Harmonic analysis of tides: Essential feature and disturbing influences

Roarn ól - Oz-Lrl

D E L F T H Y D R A U L I C S L A B O R A T O R Y

H a r m o n i c A n a l y s i s o f T i d e s

E s s e n t i a l features and disturbing influences

by A . C M . van E t l e and H . J . S c h o e m a k e r

W 3

N o t a t i o n , Surn'.nary. page 1. I n t r o d u c t i o n 1 2. H a r m o n i c a n a l y s i s .'ind t h e l e a s t s q u a r e s method f o r t i d a l o ,9jialy s i 3 - • • • 3o The s t r u c t u r e o f t h e m a t r i o e a o f t h e n o r m a l e q u a t i o n s . 8 4. The i n f l u e n c e o f t h e o m i t t o d c o n s t i t u e n t s i n t h e t i d a l a n a l y s i s . 9 5. The s,ainpling i n f l u e n c e on t h e a n a l y s i s . . . 12 6, The i n f l u e n c e o f t h e n o i s e on t h e a c c u r a c y o f t h e r e s u l t s 15 7, The c o m p u t i n g p r o c e d u r e 25 8, R e c a p i t u l a t i o n o f f o r m u l a e o f s t o c h a s t i c p r o c e s s e s 26 9. C o n c l u s i o n s • • A c k n o w l e d g e m e n t . ' • ^'^

3-0 ' a j C O G . cornponento o f t h e t i d a l c o n s t i t u e n t s o f t h e supposed t i d o . ^ a j e r r o r i n a^ . ^. s i n . compo!,ents o f t h e t i d a l c o n s t i t u e n t s o f t h o s u p p o s e d t i d o . C O S . component o f t h e t i d a l c o n s t i t u e n t s o f t h e C „ , O j c ^. s i n . components o f t h e t i d a l c o n s t i t u e n t s o f t h e o b s e r v e d t i d e , s i n . com.ponent o b s e r v e d t i d e . g | 1 m a t h e m a t i c a l e > q ) e c t a t i o n (mean v a l u e ) o f a s t o c h a s t i c v a r i a b l e . f[i) supposed t i d o . o b s e r v e d t i d e . !_ \< e l o m o n t s o f t h e m a t r i x o f t h o n o r m a l e q u a t i o n s . J'^ ' numbor o f a n a l y s e d c o n s t i t u e n t s . 2-S-i-l number o f o b s e r v e d v a l u e s o f g ( t ) . r ( t ) = g ( t ) - f ( t ) r e s i d u . t i m o . s a m p l i n g i n t e r v a l . p e r i o d . v a r i a , n c e o f a s t o c h a s t i c v a r i a b l e . A t Var x ( t ) ' ' n o i s e i n g ( t x{t) s t o c h a s t i c p r o c e s s . x^{t) • c o m p l e x c o n j u g a t e o f x ( t ) . z{co ) h a r m o n i c a n a l y s i s o f x ( t ) . A I 00 ) - i S i i i L i . pov/er s p e c t n u n . d e t e r m i n a n t o f t h e n o r m a l e q u a t i o n s . A . . m i n o r o f A b e l o n g i n g t o L j j . A ' d e t e r m i n a n t b e l o n g i n g t o a g r o u p o f c o n s t i t u e n t s , an,5:ular f r e q u e n c y . 1 t i me. to a n g u l a r f r e q u e n c y .

'rhe cssenti[1l foatures of' tho cl[183ical hCtrmonic analyniD of LJ.oG are rocapitul:1tod in v-iow of the 1..130 of cligital cornputern.

Spocial att(mti.on is paLl to tho possible errors in the analysis ch18 to sampling and non-astronomic:J.l cffect3. '1'he theory of stational',), stochastio procosses is applied to the tidal phenomenon.

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i l l , 12 t h o R e c e n t l y i n a number o f p u b l i c a t i o n s l i t 6

a t t e n t i o n was d r a m i t o t h o now p o s s i b i l i t i e s o f t i d a l a n a l y . i s , w h i c h became a v a i l a b l e when t h e e l e c t r o n i c d i f _ ; i t a l c o m p u t e r made i t s e n t r y .

I n t h e p a n t t h o p r o c o s s i n ; - o f a t i d a l r e c o r d t o o k much t i m e , t h e r e f o r e t h o o l d m e t h o d s , s t i l l beinp; p a r t l y i n u s e , were d e v i s e d i n such a way

as t o r e d u c e t h e c o m p u t a t i o n as much as p o s s i b l e , b u t r e s u l t e d i n p r o c e d u r e s , w h i c h a r e n o t m a t h i e m a t i c a l l y o p t i m a l . T h i s c i r c u m s t a n o e i s now l a r f - o l y I'omovod.

The amount c f c o m p u t a t i o n n e c e s s a r y when t h e s t r a i / j l i t f o r w a r d .mothod o f l e a s t s q u a r e s i s a p p l i e d i s i n s u r m o u n t a b l e w i t h o u t t h e a i d o f t h e c o m p u t e r . I n t h i s n o t e t h e c r i t e r i u m o f t h o method o f l e a s t s q u a r e s i s assiained as a s t a r t i n g p o i n t o f t h e t i d a l a n a l y s i s . I t i s a b a s i c h : ) T o t h e 3 i 3 o f t h e t i d a l a n a l y s i s t h a t t h o a n g u l a r f r e q u e n c i e s o f t h e a s t r o n o m i c a l t i d e g e n e r a t i n g i n f l u e n c e s a r e knovm. As u s u a l we assume t h a t t h e t i d o can be r e p r e s e n t e d by • • y ( t ) = + S H. f. cos(V, + u. + OO.t - S\ ) f , , V j + U| , M] a r e c o m p l e t e l y d e t e r m i n e d by a s t r o n o m i c a l i n f l u e n c e s . H; and g; depend on t h o p a r t i c u l a r s i t e w h o r e t h e t i d e i s t o bo d e s c r i b e d . m> „ + -p; i vA ,1. o f n pi-imV.QT- o f +, h 6 s G q u a n t i t i e s by moans o f an o b s e r v a t i o n . An o b s e r v a t i o n may bo r e p r e s e n t e d by g ( t ) =' c„ + (c, cos W j t 4- d; s i n W| t ) + x ( t ) ( 1 . 1 ) VAith c^ = C j = Hi f i C 0 3 ( V i + U| - ffj ) di - n, f i s i n ( V i + u'i - gi ) x ( t ) i s t h o p a r t o f t h o r e c o r d w h i c h i s n o t d e t e r m i n e d by t i d a l i n f l u e n c e s b u t has a n o n - d e t e r m i n i s t i c c h a r a c t e r , i n p h y s i c s c a l l e d n o i s e . I n x ( t ) a r e a b s o r b e d e.g.: t h e i n f l u e n c e o f t h e c h a n g i n g m e t e o r o l o g i c a l c i r c u m s t a n c e s , t h e a c c i d e n t a l f a i l u r e s o f t h e t i d e gauge and t h o mdst.oJtos w h i c h may o c c u r d u r i n g t h o p r e p a r a t i o n o f t h e r e c o r d f o r c o m p u t e r i n p u t .

2

Because t h o number o f p o s s i b l y o c c u r r x n g c o n s t i t u e n t s i s v e r y l a r r e and m o r e o v e r some o f them have e x t r e m e l y s m a l l d i f f e r e n c e s .a ang. f..eauPncy, we have t o c o n t e n t o u r s e l v e s t o e v a l u a t e t h o most i m p o r t a n t c o n s t i t u e n t s o n l y . T h e r e f o r e we need c r i t e r i a t o be a b l e t o d e c i d e , a t l e a s t a p p r o x i m a t e l y , w h i c h c o n s t i t u e n t s may r e a s o n a b l y be computed. The h a r m o n i c a n a l y s i s i s t h e c l a s s i c a l a p p r o a c h , i t s a t i s f . e s t h e l e a s t s q u a r e s c r i t e r i u m . As t h i s c r i t e r i u m i s a c c e p t a b l e and l e a d s t o a c o n v e n i e n t way o f c o m p u t i n g , i t i s a d o p t e d h e r e . The o b s e r v e d t i d e g ( t ) i s a p p r o x i m a t e d by a f u n c t i o n f ( t ) .

f ( t ) = a„ + ( a j cos to, t + b, s i n w., t )

, ^. , V, a r e d e t e r m i n e d i n s u c h a way t h a t| ( f( t ) - g ( t ) ) ' d t w i l l and a

bo a mininiuin

..any r e c o r d s however c o n s i s t o f a sequence o f o b s e r v a t i o n s a t d i s c r e t e p o i n t s o f txme. The s a m p l i n g o f a c o n t i n u o u s s i g n a l xs n e c e s s a r y , f o r t h e use o f t h e c o m p u t e r f o r c e s us t o r e p r e s e n t g ( t ) by a sequence o f numbers.

