BOSS'79
Second International Conferenceon Behaviour of Off-Shore Structures Held at: Imperial College, London, England 28 to 31 August 1979
ENCOUNTEB PBOBABIUTy O F EXTREME STBUCTUBAL BESPGNSE VALUES BASED
ON MULTI-PABAMETEB DESCBIPTIONS OF THE PHYSICAL ENVIBONMENT
J - A . Battjes
Delft University of Technology, The Netherlands.
Summary
This paper deals with some aspects of long-term statistics of environmental
con-ditions and their use in design of offshore structures. It appears that the consequences of
describing the environmental conditions with more than one independent parameter (instead
of just one, such as a characteristic wind speed) have not been sufficiently realised in the
past. Also, the usefuhiess of specifymg environmental conditions for design, at some a
priori chosen probability level, is questioned. It is argued that it would be more logical to
work with the probabilities of response parameters. Finally, methods are described for
estimating long-term distributions (parent and extreme) of a structural response parameter
for cases of multi-parameter descriptions of the environmental conditions.
Sponsored by: Delft University of Technology, The Netherlands Massachusetts Institute of Technology, U.S.A. The Norwegian Institute of Technology, Norway University of London, England
Secretariat provided by: B H R A Fluid Engineering
Copyright: ©
609
B H R A Fluid Engineering Cranfield, Bedford, England
I n l o n g - t e r m s t a t i s t i c a l d e s c r i p t i o n s o f e n v i r o n m e n t a l f a c t o r s which a r e r e l e v a n t t o the d e s i g n o f o f f s h o r e s t r u c t u r e s , d i f f e r e n t degrees o f c o m p l e x i t y a r e b e i n g used.The s i m p l e s t one i s t o d e s c r i b e t h e (extreme) e n v i r o n m e n t a l c o n d i t i o n s t h r o u g h o n l y one independent parameter ( e . g . , a windspeed, o t h e r parameters then b e i n g supposed t o depend on t h i s ) . As more indedependent parameters a r e used, such as w i n d o r wave d i r e c t i -on, o r wave p e r i o d , m u l t i - p a r a m e t e r d e s c r i p t i o n s o f t h e environment a r e o b t a i n e d . I n the a u t h o r ' s o p i n i o n , c e r t a i n concepts which a r e m e a n i n g f u l o n l y i n t h e one-parameter case, a r e sometimes e r r o n e o u s l y used i n t h e m u l t i - p a r a m e t e r case. I t i s necessary t o be more c o n s i s t e n t l y aware o f t h e r e s t r i c t i o n s i n h e r e n t i n t h e one-parameter models. This w i l l be i l l u s t r a t e d f o r t h e n o t i o n o f r e t u r n p e r i o d o f i n d i v i d u a l wave h e i g h t s . Subsequently, l o n g - t e r m s t a t i s t i c s o f a s t r u c t u r a l response a r e d e a l t w i t h f o r t h e case of e n v i r o n m e n t a l c o n d i t i o n s d e s c r i b e d by a s e t o f p a r a m e t e r s . Both t h e l o n g - t e r m p a r e n t d i s t r i b u t i o n o f response peaks, and t h e l o n g - t e r m extreme v a l u e d i s t r i b u t i o n a r e con-s i d e r e d .
R e t u r n p e r i o d o f wave h e i g h t i n one-parameter and m u l t i - p a r a m e t e r d e s c r i p t i o n
I n t h e c o n t e x t o f d e t e r m i n i s t i c l i m i t - s t a t e d e s i g n o f o f f s h o r e s t r u c t u r e s , t h e n o t i o n of r e t u r n p e r i o d i s f r e q u e n t l y used. A d e f i n i t i o n o f r e t u r n p e r i o d f o r i n d i v i d u a l wave h e i g h t s can be g i v e n as f o l l o w s .
The r e t u r n p e r i o d o f an i n d i v i d u a l wave h e i g h t v a l u e H a t some l o c a l i t y i s s a i d t o be R y e a r s i f t h e r e i s a p r o b a b i l i t y 1/fl t h a t i n any 12-month i n t e r v a l , d r a w n a t random from t h e p o p u l a t i o n o f n o n - o v e r l a p p i n g ]2-month i n t e r v a l s , t h e r e s h a l l be a t l e a s t one. i n d i v i d u a l wave h e i g h t a t t h a t l o c a l i t y equal t o o r g r e a t e r t h a n H.
