water wave mechanics

7/27/2019 WATER WAVE MECHANICS

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M a x i m u m e n t r o p y e s t i m a t i o n o f d i r ec t io n a l w a v es p e c t r a f r o m a n a r r a y o f w a v e p r o b e sO K E Y N W O G UH y d r a u l i c s L a b o r a t o r y , N a t i o n a l R e s e a rc h C o u n c i l , O t t a w a , C a n a d a

A procedure for estimating directional wave spectra from an array of wave probes based onthe Maximum Entropy Method (MEM) is developed in the present paper. The MEM approachyields an angular spreading function at each frequency band consistent with the input cross-spectral density matrix. The method is evaluated using numerical simulations of directional seastates. The MEM is also used to analyze data obtained from the three-dimensional wave basinof the Hydraulics Laboratory, National Research Council of Canada. Finally, the MEM iscompared with the Maximum Likelihood Method (MLM) and is shown to be a powerful toolfor directional wave analysis.Key Words: directional wave analysis, maximum entropy method.

1 . I N T R O D U C T I O NOcean waves are multi-directional, and informationabout the directional distribution of wave energy isoften required for the design of coastal and offshorestructures. Of all the methods currently used to resolvedirectional sea states, the MEM has been shown to havethe highest directional resolving power. 5 The M EM hashowever only been used to analyze data from a waveprobe-current meter array or a pitch-roll buoy. Inlaboratory applications, however, current meters have arelatively high level of noise present in the signals,leading to a degradation in the resolution of the MEM(see Nwogu e t a l . : ) . Wave probes currently used inlaboratory wave basins have relatively lower noise tosignal ratios than current meters. They are also lessexpensive and easier to calibrate, making them moreconvenient for use in laboratories for estimating direc-tional wave spectra. This leads to the present develop-ment of an MEM procedure for the analysis of direc-tional wave fields using a wave probe array.The MEM has it's origins in probability theory 4 andhas also been succesfully used in spectral analysis. 2 Thespectral analysis approach is, however, based on thedefinition of the change in entropy. The directionalspreading function can be thought of as the probabilitydistribution of wave energy over direction and thedefinition of entropy used in probability theory is thusmore justifiable as the basis of the MEM for the analysisof directional waves. This differs from the definition ofchange in entropy used by Barnard t for the analysis ofdirectional wave fields.The use of the concept of a directional spreadingfunction to characterize a wave field should only applyto a spatially homogenous wave field with no corre-

Accepted January 1989. Discussion doses March 1990.

lation of the wave components travelling in differentdirections. The single summation, random phasemeth od of wave synthesis (e.g. Miles and Funke 6) pro-duces such a wave field and is used in the present paperto simulate the water surface elevation time series atdifferent wave probe locations. The cross-spectraldensity matrix is obtained from a Fast FourierTransform (FFT) of the synthesized time series. TheMEM procedure then uses the cross-spectral densitymatrix to estimate a directional spreading function foreach frequency band.

2 . T H E M A X I M U M E N T R O P Y S O L U T I O NFor a spatially homogenous wave field, the cross-spectral density of the wave elevation at two locationsXm and xn can be expressed as (see I sobe et al . 3)

Smn(~) = S(~) ex pl ik " (x,,, - x , , ) I D ( ~ , O ) d O-T r (1 )

where D(~, 0) is a nonnegative angular spreading func-tion describing the distribution of wave energy overdirection. The spreading function has to satisfy~f~ D(co, 0) = (2)O 1- -T r

The wave frequency, ~ is related to the wavenumber, kby the linear dispersion relation. The directional wavespectrum can thus be defined asS(~ , O) = S(co)D (~ , O) (3)

where S(~0) is the conventional one-dimensional fre-quency spectrum. Equation (1) can be expressed asd~j(c~) = qj(O)D(co, O) d0 j= 1 .... M+ 1 (4)

where M= N ( N - 1), Nis the number of wave probes

176 A p p l i e d O c e a n R e s e a r c h , 1 9 8 9 , V o l . 1 1 , N o . 4 , 1989 Computational Mechanics Publications


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M a x i m u m e n t r o p y e s t im a t i o n o f d i r ec t io n a l w a v e s p e ct r a: O . N w o g ua n d

q A O ) = l o s ( k r j c o s ( & - 0 ) )s l i n ( k r j c o s ( S j - 0 ) ) j = 1 . . . . M [ 2j = M / 2 + 1 . . . . M (5 )j = M + II ~ Im S tu n o 9 )e [ S m ( o o ) S . ( ~ ) ] 1/2 j = 1 . . . . M [ 2t ~ j O O= S m n 0 2)[ Sm(oa)Sn(oo) l 1 /z j = M /2 + 1 . . . . M

