S P E C I A L I S S U E PAPER 391
Post-Katrina performance of a turret-moored floating
storage and offloading unit in the Gulf of Mexico
J J M Baar
Shell I n t e r n a t i o n a l Exploration and Production, H o u s t o n , Texas, USA
ne manuscript ivas received on 25 January 2009 and was accepted after revision for publication on 15 May 2009. D O I : 10.1243/14750902JEME148
A b s t r a c t : C o n v e n t i o n a l d e s i g n o f ships a n d o f f s h o r e p l a t f o r m s relies o n p e r f o r m i n g a s h o r t -t e r m r e s p o n s e analysis f o r m e -t o c e a n c o n d i -t i o n s w i -t h a 100-year r e -t u r n p e r i o d . T h i s a p p r o a c h is e f f i c i e n t b u t n o t n e c e s s a r i l y c o n s e r v a t i v e w h e n c o m p a r e d w i t h a c o m p r e h e n s i v e l o n g - t e r m r e s p o n s e analysis t h a t c o n s i d e r s p l a t f o r m responses f o r s t o r m s e n c o u n t e r e d over t h e l i f e t i m e o f t h e p l a t f o r m . T u r r e t - m o o r e d f l o a t i n g p r o d u c t i o n , storage a n d o f f l o a d i n g s y s t e m s are sensidve t o n o n - c o l l i n e a r w i n d , waves, a n d c u r r e n t c o n d i t i o n s . D u r i n g h u r r i c a n e s i n t h e G u l f o f M e x i c o s u c h c o n d i t i o n s f r e q u e n t l y o c c u r a n d , d e p e n d i n g o n t h e r e s u l t i n g w e a t h e r v a n i n g c h a r a c t e r -istics o f a t u r r e t - m o o r e d t a n k e r , t h e y m a y h a v e a s t r o n g i m p a c t o n vessel m o t i o n s , m o o r i n g l i n e loads, a n d riser p e r f o r m a n c e . I n r e c e n t years s o m e v e r y s t r o n g h u r r i c a n e s h a v e o c c u r r e d i n t h e G u l f o f M e x i c o , e.g. K a t r i n a a n d R i t a i n 2005. W h e n i n c l u d i n g these r e c e n t s t o r m d a t a i n t h e l o n g - t e r m r e s p o n s e analysis t h e y h a v e a m a r k e d e f f e c t o n t h e e x t r a p o l a t e d 100-year l o n g - t e r m responses. T h e p o s t - K a t r i n a l o n g - t e r m r e s p o n s e s o f a g e n e r i c t u r r e t - m o o r e d f l o a t i n g s t o r a g e a n d o f f l o a d i n g u n i t are e v a l u a t e d a n d c o m p a r e d against p r e - K a t r i n a a n a l y s i s results. T h e results o f t h e analysis are u s e d t o s d p u l a t e r e s p o n s e - b a s e d d e s i g n c r i t e r i a w h i c h are s i m p l e s h o r t - t e r m d e s i g n sea states t h a t c a n r e p r o d u c e a g i v e n l o n g - t e r m r e s p o n s e (e.g. r o l l ) .
K e y w o r d s : G u l f o f M e x i c o , h u r r i c a n e s , t u r r e t - m o o r e d tanker, f l o a t i n g storage u n i t (FSU), s h o r t - t e r m r e s p o n s e a n a l y s i s , l o n g - t e r m r e s p o n s e analysis, r e s p o n s e - b a s e d d e s i g n c r i t e r i a
1 I N T R O D U C T I O N
T r a d i t i o n a l d e s i g n p r a c t i c e i n n a v a l a r c h i t e c t u r e a n d o f f s h o r e e n g i n e e r i n g relies o n e s t i m a t i n g e x t r e m e p l a t f o r m responses f r o m a s h o r t - t e r m r e s p o n s e analysis (STRA). I n STRA i t is a s s u m e d t h a t t h e e n v i r o n m e n t is s t a t i o n a r y f o r t h e s t o r m d u r a t i o n ( t y p i c a l l y 3 h ) a n d s t a n d a r d s p e c t r a l analysis t e c h n i -q u e s are used t o e s t i m a t e t h e e x t r e m e d e s i g n response o f i n t e r e s t [1, 2 ] . T o t h i s e n d t h e s t o r m e n v i r o n m e n t is n o r m a l l y c h a r a c t e r i z e d i n t e r m s o f waves, w i n d , a n d c u r r e n t c o n d i t i o n s w i t h a r e t u r n p e r i o d o f 100 years, say. T h e a d v a n t a g e s o f t h e STRA are o b v i o u s i n t h a t i t a l l o w s t h e d e s i g n e r t o r a p i d l y e s t i m a t e t h e e x t r e m e d e s i g n r e s p o n s e .
I n r e a l i t y , a s h i p o r o f f s h o r e p l a t f o r m e n c o u n t e r s m a n y sea states d u r i n g its l i f e t i m e . N o w a d a y s s t o r m
'Corresponding autlwr: SiieU Internationai Exploration and Production, 150 Nortli Daily Asliford, Houslon, TX 77079, USA. email: Jol?.Baar@>shelLcom h i n d c a s t databases are a v a i l a b l e t h a t p r o v i d e a l o n g -t e r m h i s -t o r y o f s -t o r m c o n d i -t i o n s a-t s p e c i f i c si-tes [3¬ 5). I n a l o n g - t e r m r e s p o n s e analysis (LTRA) t h e s h o r t - t e r m r e s p o n s e s are e v a l u a t e d f o r each e v e n t i n t h e h i n d c a s t d a t a b a s e a n d t h e statistics are t h e n a c c u m u l a t e d a n d e x t r a p o l a t e d to p r o d u c e a l o n g -t e r m d i s -t r i b u -t i o n f o r -t h e r e s p o n s e o f i n -t e r e s -t . T h i s l o n g - t e r m d i s t r i b u t i o n is t h e n u s e d t o e s t i m a t e t h e a c t u a l 100-year r e t u r n p e r i o d r e s p o n s e (as o p p o s e d to t h e s h o r t - t e r m r e s p o n s e o b t a i n e d b y e x p o s i n g t h e s t r u c t u r e t o o n e s i n g l e 100-year r e t u r n p e r i o d s t o r m ) .
