B E R I C H T E Aus DEM
INSTITUT FÜR MEERESKUNDE AN OER
CHRISrn^ALBRECHTS-UNIVERSITTTT KIEL
Nr . 1 1 6
OBJECTIVE ANALYSIS OF HYDROGRAPHIC DATA SETS FROM MESOSCALE SURVEYS
VON
W. HILLER & R. H. KÄSE
Kopien d i e s e r A r b e i t können bezogen werden von:
I n s t i t u t für Meereskunde an der Universität K i e l Abt. Theoretische Ozeanographie
Düsternbrooker Weg 20 23/ K i e l - FRG -
ISSN O 3 4 1 - 8 5 6 I -
Abstract
The optimal i n t e r p o l a t i o n techniques reviewed i n t h i s report have been i l l u s t r a t e d with examples o f t h e i r a p p l i c a t i o n t o the "Poseidon 86" data set i n order t o give members o f the
"Warmwassersphäre"-Research Programme, who are l i k e l y t o
use the objective analysis program package, some i n s i g h t s i n t o both p r a c t i c a l and t h e o r e t i c a l aspects o f t h i s estimation t e c h - nique i n connection with mesoscale dynamics.
The presented examples include estimation o f s c a l a r - as w e l l as v e c t o r - f i e l d s . S p e c i a l emphasis has been given t o present an approach f o r the estimation o f the s t a t i s t i c s o f the observed stochastic processes- i . e . s p a t i a l mean and covariance f u n c t i o n - i n the case where only one r e a l i s a t i o n i s a v a i l a b l e .
Zusammenfassung
Anhand der Analyse des "Poseidon 86" Datensatzes werden B e i s p i e l e für d i e Anwendung der i n diesem Bericht beschriebenen optimalen Interpolâtionstechnik gegeben. Z i e l dabei i s t es, den M i t g l i e d e r n des SFB 133 (Warmwassersphäre des A t l a n t i k s ) , welche das zugehörige
"Objektive Analyse"-Programmpaket benutzen w o l l e n , e i n i g e E i n b l i c k e i n sowohl praktische a l s auch theoretische Aspekte d i e s e r Interpo- l a t i o n s t e c h n i k zu geben - soweit s i e mit der Analyse mesoskaliger Prozesse zusammenhängen.
Es werden B e i s p i e l e der Schätzung sowohl von S k a l a r - a l s auch Vektor- f e l d e r n gegeben. Weiter werden d i e s t a t i s t i s c h e n Verfahren beschrie- ben, welche dazu dienten, für den F a l l nur e i n e r R e a l i s i e r u n g d i e räumlichen Mittelwerte und d i e Kovarianzfunkt i o n der beobachteten stochastischen Prozesse zu schätzen.
CONTENTS page
1. INTRODUCTION 1 2. OBJECTIVE ANALYSIS OF SCALAR FIELDS 7
3. STATISTICAL PREPROCESSING OF SCALAR DATA SETS 11
3.1. ESTIMATION OF THE MEAN F I E L D 11 3.2. SPATIAL COVARIANCE FUNCTIONS 13
3.2.1. ESTIMATION OF RAW COVARIANCES 14 3.2.2. NONLINEAR FITTING OF MODEL COVARIANCES 18
3.2.3. STATISTICS OF THE MESOSCALE EDDY FIELD 23
4. OBJECTIVE ANALYSIS OF VECTOR FIELDS 2 8 4.1. ANALYSIS OF QUASI-EULERIAN VELOCITY FIELDS 40
5. DISCUSSION 43 6. REFERENCES 45 7. FIGURES 46
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1 . INTRODUCTION
I n t h e framework o f t h e "Warmwassersphäre"-research p r o - gramme o f t h e I n s t i t u t für M e e r e s k u n d e h y d r o g r a p h i e map- p i n g e x p e r i m e n t s a r e b e i n g p l a n n e d a n d c a r r i e d o u t , r e - q u i r i n g s a m p l i n g schemes w h i c h w i l l g i v e i n f o r m a t i o n w i t h t h e f e w e s t p o s s i b l e s t a t i o n d e n s i t y . One i s g e n e r a l l y
a i m i n g a t a p r o p e r b a l a n c e b e t w e e n t h e c o s t s o f an e x p e r i - ment and t h e i n f o r m a t i o n g a i n e d by i t , t r y i n g t o a v o i d r e - d u n d a n t measurements as w e l l as h a v i n g a s p a r s e s t a t i o n c o v e r a g e w h i c h w i l l y i e l d an u n r e s o l v e d f i e l d . S i n c e t h e use o f o p t i m a l e s t i m a t i o n t e c h n i q u e s i s o f common i n t e r e s t f o r s e v e r a l r e s e a r c h g r o u p s , we p r e s e n t h e r e a comprehen- s i v e d e s c r i p t i o n o f t h e o b j e c t i v e a n a l y s i s method w i t h em- p h a s i s on t h e e s t i m a t i o n o f s c a l a r f i e l d s . The method i s a p p l i e d t o a d a t a s e t o b t a i n e d d u r i n g a POSEIDON c r u i s e i n t h e n o r t h e r n C a n a r y b a s i n (Käse & R a t h l e v , 1 9 8 2 ) . We w i l l a l s o g i v e a b r i e f summary o f o p t i m a l v e c t o r e s t i m a t i o n t e c h n i q u e , d i s c u s s t h e a s s u m p t i o n s i n v o l v e d a n d g i v e an e x a m p l e o f a n a l y s i n g a q u a s i - E u l e r i a n v e l o c i t y f i e l d de- r i v e d f r o m s a t e l l i t e - t r a c k e d d r i f t buoy o b s e r v a t i o n s . One o f t h e p r i m a r y o b j e c t i v e s o f t h e POSEIDON 8 6- c r u i s e i n s p r i n g 1 9 8 2 was t o map m e s o s c a l e d e n s i t y a n d g e o s t r o p h i c c u r r e n t f i e l d s o v e r an a r e a o f 5 0 0 * 5 0 0 km i n t h e C a n a r y b a s i n . As t h i s e x p e r i m e n t was d e s i g n e d t o p r o d u c e s y n o p t i c maps o f m e s o s c a l e f i e l d s as w e l l as t o d e t e r m i n e m a j o r t e r m s i n t h e l o c a l h e a t b a l a n c e e q u a t i o n , i . e . mean a d v e c - t i o n t e r m and d i v e r g e n c e o f t h e eddy f l u x , an a c c u r a t e m a p p i n g t e c h n i q u e was r e q u i r e d . F o r t h e i n t e r p r e t a t i o n o f q u a s i - s y n o p t i c d a t a s e t s o b t a i n e d f r o m an i r r e g u l a r l y s p a c e d o b s e r v a t i o n a l a r r a y , t h e t e c h n i q u e o f " o b j e c t i v e a n a l y s i s " has b e e n w i d e l y u s e d i n r e c e n t y e a r s ( B r e t h e r t o n e t a l . , 1 9 7 6 ; B r e t h e r t o n e t a l . , 1 9 8 0 ; S a r m i e n t o , 1 9 8 2 , e t c . ) .
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B a s e d on a f u n d a m e n t a l r e s u l t i n e s t i m a t i o n t h e o r y , t h e Gauss-Markov t h e o r e m ( L i e b e l t , 1 9 6 7 ) , t h e o b j e c t i v e a n a l y - s i s t e c h n i q u e i s an o p t i m a l i n t e r p o l a t i o n p r o c e d u r e i n t h a t s e n s e t h a t among l i n e a r e s t i m a t o r s on t h e a v e r a g e t h i s one h a s t h e m i n i m a l l e a s t s q u a r e e r r o r . I t a l s o y i e l d s an e s t i m a t e o f t h e r e s i d u a l u n c e r t a i n t i e s i n t h e i n t e r p o l a t e d v a l u e s .
As t h e e r r o r maps o n l y depend on t h e s t a t i s t i c s o f t h e f i e l d , t h e n o i s e l e v e l and t h e l o c a t i o n s o f t h e o b s e r v a - t i o n a l p o i n t s , t h e y c a n be c a l c u l a t e d a p r i o r i f o r d i f - f e r e n t a r r a y d e s i g n s w i t h o u t r e f e r e n c e t o any p a r t i c u l a r d a t a s e t . T h u s , i t i s p o s s i b l e t o m i n i m i z e t h e e x p e c t e d i n t e r p o l a t i o n e r r o r s p r o v i d e d t h e s t a t i s t i c s o f t h e f i e l d t o be mapped have a l r e a d y been d e t e r m i n e d .