1 j = M + I( 6 )

r j = . j ] 7 , . - x . ) z + ( Y m - - Y . ) :/ 3 j = t a n - ' [ ( y m - - y n ) / ( X m - - X n )] (7 )

T h e e n t r o p y a s s o c i a t e d w i t h t h e s p r e a d i n g f u n c t i o n c a nb e e x p r e s s e d a s

f TrE = - D(~o, 0) In D ( o o , 0) dO (8)--TrM a x i m i z i n g E s u b j e c t t o t h e c o n s t r a i n t s i m p o s e d b y th ec r o s s - s p e c t r a l d e n s i t y m a t r i x ( 4 ) y i e l d s

D ( w , 0 ) - - - e x p - 1 + ~ v j q j ( O ) (9 )j= l

w h e r e v j L a g r a n g e m u l t i p li e r s c h o s e n t o e n s u r e t h ee s t i m a t e o f t h e s p r e a d i n g f u n c t i o n i s c o n s is t e n t w i t he q u a t i o n ( 4 ). S u b s t i tu t i o n o f t h e m a x i m u m e n t r o p ys o l u t i o n ( 9 ) i n t o e q u a t i o n ( 4 ) r e s u l t s i n a n o n l i n e a r s e to f e q u a t i on s

e x p - 1 + ~ vjqj(O q~(O) dO- r r j = l

= q~ i( c0 ) i = l . . . . , M + I ( 1 0)T h e p a r a m e t e r s u j c a n n o t b e e a s il y d e t e r m i n e da n a l y t i c a l l y s o o n e h a s t o r e s o r t t o a n i t e r a t i v e t e c h -n i q u e . M o s t i t e r a t i v e p r o c e d u r e s t o d e t e r m i n e t h eu n k n o w n L a g r a n g e m u l t i p l i e r s uj w o u l d h a v e p r o b l e m sw i t h c o n v e r g e n c e s i n c e s o m e o f t h e k e r n e l f u n c t i o n sq i ( O ) a n d q j ( O ) a r e c o r r e l a t e d , d e p e n d i n g o n t h e r e l a ti v es p a c in g o f t he w a v e p r o b e s . T h e p r o b l e m w o u l d h a v e t ob e r e f o r m u l a t e d i n t e r m s o f a n o r t h o g o n a l s e t o f k e r n e lf u n c t i o n s .

A c c o r d i n t g t o w e l l k n o w n t h e o r e m s o f l i n e a r a l g e b r a ,a s et o f o r t h o g o n a l f u n c t i o n s g i (O ) c an b e f o r m e d f r o mq i ( O ) a s

g = T r q (11)w h e r e T i s a n o r t h o g o n a l m a t r i x w i t h c o l u m n v e c t o r se q u a l t o t h e o r t h o n o r m a l e i g e n v e c t o r s o f th e c o v a r i a n c em a t r i x d e f i n e d b y

f TrQ i j = q i ( O ) q j ( O ) dO (12)T h e c o m p o n e n t s o f th e c o v a r i a n c e m a t r i x c a n be

e x p l i c i t l y e x p r e s s e d a s '7 r [ J o ( z ~ ) + J 0 ( z 2 ) ]

7 r [ J o ( z , ) + J 0 ( z 2 ) ]

Q~j = 27r Jo (kr j )27r Jo(kr i )21r

w h e r e

i = 1 . . . . M / 2 ;j = l . . . . . M / 2i = M / 2 + 1 . . . . M ;

j = M / 2 + I . . . . . Mi = M + 1;j = l . . . . . M / 2i = 1 . . . . M / 2 ;j = M + Ii = M + I ;j = M + I

(13)

Zl = k[ ( r i c o s ~ i - r j cos /3 j ) 2+ (ri s in t 3 i - r a s i n / 3 a )z ] 1,2z 2 = k [ ( r i cos 5 i + r j co s /3a)2+ (ri sin Bi + rj sin 13j)z ] 1,,2

a n d J o i s t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e rz e r o . A l l o t h e r c o m p o n e n t s o f t h e c o v a r i a n c e m a t r i x a r ei d e n t i c a l l y z e r o .T h e e x i s t e n c e o f a z e r o e i g e n v a l u e o f t h e m a t r i x Qi n d i ca t e s t h a t o n e o f t h e f u n c t i o n s q i ( O ) i s e i t h e r i d en t i -c a l ly z e r o , o r c o r r e l a t e d w i t h t h e o t h e r f u n c t i o n s q j ( O ) .