Through t h e use o f t h e c l a u s e " a t l e a s t " , t h e d e f i n i t i o n g i v e n above i n e f f e c t r e f e r s t o s e p a r a t e occasions o f heavy weather ( s t o r m s ) , w i t h o u t b e i n g a f f e c t e d by t h e number of h i g h waves i n each o f those. Other d e f i n i t i o n s a r e p o s s i b l e , e.g. those i n w h i c h i n d i v i d u a l wave h e i g h t s a r e c o n s i d e r e d i r r e s p e c t i v e o f t h e i r o c c u r r e n c e i n s t o r m s . The d i f f e r e n c e s between these d e f i n i t i o n s a r e n o t r e l e v a n t t o t h e f o l l o w i n g d i s c u s s i o n , which c e n t e r s on a c o m p l i c a t i o n a r i s i n g i n t h e t r a n s i t i o n from a one-parameter model, as r e f e r r e d t o above, t o a m u l t i - p a r a m e t e r model.
I n r e f . A, wave h e i g h t v a l u e s are g i v e n f o r a number o f l o c a t i o n s on t h e Norwegian C o n t i n e n t a l S h e l f , w i t h a r e t u r n p e r i o d o f (among o t h e r s ) 100 y e a r s , based on t h e second k i n d o f d e f i n i t i o n mentioned above. T h i s i s done n o t o n l y f o r t h e wave h e i g h t s i r r e s p e c t i v e o f t h e s h o r t - t e r m mean d i r e c t i o n , b u t a l s o p e r d i r e c t i o n a l s e c t o r JU ; For each d i r e c t i o n a l s e c t o r , t h e same procedure was used t o c a l c u l a t e t h e s o - c a l l e d
100-year wave h e i g h t f r o m t h e data p e r t a i n i n g t o t h a t s e c t o r as was used t o c a l c u l a t e the o v e r a l l 100-year wave h e i g h t from t h e d a t a summed over a l l d i r e c t i o n s . Thus, t h e
100-year wave h e i g h t v a l u e per s e c t o r i s o b t a i n e d as t h e wave h e i g h t w i t h a p r o b a b i l i t y of exceedence, g i v e n t h a t t h e wave d i r e c t i o n i s w i t h i n t h e s p e c i f i e d s e c t o r , e q u a l t o t h e r a t i o o f t h e mean wave p e r i o d t o a d u r a t i o n o f 100 y e a r s . T h i s p r o c e d u r e i g n o r e s the f a c t t h a t t h e wave d i r e c t i o n i s w i t h i n t h e s p e c i f i e d s e c t o r o n l y p a r t o f t h e t i m e . The r e s u l t i n g numbers a r e t h e r e f o r e n o t v e r y m e a n i n g f u l .
The essence of t h e problems o f t h e k i n d n o t e d above ( o t h e r examples c o u l d be g i v e n ) i s t h e f a c t t h a t i t has n o t been s u f f i c i e n t l y noted t h a t t h e wave h e i g h t d i s t r i b u t i o n per d i r e c t i o n a l s e c t o r i s a c o n d i t i o n a l d i s t r i b u t i o n , as opposed t o t h e m a r g i n a l d i s
-t r i b u -t i o n o f a l l -t h e wave h e i g h -t s , i r r e s p e c -t i v e o f d i r e c -t i o n . When -t r a n s f o r m i n g con-d i t i o n a l p r o b a b i l i t i e s i n t o time con-d u r a t i o n s , t h e f r a c t i o n o f t i m e con-d u r i n g w h i c h t h e
assumed c o n d i t i o n s o c c u r must he taken i n t o account i n o r d e r t o g e t ' " ^ ^ " ^ ^ S f u l r e s u t s . For example, take a wave c l i m a t e i n which wave h e i g h t and d i r e c t i o n a r e s t o c h a s t i c a l l y independent. The 100year wave h e i g h t s f o r t h e d i f f e r e n t d i r e c t i o n a l s e c t o r s , c a l c u l a -ted w i t h t h e method used i n r e f . A, would a l l be e q u a l , even i f t h e waves would come from one such 30°-sector f o r o n l y 1% o f t h e t i m e , say.
The m o r a l o f t h e above i s t h a t concepts and methods which a r e v a l i d i n t h e °^e-para-meter case (wave h e i g h t s o n l y ) should n o t be used w i t h o u t p r o p e r e x t e n s i o n and g e n e r a l
i z a t i o n s i n a m u l t i - p a r a m e t e r case (wave h e i g h t and d i r e c t i o n ) . The q u e s t i o n whether and how a r e t u r n p e r i o d f o r m u l t i - p a r a m e t e r l o n g - t e r m e n v i r o n m e n t a l s t a t i s t i c s can be used i n d e s i g n procedures i s c o n s i d e r e d i n t h e f o l l o w i n g s e c t i o n .
