y , t h e P3P v a l u e varies f r o m 6.2% at t h e coast o f P o r t u g a l (Sines) t o 27.8% a r o u n d t h e A z o r e s . The ratio b e t w e e n t h e a m o u n t o f d o u b l e - p e a k e d spectra a n d m u l t i p e a k e d spectra also varies o v e r t h e ocean. For e x a m p l e , at t h e N E - p a r t o f N o r t h A t l a n t i c t h e P2P is t h r e e t i m e s g r e a t e r t h a n P^p. A t t h e same t i m e , at t h e A z o r e s t h e

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138 AV. Boukhanovsky. C. Guedes Soares / AppUed Ocean Research 31 (2009) 132-141

0

. / : H Z

f .

Hz / H z / H z b ( l ) ƒ Hz b(2) / H z b(3) / Hz b(4) / Hz c ( l ) / Hz c(2) / H z c(3) / f Iz c(4) / Hz 360 270 90 270 180 90 270 90 2 7 0 180 90 O 0.05 0.10 0.15 0 . 2 0 0 . 3 0 0.40 0.05 0,10 0.15 0 . 2 0 0 . 3 0 0.40 0.05 0.10 0.15 0.20 0.30 0.40

Fig. 3. Comparison of explicit, semi-explicit and i m p l i c i t approaches to fit of multipeaked directional wave spectrum. Rows: 1 - i n i t i a l data example ( N o r t h Atlantic coast of

Portugal, HIPOCAS data), 2 - e x p l i c i t approach, 3 - s e m i - e x p l i c l t approach, 4 - i m p l i c i t approach by Eqs. ( 1 3 ) - ( 1 6 ) . Columns: ( a ) - t h e case of well-separation (all techniques are adequate), ( b ) - t h e case of moderate prevalence of i m p l i c i t approach by Eqs. ( 1 3 ) - ( 1 6 ) , ( c ) - t h e case of ultimate prevalence of implicit approach by Eqs. ( 1 3 ) - ( 1 6 ) .

values o f Pjp a n d P^P are v e r y close due t o the h i g h p o s s i b i l i t y t o observe several s w e l l systems s i m u l t a n e o u s l y .

The p a r a m e t e r K i n Table 2 d e n o t e s t h e i n d i c a t o r o f w i n d sea p r e v a l e n c e i n a c o m p l e x sea s p e c t r u m , K

= hw/hs =

V S S Ë R , w h e r e hs is c o m p u t e d o v e r all s w e l l s y s t e m s i n a c o m p l e x sea s p e c t r u m , a n d SSER is t h e s e a - s w e l l e n e r g y r a t i o [ 1 5 ] . The value o f K is i d e n t i f i e d o n l y for m u l t i p e a k e d c o m p l e x sea spectra w i t h t h e t o t a l p r o b a b i l i t y P2P + Pap.

A d i f f e r e n t b e h a v i o r o f K w i t h respect to w a v e h e i g h t for d i f -f e r e n t p o i n t s is clearly seen. For e x a m p l e , -for t h e NE p a r t o -f t h e N o r t h A t l a n t i c a n d the Portuguese coast, t h i s r a t i o is r a t h e r stable i n t h e range 1.0-1.2. For t h e Azores zone a n i n c r e a s i n g t e n d e n c y , a n d for Canaries zone a d e c r e a s i n g t e n d e n c y , are o b s e r v e d . A n a l o -g o u s l y , t h e r e -g i o n a l differences reflect t h e s t r u c t u r e o f t h e severe w a v e s i n s t o r m s . For t h e Canaries a n d t h e Azores f o r hs > 1 m e t e r s , o n l y s i n g l e - p e a k e d w a v e spectra are o b s e r v e d , b u t for t h e N o r t h e r n coast o f B r i t a i n a n d the Portuguese coast i t is also l i k e l y t h a t severe c o m p l e x sea s i t u a t i o n s w i t h m u l t i p e a k e d spectra w i l l occur.