Due t o t h e f a c t t h a t o u r o p t i m a l e s t i m a t o r i s l i n e a r , i . e . a w e i g h t e d sum o f a l l o b s e r v a t i o n s , t h e o b j e c t i v e a n a l y s i s p r o c e d u r e w i l l p r o d u c e a smoothed v e r s i o n o f t h e o r i g i n a l f i e l d w i t h a t e n d e n c y t o u n d e r e s t i m a t e t h e t r u e f i e l d b e - cause o f t h e s p e c i f i c a s s u m p t i o n s i n v o l v e d i n o u r t r e a t - ment o f measurement n o i s e and s m a l l s c a l e s i g n a l s u n r e s o l - ved by t h e a r r a y .
The a p p l i c a t i o n o f t h e G a u s s - M a r k o v t h e o r e m i s s t r a i g h t - f o r w a r d , p r o v i d e d t h e f i r s t and s e c o n d moments o f t h e s t o - c h a s t i c p r o c e s s t o be e s t i m a t e d a r e known, and f u r t h e r m o r e t h a t t h e s e c o n d moment m a t r i x o f a l l o b s e r v a t i o n s f u l f i l s t h e c o n d i t i o n o f b e i n g p o s i t i v e d e f i n i t e . However, t h e d e - t e r m i n a t i o n o f t h e s e moments f r o m d a t a c a n p r o v e t o be a d i f f i c u l t p r o b l e m , e s p e c i a l l y when t h e j o i n t p r o b a b i l i t y f u n c t i o n o f t h e p r o c e s s v a r i e s w i t h r e s p e c t t o s p a c e and
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t i m e , i . e . i n s t a t i o n a r i t y o r i n h o m o g e n e i t y o f t h e o b s e r v e d f i e l d . T h e r e f o r e t h e most f u n d a m e n t a l a s s u m p t i o n on w h i c h t h e o b j e c t i v e a n a l y s i s method u s u a l l y i s b a s e d , i s t h e s t a t i o n a r i t y o f t h e s t a t i s t i c s o f t h e f i e l d t o be mapped.
The a s s e r t i o n t h a t t h e s t a t i s t i c s a r e s t a t i o n a r y c a n n e v e r be deduced f r o m o b s e r v a t i o n s , b e c a u s e t h i s w o u l d i n v o l v e t h e v e r i f i c a t i o n t h a t a l l s t a t i s t i c p r o p e r t i e s o f t h e s t o - c h a s t i c p r o c e s s a r e i n v a r i a n t w i t h r e s p e c t t o t i m e and s p a c e t r a n s l a t i o n s . I t c a n o n l y be p o s t u l a t e d as a w o r k i n g h y p o t h e s i s . T e s t s s h o u l d be made t o d e c i d e w h e t h e r t h i s a s s u m p t i o n i s s i g n i f i c a n t l y i n c o n s i s t e n t w i t h o b s e r v a t i o n s . F i e l d o b s e r v a t i o n s o f t e n show t r e n d s i n t h e mean w i t h o u t e x h i b i t i n g any f o r m o f n o n - s t a t i o n a r i t y o f more c o m p l i c a t e d c h a r a c t e r . I f t h e r e i s o n l y one r e a l i z a t i o n a v a i l a b l e , t h i s p r o b l e m c a n n o t be s o l v e d by t a k i n g t h e e n s e m b l e a v e r a g e . I n t h i s c a s e , no p r o g r e s s c a n be made w i t h o u t r e f e r e n c e t o some a p r i o r i p r e j u d i c e . I n t h e p r e s e n t a n a l y s i s o f t h e me- s o s c a l e f i e l d s measured d u r i n g "POSEIDON 8 6 " , t h e assump- t i o n was made t h a t t h e h o r i z o n t a l t r e n d i n t h e mean c a n be a p p r o x i m a t e d by a t w o - d i m e n s i o n a l l i n e a r f u n c t i o n f i t t e d t o t h e d a t a u s i n g a m u l t i p l e r e g r e s s i o n scheme. I n s p e c t i o n o f h i s t o r i c a l d a t a p r o v e d t h a t a l i n e a r a p p r o x i m a t i o n o f t h e mean f i e l d , w h i c h i s p a r t o f t h e s u b t r o p i c a l g y r e , w o u l d be a p p r o p r i a t e . The r e m o v a l o f a l i n e a r t r e n d i s an i m p o r t a n t s t e p i n t h e p r o c e s s o f e s t i m a t i o n o f t h e f i e l d s t a t i s t i c s . O t h e r w i s e , due t o t h e f a c t t h a t t h e o b s e r v a - t i o n a l a r r a y h a s f i n i t e l e n g t h , n e g l e c t i o n o f t h e t r e n d w o u l d i n c o n s e q u e n c e r e d i s t r i b u t e power f r o m l a r g e s c a l e s t h r o u g h t h e w h o l e wavenumber s p a c e .
The d e t e r m i n a t i o n o f t h e c o v a r i a n c e f u n c t i o n was a m a j o r t a s k i n p r e p a r i n g t h e "POSEIDON 8 6 " d a t a s e t f o r o b j e c t i v e
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a n a l y s i s . The h i s t o r i c a l d a t a a v a i l a b l e f o r t h e r e g i o n o f i n t e r e s t (25° W - 18°W / 31°N - 36°Nj a r e n o t s u f f i c i e n t t o y i e l d s i g n i f i c a n t e s t i m a t e s f o r t h e c o v a r i a n c e s . The e s t i - m a t i o n o f t w o - p o i n t s t a t i s t i c s f r o m u n e q u a l l y s p a c e d d a t a i s a d i f f i c u l t p r o b l e m ( s e e B r e t h e r t o n , 1 9 7 6 ; B r e t h e r t o n ,
1 9 8 0 ) . The l o c a t i o n s o f d a t a p o i n t s were c h o s e n more o r l e s s i r r e g u l a r l y f r o m a 3 0 x 30 nm g r i d ( s e e f i g . 1 ). As a c o n s e q u e n c e t h e e f f e c t i v e a l i a s c l a s s o f u n m e a s u r a b l e s p e c t r a l f e a t u r e s has a more c o m p l i c a t e d s t r u c t u r e t h a n i n t h e c a s e o f u n i f o r m l y s p a c e d a r r a y s where a f i x e d N y q u i s t wave number e x i s t s . However, i n p r a c t i c e o t h e r a s p e c t s s u c h as l o g i s t i c s , s y n o p t i c d e c a y o f t h e f i e l d and t h e
amount o f a v a i l a b l e s h i p t i m e have t o be t a k e n i n t o a c c o u n t ; and o n l y i f t h e c h a r a c t e r i s t i c s o f t h e wave number s p e c t r u m a r e known w i t h s u f f i c i e n t p r e c i s i o n a p r i o r i , an o p t i m a l e f f i c i e n t a r r a y can be p l a n n e d w h i c h n o t o n l y s a t i s f i e s t h e r e q u i r e m e n t s f o r s y n o p t i c m a p p i n g , b u t a l s o g i v e s s u f f i c i e n t i n f o r m a t i o n f o r t h e e s t i m a t i o n o f t h e c o v a r i a n c e f u n c t i o n o r e q u i v a l e n t l y o f t h e wave number s p e c t r u m . As t h i s was n o t t h e c a s e w i t h "POSEIDON 8 6 " , o u r main o b j e c t i v e was t o a i m a t an e c o n o m i c s a m p l i n g scheme w h i c h p e r m i t s q u a s i s y n o p t i c mapping. O n - l i n e d a t a p r o c e s s i n g f a c i l i t i e s on b o a r d e n a b l e d t h e c o n c e n t r a t i o n o f d a t a p o i n t l o c a t i o n s i n d y n a m i c a l l y i n - t e r e s t i n g r e g i o n s and t h e use o f a w i d e r and t i m e - s a v i n g s a m p l i n g scheme i n r e g i o n s o f l e s s a c t i v i t i e s .
U l t i m a t e l y , a l l c o v a r i a n c e e s t i m a t i o n p r o c e d u r e s a r e b a s e d on raw e s t i m a t e s o f p a i r - w i s e c o v a r i a n c e s a v e r a g e d o v e r a number o f r e a l i z a t i o n s . G i v e n a l a r g e e n s e m b l e o f r e a l i z a - t i o n s , t h e e n s e m b l e a v e r a g e o f t h e raw c o v a r i a n c e s s h o u l d c o n v e r g e t o t h e t r u e c o v a r i a n c e f u n c t i o n . However, i f o n l y
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one r e a l i z a t i o n i s a v a i l a b l e , t h e raw e s t i m a t e s s u f f e r s t r o n g l y f r o m a random s c a t t e r w h i c h l i m i t s t h e i n f e r e n c e s t h a t c a n be drawn f r o m them. I n o r d e r t o r e d u c e t h e n o i s e o f t h e raw e s t i m a t e s , p a i r s w i t h s i m i l a r d i s t a n c e v e c t o r r o r - r ( a s s u m p t i o n o f s t a t i o n a r i t y ) o r | r | ( a s s u m p t i o n o f i s o t r o p y ) were a v e r a g e d . The t w o - d i m e n s i o n a l s p a t i a l raw c o v a r i a n c e s d e r i v e d f r o m t h e "POSEIDON 8 6 " d a t a s e t seem t o i n d i c a t e n o n i s o t r o p i c f e a t u r e s i n t h e d a t a and an e a s t - w e s t o s c i l l a t i o n o f 5 0 0 km w a v e l e n g t h .