I s r e t u r n p e r i o d o f e n v i r o n m e n t a l c o n d i t i o n s u s e f u l i n design?
I t i s common i n d e t e r m i n i s t i c l i m i t - s t a t e d e s i g n procedures f o r o f f s h o r e s t r u c t u r e s t o s e l e c t a v a l u e o f an e n v i r o n m e n t a l parameter, u s u a l l y wind speed o r wave h e i g h t , such t h a t i t s o c c u r r e n c e o r exceedence has a s u i t a b l y l o n g r e t u r n p e r i o d , e.g. 50 o r 100 y e a r s .
Two p o i n t s a r e noted w i t h r e s p e c t t h i s approach.
F i r s t , t h e procedure o f u s i n g a d e s i g n c o n d i t i o n o f which t h e r e t u r n p e r i o d i s chosen a p r i o r i can be c o n s i d e r e d as a s h o r t - c u t i n l i e u o f a more e l a b o r a t e , p r o b a b i l i s t i c approach, c o n s i s t i n g o f some o p t i m i z a t i o n p r o c e d u r e , i n which i n i t i a l i n v e s t m e n t s are c o n s i d e r e d , expected damages, down-time l o s s e s , human and e n v i r o n m e n t a l s a f e t y , and so f o r t h . The m e r i t s and shortcomings o f t h i s s h o r t - c u t , compared w i t h t h e more e l a b o r a t e p r o c e d u r e s , w i l l n o t be d i s c u s s e d here.
Second, t h e procedure o f u s i n g an e n v i r o n m e n t a l parameter t o c h a r a c t e r i z e a d e s i g n cond i t i o n i s i n some sense n o t l o g i c a l , s i n c e t h e cond e s i g n i s u l t i m a t e l y basecond on o t h e r c r i -t e r i a , r e l a -t e d -t o -t h e s -t r u c -t u r a l i n -t e g r i -t y and -t h e a b i l i -t y -t o c a r r y o u -t -t h e necessary o p e r a t i o n s . I n o t h e r words, i t i s n o t so much t h e e n v i r o n m e n t a l c o n d i t i o n s themselves w h i c h a r e o f u l t i m a t e c o n c e r n , b u t t h e s t r u c t u r e ' s response t o them. I f - w i t h i n t h e c o n t e x t o f a d e t e r m i n i s t i c d e s i g n p r o c e d u r e - a r e t u r n p e r i o d i s chosen then t h i s s h o u l d l o g i c a l l y r e f e r t o a r e l e v a n t response parameter, r a t h e r t h a n t o a c a u s a t i v e f a c t o r such as wind speed o r wave h e i g h t .
The p r a c t i c e o f u s i n g some a p r i o r i chosen v a l u e f o r a d e s i g n wind speed o r wave h e i g h t i s a t most j u s t i f i a b l e t o t h e e x t e n t t h a t t h e r e i s a one-to-one correspondence between such e n v i r o n m e n t a l parameter and t h e reponse c o n s i d e r e d . I t goes w i t h o u t s a y i n g t h a t such one-to-one correspondence i s p o s s i b l e o n l y as l o n g as t h e e n v i r o n m e n t a l con-d i t i o n s r e l e v a n t t o t h e con-d e s i g n a r e con-d e s c r i b e con-d t h r o u g h a s i n g l e incon-depencon-dent parameter. (There i s no r e s t r i c t i o n on t h e number o f dependent e n v i r o n m e n t a l p a r a m e t e r s , such as wave p e r i o d as a f u n c t i o n o f wave h e i g h t . ) However, when t h e e n v i r o n m e n t a l c o n d i t i o n s a r e d e s c r i b e d w i t h m u l t i p l e independent p a r a m e t e r s , such as wind speed and d i r e c t i o n , or wave h e i g h t and p e r i o d , t h e n t h e r e a r e s e v e r a l c o m b i n a t i o n s o f them w h i c h w i l l g i v e the same v a l u e o f t h e c o n s i d e r e d s t r u c t u r a l response ( e . g . , s t r e s s i n a j o i n t ) . I t i s then no l o n g e r p o s s i b l e t o t r a n s l a t e a d e s i g n v a l u e o f such response i n t o a s i n g l e en-v i r o n m e n t a l c o n d i t i o n . The r e t u r n p e r i o d o f any one o f t h e e n en-v i r o n m e n t a l parameters has l o s t i t s u s e f u l n e s s . The n a t u r a l approach would t h e n seem t o be t o work n o t w i t h v a l u e s o f e n v i r o n m e n t a l parameters a t a p r o b a b i l i t y l e v e l chosen a p r i o r i , b u t w i t h a range o f them, c o n t r i b u t i n g t o t h e p r o b a b i l i t i e s o f exceedence o f t h e response b e i n g c o n s i d e r e d . Such approach i s c a l l e d f o r e.g. i n case o f n o n - n e g l i g i b l e dynamic response of t h e s t r u c t u r e , w h i c h v a r i e s s i g n i f i c a n t l y b o t h w i t h wave h e i g h t and wave p e r i o d . I n t h e f o l l o w i n g , a few methods a r e d e s c r i b e d f o r t h e c a l c u l a t i o n o f l o n g - t e r m respon-se d i s t r i b u t i o n s f r o m a m u l t i - p a r a m e t e r model o f t h e environment.