Table 2 also p r e s e n t s the values o f m i s c l a s s i f i c a t i o n p r o b a -b i l i t i e s Pm for d i f f e r e n t regions a n d levels o f w a v e h e i g h t . This p r o b a b i l i t y is associated w i t h the o c c u r r e n c e o f m u l t i p e a k e d ( d o u b l e - t h r e e - o r m o r e ) spectra, w h e n the n u m b e r o f peaks i n t h e d i r e c t i o n a l s p e c t r u m is g r e a t e r t h a n the ones o f t h e c o r r e s p o n d i n g f r e q u e n c y s p e c t r u m . This l a t t e r one w o u l d o n l y i n c l u d e t h e c o m p o -n e -n t s d e t e c t e d b y t h e e a r l i e r m e t h o d t h a t w o u l d -n o t c o -n s i d e r t h e d i r e c t i o n a l i n f o r m a t i o n . These results w i l l be discussed i n d e t a i l in t h e S e c t i o n 5.

Thus, t h e s t u d y o f t h e features o f s p e c t r a l data f o r t h e d i f f e r e n t r e g i o n s o f the N o r t h A t l a n t i c s h o w s t h a t i t is i m p o s s i b l e t o f o r m u -late g e n e r a l and exact a s s u m p t i o n s a b o u t the s p e c t r a l c o m p o n e n t n u m b e r N . Hence, t h e s t r u c t u r e o f Eq. ( 1 ) m u s t be f l e x i b l e , w h i c h is r e f l e c t e d by t h e a u t o m a t e d i t e r a t i v e p r o c e d u r e f o r r e v e a l i n g o f s p e c t r a l c o m p o n e n t s by means o f t h e i m p l i c i t a p p r o a c h by Eqs. ( 1 3 ) - ( 1 6 ) . I n Table 3 t h e r e is a n o t h e r k i n d o f w a v e spectra p a r a m e t e r f o r t h e above m e n t i o n e d p o i n t s : t h e f i t t i n g e r r o r by Eq. ( 1 7 ) a n d t h e

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A.V. Boukhanovsky. C. Guedes Soares / Apphed Ocean Research 31 (2009) 132-141 139

Table 3

Mean values ofspectral shape parameters for the (Gamma-2s)- and 0ONSWAP-2s)-based models of complex sea directional spectra.

hs, m (Gamma-2s)-approximation 0ONSWAP-2s)-approximation

Error Wind sea Swell Error Wind sea Swell

n s n s _Y s Y s

North of the Britain (64N, 6 W )

0-2 0.08 4.8 2.1 5.9 5.7 0.08 1.6 2.1 2.1 5.7 2-4 0.30 5.1 2.6 6,2 5.6 0.31 1.7 2.9 2.2 5.8 4-7 0.17 5.2 2,9 6.1 5.4 0.18 1.8 3.2 2.2 5.5 >7 0.05 5.8 3.0 6.2 4.8 0.05 2.0 3.5 2.2 4.8 Azores (40N, 26W) 0-2 0.16 4.5 2.0 5.9 7.2 0.17 1.4 2.1 2.1 7.5 2-4 0.33 4.6 2.5 6.4 9.1 0.34 1.5 2.6 2.1 9.2 4-7 0.12 4.7 3.4 5.9 9.8 0.12 1.6 3.6 1.7 9.8 >7 n ni

fi

7 dfi nn9 n A i a — . . . .

Portuguese coast, Sines (38N, 9W)

0-2 0.39 6.2 2.9 5.6 7.3 0.40 2.1 3.0 1.9 7.2 2-4 0.29 4.3 3.0 6.1 7.1 0.29 1.2 3,0 1.9 7.5 4 - 7 0.02 3.9 3.2 6.5 9.8 0.03 1.2 3.2 2.1 10.3 > 7 0.01 3.7 3.6 5.4 8.7 <0.05 1.0 3.3 1.9 12.7 Canaries (29.25N, 15.25W) 0-2 0.16 4.8 2.2 5.9 10.2 0.17 1.5 2.3 2.0 10.2 2-4 0.44 4.7 2.4 6.0 11.2 0.45 1.5 2.5 2.0 11.1 4-7 0.07 5.2 3.1 6.7 10.7 0.08 1.8 4.0 2.1 11.2 >7 0.01 6.2 5.4 - - - 2.2 5.9 - -average e s t i m a t e s o f p a r a m e t e r s ( n , s) a n d ( y , s) f o r w i n d sea a n d s w e l l c o m p o n e n t s separately. T h e ( G a m m a - 2 s ) a p p r o x i m a t i o n b y Eqs. ( 9 ) a n d ( 1 0 ) a n d t h e 0ONSWAP-2s) a p p r o x i m a t i o n b y Eqs. ( 8 ) a n d ( 1 0 ) are c o n s i d e r e d c o n c u r r e n t l y .