F u n c t i o n s w i t h z e ro e i g e n v a lu e s a r e th u s d r o p p e d a n dq u e s t i o n s a r i s e a s t o t h e r e l a t i v e i m p o r t a n c e o f e i g e n -v a l u e s c l o s e t o z e r o . I f a l l t h e n o n - z e r o e i g e n v a l u e s a r er e t a i n e d , t h e s o l u t i o n p r o c e d u r e t e n d s t o b e c o m eu n s t a b l e . A c r i t e r i o n u s e d t o s e l e c t a n u m b e r o f ' i m -p o r t a n t ' e i g e n v a lu e s , L , i s b a s e d o n t h e r a t i o o f e a c he i g e n v a l u e t o t h e l a r g e s t e i g e n v a l u e . I n t h e p r e s e n ta p p l i c a t i o n , e i g e n v a l u e s g r e a t e r t h a n 0 .1 7o o f t h e l a r g e s te i g e n v a l u e w e r e c o n s i d e r e d t o b e i m p o r t a n t . T h e p r o b -l e m n o w is r e d u c e d t o d e t e r m i n i n g t h e p a r a m e t e r s iz ,j = 1 . . . . L w h i c h s a t i s f y t h e n o n l i n e a r s e t o f e q u a t i o n s .

t [i = g i ( O ) e x p - 1 + # j g j ( O ) d O i = 1 . . . . L-Tr j= l (14)w h e r ep = T r c b (15)

T h e L e v e n b e r g - M a r q u a r d t a l g o r i th m ~s u s e d t o s ol v et h e s y s t e m o n n o n - l i n e a r e q u a t i o n s f o r t he p a r a m e t e r s ~ ja 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 f u n c t i o n i s n o w g i v e n b y

D (c0 , 0 ) = ex p - 1 ~ j g j ( O ) (16).=A s u i t a b l e i n i ti a l g u e s s f o r th e p a r a m e t e r s ~ j c a n b ed e t e r m i n e d f r o m t h e l i n e ar a p p r o x i m a t i o n o f t h ee x p o n e n t i a l f u n c t i o n a s

ex p - 1 + ~ j g j ( O ) = ~ ~ j g j ( O ) (17)j = l j = l

l e a d i n g t o i n i t i a l g u e s s e sI t j = P j j = l . . . . . L 0 8 )

T h e c o n v e r g e n c e c r i t e r i a u s e d t o s t o p t h e i t e r a t i v e p r o -c e d u r e r e q u i r e s t h a t e i t h e r t h e u p d a t e d v a l u e s o f t h e / ~ ' sa g r e e w i t h t h e c u r r e n t v a l u e s o f t h e # ' s t o a s p e c i f i e d a c -c u r a c y , o r t h a t d i f f e r e n c e s b e t w e e n t h e l e f t a n d r i g h th a n d s i d e s o f e q u a t i o n ( 1 4 ) b e le s s t h a n a s p e c i f i e dt o l e r a n c e . D u e t o t h e p r e s e n c e o f n o i s e a n d t r u n c a t i o no f i n f o r m a t i o n p r o v i d e d b y t h e o r i g in a l d a t a s e t , th e

A p p l i e d O c e a n R e s e a r c h , 1 9 8 9, V o l. 1 1 , N o . 4 1 " /7


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M a x i m u m e n t r o p y e s t i m a t i o n o f d i re c t io n a l w a v e sp ec tr a." O . N w o g ua b o v e t w o c r i t e r i a c a n n o t b e m e t i n s o m e c a s e s a n d t h es o l u t i o n is d e t e r m i n e d a s t h a t w h i c h m i n i m i z e d t h es q u a r e e r r o r i n t h e i te r a t i v e p r o c e d u r e . T h e s q u a r e e r r o rE Q i s g i v e n b y

E Q =P i - g i ( O ) e x p - I + / ~ I~jgj(O)

L i = 1d o ] ]

(19)T h e e i g e n v a l u e a n a l y s i s i s a l s o u s e f u l f o r i n v e s t i g a t i n gd i f f e r e n t a r r a y s p a c i n g s a n d c o n f i g u r a t i o n s s i n c e i t i n -d i ca t es t h e a m o u n t o f r e d u n d a n t i n f o r m a t i o n i n t h e

a v a i l a b l e d a t a . A s a n e x a m p l e , i f a f iv e p r o b e a r r a y a n da te n p r o b e a r r a y h a v e t he s a m e n u m b e r o f i m p o r t a n te i g e n v a l ue s , t h e a d d i t i o n a l i n f o r m a t i o n p r o v i d e d b y t h ea d d i t i o n a l f iv e p r o b e s c a n b e c o n s i d e r e d t o b er e d u n d a n t .