STRA a n d LTRA d o n o t necessarily y i e l d e q u i v a l e n t e s t i m a t e s . T h i s is t h e case i n p a r t i c u l a r f o r s t r u c t u r e s t h a t are d y n a m i c a l l y s e n s i t i v e t o v a r i a r i o n s i n e.g. w a v e p e r i o d o r n o n - c o U i n e a r w i n d , w a v e , a n d c u r r e n t d i r e c t i o n s . S u c h c o n d i t i o n s c o m m o n l y o c c u r i n t h e G u l f o f M e x i c o ( G O M ) d u r i n g t h e passage o f t h e eye o f a h u r r i c a n e . E x a m p l e s o f d y n a m i c a l l y s e n s i t i v e p l a t f o r m s i n c l u d e t e n s i o n l e g p l a t f o r m s [6, 7] a n d t u r r e t - m o o r e d ( w e a t h e r v a n i n g ) t a n k e r - b a s e d
392 I J M Baar
f l o a t i n g p r o d u c t i o n , storage, a n d o f f l o a d i n g (FPSO) s y s t e m s [ 8 - 1 0 ] .
O n c e t h e a c t u a l l o n g - t e r m response has b e e n e s t i m a t e d , there are several t e c h n i q u e s a v a i l a b l e to p r o d u c e a s o - c a l l e d r e s p o n s e - b a s e d d e s i g n s t o r m (RBDS). A RBDS is s i m p l y a n a r t i f i c i a l s t a t i o n a r y sea state t h a t is m o s t l i k e l y t o r e p r o d u c e the l o n g - t e r m response e s t i m a t e f r o m a s i m p l i f i e d STIIA. As s u c h it is d e p e n d e n t o n t h e s t o r m s a n d sile(s) selected f r o m t h e s t o m i h i n d c a s t database, o n t h e p l a t f o r m p a r r i -culars (e.g. h u l l g e o m e t r y , l o a d i n g c o n d i t i o n , m o o r i n g system), a n d o n t h e d y n a m i c characteristics o f t h e response o f interest. T h a t is, a RBDS s h o u l d o n l y b e u s e d f o r a p a r t i c u l a r p l a t f o m i at a g i v e n site a n d f o r w h a t e v e r response t o w h i c h i t p e r t a i n s . T w o d i f f e r e n t responses, e.g. r o l l a n d m o o r i n g l i n e t e n s i o n , t y p i c a l l y h a v e t w o d i f f e r e n t RBDSs associated w i d i t h e m . I t is also possible t h a t f o r a g i v e n response t h e r e m a y be m o r e t h a n o n e RBDS. A L T R A is m o r e c u m b e r s o m e a n d t i m e - c o n s u m i n g t h a n a STRA. H o w e v e r , w i t h t h e s i g n i f i c a n t l y i m p r o v e d q u a l i t y o f p r e s e n t - d a y s t o r m h i n d c a s t databases a n d u n c o n s t r a i n e d c o m p u t i n g p o w e r , i t has e f f e c t i v e l y b e c o m e a n a c c e p t e d p a r t o f Shell's d e s i g n p r a c t i c e f o r d e e p w a t e r f l o a t i n g o f f s h o r e s t r u c t u r e s to p e r f o r m a c o i u p r e h e n s i v e L T R A . Ex-p e r i e n c e has t a u g h t t h a t i n m o s t cases t h e t r a d i t i o n a l s h o r t - t e r m d e s i g n r e c i p e b a s e d o n e.g. 100-year w a v e h e i g h t t e n d s t o be s l i g h f l y c o n s e r v a t i v e w h e n c o m p a r e d t o a c t u a l 100-year l o n g - t e r m responses. RBDSs c a n p r o v i d e a n e f f e c t i v e m e a n s o f r e d u c i n g t h e d e s i g n cycle t i m e i n t h a t t h e y a v o i d h a v i n g t o recycle a c o m p l e t e LTRA f o r Ü i e case o f a c h a n g e i n o n e o f t h e d e s i g n p a r a m e t e r s . T h e y c a n i n p r i n c i p l e also be u s e d t o r e d u c e d e s i g n c o n s e r v a t i s m , a l t h o u g h c a u t i o n s h o u l d be u s e d i n v i e w o f t h e m a n y v a r i a b l e s t h a t p l a y a r o l e , n o t a l l o f w h i c h c a n b e r e a d i l y c o n t r o l l e d . T h i s p a p e r i l l u s t r a t e s t h e a b o v e c o n c e p t s a n d a p p l i e s t h e m t o a t u r r e t - m o o r e d f l o a t i n g s t o r a g e a n d o f f l o a d i n g u n i t (FSU) i n t h e G O M . W e a t h e r v a n i n g t a n k e r - s h a p e d vessels a s s u m e a h e a d i n g o f least resistance (i.e. m i n i m u m y a w p o t e n t i a l energy) w i t h respect t o t h e p r e v a i l i n g w i n d , w a v e , a n d c u r r e n t c o n d i t i o n s . H o w e v e r , c e r t a i n n o n c o l l i n e a r c o m b i -n a t i o -n s o f w i -n d , waves, a -n d c u r r e -n t m a y a c t u a l l y expose t h e vessel t o q u a r t e r i n g o r e v e n b e a m seas c o n d i t i o n s , r e s u l t i n g i n s i g n i f i c a n t levels o f r o l l m o t i o n , n e t v e r t i c a l m o t i o n s at t h e t u r r e t , m o o r i n g l i n e t e n s i o n s , a n d riser l o a d s . A f t e r a b r i e f r e v i e w o f t h e t h e o r y o f LTRA, t h e e v o l u t i o n o f d e s i g n w a v e h e i g h t i n t h e p r e a n d p o s t -K a t r i n a era is i l l u s t r a t e d a n d discussed. T h e l u e t h o d is t h e n a p p l i e d to a t u r r e t - m o o r e d F S U . T h e i n v e s t i g a t i o n focuses o n t h r e e responses: h e a v e (i.e. t h e net v e r t i c a l m o t i o n at t h e t u r r e t ) , r o l l , a n d p i t c h m o t i o n . These responses are c o n s i d e r e d s y m p t o -m a t i c o f -m o o r i n g l i n e a n d riser loads, b u t easier t o evaluate. B o t h STRA a n d L T R A results are p r e s e n t e d , as w e l l as a n i n d i c a t i v e RBDS Ü i a t c a n be u s e d f o r p r e l i m i n a r y d e s i g n s c r e e n i n g p u r p o s e s . 2 T H E O R Y T h e basis f o r t h e LTRA m e t h o d is a s t o r m h i n d c a s t database c o n t a i n i n g t h e h i s t o r y o f N s t o r m s at M sites o r g r i d p o i n t s . For e a c h s t o r m / = ] , A ^ ^ i a n d site 7 = l , . . . , W g p t h e d a t a b a s e c o n t a i n s a h i s t o r y o f k=l, ...,Kii,j) p a r a m e t e r s d e s c r i b i n g t h e t e m p o r a l v a r i a t i o n o f w a v e h e i g h t a n d p e r i o d , w i n d speed, a n d c u r r e n t speed, a n d w a v e , w i n d , a n d c u r r e n t d i r e c -t i o n s . C o n s i d e r , f o r e x a m p l e , -t h e d i s -t r i b u -t i o n o f w a v e h e i g h t . T h e basic a s s u m p t i o n is t h a t t h e f u t u r e w i l l b e s i m i l a r t o t h e p a s t or, i n o t h e r w o r d s , t h a t t h e s a m p l e d i s t r i b u t i o n c a l c u l a t e d f r o i n t h e h i n d c a s t s w i l l a p p r o x i m a t e t h e t r u e d i s t r i b u t i o n o f w a v e h e i g h t s . T h e w a v e h e i g h t d i s t r i b u t i o n c a n t h u s b e averaged over sites t o p r o d u c e a s m o o t h d i s t r i b u t i o n f o r a n a r b i t r a r y site [11]. T h e s a m e a s s u m p t i o n s m a y also be a p p l i e d t o a n y r e s p o n s e Y{X) w h i c h is d e p e n d e n t o n t h e e n v i r o n m e n t X.