Long-term d i s t r i b u t i o n o f response peaks
I n t h i s s e c t i o n , t h e l o n g - t e r m d i s t r i b u t i o n o f i n d i v i d u a l peak v a l u e s o f a response w i l l be c o n s i d e r e d , r e g a r d l e s s o f t h e sequence i n w h i c h they o c c u r . T h e i r l o n g - t e r m
ex-treme v a l u e d i s t r i b u t i o n i s d e a l t w i t h i n t h e f o l l o w i n g s e c t i o n .
Consider a g i v e n s t r u c t u r e , o r a g i v e n d e s i g n t h e r e o f , i n a g i v e n e n v i r o n m e n t a l c o n d i -t i o n , as d e s c r i b e d by a s e -t o f parame-ters f o r e.g. w i n d , waves, c u r r e n -t , sea l e v e l , and so on. For b r e v i t y , and w i t h o u t l o s s o f g e n e r a l i t y , we s h a l l i n t h e f o l l o w i n g r e f e r t o sea s t a t e parameters o n l y .
Let us assume f o r d e f i n i t i v e n e s s t h a t each sea s t a t e i s d e s c r i b e d by t h r e e p a r a m e t e r s , a c h a r a c t e r i s t i c wave h e i g h t ( e . g . , t h e s i g n i f i c a n t wave h e i g h t ) , a c h a r a c t e r i s t i c wave p e r i o d T , and a c h a r a c t e r i s t i c wave d i r e c t i o n 0^. I n t h e l o n g t e r m , these are^
random v a r i a b l e s , which i s denoted w i t h an u n d e r s c o r e . T h e i r j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n i s w r i t t e n as p(H,T,9).
We c o n s i d e r t h e maxima rj,, o f some response £(t) o f a g i v e n s t r u c t u r e i n a g i v e n sea s t a t e . We s h a l l assume t h a t r ^ t h e n i s R a y l e i g h - d i s t r i b u t e d , w i t h mean square v a l u e equal t o 2a^, and t h a t t h e average number of maxima p e r u n i t t i m e i s X^., Both Oj. and Xr a r e f u n c t i o n s of t h e sea s t a t e parameters, f o r a g i v e n s t r u c t u r e . I n t h e p r e s e n t c o n t e x t they a r e supposed t o be known, e.g. through t h e a p p l i c a t i o n o f s p e c t r a l t r a n s -f e r -f u n c t i o n s .
Under t h e assumptions s t a t e d , t h e c o n d i t i o n a l d i s t r i b u t i o n o f r ^ , f o r g i v e n v a l u e s
(H,T,e)
o f (Hc,Tc,Sc)j can be w r i t t e n asP r ( r > r|H,T,6) = exp( — ^ )
20^(H,T,e)
The f r a c t i o n o f t i m e d u r i n g w h i c h H^, T^ and 6^ s i m u l t a n e o u s l y a r e i n t h e ranges (H, H + dH), ( T , T + dT) and ( 6 , 6 + dS) r e s p e c t i v e l y , i s g i v e n by p(H,T,e)dHdTde, as f o l l o w s f r o m t h e d e f i n i t i o n o f
p(H,T,e).
The expected number o f maxima o f r ( t ) t under t h e s e c o n d i t i o n s i sXj.(H,T,e),
o f which a f r a c t i o n givèn by ( I ) exceeds t h e l e v e l r . I t f o l l o w s t h a t t h e expected number o f events ( r ^ > r ) p e r u n i t t i m e , i r r e s p e c t i v e of t h e v a l u e s o f He, Tc and £c. equals ^ ^^-^fff P r ( r j i i > r|H,T,e)Xr(H,T,e)p(H,T,e)dHdTde , (2)
i n which t h e i n t e g r a t i o n i s c a r r i e d o u t over a l l p o s s i b l e v a l u e s o f
(H,T,e).