3 . A N A L Y S I S O F S I M U L A T E D A N D M E A S U R E DD A T A

3 . 1 N u m e r i c a l l y s i m u l a t e d d a taT h e t i m e s e r ie s o f t h e w a t e r s u r f a c e e l e v a t i o n a t d i f -f e r e n t w a v e p r o b e l o c a t i o n s w e r e s y n t h e s i z e d u s i n g t h es i ng l e s u m m a t i o n m o d e l d e s c r i b e d i n M i le s a n d F u n k e . 6

T h e w a t e r s u r f a c e e l e v a t i o n is g i v e n b yr /( x , y , t ) =

,% 'A i co s[ w i t - k i ( x cos O i + y s i n O i) + e i] (20)

i = 1w h e r e A i a n d e i a r e t h e a m p l i t u d e s a n d p h a s e s o f t h ew a v e c o m p o n e n t s g i v e n b y th e r a n d o m p h a s e m e t h o d a s

A i = ~ ' 2 S ( w i )D ( w i , O i) A w A O (21)ei = 2 7rU [0 , 1 ]

w h ere ~ 0 i= i (Ao~/J) a n d U [ 0 , 1 ] i s a u n i f o r m d i s t r i-b u t i o n f r o m z e r o t o o n e . T h e a n g l e s 0~ a r e c h o s e n b y ap r o c e d u r e s u c h t h a t t h e s p r e a d i n g f u n c t i o n i s a p p r o x i -m a t e d b y J w a v e a n g le s i n e a c h f r e q u e n c y b a n d o f w i d t h2 , . T h e a n g le s c a n e it h e r b e c h o s e n r a n d o m l y f o r e a c hf r e q u e n c y c o m p o n e n t o r c h o s e n t o v a r y l i n e a r l y w i t hf r e q u e n c y . T h e w a v e f i el d p r o d u c e d b y t h e s i ng l e d ir e c -t i o n p e r f r e q u e n c y m o d e l i s s p a t i a l l y h o m o g e n o u s a n di s t h u s m o r e a p p r o p r i a t e t o u s e f o r c h e c k i n g d i r e c t i o n a lw a v e a na l y si s p r o g r a m s t h a n t h e d o u b le s u m m a t i o nm o d e l .T h e f i rs t e x a m p l e c o n s i d e r e d is a J O N S W A P i n c i de n tw a v e s p e c t r u m w i t h f p = 0 . 4 H z a n d 3 ~= 3 .3 . T h ea n g u l a r s p r e a d i n g i s g i v e n b y t h e f r e q u e n c y i n d e p e n d e n tc o s i n e s p r e a d i n g f u n c t i o n

1 t ' ( s + 1 ) c o s 2 ' 0 ) 1 01 < ~ r / 2D ( 0 ) = ~ r ( s + ~)x=TF= 0 o t h e r w i s e ( 2 2)

w h e r e E is t h e g a m m a f u n c t i o n a n d t h e p a r a m e t e r s isa s p r e a d i n g i n d e x d e s c r i b i n g t h e d e g r e e o f w a v e s h o r t -c r e s t e d n e s s w i t h s --* oo r e p r e s e n t i n g a l o n g - c r e s t e d w a v ef ie l d. T h e a r r a y u s e d f o r t h e d i r e c t i o n a l w a v e a n a l y s i sc o n s i st s o f f i v e w a v e p r o b e s a r r a n g e d i n th e l a y o u ts h o w n i n F i g . 1 . T h e c o n f i g u r a t i o n i s s i m i l a r t o t h e

1 ( 0 . 2 6 4 , 0 . 8 1 ) ~ 2 ( 0 . 2 6 4 , -0 ,8 1 )y ~ / ~ ../~ .- - /~ . .._ _ ._ _ _ . . _ ~[ ~ 0 . 2 6 4\ \ , , ,) 1 0 . 6 94 ( - 0 ~ . - 0 . 5 1 -