Let X(i,j,k) d e f i n e t h e s t o r m p a r a m e t e r s (wave h e i g h t , w i n d speed, etc.) d u r i n g h o u r k o f s t o r m / at site j. Stationary' c o n d i t i o n s are a s s u m e d . T h e c u m u l a t i v e s h o r t - t e r m d i s t r i b u t i o n o f t h e r a n d o m v a r i a b l e Yis
PsT{y\X{iJ, k))=ProhiY^y\X{iJ, k)) (1)
For i n s t a n c e , w h e n t h e r e s p o n s e is, e.g. s i g n i f i c a n t w a v e h e i g h t t h e s h o r t - t e r m d i s t r i b u d o n f u n c t i o n degenerates i n t o a s i m p l e H e a v i s i d e u n i t s t e p f u n c -t i o n , w h i c h c h a n g e s a-t -t h e h i n d c a s -t v a l u e . I f , o n -t h e o t h e r h a n d , t h e i n d i v i d u a l peaks o f t h e z e r o - m e a n s t a t i o n a r y r a n d o m process y ( f ) are R a y l e i g h d i s t r i b -u t e d a n d a t o t a l o f Ny peaks o c c -u r d -u r i n g s t o r m k ( w h i c h is a s s u m e d t o h a v e a d u r a t i o n o f 3 h , say), t h e n P s ï ( y | X ( / . i , A : ) ) = 1 - e x p - (2) w h e r e CT^, is t h e s t a n d a r d d e v i a t i o n o f t h e r e s p o n s e . T h i s f o r m u l a t i o n is p e r t i n e n t t o t y p i c a l s h i p m o t i o n s , e.g. heave, r o l l , a n d p i t c h . For o t h e r r e s p o n s e s s u c h
Post-Katrina performance of a turret-moored F S U 393 as m o o r i n g l i n e t e n s i o n , t h e R a y l e i g h m o d e l i n a y n o t s u f f i c e a n d a m o r e s k e w e d e m p i r i c a l d i s t r i b u t i o n m o d e l s u c h as e.g. W e i b u l l o r G u m b e l m a y n e e d t o b e d e t e r m i n e d f r o m m o d e l tests. F o r t h e r e s p o n s e t o b e less t h a n y f o r a g i v e n s t o r m at a g i v e n site, i t m u s t be less t h a n y f o r e a c h h o u r i n t h e s t o r m . T h u s , t h e ' m e d i u m - t e r m ' d i s t r i b u t i o n f u n c t i o n f o r t h e m a x i m u m r e s p o n s e d u r i n g s t o r m / at site j is g i v e n b y a s i m p l e m u l t i p l i c a t i o n o f t h e s h o r t - t e r m 3 - h d i s t r i b u t i o n s
PMj{y\xiij)) = 'n^ Psiiymj, k)) (3)
T h i s f o r m u l a t i o n also a c c o u n t s f o r t h e l i k e l i h o o d t h a t t h e i n a x i m u m r e s p o n s e l u a y n o t o c c u r d u r i n g t h e h o u r w h e n t h e e n v i r o n m e n t is ' m o s t severe' (e.g. t h e w a v e h e i g h t is a m a x i m u m ) . T h e d i s t r i b u t i o n f u n c t i o n f o r t h e e n t i r e p e r i o d c o v e r e d b y a l l s t o r m s c o n t a i n e d i n t h e h i n d c a s t database a n d f o r a l l sites i n c l u d e d c a n b e c a l c u l a t e d b y f o r m i n g a n average over a l l N^t s t o r m s a n d A/gp sites. E v e n r e l a t i v e l y s m a l l d i f f e r e n c e s i n t h e p o s i t i o n s o f t h e sites w i t h respect t o t h e t r a c k s o f t h e s t r o n g e s t s t o r m s c a n cause t h e u p p e r t a i l s o f these d i s t r i b u t i o n s t o dift'er f r o m e a c h o t h e r . Since t h e h i s t o r i c a l s t o r m tracks are n o t e x p e c t e d to b e e x a c t l y d u p l i c a t e d i n t h e f u t u r e , t h e d i s t r i b u d o n f u n c t i o n s f r o m m a n y sites s h o u l d be averaged. T h i s p r o c e s s i n e f f e c t r a n d o m i z e s t h e tracks o f f u t u r e s t o r m s r e l a t i v e t o a p a r t i c u l a r site. T h u s t h e s i n g l e r a n d o m s t o r m d i s t r i b u t i o n is o b -t a i n e d b y -t a k i n g a n average o v e r a l l s -t o r m s a n d si-tes i n c l u d e d i n t h e h i n d c a s t d a t a b a s e ^^.SRS ( y ) = ^ E E PMriy\XilJ)) (4) •'"st ,-=1 -f^gli j^l S u p p o s e t h e h u r r i c a n e h i n d c a s t d a t a b a s e c o n t a i n s iVg, s t o r m s c o v e r i n g a p e r i o d o f / V y r s years. T o d e r i v e t h e d i s t r i b u t i o n o f m a x i m u m r e s p o n s e f o r e x p o s u r e s o t h e r t h a n Ny^^ years, i t is a s s u m e d t h a t t h e f r e q u e n c y o f a r r i v a l s o f h u r r i c a n e s , a n d h e n c e r e s p o n s e , is P o i s s o n d i s t r i b u t e d . T h e Tyear l o n g -t e r m c u m u l a -t