The f r a c t i o n o f a l l t h e maxima o f r ( t ) exceeding r i s t h e n o b t a i n e d as t h e r a t i o o f ( 2 ) to t h e expected number o f maxima o f £(t) per u n i t time ( l o n g - t e r m ) , w h i c h i s g i v e n byfff Xr(H,T,e)p(H,T,e)dHdTde . "^3)
Said r a t i o equals t h e m a r g i n a l ( l o n g - t e r m ) p r o b a b i l i t y t h a t r m s h a l l e x c e e d t h e l e v e l r . I f r i s a s t r e s s t h e n t h i s r e s u l t can be used i n e v a l u a t i n g f a t i g u e a c c o r d i n g t o t h e Palmgren-Miner r u l e .
I t may be n o t e d t h a t t h e r e c i p r o c a l o f ( 2 ) equals t h e average d u r a t i o n between succes-s i v e eventsucces-s ( r ^ > r ) . Asucces-s succes-such i t r e p r e succes-s e n t succes-s t h e r e t u r n p e r i o d o f t h a t e v e n t . However, one s h o u l d be c a r e f u l i n i t s i n t e r p r e t a t i o n , and n o t l o s e s i g h t o f t h e f a c t t h a t i t i s based on t h e f r a c t i o n o f a l l t h e maxima o f r ( t ) exceeding r , no m a t t e r i n w h i c h s e quence they o c c u r . High v a l u e s o f r ^ tend t o occur o n l y i n i s o l a t e d , rough p e r i o d s , i n each o f which s e v e r a l may exceed r . Thus, t h e r e t u r n p e r i o d o f t h e event (tm > _ r ) i s i n g e n e r a l n o t equal t o t h e average time i n t e r v a l between s u c c e s s i v e rough p e r i o d s i n w h i c h t h e event ( r j ^ > r ) occurs ( a t l e a s t o n c e ) , b u t s h o r t e r than t h a t . The e s t i m a -t i o n o f -t h e l a -t -t e r w i l l be c o n s i d e r e d i n -t h e f o l l o w i n g s e c -t i o n .
Encounter p r o b a b i l i t y o f extreme response peak
The problem t o be c o n s i d e r e d here i s t h e e s t i m a t i o n of t h e p r o b a b i l i t y t h a t a t l e a s t one response peak ( r j ^ ) s h a l l exceed an extreme l e v e l r , i n a t i m e i n t e r v a l o f ( l o n g term) d u r a t i o n L.
Since t h e exceedence l e v e l r i s g i v e n t o be extreme, i t i s f o r a l l p r a c t i c a l p u r p o s e s ^ c e r t a i n t h a t r w i l l o n l y be exceeded d u r i n g i s o l a t e d , rough p e r i o d s ( s t o r m s ) . T h i s im p l i e s t h a t t h e r e q u i r e d p r o b a b i l i t y i s v i r t u a l l y equal t o t h e p r o b a b i l i t y o f o c c u r r e n c e .
i n a time i n t e r v a l o f d u r a t i o n L, o f a t l e a s t one storm i n which a t l e a s t one v a l u e o f r j ^ exceeds r .
L e t N = N ( r ; L ) denote t h e number o f storms i n t h e d u r a t i o n L i n w h i c h a t l e a s t one r e s -ponse peak exceeds t h e l e v e l r . The r e q u i r e d p r o b a b i l i t y can then be w r i t t e n as
] - Pr(N = 0 ) .
L e t p denote the l o n g t e r m average f r e q u e n c y o f occurrence o f storms; pL then r e p r e -sents t h e expected number o f storms i n t h e d u r a t i o n L. F u r t h e r m o r e , l e t Q ( r ) be t h e p r o b a b i l i t y t h a t i n an a r b i t r a r y s t o r m a t l e a s t one response peak s h a l l exceed r :
Q ( r ) = 1 - Pr(max r j ^ <^ r | a r b i t r a r y s t o r m ) . ( 4 )
The expected v a l u e o f N i s then g i v e n by
E(N) = E { N ( r ; L ) } = y L Q ( r ) .
C o n s i d e r i n g encounter p r o b a b i l i t i e s o f extreme v a l u e s i n a l i f e t i m e o f a s t r u c t u r e , we have y L » 1 and Q ( r ) « 1. T r e a t i n g t h e storms as independendent e v e n t s , N i s t h e n v e r y n e a r l y P o i s s o n d i s t r i b u t e d , i n which case t h e r e q u i r e d p r o b a b i l i t y can be e s t i m a -ted as
1 - P r ( N = 0) = 1 - exp {-E(N)} .