I t is seen t h a t , g e n e r a l l y , f o r w i n d w a v e s n ^ 4 - 6 . For t h e s w e l l n « 5 - 7 a n d s ^ 4 - 1 1 due t o m o r e n a r r o w peak shape. M o r e o v e r , f o r w i n d w a v e s y ^ 1 . 0 - 2 . 0 , a n d f o r m o r e p e a k e d s w e l l spectra y « 2. These estimates o f peakedness are l o w e r t h a n t h e w e l l k n o w n averaged value y = 3.3 [ 2 7 ] . The reason f o r this d i f f e r e n c e m a y be r e l a t e d t o t h e fact t h a t this e s t i m a t e w a s o b t a i n e d f r o m m e a s u r e d spectra, b u t t h e Table 3 w a s based o n t h e h i n d c a s t d a t a w h i c h does n o t have such f i n e f r e q u e n c y d i s c r i m i n a t i o n . F u r t h e r m o r e , t h e value o f y = 3.3 w a s d e t e r m i n e d i n an e x p e r i m e n t a d d r e s s i n g d e v e l o p i n g seas, w h i l e t h e h i n d c a s t data w i l l i n c l u d e m a n y cases o f f u l l y d e v e l o p e d seas. The d e t a i l e d analysis o f s i m u l t a n e o u s l y h i n d c a s t a n d m e a s u r e d w a v e spectra is r e q u i r e d f o r r e l i a b l e a n s w e r i n g o n this q u e s t i o n .

For b o t h types o f m o d e l s , t h e values o f m e a n d i m e n s i o n l e s s e r r o r s b y Eq. ( 1 7 ) are v e r y close. This means t h a t , i n p r a c t i c e , f o r h i n d c a s t spectra, a s a t i s f y i n g f i t t i n g accuracy i n t h e p r o c e d u r e based o n Eqs. ( 1 3 ) - ( 16) m a y be achieved e i t h e r b y v a r y i n g o f shape p a r a m e t e r n i n Eq. ( 9 ) , o r b y v a r y i n g t h e peakedness p a r a m e t e r y i n Eq. ( 8 ) , a l t h o u g h u s i n g Eq. ( 9 ) has l o w e r c o m p u t a t i o n a l r e q u i r e m e n t s .

Analysis o f Table 3 s h o w s t h a t t h e d i m e n s i o n l e s s e r r o r b y Eq. ( 1 7 ) is decreasing w i t h t h e increase o f w a v e h e i g h t hs. This means t h a t i t is possible to o b t a i n m o r e r e l i a b l e results f o r severe s t o r m w a v e s , a n d t h e m a x i m u m u n c e r t a i n t y w i l l be f o r t h e w e a k a n d m o d e r a t e w a v e s d u e t o c o m p l e x spectral shape a n d possible i n t e r a c t i o n s b e t w e e n spectral c o m p o n e n t s , w h i c h are n o t c o n s i d e r e d i n t h e Eq. ( 1 ) .

6. Discussion

The p r o p o s e d t e c h n i q u e is based o n t h e f i t t i n g o f d i r e c t i o n a l spectra u s i n g a l l t h e i n f o r m a t i o n about peak l o c a t i o n s , b o t h i n t h e f r e q u e n c y a n d d i r e c t i o n a l d o m a i n s . This extends t h e w e l l k n o w n approaches f o r m o d e l l i n g o f d o u b l e - p e a k e d w a v e spectra o n t h e basis o f f r e q u e n c y data o n l y (see [ 4 ] ) . I n fact, t h e e x t e n s i o n o f t h e m o d e l f i t t i n g a p p r o a c h t o t h e d i r e c t i o n a l d o m a i n a l l o w s t a l d n g

i n t o account m a n y w a v e s i t u a t i o n s , i n w h i c h t h e c o m p l e x sea is c o m b i n e d b y t h e w i n d w a v e s a n d s w e l l w i t h t h e d i s t i n c t d i r e c t i o n s a n d close f r e q u e n c i e s , as s h o w n i n Fig. 4 ( a ) .