Al l d im e ns ions in m e t e rs

F i g u r e 1 . A r r a n g e m e n t o f f i v e p r o b e a r r a y

C E R C a r r a y u s e d b y P a n i c k e r a n d B o r g m a n . s T h ew a t e r s u r f a c e e l e v a t i o n t i m e s e r i es a t t h e d i f f e re n t p r o b el o c a t i o n s w e r e s y n t h e s iz e d f o r a d u r a t i o n o f 8 1 9 . 2 su s in g a f r e q u e n c y b a n d w i d t h o f 0 .0 2 H z a n d 3 2 w a v ed i r e c t i o n s p e r f r e q u e n c y b a n d .

T h e F F T a n a l y s i s w a s p e r f o r m e d u s i n g t h e s a m eb a n d w i t h u s e d i n t h e s y n t h e s i s . F i g u r e 2 s h o w s a c o m -p a r i s o n o f t h e ta r g e t J O N S W A P s p e c t r u m w i t h t h e sy n -t h e s iz e d w a v e s p e c t r u m f o r t h e c e n tr a l p r o b e . A se x p e c t e d , t h e s y n t h e s i z e d s p e c t r u m m a t c h e s t h e t a r g e ts p e c t r u m f a i r l y w e l l . T h e s l i g h t d i f f e r e n c e s a r e d u e t ot h e w i d e f r e q u e n c y b a n d u s e d . F i g u r e s 3 t o 5 s h o w t h et a r ge t , M E M a n d M L M ( se e I s o b e et al . 3) e s t i m a t e da n g u l a r s p r e a d i n g f u n c t i o n s a t t h e p e a k f r e q u e n c y f o rs = 1 , 5 a n d 2 0 r e s p e c t i v e l y . T h e s p e c i f i e d m e a n d i r e c -t i o n 0 o = 0 . I t c a n b e s e e n t h a t t h e M E M e s t i m a t e s t h es p e c if i e d s p r e a d i n g f u n c t i o n s b e t t e r t h a n t h e M L M f o ra l l c a s e s . T h e f it i s h o w e v e r b e t t e r f o r t h e h i g h e rs p r e a d i n g i n di c e s. T h e M L M a l w a y s e s t i m a t e s a b r o a d e rs p r e a d i n g f u n c t i o n b e c a u s e i t d o e s n o t s a t i s f y t h e c o n -s t r a i n t s i m p o s e d o n t h e c r o s s - s p e c t r a l d e n s i t y m a t r i x o ft h e o r i g i n a l d a t a ( e q u a t i o n ( 1 ) ) .

.=O,OS

0 . 0 3 7 S

0 . 0 2 $

T a r | e t. . . . $ y n t h e s i = e d

I .Ql2S

J I ~ " - - ~ , ]O.e0 . 0 O , S l . O I , $ 2 , 0F r e q u e n c y ( F I z }

F i g u r e 2 . C o m p a r i s o n o f t a rg e t a n d s y n t h e s i z e dJ O N S W A P s p e c tr a ( f p = 0 . 4 H z , 7 = 3 . 3 ) at th e c e n tr a lp r o b e o f t h e a r r a y

17 8 A p p l i e d O c e a n R e s e a r c h , 1 9 8 9 , V o l . 11 , N o . 4


4/7

Maximum entropy estimation of directional wave spectra: O. NwoguT h e e f f e c t o f n o i s e o n t h e r e s o l u t i o n o f t h e M E M w a se x a m i n e d b y a d d i n g a n o i s e si g n al r e p r e s e n ti n g b a n d -l i m i te d w h i t e n o i se f r o m z e r o t o t h e N y q u i s t f r e q u e n c yt o t h e s y n t h e s i z e d t i m e s e r i e s . F i g u r e 6 s h o w s a c o m -p a r i s o n o f t h e t a r g e t a n d M E M e s t i m a t e d s p r e a d in gf u n c t i o n s a t t h e p e a k f r e q u e n c y f o r s = 5 w i t h 0 % a n d