E v a l u a t i o n o f t h i s r e q u i r e s knowledge o f L, which i s g i v e n , and o f y and Q ( r ) , which must be e s t i m a t e d f r o m h i s t o r i c a l d a t a and t h e s t r u c t u r a l p r o p e r t i e s . Regarding Q ( r ) , we s h a l l f i r s t c o n s i d e r a g i v e n , s i n g l e storm.
A s i n g l e s t o r m i s supposed t o be d e s c r i b e d by t h e time h i s t o r y o f t h e s h o r t - t e r m sea s t a t e parameters
(H,T,e).
As b e f o r e , we c o n s i d e r response maxima r j ^ w h i c h f o r a g i v e ^ sea s t a t e a r e assumed t o be R a y l e i g h - d i s t r i b u t e d , w i t h mean square v a l u e e q u a l t o 2ar and w i t h mean f r e q u e n c y Aj., b o t h o f w h i c h v a r y w i t h t h e parameters(H,T,e).
C o n s i d e r i n g a s t o r m l a s t i n g f r o m t = t ] t o t = t 2 , t h e expected number o f maxima r^n which exceed ri s ~
m ( r ) = ƒ A r( t ) exp { - i r^ / a j c t ) } d t ,
•^1
where i t should be understood t h a t and Oj. v a r y w i t h t i m p l i c i t l y , t h r o u g h t h e i r de-pendence on t h e t i m e - v a r y i n g v a l u e s
(H,T,e).
On the b a s i s o f t h e Poisson d i s t r i b u t i o n f o r t h e number o f exceedences o f t h e l e v e l r by r j ^ , t h e p r o b a b i l i t y t h a t i n t h e g i v e n storm no r j ^ - v a l u e s h a l l exceed r i s e s t i m a t e d asR(r) = Pr(max r j ^ <_ r | g i v e n storm) = exp {- m ( r ) } . ( 8 )
The p r o b a b i l i t y (8) i s c o n d i t i o n a l i n t h a t i t i s g i v e n t h a t a storm occurs w i t h a g i v e n t i m e h i s t o r y o f
(H,T,e).
I n t h e l o n g - t e r m v i e w , t h e s t o r m may o r may n o t o c c u r , and i t s parameters a r e random v a r i a b l e s , w i t h an a s s o c i a t e d m u l t i d i m e n s i o n a l p r o b a b i -l i t y d i s t r i b u t i o n . Moreover, t h e v a r i a b -l e p a t t e r n o f t h e i r v a r i a t i o n w i t h t i m e d u r i n g a s t o r m s h o u l d be t a k e n i n t o account. A l l o f t h i s g r e a t l y c o m p l i c a t e s t h e t r a n s i t i o n from t h e c o n d i t i o n a l p r o b a b i l i t y ( 8 ) t o t h e n o n - c o n d i t i o n a l p r o b a b i l i t y ( 4 ) , which i s r e q u i r e d .A f i r s t s t e p i n g e t t i n g around t h i s d i f f i c u l t y i s t o r e c o g n i z e t h a t t h e e n v i r o n m e n t a l (sea s t a t e ) parameters have an e f f e c t on m o n l y t h r o u g h Xj- and 0^ and t h e i r v a r i a t i o n w i t h t i m e . (Needless t o say, t h i s i s a r e d u c t i o n i n c o m p l e x i t y o n l y i f t h e number of e n v i r o n m e n t a l parameters was more than two.) F u r t h e r m o r e , f o r l a r g e r o n l y a r e l a t i v e l y s m a l l time i n t e r v a l around t h e peak o f t h e response i n t e n s i t y Oj c o n t r i b u t e s s i g n i f i -c a n t l y t o m. (The a -c t u a l v a l u e s o f t j and t2 i n (7) are then i m m a t e r i a l , p r o v i d e d they are s u f f i c i e n t l y f a r away f r o m t h e t i m e o f the maximum response i n t e n s i t y . ) T h i s im-p l i e s t h a t m i s m a i n l y determined by t h e storm's maximum a ^ - v a l u e , and by t h e number of r u - v a l u e s around t h a t maximum w h i c h c o n t r i b u t e s i g n i f i c a n t l y t o m. Borgman ( r e f s . 2 and 3) has shown t h a t f o r the p r e d i c t i o n o f extreme v a l u e s t h e a c t u a l t i m e h i s t o r y of X].(t) and a j . ( t ) can be r e p l a c e d by one of c o n s t a n t i n t e n s i t y (a) and an e f f e c t i v e t o t a l number o f c y c l e s ( n ) , such t h a t the f o l l o w i n g a p p r o x i m a t i o n h o l d s f o r n o t t o o s m a l l v a l u e s o f r :
m ( r ) E ƒ Xy.(t) exp {- ir^/a^MJdt =n exp (- k ^ / a ^ ) . (9)
Borgman a l s o g i v e s methods f o r e s t i m a t i n g t h e v a l u e s o f O and n f r o m t h e t i m e h i s t o r i e s o f X j . ( t ) and aj.(.t). (Borgman a c t u a l l y d e a l s w i t h wave h e i g h t s i n s t e a d o f response peaks b u t t h a t does n o t a f f e c t h i s method.)