An i t e r a t i v e n o n l i n e a r p a r a m e t e r f i t t i n g p r o c e d u r e ( G a u s s - Newton method) was t h e n a p p l i e d t o a d a p t t h e s e raw c o v a r i - a n c e s t o one o f a f a m i l y o f model c o v a r i a n c e f u n c t i o n s
(METZLER e t a l . , 1 9 71* ) . A s i m i l a r p r a c t i c a l a p p r o a c h i s w i d e l y u s e d by m e t e o r o l o g i s t s ( ' b i n method', J u l i a n and
C l i n e , 1 9 7 4 ) i n e s t i m a t i n g t h e s p e c t r a l d e n s i t y E ( K ) w h i c h u n d e r t h e a s s u m p t i o n o f s p a t i a l s t a t i o n a r i t y i s e q u i v a l e n t t o t h e s p e c i f i c a t i o n o f t h e s p a t i a l c o v a r i a n c e f u n c t i o n .
( F o r a more d e t a i l e d d i s c u s s i o n see B r e t h e r t o n and Mc W i l l i a m s 1 9 8 0 , § 3 ) .
As t h e r e was o n l y one r e a l i z a t i o n a v a i l a b l e , t h e n o i s e i n - h e r e n t i n t h e c o v a r i a n c e raw e s t i m a t e s r e m i n d e d us o f t h e d a n g e r o f d r a w i n g s t a t i s t i c a l i n f e r e n c e s f r o m t o o l i t t l e d a t a . C o n s e q u e n t l y t h e model c l a s s c h o s e n f o r t h e f i n a l a n a l y s i s was a G a u s s i a n f u n c t i o n w h i c h s u p p r e s s e s t h e o s - c i l l a t i o n s b u t t a k e s a c c o u n t o f t h e n o n i s o t r o p i c b e h a v i o u r o f t h e raw c o v a r i a n c e s , e s p e c i a l l y o f t h e r a p i d d e c r e a s e o f c o r r e l a t i o n i n t h e e a s t - w e s t d i r e c t i o n .
N e x t t o t h e t a s k o f s y n o p t i c mapping o f m e s o s c a l e f i e l d s one o f t h e main o b j e c t i v e s i n t h e g a t h e r i n g o f t h e "POSEI- DON 86" d a t a s e t was t h e e s t i m a t i o n o f t e r m s i n t h e l o c a l
- 6 -
t e m p e r a t u r e c o n s e r v a t i o n e q u a t i o n . T h i s h a s b e e n d e s c r i b e d e l s e w h e r e (Käse e t a l . , 1 9 8 3 ) , so we w i l l o n l y o u t l i n e t h e p r o c e d u r e s a d o p t e d so f a r as t h e e s t i m a t i o n o f t h e s t a t i s - t i c s o f t h e o b s e r v e d m e s o s c a l e eddy f i e l d i s c o n c e r n e d . The g e o p o t e n t i a l anomaly f i e l d shown i n f i g . 2a r e v e a l s t h a t t h r e e s e p a r a t e h o r i z o n t a l s c a l e s a r e p r e s e n t i n t h e o b s e r v a t i o n a l a r e a . F i r s t t h e r e i s t h e b o x - w i d e s c a l e a l - r e a d y m e n t i o n e d t h a t i s r e l a t e d t o t h e mean o r g y r e c i r c u - l a t i o n . Second t h e r e i s a s m a l l e r s c a l e a s s o c i a t e d w i t h a m e a n d e r i n g f l o w . T h i r d t h e r e i s t h e m e s o s c a l e eddy f i e l d . I n f i g . 2b t h e anomaly f i e l d a f t e r r e m o v a l o f t h e l i n e a r t r e n d i s shown where a l a r g e c y c l o n i c anomaly i n t h e c e n t r e and an a n t i c y c l o n i c anomaly i n t h e e a s t a r e f o u n d due t o t h e m e a n d e r i n g f l o w . T h i s e n e r g e t i c f e a t u r e d o m i - n a t e s much o f t h e f l o w and p r o p e r t y d i s t r i b u t i o n and must be removed t o r e v e a l t h e u n d e r l y i n g s m a l l e r s c a l e v a r i a b i - l i t y , f r o m w h i c h t h e eddy f l u x d i v e r g e n c e i s d e r i v e d . Due t o t h e o r e t i c a l r e a s o n s and t h e o b s e r v a t i o n o f numerous d r i f t buoys ( f i g . 13) we h y p o t h e s i z e d t h e e x i s t e n c e o f a R o s s b y wave c o n t r i b u t i o n (Käse e t a l . , 1 9 8 3 ) . A f t e r s u b - t r a c t i o n o f t h e s p a t i a l l i n e a r t r e n d a s i m p l e f i r s t mode R o s s b y wave model
i f iR( x ) = P • s i n ( n y - 4^) c o s (KX - 4>2)
was f i t t e d . The c o m p o s i t e mean f i e l d i s d i s p l a y e d i n f i g . 2c . Removal o f t h e wave f i e l d r e d u c e s t h e m e a n d e r - s c a l e s t r u c t u r e c o m p l e t e l y and r e v e a l s a m e s o s c a l e eddy f i e l d ( f i g . 2 d ) . From t h i s f i e l d raw c o v a r i a n c e s were d e r i v e d and w i t h t h e a i d o f t h e Gauss-Newton method a d a p t e d t o one o f a f a m i l y o f model c o v a r i a n c e f u n c t i o n s . I n t h i s c a s e a G a u s s i a n i s o t r o p i c f u n c t i o n p r o v e d t o be t h e most s u i t a b l e one.
2 . OBJECTIVE ANALYSIS OF SCALAR FIELDS
The o b j e c t i v e a n a l y s i s t e c h n i q u e s p r e s e n t e d h e r e have been de- v e l o p e d f o r b o t h m e t e o r o l o g i c a l ( G a n d i n , 1 9 6 5 ) and océanogra- p h i e a p p l i c a t i o n s ( B r e t h e r t o n e t a l . , 1 9 7 6 ; B r e t h e r t o n , 1 9 8 0 ) i n t h e p a s t . U l t i m a t e l y , t h e y a l l d a t e b a c k t o t h e d a y s o f Gauss i n b e i n g a p p l i c a t i o n s o f t h e f u n d a m e n t a l G a u s s - M a r k o v t h e o r e m ( L i e b e l t , 1 9 6 7 ) . We s h a l l g i v e a b r i e f summary o f t h i s t e c h n i q u e h e r e t o g e t h e r w i t h a more d e t a i l e d a c c o u n t o f t h e s t a t i s t i c a l and n o n - s t a t i s t i c a l a s s u m p t i o n s i n v o l v e d . G i v e n a s e t o f measurements <j>^ o f a s c a l a r v a r i a b l e a t d a t a p o i n t s x^, 1 < i < N , we want t o e s t i m a t e a t t h e g e n e r a l p o i n t x i n o u r o b s e r v a t i o n a l a r e a . We assume t h a t t h e m e a s u r e d v a l u e <JK i s composed o f t h e t r u e v a l u e 'l'(x^) and t h e random n o i s e
( 1 ) 4>± = 4>(x±) + e1
The random n o i s e c o n s i s t s o f measurement e r r o r s and s m a l l -
s c a l e f l u c t u a t i o n s u n r e s o l v e d by t h e a r r a y ( e . g . i n t e r n a l waves, small s c a l e t u r b u l e n c e ) , w h i c h we want t o s u p p r e s s i n o u r a n a l y s i s .
We assume t h a t t h e e r r o r s ei a r e n o t c o r r e l a t e d w i t h t h e f i e l d and w i t h e a c h o t h e r and have a known v a r i a n c e
( 2 ) E {e.} = 0
E {e.E.} = oz-S.. f o r 1 s i , j < N
1 J £ -'-J
E { * ei) = 0
T h i s t r e a t m e n t o f t h e n o i s e f i e l d i s r i g o r o u s and i n e s s e n c e c o r r e c t o n l y f o r t h e t r u e measurement n o i s e . However, i f a s m a l l s c a l e s i g n a l i s p r e s e n t w h i c h i s u n r e s o l v e d by t h e o b s e r v a t i o n a l a r r a y , we assume t h a t t h e s c a l e o f t h i s n o i s e i s s m a l l compared t o t h e s c a l e o f t h e o b s e r v e d f i e l d .