I n t h e s e c a s e s t h o c r i t e r i u m becomes: 2 I ( f ( t ^ ) - ) ) = minimum. k=1 I n t h i s n o t e t h e m a t r i x o f t h e n o r m a l e q u a t i o n s , w h i c h a r i s e by e l a b o r a t i n g t h e above m e n t i o n e d c r i t e r i u m , i s s t u d i e d . I t i s p o s s i b l e t o draw c o n -c l u s i o n s -c o n -c e r n i n g t h e d u r a t i o n o f t h e o b s e r v e d t i d e g ( t ) , t h e t i m e ... b e t w e e n s u c c e s s i v e o b s e r v a t i o n s , t h e i n f l u e n c e o f c o n s t i t u e n t s •n . ( t ) w h i c h a r e n o t a n a l y s e d i n t h e h y p o t h e t i c a l t i d e f ( t ) and t h e i n f l u -e n c ^ o f t h -e n o i s -e x ( t ) on t h -e a c c ^ a c y o f t h -e r -e s u l t s o f t h -e t i d a l a n a l y s -e s . „ v;+ [?! o r 207 - 215, a p e r i o d i c f u n c t i o n Under g e n e r a l c o n d i t i o n s , l i t [('J pp. f y ( t ) may be e x p r e s s e d a s : • (2 y ( ^ ) ^ + £ ( C O S n t + c5n s i n n t ) n=1 I t i = i n f a c t .l»ay= p o = r , i b l o t o t r a n . f o r n : t h o p o r i o d e t o t h e I n t e r v a l (-.,

4 = ^ / " y l O s i n n t d t The v a l u e s o f ' yn ' '^n ^"oHow f r o m an a p p l i c a t i o n o f t h e l e a s t s q u a r e s method. y ( t ) i s t o be a p p r o x i m a t e d b y a f u n c t i o n y ( t ) ~ ~ + I ( a cos nt + A s i n n t ) such t h a t 2 Y » / 7 y( t ) - y ( t ) ) d t = minimum. « • Y depends upon , . . . Y i s a minimum i f t h o e q u a t i o n s •hoc o — ^ = 0 n = 1,2,. . .. . a r e s a t i s f i e d . B e f o r e d o i n g t h i s i t s h o u l d be r e c a l l e d t h a t f-TC 1 GOO i t c o s j t d t ='0 i / j J-1X = n i = j

/

/ s i n i t c o s j t d t = 0 ( 2 . 3 ) i , j i n t e g e r s . I s i n i t s i n j t d t = 0 i / j = n i = j Prom ^ --^ 0 f o l l o v / s / ^ ( y ( t ) y n , ( t ) ) d t 0 A f t e r s u b s t i t u t i o n o f t h e e x p r e s s i o n o f ym('t) i t f o l l o w s t h a t « - / " y ( t ) d t = y v-'ith ^ = 0 f o l l o w s r ( y ( t ) ~ y ^ ( t ) ) ^ d t = 0 J l/_7t ^

4

thnBf\y{t) - y^ii)) l ^ ^ ^ d t = / ^ ( y ( t ) - y j I ) ) con j t d t

i n t r o d u c i n g t h o e x p r e s n i o n f o r y ( t ) and u a i n g t h e r e l a t i o n s ( 2 , 3 ) we o b t a i n f i n a l l y

«j " ~ / y ( t ) C 0 3 d t

t h u s a j = j^j j = 1, . . . m

I n t l i e same way we o b t a i n a l s o yh.^tS.^ j 1 , . , , . m.

Tho same method can bo a p p l i e d f o r t h e a | ) p r o x i m a t i o n o f - g ( t ) by

means o f f ( t ) . Two d i f f e r e n t s i t u a t i o n s a r e d i s c e r n e d . I n t h e f i r s t p l a c e g ( t ) may be a c o n t i n u o u s r e c o r d on t h e i n t e r v a l [ - i: , ~) - In t h e second

2 (L

p l a c e g ( t ) may be o b s e r v e d a t more o r l e s s r e g u l a r i n t e r v a l s . However, i n t h e f o l l o w i n g i t i s a l w a y s u n d e r s t o o d t h a t t h e t i m e b e t w e e n s u b s e q u e n t o b s e r v a t i o n s i s a c o n s t a n t . So ( 2 N + l ) v a l u e s o f g ( t ) a r e a v a i l a b l e a t t h e p o i n t s o f t i m e : - NAt , ( - N + l ) A t , , NAt. Though i n p r a c t i c a l c i r c u m s t a n c e s when a d i g i t a l c o m p u t e r i s u s e d g ( t ) w i l l a l w a y s he a v a i l a b l e as a sampled r e c o r d , t h e c o n t i n u o u s case i s s t u d i e d as w e l l , because t h e d i f f e r e n t f o r m u l a e w h i c h a r i s e g i v e a good i n s i g h t i n t h e c o n s e q u e n c e s o f s a m p l i n g , by w h i c h v/e c a n e a s i l y examine w h i c h c o n d i t i o n s s h o u l d be f u l f i l l e d b y t h e s a i i i p l i n g i n t e r v a l A t . 0.) g ( t ) c o n t i n u o u s f v t ) = a^ + 1 ( a j cosiOj t + b j s i n oJ-^t) " i =

1

f o l l o w i n g t h e l e a s t s q u a r e s method a ...b s h o u l d be chosen svich t h a t

F = ( f ( t ) - g ( t ) ) d t - minimum "T " t h e r e f o r e t h e f o l l o w i n g e q u a t i o n s s h o u l d be s a t i s f i e d : -a F =• 0 = 0 j = 1,2, m -?>a 0 5 f -

.

0 From ^ = 0 i t f o l l o w s t h a t / ^ ( f ( t ) - g ( t ) ) — =• 0 m

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V "bf ' d i v i d i n g b y — , s u b s t i t u t i n g t h e e x p r e s s i o n o f f ( t ) and w i t h = 1 we g e t s 2 'i>%

E l a b o r a t i o n o f t h o l e f t member l o a d s t o a„=lo,o + a^-^t^^^ = | / 2 g ( t ) d t , ^ I

2

T s m w, w i t h ^ 0," ' 9 From •baj T 0 i t f o l l o w s by s u b s t i t u t i n g f ( t ) , d i v i d i n g by ~ and w i t h ^ f = c o s iOjt t h a t ; ^ aj J 2 | - j^ ( a ^ + . E ( a . cos 6Jj t + bj s i n W j t ) ) c o s w . t d t = ^ g ( t ) c o s ^ j t d t .

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s i n Wj w i t h ^=1.^- 2 2 s i n 60. T 'Sd= ^-^ T \ 1' s i n ( W i c^Jj ) 3 i n ( 6J| + ^Oj ) -( 2 . 5 ) ( W i - ^ O j ) i ( a ) i + W j ) l I n t h i s way t h e f o l l o w i n g s y s t e m o f (m+1) e q u a t i o n w i t h ( m + l ) unknowns i s f o u n d 2 2 ^m't^,m ^ I l' s ( t ) c o s ^ , ( t ) d t '2

Because t 0 has been c h o s e n as t h e m i d d l e o f t h e o b s e r v a t i o n , we g e t two d i f f e r e n t systems f o r t h e imlcnowns a a n d b .

6

The a t t e n t i o n i s d r a w n t o t h e f a c t t h a t , i f t h o z e r o p o i n t i s n o t

c h o s e n i n t h e m i d d l e o f t h e r e c o r d t h e u n k n o v m B c a n n o t bo s e p a r a t e d i n t h o d e s c r i b e d way. The same h o l d s a l s o i f t h e r e c o r d has i r r e g u l a r g a p s .