As r e s u l t o f t h e s u m i n Eq. (1), t h e f r e q u e n c y s p e c t r u m SiJ) can l o o k s i n g l e - p e a k e d , e v e n w h e n t h e d i r e c t i o n a l d i s t r i b u t i o n Q ( / , 6)

is m u l t i p e a k e d . T h u s , t h e a u t o m a t e d analysis o f such s p e c t r u m i n a f r e q u e n c y d o m a i n o n l y leads to misclassification, w h e n t h e s i n g l e - p e a k e d m o d e l is f i t t e d t o t h e c o m b i n a t i o n o f t w o o r m o r e spectral c o m p o n e n t s . T h e effect o f m i s c l a s s i f i c a t i o n is i l l u s t r a t e d i n Fig. 4 ( b ) - ( d ) , w h e n t h e m o d e l w i n d w a v e spectra b y Eq. ( 9 ) w i t h

hs = 3 m, Tp = 7.5 s a n d n = 5 are s h o w n i n c o m b i n a t i o n w i t h

s w e l l spectra, also a p p r o x i m a t e d b y Eq. ( 9 ) w i t h hs = 3 m, a n d n = 6. The s w e l l peak p e r i o d is v a r i e d f r o m 1 2 s ( b ) d o w n t o l O s ( d ) . Also, i n Fig. 4 , t h e r e s u l t o f fitting b y Eqs. ( 1 3 ) - ( 1 6 ) o f a s u m m a r i z e d w a v e s p e c t r u m b y t h e s i n g l e - p e a k e d m o d e l , Eq. ( 9 ) , is s h o w n . It is seen t h a t t h e peak p o s i t i o n o f t h e fitted spectra is s h i f t e d to a h i g h - f r e q u e n c y d o m a i n , a n d t h e shape p a r a m e t e r n ' o f fitted spectra becomes less t h a n t h e c o r r e s p o n d i n g p a r a m e t e r s f o r w i n d w a v e s a n d t h e s w e l l . This means t h a t t h e fitted spectra b e c o m e w i d e r t h a n t h e o r d i n a r y spectra o f w a v e s . Hence, t h e e f f e c t o f m i s c l a s s i f i c a t i o n m i g h t r e f l e c t o n t h e w r o n g e s t i m a t i o n o f t h e w a v e l o a d i n g s o n t h e m a r i n e s t r u c t u r e s , because t h e w i d e r s p e c t r u m leads t o d i f f e r e n t r e s o n a n t c o n d i t i o n s t h a n t h e n a r r o w e r one. I n Table 2 t h e m i s c l a s s i f i c a t i o n p r o b a b i l i t i e s are p r e s e n t e d f o r d i f f e r e n t regions a n d levels o f w a v e h e i g h t . This means t h a t t h e o c c u r r e n c e o f s u c h k i n d o f m u l t i p e a k e d ( d o u b l e - t h r e e - o r m o r e ) spectra, w h e n t h e peak n u m b e r N o f d i r e c t i o n a l spectra

S ( f , 9) is g r e a t e r t h a n N ' f o r t h e f r e q u e n c y spectra S ( f ) o n l y . For e x a m p l e , i n t h e Fig. 1(a) s u c h k i n d o f t h r e e - p e a k e d spectra is s h o w n , w h e n N' = 2 f o r c o r r e s p o n d e n t d i r e c t i o n a l spectra. It is c l e a r l y seen t h a t t h e m i s c l a s s i f i c a t i o n p r o b a b i l i t y v a l u e Pm varies f r o m 9.9% at t h e Portuguese coast o f N o r t h A t l a n t i c t o 46.8% at t h e Azores. C o n s i d e r i n g t h e r a t i o Pm/iPjp - f Pap) t o t h e w h o l e a m o u n t o f m u l t i p e a k e d spectra, this m e a n s t h a t a t least 30%-80%

( f o r d i f f e r e n t r e g i o n s ) o f m u l t i p e a k e d spectra are possible to be misclassifled o n t h e base o n t h e f r e q u e n c y s p e c t r a o n l y .