- 8 -
As t h e s c a l e o f t h e n o i s e a p p r o a c h e s z e r o , t h e n o i s e c o v a r i a n c e a p p r o a c h e s a d e l t a f u n c t i o n o f v a l u e . T h u s , i n t h e l i m i t b o t h i n s t r u m e n t a l n o i s e and s m a l l s c a l e s i g n a l w o u l d become i n d i s t i n g u i s h a b l e . I n t h e p r e s e n t a n a l y s i s t h e e s t i m a t e d e r r o r v a r i a n c e came up t o 5-35 % o f t h e t o t a l v a r i a n c e o f t h e measurements d e p e n d i n g on t h e d e p t h and w h i c h p r o p e r t y f i e l d was t o be mapped h o r i z o n t a l l y .
The most g e n e r a l l i n e a r e s t i m a t o r f o r a s c a l a r v a r i a b l e w i t h s p a t i a l l y dependent mean v a l u e s has t h e f o r m
N
M i n i m i z a t i o n o f t h e mean s q u a r e e r r o r
( 4 ) e2 ( x ) = E { ( • ( x ) - $ ( x ) )2>
w i t h r e s p e c t t o t h e c o e f f i c i e n t s <*xi , 6x i n ( 3 ) y i e l d s
( 5 ) a
N
= r c
y 1 < i < N
X I
N 8 = M - I
~ ~ .1 x x
where
( 6 ) M = E {* ( x ) }
a r e t h e mean v a l u e s o f *
- 9 -
and
( 7 ) ( A ) ~l = (C + R )- 1
i s t h e i n v e r s e m a t r i x o f t h e sum o f t h e p o s i t i v e d e f i n i t e N x N m a t r i x o f c o v a r i a n c e s
C±. = CCV ( x . ) , * ( x ) ) , 1 < i , j < N
o f ip a t t h e p o i n t s x., x. and t h e e r r o r c o v a r i a n c e m a t r i x
R. . = COV (e.,e . ) , 1 < i , j < N
a t p o i n t s x^, •
( 8 ) C . = COV (• ( x ) , A . )
i s t h e c o v a r i a n c e o f <p a t t h e i n t e r p o l a t i o n p o i n t x and t h e o b s e r v a t i o n A- a t p o i n t x. .
j J
N o t e t h a t w i t h t h e a s s u m p t i o n o f ( 2 ) t h e e r r o r c o v a r i a n c e m a t r i x r e d u c e s t o a d i a g o n a l m a t r i x w i t h d i a g o n a l e l e m e n t s o2
( 9 ) R±i = E {E i £ j} = o | . 6i < f
- 10 -
E s t i m a t e s o f t h e r e s i d u a l u n c e r t a i n t i e s i n t h e i n t e r p o l a t e d v a l u e s a r e g i v e n by t h e e r r o r c o v a r i a n c e f u n c t i o n
(10) E { ( * ( x ) - $ ( x ) ) (* (Z) - * ( y ) ) } N N
= C - E E C . ( A "1 ) • • C .
xx i = i j = i x l ;I J y j w h i c h y i e l d s an e x p e c t e d rms e r r o r
N N 1 7
(11) e ( x ) = (C - I E C . ( A "1) • • C .) 2
^ i = l j = l X 1 1J £J
where
C = COV dp ( x ) , * (y_)) .
I n o r d e r t o a c c o u n t f o r t h e o v e r a l l h o r i z o n t a l t r e n d i n t h e mean we assumed t h a t t h e mean f i e l d can be a p p r o x i m a t e d by a t w o - d i m e n s i o n a l l i n e a r f u n c t i o n w h i c h was f i t t e d t o t h e d a t a by u s i n g a m u l t i p l e r e g r e s s i o n scheme. D e t a i l s o f t h i s f i t t i n g p r o c e d u r e w i l l be d i s c u s s e d b e l o w .
S i n c e t h e computed e r r o r maps u s i n g e x p r e s s i o n ( 1 1 ) c a n n o t r e f l e c t t h e u n c e r t a i n t i e s i n h e r e n t i n t h e e s t i m a t i o n o f t h e mean o f ifr, we assumed f u r t h e r t h a t t h i s e r r o r c a n be n e g l e c - t e d . I t s h o u l d be p o i n t e d o u t h e r e t h a t - i f t h e mean
E {<J» ( x ) } , t h e c o v a r i a n c e f u n c t i o n C and t h e n o i s e c o v a r i - ance R^. a r e known - t h e c o m p u t a t i o n o f e r r o r maps o f t h e f i e l d to be o b s e r v e d i s s t r a i g h t f o r w a r d s h o w i n g t h e a b i l i t y o f
d i f f e r e n t s a m p l i n g schemes w i t h o u t r e f e r e n c e t o any p a r t i c u - l a r d a t a s e t .
- 11 -
3. STATISTICAL PREPROCESSING OF SCALAR DATA SETS 3.1. ESTIMATION OF THE MEAN FIELD
E q u a t i o n s ( 3 ) and ( 5 ) , ( 6 ) show t h a t o u r o p t i m a l e s t i m a t o r c a n n o t be a p p l i e d w i t h o u t k n o w l e d g e o f t h e mean v a l u e s o f • a t t h e o b s e r v a t i o n a l p o i n t s and t h e p o i n t s where ^ i s t o be e s t i m a t e d .
I n t h e a n a l y s i s o f t h e "POSEIDON 86" d a t a s e t t h e f i e l d ob- s e r v a t i o n s c l e a r l y show a s p a t i a l dependence i n t h e mean f i e l d s w h i c h i s b a s i c a l l y a g e n e r a l m e r i d i o n a l t r e n d . A p r a c t i c a l a p p r o a c h t o m i n i m i z e t h e e f f e c t o f t h e unknown mean v a l u e s i s t o a p p r o x i m a t e t h e s p a t i a l mean f i e l d by a t w o - d i m e n s i o n a l l i n e a r f u n c t i o n M ( x ) w h i c h i s d e t e r m i n e d by a m u l t i p l e r e g r e s s i o n a n a l y s i s u s i n g t h e l e a s t s q u a r e s c r i - t e r i a
N
( 1 2 ) Z (<j>. - M (x. ) )2 = M i n i = l
where
( 1 3 ) M ( x . ) = A •Xi + B , A = Ux, Ay) .
I n a d d i t i o n t o t h e u s u a l a n a l y s i s o f v a r i a n c e , an o v e r a l l F - t e s t ( t e s t i n g t h e n u l l h y p o t h e s i s t h a t a l l r e g r e s s i o n p a r a - m e t e r s a r e z e r o ) and p a r t i a l F - t e s t s ( d e l e t i o n o f one r e - g r e s s i o n p a r a m e t e r f r o m t h e m o d e l ) were u s e d t o c h e c k o u r a s s u m p t i o n s f o r c o n s i s t e n c y w i t h t h e d a t a .
- 12 -
F o r t h e d i f f e r e n t p r o p e r t y f i e l d s t h e p e r c e n t a g e o f v a r i a t i o n e x p l a i n e d by m u l t i p l e r e g r e s s i o n was i n t h e r a n g e o f 53% - Some examples o f t h e r e s u l t i n g o u t p u t i n f o r m a t i o n f o r t h e m u l t i p l e r e g r e s s i o n a n a l y s i s f o r e a c h p r o p e r t y f i e l d a r e shown i n f i g . 14.
- 13 -
3-2. SPATIAL COVARIANCE FUNCTION
T h e r e a r e two main o b j e c t i v e s w h i c h have t o be a c c o m p l i s h e d when e s t i m a t i n g t h e c o v a r i a n c e f u n c t i o n s : t h e f i r s t and most e s s e n t i a l p o i n t i s t h a t e v e r y moment m a t r i x
A „ = E U $ }
r s yrys
drawn f r o m t h e e s t i m a t e d c o v a r i a n c e f u n c t i o n o f must be a n o n - n e g a t i v e d e f i n i t e m a t r i x , i . e . none o f i t s e i g e n v a l u e s a r e n e g a t i v e . T h i s i s a c h a r a c t e r i s t i c o f t h e t r u e moment m a t r i x
A = E {<)><{> }
r s r rYs
on w h i c h t h e method t o m i n i m i z e t h e mean s q u a r e e r r o r ( 4 ) i n t h e p r o o f o f t h e Gauss-Markov t h e o r e m h i g h l y d e p e n d s . The s e c o n d p o i n t i s t h a t t h e e s t i m a t e d c o v a r i a n c e f u n c t i o n a p p r o x i m a t e s t h e t r u e c o v a r i a n c e f u n c t i o n o f <J> .