A s i m p l i f y i n g a p p r o x i m a t i o n , on t h e c o n s e r v a t i v e s i d e , c o n s i s t s o f e q u a t i n g a, t h e e f f e c t i v e v a l u e o f Oj., t o the maximum a^-value o c c u r r i n g d u r i n g a s t o r m . The use -of t h i s a p p r o x i m a t i o n o b v i a t e s t h e need o f c o m p l e t e l y f o l l o w i n g t h e procedure f o r a more p r e c i s e e s t i m a t e of a.
Using ( 9 ) , the response's t i m e h i s t o r y d u r i n g the s t o r m i s d e s c r i b e d by j u s t two con-s t a n t con-s , a t l e a con-s t i n con-s o f a r acon-s i t a f f e c t con-s m ( r ) and R ( r ) . We can w r i t e t h e r e f o r e
m = m ( r ; n , a ) C^)
and i n t e r p r e t R as a p r o b a b i l i t y which i s c o n d i t i o n a l on t h e o c c u r r e n c e of s p e c i f i c v a l u e s (n,a) of t h e l o n g - t e r m random v a r i a b l e s ( n , o ) :
R = R ( r | n , a ) = Pr(max r^^^ <_ r | n = n, £ = a) . ( ' 0
For each s t o r m , one p a i r of v a l u e s ( n , ^ ) can be c a l c u l a t e d . I n p r i n c i p l e , t h e j o i n t p r o b a b i l i t y d e n s i t y p ( n , a ) of (n,a) can t h e r e f o r e be e s t i m a t e d f r o m p a s t s t o r m s . Com-pounding t h i s w i t h t h e c o n d i t i o n a l p r o b a b i l i t y R ( r | n , a ) , t h e p r o b a b i l i t y t h a t a t l e a s t one response peak s h a l l exceed r i n a storm p i c k e d a t random f r o m t h e p o p u l a t i o n o f storms becomes
Q(r) = 1 - ƒƒ R ( r l n , a ) p ( n , 0 ) d n d a
= 1 - ƒƒ e""'^'''"'°^(n.0)dnda . ('2)
Together w i t h (5) and ( 6 ) , t h i s i s a s o l u t i o n t o the problem which was posed. I n most p r a c t i c a l cases, the v a r i a b i l i t y o f n has much l e s s e f f e c t on Q ( r ) t h a n t h e v a r i a b i l i t y o f a. A f u r t h e r a p p r o x i m a t i o n can t h e n be made by a s s i g n i n g some mean v a l u e t o n, which may ( b u t need n o t ) v a r y w i t h o. I t i s then s u f f i c i e n t t o w o r k w i t h t h e p r o b a b i l i t y d e n s i t y o f a o n l y , i n s t e a d of the j o i n t d e n s i t y of n and o. I n t h i s case, we have
m = m { r ; n ( a ) , a } , R = R { r | n ( a ) , a } (13)
and
Q ( r ) = ] - ƒ R { r | n ( 0 ) , a } p ( a ) d a
= 1 - ƒ e-'°^^5^(^>''^>p(a)do.