Thus, t h i s r e s u l t s h o w s t h e necessity t o i d e n t i f y a c o m p l e x sea spectral m o d e l b y Eq. ( 1 ) b o t h i n f r e q u e n c y a n d d i r e c t i o n a l d o m a i n s , a n d r e q u i r e t h e a p p l i c a t i o n o f t h e r a t h e r c o m p u t a t i o n a l l y i n t e n s i v e t e c h n i q u e based o n Eqs. ( 1 3 ) - ( 1 6 ) .

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MO A.V, Boukhanovsky, C. Guedes Soares/Applied Ocean Research 31 (2009) 132-141

b

14r 12 / , H z Ajr=0.042Hz O 0.05 14 12 10 g 6 4 2 O / , H z 5 ( / ) , ' n A 1 1 1 - V Ay^=0.033Hz \ ii'=4.6

1

\ O 0.05 0.1 0.15 0.2 0.25 / , H z - 3

Fig. 4. Effect o f misclassification of double-peaked complex sea spectrum in a frequency domain. ( a ) - i l ! u s t r a t i o n i n directional d o m a i n ; ( b - d ) - e f f e c t of spectral fitting i n frequency domain only. Here 1 - i n i t i a l spectra of w i n d sea and the s w e l l ; 2 - c o m p l e x sea spectrum by Eq. (1) w i t h N = 1 ; 3 - f i t t i n g of single-peaked spectrum model by Eq.(9).

7. Conclusions

A n a p p r o a c h for m o d e l l i n g m u l t i p e a k e d d i r e c t i o n a l w a v e s p e c -t r a is p r o p o s e d . This a p p r o a c h is based o n a p a r a m e -t r i c d e s c r i p -t i o n , Eq. ( 1 ) , o f t h e d i r e c t i o n a l w a v e s p e c t r u m . A n u m e r i c a l o p t i m i z a t i o n p r o c e d u r e u s i n g t h e r a n d o m l i n e a r search a l g o r i t h m is p r o p o s e d f o r e s t i m a t i o n o f the s p e c t r a l p a r a m e t e r s . The n u m b e r o f w a v e s y s -t e m s i n -t h e s p e c -t r a is e v a l u a -t e d by a n i -t e r a -t i v e p r o c e d u r e .

The p r o p o s e d t e c h n i q u e , based o n Eqs. ( 1 3 ) ( 1 6 ) , a l l o w s f i t -t i n g o f s p e c -t r a l m o d e l s -t o m e a s u r e d o r h i n d c a s -t d a -t a . T h e HIPOCAS h i n d c a s t d a t a f o r N o r t h A t l a n t i c w e r e used f o r t h e case s t u d y . The analysis s h o w e d a s i g n i f i c a n t v a r i a b i l i t y o f s p e c t r a l p a r a m e t e r s f o r d i f f e r e n t r e g i o n s o f t h e o c e a n , w h i c h r e q u i r e s t h e a p -p l i c a t i o n o f a r a t h e r c o m -p u t a t i o n a l l y i n t e n s i v e t e c h n i q u e b y u s i n g E q s . ( 1 3 H 1 6 ) . The q u a l i t a t i v e a n a l y s i s o f t h e o c c u r r e n c e s o f m u l t i p e a k e d w a v e s p e c t r a f o r d i f f e r e n t r e g i o n s o f t h e oceans s h o w e d t h a t 30% t o 80% o f m u l t i p e a k e d w a v e s p e c t r a m a y be c o r r e c t l y m o d e l l e d b o t h i n f r e q u e n c y a n d d i r e c t i o n a l d o m a i n s t o g e t h e r . This is the m a i n a d v a n t a g e o f t h e a b o v e m e n t i o n e d a p p r o a c h i n c o m p a r i s o n w i t h t h e t e c h n i q u e s based o n l y o n t h e f r e q u e n c y s p e c t r u m s e p a r a t i o n .

Acknowledgment

The f i r s t a u t h o r has been f u n d e d b y t h e P o r t u g u e s e F o u n d a t i o n f o r Science a n d T e c h n o l o g y (FCT) u n d e r g r a n t nr. SFRH/BPD/ 2 0 9 8 1 / 2 0 0 4 .

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