P r o c e e d i n g i n t h e same way f o r t h e b's a s y s t e m o f m e q u a t i o n s w i t h m \ml<:novms i s o b t a i n e d : b ' l ' + ... + b - f^g{t) Bin to, t d t I 1,1 m m,i j f ^ '

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b ' l ' + + K^^L, = Rs{^) s i n ^ o t d t

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A s y s t e m o f m e q u a t i o n s w i t h m imlcnowns. c^, 3 i n ( ^ i - t ^ j ) | s i n( ^ i + ^ j ) I j . » _ ^ _ ^ \ ' / c., . sln^Oj?"" b ) g ( t ) i s a sampled f u n c t i o n a t t h e p o i n t s o f t i m e t = k A t k - -N, -N+1,...,0, N I n t h i s case t h e l e a s t s q u a r e s c r i t e r i u m i s P = Z ( f ( k A t ) - g ( k A t ) ) ^ minimum. Ppon, H L =. 0 i t f o l l o w s t h a t z ( f ( k A t ) ~ g ( k A t ) ) ~ ™ = 0 k = -N t f D i v i d i n g by N+é and s u b s t i t u t i n g f ( k A t ) and = 1

o

i s o b t a i n e d : 1^

Z ( a + E (a. co8(0; k A t + b; s i n to, k A t ) ) = E g ( k A t )

^"•-^ti krr-N ° i = 1 ' k = -N E l a b o r a t i o n o f t h e l e f t s i d e o f t h i s e q u a t i o n l e a d s t o s e r i e s l i k e I c o s 60. k A t = 1 + 2 E c o s (Oj k A t k = -N k=1 ? E cos^o- k A t = E 2 s i n — - A t c o s w i k A t = s i n ^ A t 2 c o s Wi A t s i n A t + 2 c o s 2 iO| A t s i n ~ - A t + 3xn At

a i n ^ At n i i i ( N + ; ; ) <u)| A t - B i n ~ - A t u n N 10 t h a t E coo Wj k A t 3 l n (N + i ) Wi A t 3 i n A moro e l e g a n t d e r i v a t i o n can be g i v e n i n t h e c o m p l e x n o t a t i o n . The e q u a t i o n becomes 1 g ( k A t ) a I + ... + a I ~ TTTT k=-N s i n ( N + ; j ) wj A t B i n ( 2 . 7 ) ^ a: = 0 g i v e s c o n t i n u i n g i n t h e same way. N4 ^-r- 1 g ( k A t ) c o s (W|kAt. ^ k=-N ' Thus a s y s t e m i s o b t a i n e d o f ( m + l ) e q u a t i o n s , t h e so c a l l e d n o r m a l e q u a t i o n , w i t h ( m + 1 ) u i f k n o r a s . I N , • E g,(kAt) =< r~-r I g ( k A t ) c o s ^ ) k A t . m ( 2 . 8 ) •J'J o i n ( N + g ) ( AJj - Wj ) A t 12N + Ï ) s i n i { MI - j ' l A t ' s i n ( 2 N + l ) w j A t s i n ( N + i ' ) ( (^i + t^ij ) A t (2Kt 1 )3inA(£0i H-^Oj ) A t = 1 + 2 K + 1 ) s i n A t ( 2 . 9 )

Y-ith — = 0 t h e n o r m a l e q u a t i o n s f o r t h e unknowns b j j - l , . . . m becomes •Jibj ain{ll+h){^] ~ ) A t •(2N+1) s i n l . ( A); - tOj")aT s i n ( 2 N + l ) i O j A t X W ï T T o i r r M f A T ' E g ( k A t ) s i n iOj k A t k=-N ~r ^ g ( k A t ) s i n A i ^ k A t . s i n (K + i ) ( tO; + ) A t (2N + 1 ) s i n i ( «Oj + iOj ) a t ( 2 . 1 0 )

8

3. The s t r u c t u r e o f t h o r n a l r i c e s o f t h o n o r m a l o g u a t i o n a .

As t h e r e i s no e s s e n t i a l d i f f e r e n c e i n t h e s t r u c t u r e o f t h o m a t r i c e s o f t h e n o r m a l e q u a t i o n s o f t h o unknowns a,j and b j r e s p e c t i v e l y , i n t h e f o l l c w i n g o n l y t h o unknowns Oj a r e c o n s i d o r e d . A l l c o n c l u s i o n s w h i c h a r e dravm a r e v a l i d f o r b o t h s y s t e m s .

The a n g u l a r f r e q u e n c i e s o f t h e t i d a l c o n s t i t u e n t s a r e d i s t r i b u t e ! ! i n a c h a r a c t e r i s t i c way, F i r s t o f a l l one may make a d i v i s i o n b e t w e e n t h o d i f f e r e n t s p e c i e s . T h e r e a r e t h o l o n g - p e r i o d i c c o n s t i t u e n t s , w h i c h have p e r i o d e s l o n g e r t h a n 1'1 d a y s , t h e n t h e c o n s t i t u e n t s h a v i n g p e r i o d e s i n t h e n e i g h b o u r h o o d o f one day, t h e n d a y , j day e t c . down to ^ day and even

s h o r t e r .

I n t h e s e d i f f e r e n t s p e c i e s t h e m s e l v e s g r o u p s may be d i s t i n g u i s h e d whose c o n s t i t u e n t s have a n g u l a r f r e q u e n c i e s v e r y n e a r t o each o t h e r e.g. among the d i u r n a l s p e c i e s t h e r e a r e t h e c o n s t i t u e n t s , , R2 1^2 ^^'^ a n g u l a r f r e q u e n c i e s o f w h i c h d i f f e r s u b s e q u e n t l y by 0,04''/h = r a d / h . T h i s d i s t r i b u t i o n o f a n g u l a r f r e q u e n c i e s l e a d s t o a c h a r a c t o r i s t i c f o r m o f t h e m a t r i x whose e l e m e n t s a r e g i v e n by ( 2 . 9 ) and ( 2 . 1 0 ) . E x c e p t a l l t h e e l e m e n t s on t h o p r i n c i p a l d i a g o n a l w i l l be a b o u t one, w i t h a p o s s i b l e e x c e p t i o n f o r t h e l o n g p e r i o d i c c o m p o n e n t s , i f t h e d u r a t i o n o f t h e r e g i s t r a t i o n i s so t h a t ( N+ ^ O- ^ j A t « n. The e x p r e s s i o n f o r t h e o t h e r e l e m e n t s o f t h e m a t r i x s p l i t s i n t o two p a r t s . As f o r e v e r y t i d a l a n a l y s i s t h e d u r a t i o n o f t h e r e g i s t r a t i o n w i l l bo 3 0 l o n g t h a t ( 2 H + l ) A t i s a l a r g e number, ( N + i ) A t ( + ) w i l l be much g r e a t e r t h a n n, t h e r e f o r e g e n e r a l l y

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r r r « 1 w i 11 ho I d . (2N+1 j s i n -;j(60| + 6v5j)At I n t h i s p a r a g r a p h t h e case i s e x c l u d e d t h a t + ) A t l i e s i n t h e n e i g h b o u r h o o d o f n o r o f an i n t e g r a l m u l t i p l e o f n. T h i s s i t u a t i o n . w i l l be s t u d i e d i n p a r a g r a p h 5. The same h o l d s f o r r e l a t i n g t o p a i r s o f c o n s t i t u e n t s w i t h d i f f e r e n c e i n an,'?ular f r e q u e n c y such t h a t (N+g ) ( ^ j - Wj) A t > I f t h e r e g i s t r a t i o n i s so l o n g t h a t even t h e g r o u p , 3^ , » ^^2 s a t i s f i e s t h i s c o n d i t i o n , a l l e l e m e n t s o f t h e m a t r i x l| j i j w i l l be s m a l l w i t h r e s p e c t t o t h e e l e m e n t s o f t h e p r i n c i p a l d i a g i n a l . Per t h i s group t h e d u r a t i o n o f t l i e measurement s h o u l d be a b o u t one y e a r .

I n many c a s e s t h i s c o n d i t i o n w i l l n o t be s a t i s f i e d , so t h a t f o r the c o n s t i t u e n t s w h i c h a r e v e r y n e a r t o each o t h e r I j ^ j w i l l be o f t h e same o r d e r o f m a g n i t u d e as t h e e l e m e n t on t h o p r i n c i p a l d i a g o n a l . The m a t r i x l o o k s l i k e t h e f o l l o w i n g :

On t h e p r i n c i p a l d i a g o n a l numbers I j j z 1 , l-o,o " a b o u t t h i s d i a g o n a l a number o f b l o c k s c o n s i s t i n g o f nujiibers aomewliat s m a l l e r t h a n one. The r e m a i n i n g e l e m e n t s a r e v e r y much s m a l l e r .