1 0 % n o i s e r e s p e c ti v e ly . T h e a d d i t i o n o f 1 0 % n o i s e d o e sn o t s e e m t o a f f ec t t h e r e s o l u t i o n o f th e M E M a t t h ep e a k f r e q u e n c y a p p r e c i a b le . F i g u r e s 7 a n d 8 s h o w p l o t so f t h e m e a n d i r e c t i o n a n d s t a n d a r d d e v i a t i o n o f t h ew a v e p r o p a g a t i o n a n g l e r e la t iv e t o t h e e s t im a t e d m e a nd i r e c t io n o v e r t h e f r e q u e n c y r a n g e 0 . 6 5 f p t o 2fp. B o t h

l , ? ST a r l e t

I . I . . . . M E M. . . . . . . . M L M

0 . 4 5

1 .3

l . I $ // ~

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- 1 8 1 . I - l l l . l - 4 1 . I O.O l O , l I I 0 , I 1 8 1 . ID i r e c t i o n ( d e l )

Figure 3. Comparison of target, MEM, and ML Mdirectional spreading fun cti ons (s = 1, f = fp)

I .ST l r l e tM I M ( O m N o l l e )M I M (t O ~ N o l l e )

I. Z

0. 1A

l . l

+ .3

I t. 0 I ~ L- t | 0 , 0 - | 2 l , O - 1 0 . l O .O 1 0 , l I Z O . l 1 6 1 , 0

D i r e c t i o n ( d e | )

Figure 6. Comparison of target and ME M directionalspreading functions with noise (s = 5, f = fp)

1. 5T a r l e t

I . | . . . . M E I dM L M

I . I I

$ .4

1 . 0 ' + ~ ' - l- 1 8 1 . - I I I . I - I I . I I . I I I . I I Z I . I I I I , |D i r e c t i o n ( l e g )

Figure 4. Comparison o f target, MEM , and ML Mdirectional spreading fun cti ons (s = 5, f = fp )

I 0 . 0

- :10 .11

T t r z e t. . . . M | g ( o ~ N o r s e ). . . . .. . . M | I I ( I O X N o i s e )

- 4 0 . 0 I I r l I0 , 2 0 . 3 0 . 4 O . S 0 . 4 0 . 7

F r e q u e n t 7 ( H z )I . I

Figure 7. Comparison of target and ME M estimatedmean direction o f wave progagation (s = 5)

3 . 0

1 .4

1. 0

1. 2

0 .1 L + / ~ " ' - . , . I

T t r l e t. . . . I d g M. . . . . . . . M L M

0 + 0 I- L 8 0 . 0 - I l O , g - I 0 . 0 0 . 0 6 4 1 . t t l O . I I I 1 . 0

D i r e c t i o n ( ( l e E )

Figure5. Comparison of target, MEM , and ML Mdirectional spreading fun cti ons (s = 20, f = fp)

4 0 . 0

3 0 . 0

2 0 . 0

1 0 4

T a r l e t. . . . I i l I M ( O X N o i s e ). . . . . . . . l i e u ( t O Z N o i s e )

0 , 0 I I I I I0 , 2 0 . ~ 0 . 4 0 . 5 O . I 0 . 7 Q . O

F r e q u e n c y ( H I )

Figure 8. Comparison of the standard deviations of thedirectional spreading fun cti ons (s = 5)

Applied Ocean Research, 1989, Vol. 11, No. 4 179


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Maximum entropy estimation of directional wave spectra: O. Nwogu0 , 7 5

T a r s s t

0 ,8 . . . . MEM. . . . . . . . ldLld

0 , 4 5

0 3

/1 . 1 5

/ t

0 .0 : ' L ' ; ; - ; ~ . . . . . . . . . . . . . . . . .- l l 0 . 0 - [ t l , I - 6 0 . 0 I , I I 6 0 .Q I l e . g I I 0 . ID i r e c t i o n ( d e g )

Figure 9. Comparison of target, MEM, and M L Mspreading funct ions fo r a bidirectional sea state ( f = fp )

quantities are not noticeably affected by the 10% noisein the specified frequency range.A bidirectional sea state with the same JONSWAPspectrum as the previous example was simulated by in-terlacing the frequencies of the two principal wavetrains so as to produce a spatially homogenous wavefield. One wave train had a princi pal direction 0o = 60 and a spreading index s = 5 while the other wave trainhad 0o = - 6 0 and s = 1. Figure 9 shows a comp ari sonof the target, MEM and MLM estimated spreadingfunctions at the peak frequency. The MEM resolvessuch a bidirectional sea state much better than theMLM.