The s t a n d a r d a p p r o a c h c h o s e n h e r e t o meet t h e s e two r e q u i r e - ments was t o d e r i v e raw c o v a r i a n c e s f r o m t h e d a t a and t o f i t a smooth c u r v e w h i c h i s a member o f a c l a s s o f model c o v a r i - ance f u n c t i o n s and a p p r o x i m a t e s t h e raw e s t i m a t e s .
S u c h a c l a s s o f model c o v a r i a n c e s
( 1 4 ) £ = ( F ( r ; P ) , P = ( P1, . . . , P , ) }
- l i l -
w i t h t h e l a g v e c t o r r = ( r , r ) and f r e e p a r a m e t e r s
_ x y
pi > - - ' » P i i s chosen a p r i o r i t o g e t h e r w i t h a f i r s t guess
P = ( P1, . . . , P1) f o r t h e u n d e t e r m i n e d p a r a m e t e r s . A n o n - l i n e a r f i t t i n g p r o c e d u r e (METZLER e t a l . , 1971*) was u s e d t o compute an e s t i m a t e P f o r the p a r a m e t e r v e c t o r P w h i c h m i n i m i z e s t h e sum o f w e i g h t e d s q u a r e s o f d e v i a t i o n s i n t h e v a l u e s o f t h e raw c o v a r i a n c e s and t h e model c o v a r i a n c e f u n c t i o n
( 1 5 ) I (C ( r . ) - P ( r . , P ) )2- w . = MIN
where
C ( r . ) i s t h e raw c o v a r i a n c e f u n c t i o n (see d i s c u s s i o n b e l o w ) and W. a r e a p p r o p r i a t e w e i g h t s d e r i v e d f r o m t h e 95 % c o n f i - dence l i m i t s o f the raw c o v a r i a n c e e s t i m a t e s .
3.2.1. ESTIMATION OF RAW COVARIANCES
T h e o r e t i c a l l y , t h e e s t i m a t i o n o f raw c o v a r i a n c e s w o u l d i n - v o l v e t h e c a l c u l a t i o n o f t h e p r o d u c t s
(16) Pi j. = (*i - E {*. - E {$.})
f o r e a c h p a i r o f o b s e r v a t i o n a l p o i n t s , x. and a v e r a g i n g o v e r a l a r g e number m o f r e a l i z a t i o n s .
- 15 -
i n o r d e r t o e s t i m a t e t h e t r u e c o v a r i a n c e
( 1 8 ) C. . = E { ( * . - E {*.}) - E {*.})}
b e t w e e n t h e s e p o i n t s .
However, a d i f f e r e n t a p p r o a c h h a d t o be c h o s e n i n v i e w o f t h e f a c t t h a t o n l y one r e a l i z a t i o n was a v a i l a b l e . A f t e r s u b t r a c t i o n o f t h e e s t i m a t e d mean f i e l d ( s e e 3 . 1 . ) f r o m t h e o b s e r v a t i o n s , a d a t a s e t w i t h s p a t i a l l y q u a s i - s t a t i o n a r y s t a t i s t i c s was o b t a i n e d where t h e c o v a r i a n c e f u n c t i o n was o n l y a f u n c t i o n o f t h e l a g v e c t o r r .
L e t
(19)
*
±--
^ ~ M.where
( 2 0 ) M\ = A • x± + B
a r e e s t i m a t e s o f t h e mean f i e l d g i v e n by t h e f u n c t i o n a l r e - p r e s e n t a t i o n o f t h e mean.
F o r e a c h p a i r o f o b s e r v a t i o n a l p o i n t s t h e p r o d u c t s
( 2 1 ) P ( r ) = (*± - M) (%• - M) , = x1 + r
- 16 -
were computed where
- 1 N ~
( 2 2 ) M = i I <j>.
1 = 1 1
t h e s a m p l e r e s i d u a l mean c a u s e d by f l u c t u a t i o n s i n t h e anomaly f i e l d i s a p p r o x i m a t e l y z e r o .
These p r o d u c t s were a v e r a g e d o v e r a l l p a i r s o f o b s e r v a t i o n a l p o i n t s i n t h e a r r a y w i t h an i d e n t i c a l l a g v e c t o r r o r - r
1 m ~ H 1 ( 2 3 ) C ( r ) = - i Z P U ; ( r )
N i = 1
t o r e d u c e t h e n o i s e i n t h e raw e s t i m a t e s ( 2 1 ) .
A l t e r n a t i v e l y - b u t e q u i v a l e n t l y - t h e s e raw e s t i m a t e s c a n be computed by a g e n e r a l i z a t i o n t o t w o - d i m e n s i o n a l s p a c e o f t h e f o l l o w i n g e s t i m a t o r s f o r t h e a u t o - c o v a r i a n c e f u n c t i o n o f t i m e s e r i e s o f a s t a t i o n a r y s t o c h a s t i c p r o c e s s i(( ( t ) w i t h o b s e r v a t i o n s <f> ( t )
* 1 T" l rl
( 2 4 ) C ( r ) = i / U ( t ) - ? ) (• ( t + | r | ) - •) d t
^ o o r
/ - T - | r |
( 2 5 ) ( r ) = rpzjjj I (• ( t ) - •) U ( t + | r | ) - *) d t
where <j> i s t h e sample mean o f •. Note that our estimator (23) which i s equivalent t o (24) i s a biased estimator, whereas (25) i s unbiased but has a greater mean square error than (24 ).
- 17 -
A p p a r e n t l y , a l l t h e s e e s t i m a t o r s have i n t u i t i v e a p p e a l , i . e . t h e y a r e n o t o p t i m a l e s t i m a t o r s i n any known s e n s e . F o r e x - a m p l e : no maximum l i k e l i h o o d e s t i m a t o r i s known t o e x i s t f o r t h e a u t o c o r r e l a t i o n f u n c t i o n o f an o b s e r v e d t i m e s e r i e s . A s s u m i n g t h a t t h e 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 n o r m a l , t h e l i k e l i h o o d f u n c t i o n can be d e r i v e d . But t h e s e t o f e q u a t i o n s o b t a i n e d by d i f f e r e n t i a t i o n i s i n t r a c t a b l e ( J e n k i n s & W a t t s , 1 9 6 8 ) .
H e n c e , we a r e l e f t w i t h t h e s e a d m i t t e d l y i n t u i t i v e e s t i m a t o r s w h i c h , o f c o u r s e , may be compared a c c o r d i n g t o c r i t e r i a s u c h as minimum mean s q u a r e e r r o r o r b i a s o f t h e e s t i m a t e s .
I n f i g . 3a t h e t w o - d i m e n s i o n a l raw c o v a r i a n c e s f o r geopotential anomaly (25/1500 d b a r ) a r e shown, whereas t h e c o r r e s p o n d i n g c o n f i d e n c e l i m i t s and l e v e l s o f z e r o s i g n i f i c a n c e a r e d i s - p l a y e d i n f i g . 3 b , c .
As c a n be s e e n , t h e computed raw c o v a r i a n c e s show a c o n s i d e r - a b l e amount o f random s c a t t e r . I n s p e c t i o n o f t h e 95 % c o n f i d e n c e l i m i t s and l e v e l s o f z e r o s i g n i f i c a n c e ( f i g . 3 b , c ) shows t h a t
w i t h t h e p r e s e n t amount o f d a t a t h e q u e s t i o n i s l e f t u n a n s w e r - ed w h e t h e r t h e e a s t - w e s t o s c i l l a t i o n i s p r i m a r i l y due t o r e a l p h y s i c s o r r a t h e r a r e s u l t o f a random s a m p l i n g e r r o r i n h e r e n t i n t h e d a t a . Even t h e c o v a r i a n c e s computed u n d e r t h e assump- t i o n o f i s o t r o p y ( f i g . 4 ) , w i t h i n c r e a s e d d e g r e e s o f f r e e d o m , a r e n o t s i g n i f i c a n t enough t o g i v e c l e a r e v i d e n c e c o n c e r n i n g t h i s p o i n t . A d d i t i o n a l l y i t m i g h t be p o i n t e d o u t t h a t t h e raw c o v a r i a n c e s a r e i n f l u e n c e d by t h e u n c e r t a i n t i e s i n h e r e n t i n t h e a p p r o x i m a t i o n o f t h e t r u e mean f i e l d . E s p e c i a l l y t h e z e r o c r o s s i n g p o i n t s a r e s e n s i t i v e t o changes i n t h e mean f i e l d .