Based on t h e above m e n t i o n e d c o n s i d e r a t i o n i n t h e p a r a g r a p h s 4 and 6 t h e f o l l o w i n g a p p r o x i m a t i o n w i l l be u s e d , l i i ~ 1 " 3 i n( N + è ) ( w ; - ) A t ^ i ' i X2ÏÏ+rj7ïnS{^i - ^ j ) A t 4c. The i n f l u e n c e o f t h e o m i t t e d c o n s t i t u e n t s i n t h e t i d a l a n a l y s i s . I n g e n e r a l a t i d a l r e g i s t r a t i o n a t a p a r t i c u l a r s i t e cam be r e p r e s e n t e d by ( l . l ) . The number o f t h e o c c u r r i n g c o n s t i t u e n t s i n t h e r e g i s t r a t i o n i s v e r y l a r g e . The an^gular f r e q u e n c i e s o f a nuniber o f t h e c o n s t i t u e n t s a r e v e r y near each o t l i e r . I f a l l t h e s e c o n s t i t u e n t s w o u l d be a c c o u n t e d f o r i n t h e a n a l y s i s , t h e g e n e r a t e d m a t r i c e s w o u l d show l a r g e b l o c k s c o n s i s t i n g o f e l e m e n t s a l l b e i n g v e r y n e a r one. These m a t r i c e s a r e i l l - c o n d i t i o n e d i n t h e sense t h a t a s o l u t i o n o f t h e s y s t e m becomes h i g h l y i n a c c u r a t e . A p a r t f r o m t h i s i n many c a s e s no a p r i o r i i n f o r m a t i o n i s a v a i l a b l e a b o u t t h e r e l e v a n t c o n s t i t u e n t s . I t i s t h e g e n e r a l s i t u a t i o n t h e r e f o r e t h a t i t can be e x p e c t e d t h a t t h e o b s e r v e d t i d e g ( t ) c o n t a i n s more c o n s t i t u e n t s t h a n w i l l be l a i d i n t h e supposed t i d e f ( t ) . T h e r e f o r e t h e i n f l u e n c e o f t h e s e c o n s t i t u e n t s on t h e a n a l y s i s w i l l be i n v e s t i g a t e d . The i n f l u e n c e o f x ( t ) w i l l be s e p a r a t e l y s t u d i e d i n p a r a g r a p h 6. A g a i n o n l y t h e unknowns a j a r e c o n s i d e r e d . L o t f ( t ) a + z ( a ; COB

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t + b j s i n (Oj t ; ° 1 = 1 _ ( 4 . 1 ) g ( t ) = c, + ( c ; c o 8( 0 ; t + b j s i n (Oj t ) + c^^^ COS 60^^^ t + d^^^ s i n w ^ ^ ; t

F i r s t t h e s i t u a t i o n w i l l be c o n s i d e r e d i n w h i c h f ( t ) has been composed o f s u c h c o n s t i t u e n t s t h a t a l l l ; j « 1 , so t h a t t h e y c a n be n e g l e c t e d th By means o f ( 4 . I ) and ( 2 . 8 ) , t h e r i g h t s i d e o f t h e j - e q u a t i o n becomes! ^

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10 W i t h t h o a c c e p t o d n e g l e o L a t h e j e q x i a t i o n becomes 30 t h e r e a p p e a r s an e r r o r 6a; _{•J m+1 "^m+l} li,m+1 = r c o s 60 k A t c o s k A t « + 2 kr-N m + 1 •* a i n ( N + i - ) ( 6 0 j - W ^ j A t r 2 N+ l) ' 7 T^ H ^ " - " m+lMt The e r r o r 6aj i s o f no i n t e r e s t i f : 1) + 1 « c j

2 ) m+1 « 1» t h i s case c j may be l a r g e compared w i t h C j , I j_m+1 « 1 i f {11+^) ( W j - Wr„.,i)At » n .

S e r i o u s e r r o r s w i l l o c c u r i f o^^^ has t h e same o r d e r o f m a g n i t u d o a s C j and (M+;) ( ^ j -

(0^^i

) A t << n.

A s l i g h t l y more c o m p l i c a t e d s i t u a t i o n a r i s e s i n caae t h e r e i s a b l o c k c o n s i s t i n g o f two e q u a t i o n s i n t h e m a t r i x .

Under t h e same s u p p o s i t i o n ( 4 ' 1 ) f o r g ( t ) . t h e s e e q u a t i o n s become, + l i j " °|li,i + °j l i . j + °m + 1 ll,m+1 a; I ; ; ^ j j " °-' _{I J''} The s o l u t i o n f o r a; i s : + °j ^ j J °m+1 ^j,m+1 c. L.. + c. I , + c L. I 1,1 J J,J m+1 I,m+1 c. L . + c. L , -f c L. I j , i J JJ m+1 J, m+1 J. J L. i.i I . . J, J = c.+c ' m+1 i,m+ 1 j > + 1 L. . =:Ci + é . a . •1,1

Tho m o a n i n g o f t h i s o x p r o a s i m . may be i l l u s t r a t e d by means o f an example. L e t t h o cons t i tuents,: and K j . ( a n g u l a r f r e q u e n c i e s o f 30,0(f/h and 3 0 . ü 8 7 h r o s p o c t i v o l y ) , be a n a l y s e d on t h e i n f l u e n c e o f t h e o m i t t e d T has an a n g o i l a r f r e q u e n c y o f 29,96°/h. 2 The d i f f e r e n c e o f t h o a n g u l a r f r e q u e n c y o f K2 and s i s 0,08°/h Let (H+z. ) A t r a d / h 4500 X, t h e unknowns Hj and «2 r o s p . Because o f t h e s m a l l v a l u e o f [6^; - t o ; a r o i d e n t i f i e d w i t h en ,' f o l l o w s sin — ( uij-Wj) At ~ F u r t h e r i t f o l l o w s t h a t : s i n( N+ A ) ( - ) A t ' J ~ ( 2 I i + l ) 3 i n ^ ( w; - tOj ) A t s i n 2x ~ 1 _{3 l} The m a t r i x e l e m e n t l- • becomes 1 - 4e w i t h y r = ^ s i n ( N + é ) ( w i U t I rrvf T s m X X s i n( N + ; i ) ( - "m^.i)At ^»"'+'' ~ l 2 N T l^ s i n : , ( W i ^ ) s i n 3x 3x Thus 6a ~ c

m +

1 6a; ~ 1,5 c 1 - e 1 - 4e 1 - 9e 1 1 1 - 4e 1 - 4e 1 m+ 1 So i n t h o e r r o r c a u s e d by i n t h e c o m p u t a t i o n o f t h e r e i s a m u l t i p l i c a t i o n f a c t o r 1,5. A s i m i l a r e x p r e s s i o n can be g i v e n f o r K2

-I n t h e same way i t can bo shown t h a t s l o w l y c h a n g i n g a v e r a g e s can be a source o f v e r y s e r i o u s e r r o r s , i f t h e a n a l y s i s i s c a r r i e d w i t h o u t s p e c i a l p r e c a u t i o n s . I f t h e c o n s t i t u o n t l ^ i a t a k e n i n t o a c c o u n t t h e d e t e r m i n a n t s i n both t h o n o m i n a t o r and d e n o m i n a t o r become v e r y s m a l l , t h e r e f o r e t h e e l a b o -r a t i o n o f t h e system o f t h e e q u a t i o n s w i l l become most d i f f i c u l t . On t h o o t h e r hand a l s o t h e i n f l u e n c e o f t h e n o i s e becomes v e r y s e r i o u s as w i l l be

shorn i n i ) a r a g r a p h 6.

In f a c t t h e p r a c t i c a l l o w e r l i m i t o f t h e v a l u e s o f t h e d e t e r m i n a n t s d e t e r m i n e o the r e s o l v i n g power o f t h e a n a l y s i s .