Another method o f e s t i m a t i n g Q ( r ) , i n which a l s o a r e d u c t i o n i s o b t a i n e d t o o n l y one l o n g - t e r m random v a r i a b l e , b u t w i t h o u t t h e a p p r o x i m a t i o n s i n h e r e n t i n (9) and i n t h e n e g l e c t o f t h e v a r i a b i l i t y o f n f o r g i v e n a, has been p r e s e n t e d by B a t t j e s ( r e f . Ï ) . I n t h i s approach, m ( r ) i t s e l f i s t r e a t e d as t h e b a s i c random v a r i a b l e . For a g i v e n v a l u e o f t h e l e v e l r , one v a l u e o f m ( r ) can be c a l c u l a t e d p e r s t o r m , u s i n g ( 7 ) . I n p r i n c i p l e , t h e l o n g - t e r m p r o b a b i l i t y d e n s i t y o f m c a n t h e r e f o r e be e s t i m a t e d f r o m p a s t storms, w i t h r as a parameter; i t i s w r i t t e n as p ( m ; r ) . I n t h i s approach, t h e p r o -b a -b i l i t y R i s c o n d i t i o n a l on t h e occurrence o f a s p e c i f i c v a l u e o f m ( r ) :
R = R(m) = Pr(max r j ^ < r | m ( r ) = m) = e~™ . (15)
Note t h a t t h e dependence on r i s absent i n t h i s c o n d i t i o n a l p r o b a b i l i t y . I t r e - e n t e r s the problem t h r o u g h t h e l i k e l i h o o d o f t h e event m ( r ) = m, o r , r a t h e r , o f t h e event (m < m ( r ) < m + dm), w h i c h i s used i n t h e c a l c u l a t i o n o f t h e n o n - c o n d i t i o n a l p r o b a b i l i t y Q ( r ) :
Q ( r ) = I - ƒ R(m)p(ra;r)dm
= 1 - ƒ e '"p(m;r)dm, ( j g )
An advantage o f t h i s approach i s t h a t one deals t h r o u g h o u t w i t h o n l y one l o n g - t e r m random v a r i a b l e , w h i l e s t i l l t a k i n g account o f t h e e f f e c t s o f t h e j o i n t v a r i a b i l i t y of a l l t h e e n v i r o n m e n t a l parameters which a r e needed i n t h e problem.
I t i s p o i n t e d o u t t h a t t h e methods sketched above, l e a d i n g t o eqs. 12, 14 and 16 r e s -p e c t i v e l y , d i f f e r o n l y i n t h e e s t i m a t i o n o f Q ( r ) , i . e . t h e -p r o b a b i l i t y t h a t i n a s t o r m p i c k e d a t random a t l e a s t one response peak s h a l l exceed r . The subsequent c a l c u l a t i o n o f t h e l o n g - t e r m encounter p r o b a b i l i t y i s t h e same (eqs. 5 and 6 ) .
The c a l c u l a t i o n s o f Q ( r ) can be c a r r i e d o u t f o r d i f f e r e n t v a l u e s o f t h e l e v e l r . Thus, u s i n g ( 5 ) , ^ t h e f u n c t i o n E { N ( r ; L ) } can be e v a l u a t e d . The v a l u e o f r f o r which
E { N ( r ; L ) } - 1, w r i t t e n as r ( L ) , i s t h e response v a l u e w i t h t h e r e t u r n p e r i o d L, i n t h e sense t h a t storms, i n w h i c h a t l e a s t one response peak exceeds f ( L ) , occur on t h e average once m a d u r a t i o n L. The v a l u e f ( L ) i s a p p r o x i m a t e l y equal t o t h e most proba-b l e v a l u e o f t h e l a r g e s t r^^ i n a d u r a t i o n L, and t h e p r o proba-b a proba-b i l i t y t h a t i n a d u r a t i o n L a t l e a s t one storm occurs i n w h i c h a t l e a s t response peak r„ exceeds r ( L ) , i s
1 - exp (-1) 0.63 ( r e f . 5 ) . ^ -m
Throughout t h e developments g i v e n above, storms have been t r e a t e d as independent events w i t h r e s p e c t t o t h e times o f t h e i r occurrence and w i t h r e s p e c t t o t h e i r parameters. A l -though t h i s may n o t be c o r r e c t i n t h e s t r i c t sense, i t has so f a r been g e n e r a l l y
accepted as a b a s i s f o r a n a l y s i s and p r e d i c t i o n . A t any r a t e , t h e r e i s a t p r e s e n t i n s u f f i -c i e n t knowledge about p o s s i b l e dependen-cies between storms t o i n -c o r p o r a t e t h a t i n a model. Moreover, i f i n f a c t some dependence i s p r e s e n t t h e n t h e independence assumption i s c o n s e r v a t i v e , i n t h e sense t h a t i t then o v e r p r e d i c t s t h e encounter p r o b a b i l i t i e s o f
Concluding remark
I n t h i s paper, some g e n e r a l aspects have been c o n s i d e r e d o f l o n g - t e r m s t a t i s t i c s o f e n v i r o n m e n t a l f a c t o r s and t h e i r use i n the d e s i g n o f o f f s h o r e s t r u c t u r e s . A p p l i c a t i o n s t o s p e c i f i c cases have n o t been presented h e r e i n ; i n s t e a d , the p h i l o s o p h y o f approach and f o r m a l i s m s f o r c a l c u l a t i o n s have been emphasized because i n the a u t h o r ' s o p i n i o n these o f t e n r e c e i v e i n s u f f i c i e n t a t t e n t i o n .
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