- 18 -
3-2.2. NONLINEAR FITTING OF MODEL COVARIANCES
S i n c e o n l y l i t t l e p r i o r i n f o r m a t i o n on t h e s t a t i s t i c s o f t h e m e s o s c a l e v a r i a b i l i t y i n t h e Canary b a s i n was a v a i l a b l e , s e v e r a l model c l a s s e s had t o be t e s t e d f o r t h e i r a b i l i t y o f r e p r e s e n t i n g t h e raw c o v a r i a n c e s . The m o t i v a t i o n f o r t h e de- s i g n o f t h e d i f f e r e n t model c l a s s e s p a r t l y came f r o m t h e o r e - t i c a l c o n s i d e r a t i o n s , ( c a s e 4 i n t a b l e 1 f o r i n s t a n c e r e p r e - s e n t s a t h e o r e t i c a l c o v a r i a n c e f u n c t i o n f o r p r o p a g a t i n g R o s s b y waves g e n e r a t e d a t an e a s t e r n b o u n d a r y ) , o r was i m - p l i e d s i m p l y b e c a u s e o f a n a l y t i c r e a s o n s ( i . e . b e s t f i t t o t h e raw c o v a r i a n c e s w i t h o u t s y s t e m a t i c d e v i a t i o n s ) .
R e s u l t s o f t h e n o n l i n e a r f i t t i n g p r o c e d u r e f o r t h e d i f f e r e n t m o d e l c l a s s e s a r e shown i n t a b l e 1, w h i c h l i s t s t h e v a l u e s o f t h e weighted sum o f s q u a r e d d e v i a t i o n s (WSS), c o r r e l a t i o n b e t w e e n t h e raw c o v a r i a n c e s and p r e d i c t e d v a l u e s (COR), and p a r a m e t e r s w i t h 95 % c o n f i d e n c e l i m i t s f o r e a c h model f u n c - t i o n c l a s s .
I n f i g . 5 p l o t s o f t h e c o r r e s p o n d i n g f i t t e d f u n c t i o n s a r e shown. A l t h o u g h i n some c a s e s a good f i t t o t h e d a t a was ob- t a i n e d by u s i n g WSS o r COR as a measure o f t h e goodness o f f i t , t h e p l o t s showed s y s t e m a t i c d e v i a t i o n s i n d i c a t i n g a w r o n g model c l a s s ; e.g. i n case 6 f o r i n s t a n c e t h e c o v a r i - ance f u n c t i o n does n o t decay f a s t enough l e a d i n g t o s y s t e - m a t i c n e g a t i v e c o r r e l a t i o n s f o r l a r g e e a s t - w e s t s e p a r a t i o n s . On t h e o t h e r h a n d , sometimes a l a r g e v a r i a b i l i t y i n t h e e s - t i m a t e s o f t h e p a r a m e t e r s o c c u r e d , i n d i c a t i n g t h a t t h e
weighted sum o f s q u a r e s was n o t v e r y s e n s i t i v e t o changes i n t h e p a r a m e t e r s . I n t h e s e c a s e s t h e model c l a s s was r e j e c t e d f r o m f u r t h e r a n a l y s i s .
- 19 -
F o r t h e s y n o p t i c mapping o f t h e d a t a a n o n i s o t r o p i c G a u s s i a n f u n c t i o n was c h o s e n :
w h i c h s u p p r e s s e s n e g a t i v e c o r r e l a t i o n s .
T h i s was m o t i v a t e d by t h e f a c t t h a t even u n d e r t h e assump- t i o n o f i s o t r o p y t h e r e was no s i g n i f i c a n t n e g a t i v e c o r r e l a - t i o n ( s e e d i s c u s s i o n a b o v e ) . As can be s e e n f r o m t a b l e 1, t h e c o v a r i a n c e f u n c t i o n ( 2 6 ) s t i l l i s a good f i t t o t h e raw c o v a r i a n c e s ( t a b l e 1, c a s e 7) a n d t a k e s a c c o u n t o f t h e n o n - i s o t r o p i c f e a t u r e s i n t h e d a t a , e s p e c i a l l y t h e r a p i d d e - c r e a s e o f c o r r e l a t i o n i n t h e e a s t - w e s t d i r e c t i o n .
Some examples o f the influences on the c a l c u l a t i o n o f the maps using c o v a r i - ance f u n c t i o n s o f t h e r e m a i n i n g model c l a s s e s ( w h i c h r e p r e - s e n t a good f i t and were n o t r e j e c t e d ) may be s e e n by com- p a r i n g f i g . 6 a ) - c ) . The v i s u a l d i f f e r e n c e s b e t w e e n t h e s e maps a r e q u i t e n o t i c e a b l e b u t r e m a i n w i t h i n t h e r a n g e o f t h e p r e d i c t e d rms e r r o r f i e l d s . ( f i g . 7a)
The c o r r e l a t i o n s c a l e i n t h e e a s t - w e s t d i r e c t i o n \^ = 32.0 nm i s c o m p a r a b l e t o t h e a v e r a g e s e p a r a t i o n b e t w e e n t h e s t a t i o n s o f o u r a r r a y ( f i g . 1 ).
However, a s we have n o t c o n s t r a i n e d o u r g e n e r a l e s t i m a t o r ( 3 ) t o have z e r o b i a s by r e q u i r i n g
( 2 6 ) = 48.0 nm
N
( 2 7 ) 2 o s
7 . X I
1 = 1 "
= 1
i t w i l l g i v e a s t r o n g e r w e i g h t t o t h e l i n e a r r e p r e s e n t a t i o n o f t h e mean f i e l d i n t h e r e g i o n s o f l o w e r s t a t i o n d e n s i t y . As t h e g e n e r a l s t a t i o n c o v e r a g e was g o o d , t h i s o n l y h a p p e n e d
- 20 -
on t h e b o u n d a r i e s o f o u r o b s e r v a t i o n a l a r e a where t h e p r e - d i c t e d e r r o r v a r i a n c e comes up t o a r o u n d 50 % o f t h e v a r i - ance o f t h e anomaly f i e l d , as c a n be s e e n f r o m f i g . ja.
F u r t h e r e x a m p l e s o f o b j e c t i v e mapping o f d i f f e r e n t p r o p e r t y f i e l d s b a s e d on t h e p r i n c i p l e s o u t l i n e d so f a r , a r e d i s p l a y e d i n f i g . 15a) - d ) . The n o n - i s o t r o p i c c o v a r i a n c e f u n c t i o n
(26) was u s e d a g a i n , w i t h t h e p a r a m e t e r s Xx, Xy d e t e r m i n e d f r o m t h e f i t o f t h i s model c l a s s t o t h e raw c o v a r i a n c e s o f t h e d i f f e r e n t f i e l d s . The summary o f t h e c o r r e s p o n d i n g m u l t i p l e r e g r e s s i o n a n a l y s i s f o r t h e mean f i e l d s i s g i v e n i n f i g . 14.
TABLE 1
case m o d e l c l a s s wss COR p a r a m e t e r s nm o r (nm) resp
l o w e r a n d u p p e r c o n f i d e n c e l i m i t s
1. \X2 X2/
F ( r ) = e x y /. c o s ( k | r |+l|r |) x y
1 . 8 8 0 . 8 0 7
Ax = 3 1'7 Xy = 4 7 . 9
k * 0
L a 0
2 7 . « - 3 5 . 6 4 3 . 1 " 5 2 . 7 - 0 . 5 2 - 0 . 5 2 - 2 6 0 . 0 ~ 2 6 0 . 0
2«
J u l ) *
P (r) * e V X y • (l-k|rv|) 0 . 6 6 7 0 . 9 1
Xx = 7 7 . 3
Ay = 4 4 . 0 k = 0 . 1 5 1 ' 1 0 " *
no e s t i m a t e o f e r r o r
3 . . f e l l ) VX2 X2/
P (r) = e X x y /. ( l- k| rx| ) 1.07 0.884
Xx = 9 9 . 9
xy = 4 9 . 0 k = 0 . 1 6 « 1 0 "1
9 2.8 - 1 0 6 . 9 4 5 . 6 - 5 2 . 3 0 . 1 5 - 1 0 "1 - 0 . 1 7 » 1 0_ 1
4. " U2 +X2/ s i n ( k | r 1)
P ( * ) = e k | r / 1.87 0 . 8 0 7
xx = 3 1 . 7
Xy = 4 7 . 9 k = 0 . 5 - 1 0 "3
27.8 - 3 5 . 6
4 3 . 1 - 5 2 . 7 - 2 5 2 . 0 - 2 5 2 . 0
5.
/IrJ
M
1.73 0 . 8 3 5
xx = 6 9 . 4
Ay = 4 4 . 8 k = 0. 4 2- 1 0- 1
i
39.4 - 9 9 . 3
3 8 . 6 - 5i. o
0 . 3 5 * 1 0 *1 - 0 . 5 0 « 1 0 "1
TABLE 1 ( c o n t i n u e d ) case m o d e l c l a s s wss COR p a r a m e t e r s
— l
nm o r (nm) resp,
l o w e r and u p p e r c o n f i d e n c e l i m i t s
6 .