The f o r m u l a e f o r t h e o l e m e n t o o f t h e m a t r i x show t h a t t h e r e s o l v i n g power i s e x c l u s i v e l y d e t e r m i n G d b y t h e t o t a l d u r a t i o n o f t h e o b s e r v a t i o n , p r o v i d e d t h a t ( s e e p a r a g r a p h 3 ) A t s a t i s f i e s t h e c o n d i t i o n t h a t

l{u)\ + ' ^ j ) A t < 7;. I n f a c t ( K + * ) ( W i -u)j)ht « J^ S - ) "^^^ have a c e r t a i n v a l u e , so t h a t o b v i o u s l y i t i s n o t p o s s i b l e t o o b t a i n a b e t t e r r e s o l v i n g power b y d i m i n i s h i n g t h e s a m p l i n g i n t e r v a l A t .

5. The s a m p l i n g i n f l u e n c e on t h e a n a l y s i s .

By c o m p a r i s o n o f t h e f o r m u l a e ( 2 . 5 ) and ( 2. 6 ) on t h e one s i d e and ( 2 . 9 ) and ( 2 . 1 0 ) on t h e o t h e r hand t h e a p p e a r a n c e o f t h o s i n u s f u n c t i o n ' i n t h e d e n o m i n a t o r s t r i k e s i n c o n n e c t i o n w i t h t h e sampled s i g n a l .

The f o r m u l a e ( 2 . 9 ) and ( 2 . 1 0 ) i n t r o d u c e d i n p a r a g r a p h 2 have b e e n s i m p l i f i e d to ( 3 . 1 ) en ( 3 . 2 ) i n p a r a g r a p h 3 u n d e r t h e c o n d i t i o n t h a t -i( 6 0 ; + ) A t has not a v a l u e i n t h e n e i g h b o u r h o o d o f 11 o r an i n t e g r a l m u l t i p l e o f T U V/ith c o n t i n u o u s f u n c t i o n s t h i s r e s t r i c t i o n i s n o t n e c e s s a r y . Tho o c c u r e n c e of t h e e i n u s f u n c t i o n s i n t h e d e n o m i n a t o r t h e r e f o r e shows t h e c o n s m u g n c e s of t h o s a m p l i n g on t h o t i d a l a n a l y s i s -As became e v i d e n t i n p a r a g r a p h 4 w i t h t h e h e l p o f a s i m p l e example t h e o c c u r r e n c e o f l a r g e v a l u e s b e s i d e t h e p r i n c i p a l d i a g o n a l l e a d s t o a l a r g e s o n s i t i v e n e s s f o r o m i t t e d c o n s t i t u e n t s and n o i s e . T h i s was due t o t h e f a c t t h a t t h e e l e m e n t s o f t h e m a t r i x b e s i d e t h e p r i c i p a l d i a g o n a l w e r e no l o n g e r small as compared t o one. Jn t h e case o f t h e most c o m p l e t e i n f o r m a t i o n

(a c o n t i n u o u s r e c o r d o f g ( t ) ) t h i s o c c u r s o n l y when - ( tOj - ) i s s m a l l . I n t h e case o f a sampled record,hovrever» t h i s may a l s o o c c u r b y an i n a d e q u a t e c h o i c e o f t h e s a m p l i n g i n t e r v a l A t .

Formulae ( 2 . 9 ) and ( 2 . 1 0 ) show t h a t t h i s happens when t h e d e n o m i n a t o r s become s m a l l ? t h u s i f l{uj- _ ujj ) A t ~ K^rt i ( ' ^ i + " ^ j ) A t ~ K2 « » K2 i n t e g e r s I f one v d s h e s t o a v o i d t h i s f o r e v e r , n o t v d t h s t a n d i n g t h e d i s t r i b u t i o n o f t h e a n g u l a r f r e q u e n c i e s , A t s h o u l d be c h o s e n i n s t i c h a way t h a t ^ ^ j g ^ ^ ^ ^ t < ^' ^ b e i n g t h e h i g h e s t f r e q u e n c y o c c u r r i n g i n f ( t ) . max ThUBI At < 60 max As t o how f a r t h i s l i m i t c a n be a p p r o x i m a t e d t h i s d e p e n d s b o t h on t h e d u r a t i o n o f t h e r e g i s t r a t i o n as on w h i c h v a l u e s o f t h e e l e m e n t s b e s i d e the p r i n c i p a l d i a g o n a l a r e s t i l l a c c e p t a b l e .

V e r y o f t e n t h e c o n d i t i o n i s posed t h a t {li'i)iu)- - u ) j ) A t > i\ f o r a * good a n a l y s i s . I n t h a t c a s e n a m e l y s i n ( K + l.) ( Wj - 60j ) A t 7 — ^—-T-7 -~ , < 0.11 w i l l h o l d ( 2 N * l ) s i n i ( wi - wj ;At I f one a d o p t s t h i s c r i t e r i u m t h e a c c e p t a b l e u p p e r l i m i t At^, i s d e t e r m i n e d . L e t W At, » n - a max > s i n ( 2 K n ) ^„^^At^ ^ 3 i n ( 2 K a ) ( i i - a ) „ s i n ( 2 N n ) a (2K' l ) s i n w ^ A t ^ " ( 2 N n ) 8 i n K-a ( 2 N Ö ) a ^ t h u s ( 2 1 U l ) a > 71 o r « > 2 J } ^ so t h a t At s h o u l d be c h o s e n i n s u c h a way t h a t ^1^(1 " — ) max

Because N i s a l w a y s a l a r g e number t h e v a l u e ~ — may be c l o s e l y a p p r o x i -max mated, i n f a c t t h e d u r a t i o n o f t h e r e g i s t r a t i o n has b u t l i t t l e i n f l u e n c e . In v i e w o f t h e c o n c l u s i o n s o f t h e p r e c e d i n g p a r a g r a p h we can s t a t e t h a t t h i s c r i t e r i u m b e i n g s a t i s f i e d , i t i s o f no u s e t o t a k e At s m a l l e r , e x c e p t i n t h e case o f an e x o p t i o n a l h i g h n o i s e l e v e l , w h i c h w i l l be t r e a t e d i n p a r a g a p h 6. Yet t h e c o n c e q u e n c o s o f t h e s a m p l i n g have a n o t h e r a s p e c t . g ( t ) c o n t a i n s c o n s t i t u e n t s w h i c h a r e n o t t a k e n i n t o a c c o u n t i n t h e a n a l y s i s . F o r two

simple c a s e s has been l o o k e d a t t h e c o n s e q u e n c e s o f t h i s f a c t i n p a r a g r a p h 4. Now t h e e f f e c t o f t h e s a m p l i n g p r p c e d u r e has t o ' b e i n v e s t i g a t e d .

Lot g ( t ) - c„ + E (c; c o s o J i t + d i s i n W i t ) + E ( c ^ ^ cos (0 ^ ^ t + d ^ ^ s i n ^ ^ ^ t ) . Tho r i g h t s i d e o f t h e i^^- e q u a t i o n becomes

As v/as shown b e f o r e t h e r e may be an i n f l u e n c e on t h e c o n s t i t u e n t j by c o n s t i t u e n t s m+r h a v i n g an a n g u l a r f r e q u e n c y s u c h t h a t I j ^ ^ . ^ i s l a r g e

3 i n ( N 4 ) ( ' - ' j " W ^ j A t 3 i n ( N H ) ( ^ j ^^m+r)-^*

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A(601 + i o j A t ~ K j n ,'^2 I ' ^ t e g e r s p o s i t i v e o r n e g a t i v e as b e f o r e . So t h e r e a r e two g r o u p s o f o o n s t i t u o n t e w h i c h a f f e c t t h o c o n s t i t u e n t j , namely t h e c o n s t i t u e n t s f o r w h i c h h o l d s U)