i v e d i s -t r i b u -t i o n is o b -t a i n e d f r o m P L T ( y ) = e x p | - r r ( l - P s R s ( . y ) ) ] (5) w h e r e v=7Vst/Wyrs is t h e average a r r i v a l rate o fs t o r m s . I t c a n be s h o v m [12] t h a t t h e ' m o s t p r o b a b l e m a x i m u m ' ( M P M ) Tyear l o n g t e r m r e s p o n s e c o r r e -s p o n d -s t o r i -s k level-s o f 63.2 p e r cent, o r a c u m u l a t i v e p r o b a b i l i t y o f 36.8 p e r c e n t P L T ( y M P M ) = e ^ (6) T h e e n t i r e p r o c e s s c a n also be ' i n v e r t e d ' t o o b t a i n t h e ( 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 o f t h e e n v i r o n m e n t XiY), c o n d i t i o n a l o n a g i v e n r e s p o n s e Y, a n d h e n c e d i e ' r e s p o n s e b a s e d ' e n v u -o n m e n t X w h i c h is m -o s t l i k e l y t -o r e p r -o d u c e ( i n a s h o r t - t e r m sense) t h e M P M T-year l o n g - t e r m re-s p o n re-s e . T h e f o r m a l w a y to a c c o m p l i re-s h t h i re-s ire-s b y i n v o k i n g p r o b a b i l i s t i c m e t h o d s , e.g. firstorder r e l i a -b i l i t y m e t h o d a n d s e c o n d - o r d e r r e l i a -b i l i t y m e t h o d o r M o n t e C a r l o s i m u l a t i o n s . A m o r e h e u r i s t i c a p p r o a c h c a n b e e m p l o y e d t o achieve s i m i l a r r e s u l t s . T h e d a t a b a s e o f s h o r t - t e r m M P M r e s p o n s e m a x i m a f o r all s t o r m s a n d a l l sites c a n u s u a l l y be filtered d o w n t o a f a i r l y l i m i t e d n u m b e r o f ' s i m ü a r ' s t o r m s t h a t p r o d u c e t h e 100-year l o n g - t e r m r e s p o n s e t o w i t h i n c e r t a i n a c c u r a c y . T h e m o s t l i k e l y RBDS c a n t h e n be o b t a i n e d b y s i m p l y a v e r a g i n g t h e p a r a m e t e r s c h a r -a c t e r i z i n g t h e s e ' s i m i l -a r ' s t o r m c o n d i t i o n s . I t is i m p o r t a n t t o be aware t h a t a R B D S is d e p e n d e n t o n t h e s t o r m s a n d site(s) i n c l u d e d i n t h e h u r r i c a n e d a t a b a s e a n d o n t h e d y n a m i c c h a r -a c t e r i s t i c s o f t h e r e s p o n s e c o n s i d e r e d , e.g. h u l l size, l o a d i n g c o n d i r i o n , a n d m o o r m g s y s t e m . T h a t is, a RBDS s h o u l d o n l y be u s e d f o r a p a r t i c u l a r p l a t f o r m a t a g i v e n site a n d t h e r e s p o n s e t o w h i c h i t p e r t a i n s . T w o d i f f e r e n t responses, e.g. r o l l m o t i o n a n d m o o r -i n g l -i n e t e n s -i o n , t y p -i c a l l y have t w o d -i f f e r e n t RBDSs a s s o c i a t e d w i t h t h e m . A l s o , i t is p o s s i b l e f o r o n e r e s p o n s e to h a v e m o r e t h a n RBDSs a s s o c i a t e d w i t h i t . 3 E V O L U T I O N O F D E S I G N W A V E H E I G H T As a f i r s t e x a m p l e c o n s i d e r t h e e v o l u f i o n o f t h e 100y e a r s i g n i f i c a n t w a v e h e i g h t i n t h e p r e a n d p o s t -K a t r i n a era. T a b l e 1 c o m p a r e s t h e p a r a m e t e r s o f t h e t w o h u r r i c a n e h m d c a s t databases u s e d i n t h e p r e s e n t i n v e s t i g a t i o n . 1. T h e p r e - K a t r i n a G U M S H O E d a t a b a s e [3] i n c l u d e s 35 s t o r m s d u r i n g t h e 1900-85 p e r i o d , a n d g r i d p o i n t s have b e e n s a m p l e d r e p r e s e n t a r i v e o f t h e e n t i r e G O M . T h a t is, i t is a s s u m e d t h a t t h e h i s t o r i c a l s t o r m t r a c k s are n o t e x p e c t e d t o be d u p l i c a t e d [11]. 2. T h e p o s t - K a t r i n a G O M O S d a t a b a s e [4] i n c l u d e s 29 s t o r m s d u r i n g t h e 1 9 5 0 - 2 0 0 5 p e r i o d , a n d g r i d p o i n t s h a v e b e e n s e l e c t e d t h a t are s p e c i f i c t o t l i e sites o f Shell's e x i s t i n g M a r s a n d U r s a t e n s i o n l e g p l a t f o r m s (TLPs) s o m e 130 m i l e s SSE o f N e w
394 J J M Baar
Table 1 H u i r i c a n e hindcast database parameters a n d responses
i t e m Pie-Katrina Post-Katrina
Hurricane hindcast database Period
N u m b e r of storms Grid pooling
lOO-year significant wave height (m)