3.2 Laboratory dataTests were carried out in the multidirectional wavebasin of the Hydraulics Laboratory, National ResearchCouncil of Canada. The basin is 30 m 19.2 m and 3 m

deep. Tests were carried o ut in a water dept h o f 2 m.The basin is equipped with a 60 segment, multi-modewave generator capable of producing regular and ran-dom long- and short-crested sea states. Figure 10 showsa side view of the wave generator. Energy absorptionbeaches are located along the other three sides of thebasin.A wave probe array was used to obtain informationabout wave directionality in the basin. The wave gaugearray consisted of nine capacitance type wave probesmounted on a rigid frame attached to vertical posts asshown in Fig. 11. All nine wave probes can be calibratedat the same time by remotely moving the frame contain-

ing the probes vertically along the posts in still water.The spacing of the outer five probes is the same as thatused for the numerical simulations (Fig. 1). The innerfive probes have the same configuration but are at a halfthe radius of the enclosing circle.Tests were carried out for a J ON SW AP incident wavespectrum with significant wave height, Hm o = 0.22 m,fp = 0.5 Hz and different specified cosine power angularspreading functions. Data was collected at a samplingfrequency of 10 Hz for 819.2 s which was the recyclingperiod used for the generation of the waves. The FFTcross-spectral density estimates were smoothed overfrequency bands of width 2x = 0.04 Hz.Figure 12 shows a comparison of the specified andmeasured frequency spectrum for the central waveprobe. The spectrum of the measured water surfaceelevation matches the specified spectrum rather well.Figures 13 and 14 show a comparison of the MEMestimated spreading functions using the outer five pro-bes and all nine probes for target spreading indices s = 3and 6 respectively. It can be seen that there is very littledifference between the spreading functions estimatedusing the five probe array and the nine probe array. Theinformation provided by the additional four probes canthus be considered to be redundant. There is a slight

Figure lO. View of Segmented Wave Generator

180 Appl ied Ocean Research, 1989, Vol. 11, No. 4


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M a x i m u m e n t r o p y e s t i m a t io n o f d i r e c ti o n a l w a v e s p e ct ra : O . N w o g ushift in the mean direction of the observed data. This isprobably due to the fact that no attempt was made toalign the probe array carefully with respect to the x axisof the Basin.In general, it was observed that the CERC five probearray is adequate for the resolution of directional seastates by the MEM. It was however thought useful to

F i g u r e 1 1. V i e w o f n i n e p r o b e a r r a y

1 .2

T a r g e t .. . . . S p r o b e a r r a y0.9 ~ . . . . . . . . 9 - p r o b e a r r a y

0 .0

,9 ~ i,

. . . . . . ~ , " ,,3 .. . . . . . . . . .9 .0- 1 8 0 . 0 - 1 2 0 . 0 - 0 0 . 0 0 . 0 0 0 . 0 1 2 0 . 0 L 8 0 .0Di re c t i o n ( d e g )

F i g u r e 1 3 . C o m p a r i s o n o f ta r g e t a n d m e a s u r e d d i re c -t i o n a l s p r e a d in g f u n c t i o n s f o r a f i v e a n d n i n e p r o b earray ( s = 3 , f = f p )

Av

I, $

1 .2

0 .9

0 .6

0 .$

Y a r f a t. . . . 5 - p r o b e a r r a y. . . . . . . . 0 - p r o b e a r r a y

- 1 2 0 , 0 - 0 0 . 0 0 . 0 $ 0 . 0 1 2 0 ,00 . 0- 1 0 0 . 0 1 8 0 . 0Di re c t i o n (d e | )

F i g u r e / 4 . C o m p a r i s o n o f t a rg e t a n d m e a s u r e d d ir 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 s f o r a f i v e a n d n i n e p r o b ea r r a y ( s = 6 , f = f p )0 . 0 2

O.OIS

0 . 1 ) 1

T * r f * t. . . . M e a s u r e d

0 . 0 0 0

I I . .0 .0 0 . 0 0 . $ i . O 1 .5 2 . 0F r e q u e n c y (Hz )

F i g u r e / 2 . C o m p a r i s o n o f t ar g et a n d m e a s u r e dJ O N S W A P s p ec tr a ( f p = 0 . 5 H z , H s = 0 . 22 m ) a t t h ec e n t ra l p r o b e o f t h e a r ra y

1. 2

T e rs e r- - / ~ - -- 6 - P r o b e A r r a y

0 .0 / ? ~ f ~ I . . . . . P ro b a ~ e lr r a yA

0 .3 t ~. , . . . . . . . . . . . . . . . . . . . . . . . J , . . . . . . . .. . . . . . .. . . . . . .. . .- 100.11 -120 .0 - 0 0 . 0 0 . 0 0 0 . 0 120,11 180.0