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m+r j A t

I f wo l o o k a t ( 5 . 1 ) somewhat c l o s e r i t a p p e a r s t h a t t h e c o n s t i t u e n t s h a v i n g an a n g u l a r f r e q u e n c y h i g h e r t h a n ~ a r e p r o j e c t e d on t h e i n t e r v a l ( 0 , ~ ) o r i n o t h e r w o r d s no f r e q u e n c y h i g h e r t h a n nan be d i s t i n g u i s h e d from some f r e q u e n c y b e l o n g i n g t h o i n t e r v a l ( 0 , • I t i s t o be u n d e r s t o o d \ t h a t t h o f r e q u e n c i e s a r e c o n s i d e r e d as e s s e n t i a l l y p o s i t i v e numbers. Tho o a s i o s t way t o i n d i c a t e how t h o p r o j o c t i o n t a k e s p l a c e i s by a f i g u r e . f i g u r e 1 T h i s phenomenon i s c a l l e d a l i a s i n g o r f o l d i n g , t h e f r e q u e n c y ^ t h e f o l d i n g f r e q u e n c y o r N y q u i s t - f r e q u e n c y . I f t h e r e a r e no f r e q u e n c i e s h i g h e r than •—- p r e s e n t , t h e s a i n p l i n g has no c o n s o q u e n c o s . T h i o i s i n a c c o r d a n c e w i t h t h e s a m p l i n g t h e o r e m f r o m t h e i n f o r m a t i o n t h e o r y , K y q u i s t ' s r e s u l t , s t a t i n g t h a t w i t h a s a m p l i n g f r e q u e n c y a t l e a s t t w i c e t h o h i g h e s t f r e q u e n c y o c c u r r i n g i n a r e c o r d , no i n f o r m a t i o n w i l l be l o s t , l i t [ 3 j . I t i s a l s o p o s s i b l e t o t a k e a d v a n t a g e o f t h e f o l d i n g , l i k e M i y a z a k i l i t [ 4 ] , to d i m i n i s h t o number o f p o i n t s 2N+1 by e n l a r g i n g A t , i n o r d e r t o r o d u c o t h e Jimount o f c o m p u t a t i o n , g i v e n a c e r t a i n r e c o r d . M i y a z a k i has chosen A t - 35,5 ^ by w h i c h t h e 60 most i m p o r t a n t c o n s t i t u e n t s , w i t h a few e x c e p t i o n s , wore

I f i t i 3 r e q u i r e d t h a t no f o l d i n e w i l l a r i a e , one s h o u l d c h o s e such a A t t h a t no " r j q u e n c i o s h i g h e r t h a n - — w i l l be p r e s e n t . Assuming t h o p r e s e n c e o f e i g h t h d i u r n a l c o n s t i t u e n t s A t s h o u l d be a t most 1.5 h, u s u a l l y At = 1 h i s c h o s e n . 6. The i n f l u e n c e o f t h e n o i s e on t h e aoonr''.cv o f t h e r e s u l t s . An e x p l a n a t i o n o f t h e c o n c e p t i o n s and f o r m u l a e used i n t h i s p a r a g r a p h , n e c e s s a r y t o s t u d y t h e i n f l u e n c e o f t h e n o i s e x ( t ) , w i l l be g i v e n i n p a r a g r a p h 8. g ( t ) c o n s i s t s o f t h r o e d i f f e r e n t p a r t s , 1) t h e t i d a l c o n s t i t u e n t s w h i c h a r e d i r e c t l y a n a l y s e d . 2) t h o t i d a l c o n s t i t u e n t s w h i c h c a n n o t be computed o r a r e s i m p l y n o t i n t e r e s t i n g . 3) t h e f u n c t i o n x ( t ) , t h e n o i s e . Here t h e i n f l u e n c e o f t h e n o i s e on t h e r e s u l t s o f t h o t i d a l a n a l y s i s i s s t u d i e d . x ( t ) i s g e n e r a t e d by a l l s o r t s o f i n f l u e n c e s . These i n f l u e n c e s have an e r r a t i c c h a r a c t e r so t h a t x ( t ) has an e r r a t i c c h a r a c t e r as w e l l . D u r i n g the r e g i s t r a t i o n one o u t o f t h e many p o s s i b i l i t i e s has been r e a l i s e d . T h e r e f o r e i t can be s t a t e d t h a t g ( t ) i s a f f e c t e d by a s t o c h a s t i c p r o c e s s , o f w h i c h one r e a l i s a t i o n has been r e a l i s e d d u r i n g t h e r e g i s t r a t i o n . I t i s assumed t h a t t h e a v e r a g e c i r c u j n s t a n o e s do n o t v a r y , t h e r e f o r e x ( t ) i s

r e g a r d e d t o be 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 w i t h z e r o a v e r a g e . W i t h o u t d e t a i l e d s t u d i e s o f t h e d y n a m i c a l and p h y s i c a l c h a r a c t e r o f t h e phenomenon

t h e r e a r o no means t o know any d e t a i l s a b o u t t h e p r o p e r t i e s o f x ( t ) , so t h a t t h o h y p o t h e s i s o f s t o c h a s t i c s t a t i o n a r i t y s h o u l d be t r e a t e d w i t h v e r y g r a a t c a u t i o n . E s p e c i a l l y when t h e r e c o r d i s s h o r t t h e n o i s e i n i t may n o t g i v e a r e p r e s e n t a t i v e p i c t u r e o f i t s g e n e r a l f e a t u r e s , t h i s may i n t r o d u c e s y s t e m a t i c e r r o r s i n t h o a n a l y s i s . As t h e r e s u l t s o f t h e t i d a l a n a l y s i s a r e a f f e c t e d by e r r a t i c i n f l u e n c e s , t h e y a r e s t o c h a s t i c v a r i a b l e s t h e m s e l v e s . A n o t h e r e s s e n t i a l d i f f e r e n c e between t h o n o i s e and t h e t i d a l c o n s t i t u e n t s i s t h a t t h e n o i s e has a c o n t i n u o u s s p e c t r u m w h i l e t h e t i d a l c o n s t i t u e n t s have o n l y f r e q u e n c i e s o c u r r i n g a t some d j s c r e t o p o i n t s .

16

L o t g ( t ) " c + E ( c j c o a W | t + d'l s i n 60; t ) + x ( t )

~ m

to be a p p r u c L r o a t o d by f ( t ) a^ +.E^ ( a ; coB(A5j t t b; s i n ^0; t ) .

A p p l i o a t i o n o f t h e l e a s t s q u a r e s method g i v e s t h o n o r m a l e q u a t i o n s ( 2 . 8 ) The r i g h t s i d e o f t h e j * ^ - e q u a t i o n i s 1 N / s 1 N E g ( k A t ) c o s 6 0 i k A t - rr—r E ( ° i co3 6 0 j k A t + d j s i n c O j k A t ) + + x ( k A t ) l 003 i^j k A t 1=0 ~ E x ( k A t) c o 3W j k A t . Assuming t h a t x ( t ) i s a s t a t i o n a r y p'rooess, (8.4,^ i s u s e d . Honco, •j N 1

— E x ( k A t ) c o a to; k A t - r r - r 1 / N w k A t , A+ ^E e cos«A>| k A t d z ( W } liA \

k=-N 1 I c o s W k A t c o s 6 0 ; k A t + i E s i n W k A t c o s 60; k A t • ) N E a i n 6 0 k A t c o s W; k A t - 0 k=-N ^ 1 N L o t T T - r E oos 60kAt c o s W j k A t - X. (.W j k=-N J J sin(N+è)( oo +60j)At s i n ( N + t O ( w - ' ^ j ) A t Thus X j ( c ^ ) ( 2 N + l ) s i n -4~ A t ( 2 N + l ) s i n — ~ ^ A t (6.1) (6.2) and E: x ( k A t ) c o o 6 0 i k A t - / >• ( w )d_z(«o ) S o l v i n g a. b y means o f Cramer's r u l e i t f o l l o w s t h a t ^ ( j c o l u m n ) [ - — F E C. U + r X M d z H 0,0 |_o I o,l " •o,m "•m.o

? I i - f / X J

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m,m ' o m (6.3)

I n t h o f o l l o w i n g t h e d e t e r m i n a n t i n t h o d e n o m i n a t o r i s r e p r e s e n t e d by A and t h e m i n o r b e l o n g i n g t o I j j b y A - j Hence!