100-year MPM responses from LTRA
Heave at turret (m) Roll (degrees) Pitch (degrees) GUMSHOE 1900-85 35 GOM-wide 12.3 10.2 7.3 5.5 GOMOS 1950-2005 29 Site specific 14.7 14.4 21.0 7.5 O r l e a n s . E x c l u s i o n o f pre-1950 s t o r m s is consist e n consist w i consist h consist h e A P I B u l l e consist i n 2 I N T M E T |5] r e c o m -m e n d a t i o n , a n d t h e s i t e - s p e c i f i c p o o l i n g has b e e n c a r r i e d o u t i n a c c o r d a n c e w i t h a m e t h o d p r o -p o s e d b y H e i d e m a n a n d M i t c h e l l [13]. S i t e - s p e c i f i c p o o ü n g (rather t h a n G O M - w i d e aver-aging) r e f l e c t s t h e u n d e r s t a n d i n g t h a t t h e r e is e v i d e n c e f o r t h e r e b e i n g a r e g i o n a l d e p e n d e n c e f o r large, i n t e n s e s t o r m s , or s o - c a l l e d h u r r i c a n e alleys [5, 14]. F r o m T a b l e 1 i t c a n be seen t h a t a t y p i c a l e s t i m a t e o f t h e p r e - K a t n n a G O M - w i d e averaged 100-year s i g n i f i c a n t w a v e h e i g h t is 12.3 m versus 14.7 n i f o r t h e p o s t - K a t r i n a s i t e - s p e c i f i c e s t i m a t e , a 20 p e r c e n t increase. N o t e : these e s t i m a t e s p e r t a i n t o t h e f i n d i n g s f r o m t h e p r e s e n t s o m e w h a t s i m p l i f i e d analysis. A m o r e rigorous s t a t i s t i c a l e x t r a p o l a t i o n m e t h o d s u p p o r t s a p o s t - K a t r i n a s i t e - s p e c i f i c esti-m a t e o f 15.2 esti-m , w h e r e a s A P I B u l l e t i n 2 I N T - M E T [5] r e c o m m e n d s a v a l u e o f 15.8 m be u s e d f o r t h e C e n t r a l R e g i o n o f t h e G O M . F i g u r e 1 i l l u s t r a t e s t h e e v o l u t i o n o f d e s i g n w a v e h e i g h t f o r t h e p r e K a t r i n a ( G U M S H O E ) a n d p o s t
-K a t r i n a ( G O M O S ) era. T h e circles (triangles) i n d i c a t e t h e m a x i m u m s i g n i f i c a n t w a v e h e i g h t s o b s e r v e d d u r i n g a n y o f t h e s t o r m s i n c l u d e d i n t h e G U M S H O E ( G O M O S ) database. S o m e o f t h e m o r e severe h u r r i c a n e s have b e e n l a b e l l e d b y t h e i r d e s i g n a t e d n a m e s . N o t e t h e a p p a r e n t ( s e m i ) c e n t e n n i a l p e r i o d i -c i t y i n t h e o -c -c u r r e n -c e o f t h e m o s t severe h u r r i -c a n e s a l t h o u g h t h i s is a s u b j e c t o f o n g o i n g i n v e s t i g a t i o n [15]. T h e d a s h e d c u r v e r e p r e s e n t s t h e 100year s i g n i f i -c a n t w a v e h e i g h t e x t r a p o l a t e d f r o m t h e p r e - K a t r i n a G U M S H O E database b y c o n s i d e r i n g t l i e p e r i o d s t a r t i n g i n year 1900 a n d e n d i n g i n y e a r X . L i k e w i s e , t h e s o l i d c u r v e r e p r e s e n t s t h e 100-year s i g n i f i c a n t w a v e h e i g h t e x t r a p o l a t e d f r o m t h e p o s t - K a t r i n a G O M O S database b y c o n s i d e r i n g t h e p e r i o d s t a r t i n g i n year 1950 a n d e n d i n g i n year X. T h e average o f t h e p r e - K a t r i n a G U M S H O E 100-year s i g n i f i c a n t w a v e h e i g h t is a b o u t 12.3 m . T h e r u n n i n g average o f t h e p o s t - K a t r i n a G O M O S 100-year s i g n i f i c a n t w a v e h e i g h t is a b o u t 13.2 m . T h e r e l a t i v e l y s m a l l d i f f e r -ence b e t w e e n t h e t w o e s t i m a t e s c a n be e x p l a i n e d i n t e r m s o f e x c l u s i o n o f p r e - 1 9 5 0 s t o r m s i n t h e G O M O S database, s i t e - s p e c i f i c p o o l i n g i n d i e G O M O S ana-G U M S H O E , 1900-X. ana-G O M - w i d e averaging G O M O S . 1950-X, Mais-specitlc p o o l i n g 16 r 15 Z 14
>
e 'in 13 10 O 1 I c t o b t r l 9 1 6 Katrin a A O £ eptem ber 19 15 Betsy I v a 1 4' c 0 c O C O ille_ . / ? ~ ,» ^ p a l r C r o o ^ o Ü 6 ' Al drew — 0 — O 3 f>
b o Rit a A 1900 1920 1940 1960 1980 Y e a r XFig. 1 E v o l u t i o n of design wave height
2 0 0 0
Fosl-Katrina perfoimaiicc of a turret-moored F S U 395
lysis, as w e l l as i m p r o v e d q u a l i t y o f the G O M O S w a v e h i n d c a s t m o d e l e s t i m a t e s 14).