F ( r ) = e \ x y A ( l - k | rx| ) 0 . 9 2 6 0 . 8 8 5
xx = 91-7
Xy = 4 9 . 4 k = 0 . 1 5 - 1 0 "1
8 1 . 7 - 1 0 1 . 6 4 4 . 7 - 5 4 . 2 0. 1 4- 1 0- 1 - 0 . 1 6 - 1 0 -1
7. U2 X2/
F (r) = e y 1.87 0 . 8 0 7
\ - 3 1 . 7
xy = 4 7 . 9
28.7 - 3 4 . 5 4 3 . 1 - 5 2 . 7
8.
F ( r ) = e x x y ' 2 . 0 5 0 . 8 0 8 A = 4 5 . 6
y
2 2 . 2 - 3 1 . 5 3 8 . 3 - 5 2 . 9
9 .
( r ^ + r2) x y'
F ( | r | ) = e x 2 2 . 1 0 0 . 7 8 5 x = 3 7 . 6 35.2 - 3 9 . 9
10. M
P (|r|) » e""1" 2 . 2 0 0 . 7 9 5 X = 2 8 . 9 2 6 . 0 ~ 31.9
- 23 -
3.2.3« S t a t i s t i c s of the mesoscale eddy f i e l d
As mentioned already i n the introduction, the process of estimating the eddy flux divergence term i n the temperature conservation equat- ion implied the problem of mapping mesoscale perturbation f i e l d s where the corresponding mean f i e l d was defined through a l i n e a r s p a t i a l trend plus Rossby wave f i t .
Let
= • -(%>SSBY + W with the composite mean f i e l d <P^oSSBy+
Again, by means of the procedures described already, raw covariances of the <j>' f i e l d were derived based on different s t a t i s t i c a l assumptions, i . e . s t a t i o n a r i t y and/or without/ isotropy.
A
The two-dimensional raw correlations C(£) f o r the geopotential anomaly f i e l d 25/1500 dBar are shown i n f i g . 9 a with corresponding cross sections displayed i n f i g .9b,c . Correlations derived under the assumption of isotropy are shown i n f i g . 1 0 .
Our s t a t i o n displacement varied between 30 nm i n the region of the f r o n t a l zone and /5*30 nm i n the southern t r a n q u i l region where l e s s eddy a c t i v i t y was observed. The raw correlations show that only under the assumption o f isotropy the perturbation f i e l d i s marginally corre- lated at 30 nm.
These raw correlations were adapted to different model correlation functions with the results as shorn i n table 2 . In the case o f a non-isotropic Gaussian function, the zonal and meridional c o r r e l a t i o n scales are i d e n t i c a l within the 9 5 * confidence l i m i t s . Consequently, the c o r r e l a t i o n scale A i n the two-dimensional isotropic model
- 24 -
function i s equal to the arithmetic average of *x and Xy i n the non-isotropic case; the analysis of the one-dimensional isotropic raw covariances yields an identical correlation scale (case 3 of Table 2).
In case 4 and 5 of Table 2 model functions were f i t t e d which consider the zero crossing of the isotropic raw covariances at 43 nm with subsequent negative correlations. The resulting correlation scales are larger than i n case 3 (isotropic Gaussian function) where the positive form of the model function does not allow f o r negative correlations and - consequently - provides an underestimate of the correlation scale.
These results show that the resolution of the perturbation f i e l d was too coarse to significantly distinguish the shape of the corre- l a t i o n model function. The estimated correlation scales vary within the range of 20 - 38 nm depending on the assumed model function
(see Table 2). Therefore, - f o r the f i n a l analysis - we decided to use an "a p r i o r i " function which i s Gaussian, isotropic and has a correlation scale of 30 nm. Besides the fact that t h i s function i s within the significance limits consistent with our raw corre- l a t i o n s , i t enforces a separation of perturbations resolved by the box-grid and smaller sub-grid scale v a r i a b i l i t y .
Our approximation of the error correlation as a delta function i s only correct i f the scale of the noise i s i n f i n i t e s i m a l l y small or at least c l e a r l y to be distinguished from the main scale of
the f i e l d . Under the assumptions made with respect to the s t a t i s t i c s of the f i e l d , t h i s approximation i s v a l i d only i f the scale of
the noise i s less than 15 ran*
- 25 -
As a consequence o f the removal of the mean f i e l d
the s i g n a l t o noise r a t i o was decreased at every stage o f the f i t t i n g process. The composite mean f i e l d represents about 70% o f the t o t a l v a r i a b i l i t y - depending on the depth l e v e l and the f a c t which s c a l a r f i e l d was t o be mapped -. Even under o p t i m i s t i c as- sumptions regarding the noise s t a t i s t i c s , one f i n a l l y comes up t o 25% - J>5% o f the variance o f the mesoscale perturbation f i e l d .
Having i n mind future single ship experiments i n t h i s area these r e s u l t s underline the importance of measurement methods which can be used complementary from a ship underway - such as GEK, XCP o r geostrophic v e l o c i t i e s derived from an XBT survey through an averaged T/S r e l a t i o n s h i p , which would r e s u l t i n a f i n e r s p a t i a l r e s o l u t i o n .
B a s e d on the analysis o f s a t e l l i t e - t r a c k e d d r i f t i n g buoy observations the estimated synoptic deformation rate f o r the POSEIDON 86 survey was 50 km i n 20 days. That means that the e n t i r e survey must be f i n i s h e d w i t h i n that period. Consequently, a d d i t i o n a l l y nested f i n e - s c a l e CTD stations which are quite time consuming, prove t o be no s o l u t i o n with respect t o the s t a t i s t i c a l estimation problems
e n c o u n t e r e d . This i s enhanced by the fact that with the present sampling scheme every s t a t i o n i s s t a t i s t i c a l l y independent because o f the short c o r r e l a t i o n scale of 30 nm or l e s s . Thus, several nested f i n e - s c a l e CTD-stations would have t o be implemented i n the survey pattern at dynamically s i m i l a r regions so to have enough mesoscale events with s u f f i c i e n t degrees of freedom. This i s hard t o achieve without p r i o r information about the flow f i e l d . The above remarks are valid f o r the case that only one r e a l i z a t i o n i s a v a i l a b l e . I f there are several r e a l i z a t i o n s f o r the same ob- s e r v a t i o n a l area with s i m i l a r dynamics i n v o l v e d , the covariance
- 26 -
f i t t i n g approach used here can be replaced by the more refined spectral model f i t t i n g approach as described by Bretherton et a l .
(1980). Their estimator i s optimally e f f i c i e n t i n the sense that no other unbiased linear estimator operating on the same subspace of spectral features, can have less uncertainties. The more precise estimation of the underlying s t a t i s t i c s which - i n consequence -
improves the accuracy of reproducing the true flow or property f i e l d , must be complemented by an array design considering the expected or known spectal features. One of the fundamental properties of an i r r e g u l a r array i s the i n t r i n s i c a l i a s class,which i s the set of
spectral features for which - regardless of the number of r e a l i z a t i o n s - no information can be obtained or power between adjacent wave numbers discriminated through observations. Several examples of optimal array design f o r isotropic or non-isotropic stochastic processes are given by Bretherton & McWilliams (1979) based on the maximization of information gained according to standard theory (Middleton, I960).
I f the array i s intended to serve synoptic mapping purposes as w e l l , a more uniform distribution i n space i s desirable. For example, a rhombic two-dimensional array as described by Petersen & Middleton
(1962) i s optimal for a band-limited isotropic process.
Table 2
Case model c l a s s wss COR parameters
(ran)
lower and upper 95% confidence l i m i t s
1.
r *x r 2
~(T~T + T T )
x y
F ( r ) = e 0.93 0.77
Xx = 20.3 Xy = 23.1
17.5 - 23.0 20.5 - 25.6
2. P C = e ~ 0.94 0.77 X = 21.7 19.9 - 23.4
1-dimensional i s o t r o p i c raw c o r r e l a t i o n s :
3 .
- I d '
P ( | r | ) = e ^ 0.54 .981 X = 21.7 19.9 - 23.5
4. F ( | r | ) = (l-k»«H*>e
r
i
.037 .985 X = 35.7
k = .25-10'1
33.8 - 37.7
.245- l o' - ^ e s- i o "1
5. F(|r|) = ( l - k - l r l ) e .031 .987 X = 28.6 k = .22 1 0 '1
25.8 - 31.4
.200« l O "1— .242 l O "1
- 2 8 -
4. Objective Analysis of vector f i e l d s
I n t h i s s e c t i o n we s h a l l give a b r i e f sumnary o f the adaption o f the basic algorithm t o vector f i e l d s (Bretherton et a l . , 1 9 7 6 ) .
Some preliminary remarks w i l l serve as an introduction t o the problems which w i l l a r i s e .