L ,

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dz (w) - I

m By t ho e x p a n s i o n o f t h e d e t e r m i n a n t u s i n g e l e m e n t s o f t h e j ^ ' l c o l u m n i t f o l l o w s t h a t

6a; ^ (-1) A,_j / \{oo ) d z ( w ) 1 m

The e x p e c t e d o r mean o f t h e i n c r e m e n t s o f ) i s z e r o o r i n f o r m u l a e E - d ^ ( w) j- o ( 8 . 6 ) . I t f o l l o w s t h a t E I 6uj I ?^ ( - l ) ' ^ ^ A r j E I" ƒ' ^ A r ( w ) d z ( w ) I - 0 So t h e a v e r a g e e r r o r 6aj a r i s i n g i n a j i s z e r o . ' Our f u r t h e r i n t e r e s t i n t h e v a r i a n c e o f t h e e r r o r 6 a j « V a r . 5 a j | = E | ( 6 a j ) ^ . - E^ , 6 a j j » E . ( 6 a j ) ^ | By means o f ( 6 . 4 ) ^ t follow''' t h a t ( 6 . 4 ) ( 6 a •J ^ ( - i T ^ ^ r i / ) d z ( 6 0 ) -! 2 E l a b o r a t i n g t h i s f o r m we u s e i n f a c t 6 a j 6 a j (6a b e i n g t h e c o m p l e x c o n j u g a t e o f 6 a j ) i n s t e a d o f ( 6 a j ) , b u t t h i s i s tho same i n f a c t , because '6aj i s a r e a l q u a n t i t y ,

( 6 a j )

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ƒ3 ( u ) ) i a t h o powor s p e o t r u m o f x ( t ) . T h i s o x p r o s s i o n , w h i c h c a n n o t be e l a b o r a t e d w i t h o u t f u r t h e r a s s u m p t i o n s , i s u s e d t o s t u d y v a r i o u s c a s e s . a ) I n t h o f i r s t i n s t a n c e i t i s assumed t h a t x ( t ) i s u n c o r r e l a t o d on t h o v a r i o u s p o i n t s o f t i m e whereupon i s s a m p l e d , t h a t i s t o say E Jx(k:., At).x(k2 A t ) U, 0 / k2 M o r e o v e r l e t El'(x(kA«))^ ' » (T^» 1» t h i s c i r c u m s t a n c e / S (U) ) i s a c o n s t a n t on t h e i n t e r v a l (O, . L o t/ 5( u ) ) -/S. Tho c o n s t a n t s h o u l d be t a k e n i /i on t h o i n t e r v a l (- ^ > ^ ) i f n e g a t i v e f r e q u e n t i e s a r e t a k e n i n t o a c c o u n t . ( 8 , 2 ) , [At / A r ^ w )\g ( w )/3(to )dw=/3/ Xr (<^ ) X s ( u ) ) d w . By means o f ( 6 . 1 ) f o l l o w s N N

r i l

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1 ^ U PA

/ ( N + ^ ) " "=-N v = - N ^

^ A r Z c o s W r U A t cos tOgVAt / c o s W u A t c o s i O v A t dto ( 6 . 6 )

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. r c o s ( i ( u + v) + c o 8 n( u - v ) -vdfa ^AT ' 1 c o s t O u A t c o s i O v A t d w ^ . j .

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£ ( c o s iO^uAt c o s W j i i A t + oos (o^uAt c o s M g ( - u ) A t ) =

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T h i s r e s u l t i s s u b s t i t u t e d i n t o (6.5)

m E m , , E (-1) r = o s = o m E • s = o r.s A . A . m

4 i ^

th

has been r e p l a c e d by Ip^j r = 0 , 1 , . . , , i n . i n w o r d s by t h o s - c o l u m n . A d e t e r m i n a n t w i t h two e q u a l c o l u m n s i s z e r o . Hence: m r = o - 0 u n l e s s B»j i n t h a t case t h e r e s u l t w i l l be A 30 t h a t m Z s = o (-1)

^ A, . ^E^

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N + ii A T A ( 6 . 7 )

T h i s r e s u l t i s i n a c c o r d a n c e w i t h l i t 1 s e c t i o n 127 P 246, c o n s i d e r i n g that t h e n o r m a l e q u a t i o n s have been d i v i d e d by ( N + a ) , by w h i c h a f a c t o r (N+v,) ocours i n t h e d e n o m i n a t o r . Had t h e n o r m a l e q u a t i o n s n o t been d i v i d e d by (N+s)»

A

j j t h i s f a c t o r w o u l d n o t e x p l i c i t e l y be p r e s e n t , y e t t h e q u o t i e n t w o u l d have been ( N + ^ ) t i m e s s m a l l e r . Goncorning t h e e s t i m a t i o n o f (T by means o f t h e r e s i d u g ( k ; A t ) - f ( k A t ) , t h e OG.'jonce o f t h e c l a s s i c a l c o n c e p t i o n s a b o u t e r r o r s ( i n t h e s e r e s i d u a ) h o l d s , l i t f l l s e c t i o n 124 P 243-245. b) Now t h e w e l l - k n o w n case i s l e f t t h a t t h e s p e c t r u m o f t h e n o i s e i s a constant on t h e i n t e r v a l (O,-^™). I n t h e f o l l o w i n g i t i s assumed t h a t t h e

power s p e c t r u m i s a f u n c t i o n o f th© f r e q u e n c y , h o w e v e r , s u f f i c i e n t l y s m o o t h .

Aa has been s h o r t l y e x p l a i n e d i n p a r a g r a p h 3 t h e m a t r i x o f t h e n o r m a l 'Equations c o n s i s t s r o u g h l y o f s m a l l b l o c k s and i s o l a t e d e l e m e n t s on t h e

P^-incipal d i a g o n a l . I t i s assumed now t h a t I f j ~ 0 f o r a l l i / j w h i c h means vhat ( Wj ~ tOj ) o r T a r e so l a r g e t h a t

'ui(N+;;,)( W j - W j) A t

20 I n t h e s e c i r c m n s t a n c e s w i l l h o l d A j j ~ 0 i / j A ~ I j j A j,j ~ A j j W i t h t h e h e l p o f ( 6 . 5 ) i t f o l l o w s t h a t t Var 6a. J 3 i n ( N + T l ) ( 60 - iA)j)At

\ ; { ( ^ ) . - |2¥rr)8inl"(w - «])At ^''^ f o r w - W j a s h a r p maximum.

O u t s i d e a s m a l l i n t e r v a l a r o u n d ^o =60] t h e c o n t r i b u t i o n t o t h e i n t e g r a l i s v e r y s m a l l . A s s u m i n g /3 ( ) t o bo s u f f i c i e n t l y smooth, we p o s e /S(60 ) (60.) t h r o u g h o u t t h i s s m a l l i n t e r v a l . Ml X f ( M ) / i ( w ) d 6 0 ~/2.(W.)/ ( j ^ ) i ^ ^

'0

I n t h e same way a s . b e f o r e ( 6 . 6 ) i t w i l l f o l l o w t h a t

Thus Var I 6aj. / ^ ^ ^ ^ j ^ " 2-nA{i^i) ^6.8)

w i l l T h i s f o r m u l e i s v e r y s i m i l a r t o ( 6 . 7 ' ) w i t h t h e m a i n d i f f e r e n c e t h a t the l o c a l p o w o r d e n s i t y ^ ( 6 0 j ) i s s u b s t i t u t e d . As e x p l a i n e d i n p a r a g r a p h 3 t h e m a t r i x c o n s i s t s r o u g h l y o f b l o c k s a r o u n d t h e p r i n c i p a l d i a g o n a l , t h e s e b l o c k s a r e f o r m e d b y c o n s t i t u e n t s whoso f r e q u e n c i e s d i f f e r v e r y l i t t l e . These b l o c k s a r e p r a c t i c a l l y s e p a r a t e d aimong t h e m s e l v e s . Assximing t h a t j i s a c o n s t i t u e n t b e l o n g i n g t o s u c h a b l o c k ,Var • 6 a j bo c o m p u t e d . I n o r d e r t o a v o i d c o m p l i c a t e d i n d i c e s , we t a k e a^....a.^ f o r unkno'Wis, t h e f r e q u e n c i e s , . . . 60^ d i f f e r i n g v e r y l i t t l e c o n s e q u e n t l y . The e q u a t i o n s a r e : a, I , + +a I ^ c I + + 0 1 + T — r E x ( k A t ) c o s w, k A t

a.1-, + +a„r „ » c I +,....+ o„t + r - i y r x ( k A t ) o o 8 60^ k A t .