I n c l u s i o n o f t w o severe post-2000 h u r r i c a n e s ( I v a n a n d K a t r i n a r a n b o t h s t r a i g h t over the site o f M a r s TLP) increases t h e p o s t - K a t r i n a G O M O S e s t i m a t e o f the 100-year s i g n i f i c a n t w a v e h e i g h t n o t i c e a b l y t o 14.7 m . T h i s i n c r e a s e c a n be e x p l a i n e d by c o n s i d e r -i n g t h e d r a m a t -i c e f f e c t o f t h e -i n c l u s -i o n o f o n l y o n e r e a l l y e x t r e i u e s t o r m o n t h e tail o f t h e 100year l o n g -t e r m d i s -t r i b u -t i o n ( N o -t e : I v a n a n d K a -t r i n a are n o ' o u t l i e r s ' b u t r a t h e r a c t u a l o b s e r v a r i o n s ) . See, also. Fig. 2, w h e r e i t is seen t h a t t h e p o s t K a t r i n a d i s t r i b u t i o n is s i g n i f i c a n t l y m o r e s k e w e d t h a n its p r e -K a t r i n a c o u n t e r p a r t . C l e a r l y e x t r a p o l a ü n g t o r e t u r n p e r i o d s i n excess o f 100 years is a v e r y d i f f i c u l t p r o b l e m i n t h e absence o f m a n y , i f any, p o i n t s to d e f i n e t h e u p p e r t a i l o f the l o n g - t e r m d i s t r i b u t i o n . F i g u r e 3 i l l u s t r a t e s the 10, 50, a n d 90 p e r c e n t c o n -fidence b o u n d s o f s i g n i f i c a n t w a v e h e i g h t versus r e t u r n p e r i o d . 4 A P P L I C A T I O N T O A T U R R E T - M O O R E D F S U T h e 100-year l o n g - t e r m heave, r o l l , a n d p i t c h responses o f a t u r r e t - m o o r e d F S U h a v e b e e n e s t i m a t e d f o r b o t h t h e p r e - K a t r i n a ( G U M S H O E ) a n d p o s t - K a t r i n a ( G O M O S ) h u r r i c a n e h i n d c a s t da-tabases. M a i n p a r r i c u l a r s o f the F S U are s u m m a r i z e d i n T a b l e 2. T h e i n t e r n a l t u r r e t is l o c a t e d at 0.3L f o r u ' a r d o f a m i d s h i p s . T h e analysis has b e e n c o n -T a b l e 2 Particulars o f t u r r e t - m o o r e d FSU Item Storage capacin,-Length Beam Draft (empty) Uisplacemem Turret location (0.3Z.) I leave period Roll period Pitch period Value 200(1000 barrels 3 1 0 m 60 n i 8.8 m I.'iSOOOMT 93 m I I . O.s 13.8 s 10.4 s I 0.9 £• 0.8 3 0.7 « « O.f)
I
0.5I
" - I1
0.3 U 0.2 0.1 0 /•
/ / / / / M I'M (36.8%) / / / A-•
Post-Katrina (GOMOS) Pre-Kalrina (GtlMSHOL) l(. 12 14Significant wave height (m)
Fig. 2 L o n g - t e r m 100-year c u m u l a t i v e d i s t r i b u t i o n s o f s i g n i f i c a n t wave height
Po.st-Katrina CJOMOS 90% .50% 10% 1000 100
Return period (years)
396 ] J M Baar s e r v a t i v e l y p e r f o r m e d f o r t h e b a l l a s t d r a f t o n l y as t h i s t y p i c a l l y y i e l d s m a x i m u m heave, r o l l , a n d p i t c h responses [8, 9]. T h e h u r r i c a n e h i n d c a s t database c o n t a i n s f o r e a c h s t o r m a n d g r i d p o i n t a t i m e h i s t o r y o f v r i n d , w a v e , a n d c u r r e n t p a r a m e t e r s , i n c l u d i n g b o t h t h e i r m a g -n i t u d e a -n d t h e i r d i r e c t i o -n s . Let d e -n o t e t h e -n e t y a w m o m e n t w i t h respect t o t h e t u r r e t d u e to t h e c o m b i n e d a c t i o n o f m e a n w a v e d r i f t , w i n d , a n d c u r r e n t f o r c e s , a n d m o m e n t s . Necessary a n d s u f f i -c i e n t -c o n d i t i o n s f o r y a w e q u i l i b r i u m are i V ( ^ E o ) = 0 < 0 (7) T h e first c o n d i t i o n is t r i \ ' i a l , w h i l e t h e s e c o n d c o n d i t i o n ensures t h a t t h e vessel v d l l r e t u r n t o its o n g i n a l h e a d i n g f o l l o w i n g a s m a l l d i s t u r b a n c e . H o w e v e r , t h e r e m a y be m o r e tiian o n e h e a d i n g t h a t satisfies b o t h c o n d i t i o n s expressed b y e q u a t i o n (7). Stable a n d u n s t a b l e e q u i l i b r i u m h e a d i n g s c a n be d i s t i n g u i s h e d m o r e r e a d i l y b y c o n s i d e r i n g t h e y a w p o t e n t i a l e n e r g y N{<p)ó(t> ( 8 ) A n e x a m p l e is s h o w n i n F i g . 4 w h e r e i t is seen that: (a) t h e m o s t stable e q u i l i b r i u m h e a d i n g = 1 3 1 ° ( g l o b a l m i n i m u m PE); (b) t h e least stable e q u i l i b r i u m h e a d i n g = 2 3 0 ° ( l o c a l m i n i m u m PE); (c) tiie m o s t u n s t a b l e e q u i l i b r i u m h e a d i n g = 6° ( g l o b a l m a x i m u m PE); (d) t h e least u n s t a b l e e q u i l i b r i u m h e a d i n g = 195" ( l o c a l m a x i m u m PE). T h e heave ( c o m b i n e d v e r t i c a l m o t i o n at t h e t u r r e t ) , r o l l , a n d p i t c h m o t i o n s h a v e b e e n e v a l u a t e d f o r t h e r e l a t i v e w a v e h e a d i n g o b t a i n e d b y c o n s i d e r -i n g t h e vessel assumes -i t s m o s t stable e q u -i l -i b r -i u m h e a d i n g w i t h respect t o tiie p r e v a i l i n g v r i n d , w a v e , a n d c u r r e n t c o n d i t i o n s . M o t i o n s have b e e n e s t i -m a t e d u s i n g t h e l i n e a r s e a k e e p i n g p r o g r a -m S h i p s i m [16]. R o l l d a m p i n g has b e e n c o n s e r v a t i v e l y a s s u m e d t o a m o u n t t o 2 p e r c e n t o f c r i t i c a l d a m p i n g , regardless o f sea state. T h e c r i t i c a l r o l l d a m p i n g = A G M r p / E , w h e r e A = d i s p l a c e m e n t , G M = m e t a c e n t r i c h e i g h t a n d r ^ = r o l l p e r i o d . I n r e a l i t y r o l l d a m p i n g d e p e n d s o n t h e vessel b e a m a n d d r a f t , b i l g e r a d i u s , b i l g e k e e l d i m e n s i o n s , a n d m o r e o v e r , i t varies w i t i i sea state. F i g u r e 5 s h o w s a n e x a m p l e o f t h e t e m p o r a l v a r i a t i o n d u r i n g h u r r i c a n e I v a n o f t h e s i g n i f i c a n t w a v e h e i g h t , t h e r e l a t i v e w a v e h e a d i n g (90° = b e a m - o n ) , a n d t h e s h o r t - t e r m M P M r o l l a m p l i t u d e . N o t e t h a t m a x i m u m r o l l does n o t c o i n c i d e v r i t h m a x i m u m w a v e h e i g h t , b u t r a t h e r several h o u r s later w h e n t h e r e l a t i v e w a v e d i r e c t i o n is m o r e u n f a v o u r a b l e . F i g u r e 6 s h o w s t h e p r e - a n d p o s t - K a t r i n a 100-year l o n g - t e r m d i s t r i b u t i o n s o f r o l l a m p l i t u d e ; t h e 100-year p r e - K a t r i n a r o l l m o t i o n e s t i m a t e a m o u n t s t o 7.3° versus 2 1 ° f o r t h e p o s t -K a t r i n a analysis.