Given a s e t of observations o f a horizontal v e l o c i t y f i e l d a t N points
we can straightforward adapt the basic algorithm by introducing the observation vector
(29) % - ( • ! » • • - . • a j ) = <ui ^ i ^ - - - ' ^ ( % )}
For b r e v i t y we assume that the v e l o c i t y f i e l d has zero mean. With the a i d o f the general l i n e a r estimator ( 3 ) , we obtain optimal e s t i - mates o f the v e l o c i t y components at a general point x i n the obser- v a t i o n a l area
(28) u U±) = ( ua ( xi) , U2 (x.)) 1 < i < N
(30) <
Sl <*> - L °:
- 29 -
where
~ J l < a < 2
(3D and C ^ = E { u£ (x) ^ (x )} f o r N+l < j < 2N
are the covariances o f the £ v e l o c i t y component t o be estimated and the j observation and
(32) A.. = E {*.*.} 1 < i , j < 2N
i s the covariance between a l l p a i r s of observations. The e r r o r i n the v e l o c i t y estimates i s given by the general GauB-Markov theorem i n the f o l l o w i n g way. F i r s t , we introduce the 2 x 2 e r r o r matrix Ce
T /E {e e } , E {e e }\
C = E { ex.e} =/ 1 1 1 M
E ~ " \E {e2e1} , E {e^}]
where
e = ( ea, e2) = u - u = ( u1 - u± , 0, - a,) where u i s the true v e l o c i t y and u the estimate.
According t o the GauS-Markov theorem, the trace o f C£ i s nanimal f o r our optimal estimate (30):
E {|e|2} = E {ea 2} + E { e2 2} = t r ( C£) = M3N
Thus, we can specify the root-mean-square e r r o r e of the estimate u(x):
- 30 -
e = E {|e|2}2 = E {|u-u|2}2
(33)
I c £ - J f C ^ ( A- l) . . C ^ H t=i h i=i i,j=i «• u $ j /
where
i s the variance of the fcth velocity component.
I f we consider the velocity components as stochastic functions and assume that t h e i r joint probability function i s independent of a r b i - trary s p a t i a l translations, i . e . homogeneity, and i f we assume further that the standard assumption (2) holds, i . e . the errors inherent i n the velocity measurements are not correlated with the velocity f i e l d and with each other and have a known variance E, we can rewrite
(32) i n terms o f the velocity correlation tensor RJr)& {u.(x)u0(x+r)}:
(34) A.^. = R11( rij)+E 5iJ
Ai,j+N = R1 2(r i j} Ai+N , j = V f t j *
with
Thus, we are l e f t with the problem o f specifying the v e l o c i t y correlation tensor.
In order t o give a simple mathematical description o f R ^ C r ) , we w i l l
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adopt the standard approach used elsewhere (Bretherton et a l . , 1976;
McWilliams, 1976), and assume that the velocity f i e l d i s i s o t r o p i c . Modified f o r a 2-dim.velocity f i e l d , t h i s means that the probability d i s t r i b u t i o n i s invariant under arbitrary rotations about a v e r t i c a l axis and reflections i n any direction. The v e l o c i t y correlation
(r) i s then an isotropic second-order two point tensor and, therefore, has the form (Bachelor, 1959)
(35) R^Cr) = A(r) rk r£ + B(r) «k £ > ^ e r e r = |r|
where A, B are even scalar functions of r . (It should be noted that since the beginning of this chapter, we are ignoring time-dependence.) As our main interest concentrates on the interpretation of mesoscale v e l o c i t y measurements, t h i s - we admit - i s a poor assumption which w i l l scarcely be f u l f i l l e d i n r e a l physics. However, i t should be borne i n mind that under the least r e s t r i c t i v e symmetry condition symmetry about a plane, R j ^ r ) i s the sum of 35 terms including vector arguments (Batchelor, 1959).
The condition of homogeneity has ensured that
\iW * \*{'£>
the condition of isotropy makes Rl d l(£) f u l l y symmetrical i n the two s u f f i x e s :
\zW -~ Ri k(£} The continuity condition
V • u = 0
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has further consequences f o r R ^ r ) (summation convention f o r the tensor suffixes i s used from now on u n t i l the end of t h i s chapter):
a?" u* ( x + r> = 0
l
which y i e l d s f o r fixed k:
E { uk( x ) 3r7 u* = 3F- <E < Vx ) u* (x+£)})
= W¡ V r ) - 0
With the form of ^ ¿ ( r ) established by (35), i t follows
(36) 3A(r) + P l g s l + I ¿ B ( p ) =0
Introduction o f the convenient longitudinal and transversal v e l o c i t y correlation functions
E {u„(x) u„(x+r)}
E {u„2} E íu^íx) ujx+r)}
E { uA 2í
f o r two points x, x + r at distance r apart, gives the following r e l a t i o n s t o the scalar functions A ( r ) , B(r) of (35)
f ( r ) =
g(r) =
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P(r) = u2f ( r ) = E {U | l(x) u,,(x+r)} = r2 A(r) + B ( r ) (37)
G(r) = u2g ( r ) = E {u,(x) u,(x+r)} = B ( r )
where F ( r ) , G(r) are the l o n g i t u d i n a l and t r a n s v e r s a l covariance functions r e s p e c t i v e l y , and
(38) u2 = E {u,,2} = E U r2} = E { i ^2}
i s the t o t a l variance of the v e l o c i t y f i e l d which equals the compo- nent variance E {u^2 } because of the isotropy assumption.
I n the d e r i v a t i o n o f (36), we made use of the transformation u, = cos a • u^ + s i n a -
u± = - s i n a • u^ + cos a •
where a i s the angle between r and the x^-axis. Without l i m i t a t i o n s t o the general case, r may be assumed t o be positioned i n the ( x ^ X g ) plane as a consequence o f the isotropy assumption.
Prom (37) i t follows that
A(r) = — (P(r) - Q(r)) r2
(39)
B ( r ) = G(r)
Thus, we can rewrite R ^ i r ) i n terms o f P ( r ) and G(r)
(40) \ t M = ( P ( D - G(r)) • • 0 ( r ) - «k £
Fran (36) and (39) i t follows that, i n f a c t , we have to determine only F(r)and its f i r s t derivative:
(41) G(r) = F(r) + r - 4~ F(r)
Thus, f o r our o r i g i n a l problem of mapping a two-dimensional horizontal v e l o c i t y f i e l d , we can specify the correlation tensor used i n (39) i n terms o f F ( r ) and F ' ( r ) :
r 2
K.Ar)xx " = F(r) + - § - • F'(r) r (42) R^Cr) = R2 1( r ) • > ( r )
I f the velocity f i e l d i s known to be non-divergent, or i f non-divergence i s raised t o be an axiom of the analysis, ( i . e . i f a geostrophic view of a measured velocity f i e l d i s desired or i f low frequency currents are analysed which are i n approximate geostrophic balance,) i t i s con- venient t o introduce a stream function <Kx) with
R ^ r ) = F(r) + F'(r)
where F ' ( r ) = ^ F ( r )
a»C3S)
3 X2
u^x) =
and covariance function
C(r) = E (*(x) <Kx+r)}
From non-divergence and with the a i d of ( 4 l ) , we obtain the f o l l o w i n g r e l a t i o n s between C ( r ) , F ( r ) and G(r)
(43)
F ( r ) s _ I J L c ( r )
G(r) = - | - ( r . F ( r ) ) = - i S f c l
As pointed out by Bretherton et a l . , 1976, a d i r e c t consequence of the use of s t a t i s t i c s consistent with h o r i z o n t a l non-divergence, i s the fact that the divergence of the estimated f i e l d w i l l vanish - regardless of the nature of the measurements. This can be seen by t a k i n g the divergence of the v e l o c i t y estimates ( 3 0 ) . With weights
(44) oj = I ( A - i ) ^ * i N i = l
and 2N
= 1 i=N+l we obtain
(45)
3^(2) au^x) 3x„ 3x,
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N
2N
Thus, once the a p r i o r i constraint of non-divergence has been applied by s e l e c t i n g s t a t i s t i c s consistent with a non-divergent f i e l d , the estimated f i e l d w i l l be non-divergent even i f the data base shows marked inconsistencies with that assumption.
Consequently, i f theoretical reasons imply - f o r example - that i n the observational area the importance of the non-linear and f r i c t i o n a l terms i n the equation of motion i s small compared to the influence of the C o r i o l i s term , a geostrophic approximation may s i g n i f i c a n t l y improve the estimated velocity f i e l d , especially when associated with mesoscale motion. On the other hand, the example of vector analysis given below, i . e . estimation o f a quasi-Eulerian velocity f i e l d from averaged and low-pass f i l t e r e d d r i f t i n g buoy observations, i n some cases d i s - played marked differences between the estimated non-divergent v e l o c i t y f i e l d and the observational f i e l d i n space, showing that the assumption of non-divergence was clearly not applicable.