D i r e c t i o n ( d e s )

F i g u r e 1 5. C o m p a r i s o n o f t a r g e t a n d m e a s u r e d d i re c -t i o n a l s p r e a d in g f u n c t i o n s f o r a f i v e p r o b e a n d p r o b e -c u r r e n t m e t e r a r r a y ( s = 3 , f = f p )

A p p l i e d O c e a n R e s e a r c h , 1 9 8 9 , V o l . 11 , N o . 4 181


7/7

M a x i m u m e n t r o p y e s t im a t i o n o f d ir e c ti o na l w a v e s p e ct r a: O . N w o g udetermine for what range of wavelengths should ei therthe outer five probes or inner five probes be used. Thearrays provide an adequate resolut ion of direct ionalwave spectra when the ratio of the array radius, Ra tothe wavelength, L is in the range of 0.05 to 0.5 withopt imum resolut ion at a rat io of about 0.15.

The results from a wave probe array were also com-pared with that from a wave probe - curre nt meterarray. Figure 15 shows a comparison of the estimatedspreading functions using the two arrays for s = 3 at thepeak frequency. In the absence of measurement noise,there is very little difference between the spr eading func-t ions estimated using the probe array and probe - cur-rent meter array in a spatially homogenous wave field.

C O N C L U S I O N SA metho d for est imating direct ional wave spectra froma wave gauge array using the maximum entropy princi-ple has successfully been developed. The ME M yields aspreading funct ion of each frequency band which isconsistent with the cross-spectra of the measured watersurface elevation time series.

The MEM was compared with the MLM for differentdirectional sea states and was shown to resolve direc-t ional seas consis tent ly bet ter than the MLM, par-ticularly for bidire ction al seas. The tests also indicatethat the CERC five probe array is adequate for theresolution of most directional sea states.

Due to the t runcat ion of redunda ndant in format ionfrom the o riginal data set in the eigenvalue analysis andthe presence of noise, the MEM solut ion might in somecases not exactly satisfy the constraints imposed by the

cross-spectral densi ty matrix, but would minimize theintroduced error in a least square sense.

A C K N O W L E D G E M E N T SThis work was completed while the author was a guestworker at the Hydraul ics Laboratory, Nat ionalResearch Counci l of Canada and was part ial ly sup-ported by NRCC Contract No. 043-1397/7207. Theassistance provided by Messrs E. R. Funke, M. D. Milesand Dr. E. P. D Mans ard is grateful ly acknowledged.

R E F E R E N C E Sl Barnard, T. E. Analytical studies of techniques for the compu-tation of high-resolution wavenumber spectra, Texas InstrumentsAdvanced Array Research, Spec. Rep. No. 9, 19692 Burg, J. P. Maximum entropy spectral analysis, Proc, 37th An-nual Int. Soc. Explor. Geophys. Meeting, Oklahoma City 19673 Isobe , M., Kondo, K. and Horikawa, K. Extension of MLM forestimating directional wave spectrum, Proc, Symp. on Descrip-tion and Modelling of Directional Seas, Copenhagen, 19844 Jaynes, E. T. Information theory and statistical mechanics,

Papers l and ll, P h ys . R ev . 1957, 106, 1085 Kobune, K. and Hashimoto, N. Estimation of directional spectrafrom the maximum entropy principle, Proc. 5th Int. OffshoreMechanics and Arctic Engineering Syrup., Tokyo, Japan, Vol. I,80-856 Miles,M. D. and Funke, E, R. A comparison of methods for syn-thesis of directional seas, Proc. 6th Int. Offshore Mechanics andArctic Engineering Syrup., Houston, 1987, Vol. II, 247-2557 Nwogu, O. U., Mansard, E. P. D., Miles, M. D. and Isaacson,M. Estimation of directional wave spectra by the maximum en-tropy method, Proc. 1AHR Seminar on Wave Generation andAnalysis in Laboratory Wave Basins, Lausanne, Switzerland8 Panicker, N. N. and Borgman, L. E. Directional spectra fromwave gauge arrays, Hydrogr. Eng. Lab. Rep. HEL-16, Universityof California, Berkely, 1970

182 A p p l i e d O c e a n R e s e a r c h , 1 9 8 9 , V o l . 1 1, N o . 4

water wave mechanics

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