T h e 100-year m o s t p r o b a b l e m a x i m u m net heave, r o l l , a n d p i t c h m o t i o n s are p r e s e n t e d i n T a b l e 1. T h e p o s t - K a t r i n a e s t i m a t e s o f h e a v e a n d p h c h m o t i o n are 3 5 - 4 0 per cent h i g h e r t h a n t h e i r p r e - K a t r i n a c o u n t e r p a r t s . T h e p o s t - K a t r i n a e s t i m a t e o f t h e 100year M P M r o f i m o t i o n is a l m o s t t h r e e t i m e s t h e p r e -K a t r i n a e s t i m a t e . T h i s finding is p o s s i b l y s o m e w h a t exaggerated b y t h e f a c t t h a t a c o n s t a n t l o w r o l l d a m p i n g l e v e l has b e e n a s s u m e d i n t h e p r e s e n t analysis (see d i s c u s s i o n i n p r e v i o u s p a r a g r a p h ) w h e r e a s i n r e a l i t y r o l l d a m p i n g is d e p e n d e n t o n • Turrcl moment (tr-m) • Yaw potential energ>' (tf-m-rad) Fig. 4 0 45 90 135 180 225 270 315 360
vessel heading (degrees)
Weathervaning e q u U i b r i u m based o n m i n i m u m yaw p o t e n t i a l energy'
Post-Katrina performance of a turret-moored FSU 397 32 28 24 20 16 12 8 4 0 I l M 11
' \
1 1 360 315 270 225 180 135 ')(! 45 0 Signiricanl wave hcii;hl (m) M P M roll nniplittide (degrees) Relative wave licadiiig(degrees) I tK) 150 200 Hour 250Fig. 5 Variation o f wave height, wave iieading, a n d r o l l d t i r i n g h u r r i c a n e I v a n (2004)
'C; 0.8 'Z 0.7 0.6 c . > 0.4 s 0.3 u 0.2 0,1 0 / 1 1 1 1 t / — MPM (36.8%) / / / / / I're-Kalrina (GUMSHOE) Pcsl-Kalrina (GOMUS) 0 10 20 30
Roil amplitude (degrees)
Fig. 6 L o n g - t e r m 100-year d i s t r i b u t i o n o f roll m o t i o n
sea state a n d l i k e l y t o i n c r e a s e as t h e sea state increases. H o w e v e r , i n c l u s i o n o f r e c e n t i n t e n s e h u r r i c a n e s has a d r a m a t i c e f f e c t o n t h e e s t i m a t e d l o n g - t e r m r o l l r e s p o n s e , a n d t h e r e seems t o be n o s i m p l e w a y t o c a p t u r e t h i s b y a n y o t h e r i n e a n s t h a n p e r f o r m i n g a c o m p r e h e n s i v e l o n g - t e r m r e s p o n s e analysis. T a b l e 3 p r e s e n t s t y p i c a l RBDSs t h a t are m o s t l i k e l y t o r e p r o d u c e t h e 100-year l o n g - t e r m heave, r o l l , a n d p i t c h i n o t i o n s from T a b l e 1 i n a s i m p l i f i e d s h o r t -t e r m r e s p o n s e analysis. T h e i n -t e n s i -t y o f w i n d , w a v e s . T a b l e 3 Typical RBDSs f o r h e a v e / r o l l / p i t c h m o t i o n I t e m Pre-Katrina Post-Katrina
Hindcasl GUMSHOE GOMOS
Period 1900-!j5 1950-200.5
Sites GOM-wide Mars/Ursa
Significant wave heiglit (m) 11.8 12.4
Peak wave period (s) 13.8 14.6
W i n d sj)eed (m/s) 43 35
Current speed (m/s) 1.9 1.1
Wind-waves heading GO' 120
Current-waves heading 0 0 a n d c u r r e n t f o r t h e p r e - a n d p o s t - K a t r i n a RBDS events is c o m p a r a b l e . I n b o t h cases c u r r e n t a n d w a v e s are c o l l i n e a r . F o r t h e p r e - K a t r i n a R B D S e v e n t t h e r e l a f i v e w i n d - w a v e h e a d i n g is 60", w h e r e a s f o r t h e p o s t - K a t r i n a RBDS e v e n t t h e r e c o m m e n d e d w i n d - w a v e h e a d i n g is 1 2 0 \ I t is e m p h a s i z e d t h a t t h e RBDS events are p a r t i c u l a r t o t h e p r e s e n t i y c o n s i d e r e d FSU a n d its m a i n p a r t i c u l a r s m o t i o n c h a r a c t e r i s t i c s , as w e l l as t h e s t o r m s a n d g r i d p o i n t s i n c l u d e d i n t h e h u r r i c a n e h i n d c a s t databases. Sig-n i f i c a Sig-n t d e p a r t u r e s i Sig-n a Sig-n y o f t h e s e a s s u m p t i o Sig-n s m a y r e q u i r e t h a t t h e analysis be r e p e a t e d a n d r e s u l t i n d i f f e r e n t findings. 5 C O N C L U S I O N S M o d e r n s t o r i n h i n d c a s t databases p r o v i d e a c o m -p r e h e n s i v e l o n g - t e r m d e f i n i t i o n o f t h e c o n c u r r e n t t e m p o r a l v a r i a t i o n s o f w i n d , w a v e , a n d c u r r e n t c o n d i t i o n s at a g i v e n site. T h e y c a n be u s e d t o
398 J J M Baar p e r f o r m a LTRA a n d e s t i m a t e t l i e a c t u a l T-year r e t u r n p e r i o d response. T h u s , LTRA is a p o w e r f u l d e s i g n t o o l t h a t p r o v i d e s v a l u a b l e i n s i g h t s i n t o t h e d y n a m i c c h a r a c t e r i s t i c s o f a f l o a t i n g p l a t f o r m , its s e n s i t i v i t y to n o n - c o l l i n e a r w i n d , wave, a n d c u r r e n t c o n d i t i o n s , a n d its r o b u s t n e s s i n s u r v i v a l c o n d i t i o n s . T h e v e r y r a r e o c c u r r e n c e o f e x t r e m e h u r r i c a n e s i n t h e G O M m a k e s i t v e r y c h a l l e n g i n g t o p r o d u c e c o n s i s t e n t a n d s t a t i s t i c a l l y r e l i a b l e e s t i m a t e s o f l o n g t e r m e s t i m a t e s o f m e t o c e a n p a r a m e t e r s a n d p l a t -f o r m responses, especially -f o r l o n g r e t u r n p e r i o d s ( i n excess o f 100 years, say). T h i s c a n s i m p l y be e x p l a i n e d b y t h e p r e s e n c e o f j u s t a f e w ( i f any) data p o i n t s t o d e f i n e t h e u p p e r t a i l o f the e x t r a p o l a t e d l o n g - t e r m d i s t r i b u t i o n . T h e m e t h o d o l o g y has b e e n i l l u s t r a t e d b y c o n s i d -e r i n g t h -e n -e t h -e a v -e , r o l l , a n d p i t c h m o t i o n s o f a t u r r e t - m o o r e d F S U i n the G O M . N o n - c o l l i n e a r c o m b i n a t i o n s o f w i n d , waves, a n d c u r r e n t c o i u -m o n l y o c c u r d u r i n g t h e passage o f h u r r i c a n e s a n d c o n s e q u e n t l y the w e a t h e r v a n i n g t u r r e t - m o o r e d ves-sel m a y b e c o m e e x p o s e d t o s i g n i f i c a n t b e a m seas c o n d i f i o n s , r e s u l t i n g i n large r o l l a m p l i t u d e s . For t h i s s y s t e m a c o m p r e h e n s i v e LTRA appears essential to e n s u r e t h a t p r o p e r e s t i m a t e s o f t h e a c t u a l 100-year d e s i g n responses c a n b e o b t a i n e d .
A C K N O W L E D G E M E N T
T h i s p a p e r has b e e n wr i t t e n o n t h e o c c a s i o n o f t h e r e t i r e m e n t o f tire a u t h o r ' s P h . D . advisor, Prof. W . G e r a i n t Price. I t is also d e d i c a t e d t o t h e late P r o f " R E D " B i s h o p . T h e a u t h o r is p r i v i l e g e d t h a t Price a n d B i s h o p h a v e b e e n his teachers a n d is g r a t e f u l f o r m a i r y fine m e m o r i e s d u r i n g t h e m i d - 1 9 8 0 s at B r u n e i U n i v e r s i t y , U x b r i d g e , U K . P e r m i s s i o n f r o m Shell I n t e r n a t i o n a l E x p l o r a t i o n a n d P r o d u c t i o n , I n c . , H o u s t o n , t o p u b l i s h t h i s p a p e r is a c k n o w l e d g e d . Results p r e s e n t e d i n t h i s p a p e r are t h e f r u i t s o f Shell i n t e r n a l R & D e f f o r t s a n d are t y p i c a l o f a g e n e r i c FSU/FPSO s y s t e m .
© A u d i o r 2009
R E F E R E N C E S
1 St, Denis, M . a n d Pierson, W. J. O n the m o t i o n s of ships i n c o n f u s e d seas. Trans. SNAME, 1953, 78, 280-357.
2 Price, W. G . and Bishop, R. E . D. Probabilistic theory ofsliip dynamics, 1974 ( C h a p m a n and H a l l , L o n d o n ) .
3 Oceanvveathcr. Gulf o f M e x i c o hindcast o f oceano-graphical extremes ( G U M S H O E ) , Final report, Oceanweather, Inc., H o u s t o n , Texas, 1990.
4 Oceanweather. GOMOS — USA G u l f o f M e x i c o oceanographic study, Oceanweather, Inc., Hous-ton, Texas, 2006.
5 A m e n c a n P e t r o l e u m I n s d t u t e . Bulletin 2 I N T - M E T : i n t e r i m guidance o n h u r n c a n e s i n the Gulf o f Mexico, A m e r i c a n Petroleum I n s t i t u t e , W a s h i n g -ton, District of C o l u m b i a , 2007.
6 Forristall, G . Z . , L a i r a b e e , R. D., and Mercier, R. S. C o m b i n e d oceanographic criteria f o r deepwater sttuctures i n the G u l f o f M e x i c o . I n Proceedings o f the 23rd A n n u a l Offshore technology conference, H o u s t o n , Texas, 1991, paper no. 6541.
7 Baar, J . J . M., Howell, C . T . , Mercier, R. S., a n d Rodenbusch, G . L o n g - t e r m response o f tension leg p l a t f o r m s . I n Proceedings of d i e Seventh I n t e r n a -tional Conference o n Offshore and polar engineer-ing, H o n o l u l u , H a w a i i , 1997.
8 Baar, ] . J. M., Heyl, C . N., and Rodenbusch, G . E x u e m e responses o f t u r r e t m o o r e d tankers. I n Proceedings of the 3 2 n d Offshore technology con-ference, H o u s t o n , Texas, 2000, paper no. OTC
12147.
9 Baar, J. J. M . , Petruska, D . , Driver, D . , T h u r m o n d , B., and Schott, W. E . Response-based design c a t e n a f o r a t u r r e t - m o o r e d FSO i n the G u l f o f Mexico. I n Proceedings of the Eleventh O f f s h o r e S y m p o s i u m , H o u s t o n , Texas, 2002, S N A M E Texas Section.
10 Baar, J. J. M . FPSO's a n d the revised G u l f o f M e x i c o e n v i r o n m e n t . ASME F o r u m o n O p p o r t u n i ü e s f o r FPSO's i n the G u l f o f M e x i c o . H o u s t o n , Texas, 26¬ 27 M a r c h 2008.
11 Ward, E . G . , B o r g m a n , L . E . , and C a r d o n e , V . J . Statistics o f h u r r i c a n e waves i n the G u l f o f M e x i c o . I n Proceedings of the T e n t h A n n u a l Offshore technology conference, H o u s t o n , Texas, 1978, paper no. 3229.
12 G u m b e l , E . J . Statistics of extremes, 1958 (Dover Publications, N e w York).
13 Heideman, J. C . and Mitcltell, D . A . G r i d p o i n t p o o l i n g i n extreme value analysis o f h u r r i c a n e hindcast data. ASCE J. Waterway Port, Coastal Engng, 2009.
14 Cooper, C . K . A p r e l i m i n a r y case f o r the existence of h u r r i c a n e alleys i n d i e Gulf o f M e x i c o . I n Proceedings of tiie 2 4 t h Offshore technology con-ference, H o u s t o n , Texas, 1992, paper no. OTC 6831. 15 E m a n u e l , K . Divine wind - the liistory and science
of hurricanes, 2005 ( O x f o r d U n i v e r s i t y Press). 16 H a r t m a n , R. SInpsim user manual, 2005 (SeaSoft
Systems, Santa Barbara, California).