A' = i.n •n.n and A' . t h e m i n o r o f A b e l o n g i n g t o I . r,j hi I n th® same way as a p p l i e d f o r t h e d e d u c t i o n o f ( 6 . 5 ) i t f o l l o w s t h a t V a r f 6 a : ) - E E (-1 ) ' ' ^ ' ^ A'.A! . f x , ( W ) X ( W )/3 ( W ) d w J

J

( A ' ) 2 r=1 s=1

r j s,j r

5 / As m e n t i o n e d b e f o r e , t h e c o n t r i b u t i o n t o t h e i n t e g r a l i n ( 6 . 9 ) i s c o n f i n e d t o a s m a l l i n t e r v a l o f t h e U)-a.xe3 b o n s i s t i n g o f t h e f r e q u e n o i e s Wo assume ^{1^} t o be a c o n s t a n t o v e r t h i s i n t e r v a l . L e t / 3 ( W ) ^/J ( ) 6SJQ b e i n g some p o i n t o f t h e i n t e r v a l . I t f o l l o w s t h a t : ( 6 . 9 ) / S ( ^ J o ) " X, ( w ) X j ( w ) d w ~/S( ƒ X^ ( w )X, ( w ) d w - ^,^^4)^.^ I r,s S u b s t i t u t i n g t h i s r e s u l t i n (6.9)» we o b t a i n a r e s u l t , a n a l o g o u s t o (6,7')» Var 6a; " / l ( ^ o ) A l j j 2 n / 3 ( w j ^, The b e h a v i o u r o f

_ 1 1

TNTi)At~ A'

A'ii

_{"J J} • i s o f i m p o r t a n c e .

The e a s i e s t case i s t h e one a l r e a d y d e s c r i b e d by Munk and Hasselmann l i t 1} . They c o n s i d e r e d two a d j a c e n t c o n s t i t u e n t s . I n t h i s case we o b t a i n t h e f o l l o w i n g d e t e r m i n a n t s A' = 1,1 '2,1 1.2 '1,1 ~ s i n v N + i ) ( Ui^ - 60^)^1 J^+ h ) ( 60^ - )"M sin(N+-;|)( - ^^2)At ^ {^W)U^y - w ^ j A t ~ ~ 6 w i t h e = i - f ( N + i ) ( w , - W^)At b e c a u s e ( - • to^ ) A t i s v e r y s m a l l s i n

i (

W^- ^ 2 ^ A t

i (

A t (N+-2^) ( u j ^ - u ) 2 ) A t 1^ ' I - c t h u s A' - 1 - ( 1 - 2 e.

22 Var 6aj h e n c e : 24" /i {^o) , ( 6 . 1 0 ) T [ t ( - M^) ^ Thu3 v r i t h a g i v e n r e c o r d V a r ( b a j l i s p r o p o r t i o n a l t o c"'' t h a t i s t o say^^ w i t h ( w ^ - i ^ ^ ) " ^ . t h o s t a n d a r d d e v i a t i o n b e i n g p r o p o r t i o n a l t o ( 6 0 , - ^ ^ j ) . A w e l l - k n o w n g r o u p o f c o n s t i t u o n t s whose f r e q u e n c i e s a r e v e r y n e a r e a c h o t h e r i s f o r m e d by t h e c o n s t i t u o n t s T2 , , R2 » 2 • U s u a l l y R2 i s "'^'t a n a l y s e d , t h e r e f o r e t h e a n a l y s i s i s c o n f i n e d t o t h e v a r i a n c e o f t h o g r o u p T2 ' ^2 ' ^2 •

We talce a, f o r a^^^ » '^i "^Sz ^3 ^^'^ ""kg Tho f r e q u e n c i e s a r e : 60^= 29.96°/h » r a d / h W » 30.00°/h = ^ ^ V h W = 30.08°/h » r a d / h 3 2 ~ s i n ( N + i - ) ( i ^ , - ^ ^ ^ j A t The o r d e r o f m a g n i t u d e o f t h e d e t e r m i n a n t A' t u r n s o u t t o be , E ^ t h e r e f o r e i n t h e M a c l a u r i n e x p a n s i o n we h a v e t o t a k e t e r m s up t o c ^ i n c l u s i v e . Terms o f h i g h e r o r d e r a r e n o t n e c e s s a r y b e c a u s e t h e y do n e t c o n t r i b u t e t o t h e t e r m w i t h i n t h e e x p r e s s i o n f o r th© d e t e r m i n a n t . ^ I . s i n ( N . è ) ( ^ i - W 2 ) A t , { ( N + i ) ( ^ ) , " M 2 ) A t ] 1.2 . 1 r - — — w i t h c » 7

L , ^ 3 i n( N + i! ) ( i O , - W 3) A t 1-^ ^ ? 1 - 9 s + 24.3 e and w i t h 60^ - 60^ - 2 ( ^^J.^- t^^) I „ r - ~ 1 - 4e + 4.Ö c 2,3 ^T Ï T T K T Ö ^ ^ ^ ) A t A : A'. 1,1 3,3 1 l - c + 0 . 3 e ^ 1-9C+24.3C 1 -C+ 0 . 3G2 1 l-4e+4.8G2 1-9C+24.3c2 l - 4 e + 4 . 8 c 2 1 1 1-4C l ~ 4 c . 1 1 1-9C l - 9 e 1 1 .1-e 1-c 1 ~ 8G ~ 18c ~ 2E 57.6e Honoe V a r I 6a (6,11) V a r I 6 a ^ 2 V a r { 6a n/3( ) 8 ( N + i ) A t 57.6e^ n / i ( ) ₁₈ ( N+ 2) A t _57.6e2 n/3( <A)O ) ₂ ( N + i ) A t 57.6e2 5 n / i ( i O j l 6 0 n ^ ( U ) o ) (N^è)At[ ( N + i ) A t ( ^ ^ j ) ] * T r T ( W,- W ^ ) ] 11.25^1/3(^^0 ) 3 6 0 n y 3 ( u ) o ) ( N H O A t | ( N + É ) A t( w ^ - ^ 2 ^ ] * t | T ( t ^ ^ - W ^ 40^/3(6^)„ ) 1.25n/3( i^o ) ^ ( N + i ) A t [ ( N + i O A t( 6 o ^ " t ^ j ) ] * T ( W ^ - W ) 2''

24

Por a g i v e n r e c o r d t h e v a r i a n c e s o f t h o e r r o r s a r e p r o p o r t i o n a l t o e - ^ t h u s t o ( i^^- '^2)"^ ' s t a n d a r d d e v i a t i o n s b e i n g p r o p o r t i o n a l t o

I f t h e w h o l e g r o u p , , «2 ' ^2 a n a l y s e d i t can be shown i n t h e same, way t h a t t h e v a r i a n c e s o f t h e e r r o r s become p r o p o t i o n a l t o ( w.^ - W^)"^ >

"" 3 t h e s t a n d a r d d e v i a t i o n s p r o p o t i o n a l t o ( - W^) •

I t s h o u l d be n o t i c e d however t h a t i n v i e w o f t h e n a t u r e o f t h e e m p l o y e d e x p a n s i o n i t was assumed t h a t e « 1 e.g. ^<JQ "that i s t o say

( N + i ) A t ( 60^-6^2)

< 0 . 8 .

W i t h ( e-i^-t^2^

= 4iöÖ

^^"^/^ "^^"^ amount t o a measurement o f a b o u t 100 d a y s .

c ) The f o r m u l a e ( 6 . 7 ' ) , ( 6 . 8 ) , (6.IO) and (6.11) d e s c r i b e t h e dependence o f t h e v a r i a n c e o f t h e r e s u l t s on t h e n o i s e - l e v e l/ 3 ( 6 J ) and on t h e t o t a l d u r a t i o n o f t h o measurement T, Per t h e c a s e o f " w h i t e n o i s e " , (6.7) g i v e s

t h e dependence on cr^ b e i n g t h e t o t a l e n e r g y o f t h e n o i s e x ( t ) , on t h e t o t a l d u r a t i o n o f t h e o b s e r v a t i o n T and on t h e s a m p l i n g i n t e r v a l A t . I n w h i c h way t h e f r e q u e n c i e s h i g h e r t h a n ^ a r e p r o j e c t e d on t h e i n t e r v a l (0. — ) has been d e s c r i b e d i n p a r a g r a p h 5. The same p r o j e c t i o n t a k e s p l a c e c o n c e r n i n g t h e f r e q u e n c i e s o f x ( t ) h i g h e r t h a n ^ . I f A t has been c h o s e n i n such a way t h a t x ( t ) d o e s n o t c o n t a i n e n e r g y i n t h e f r e q u e n c y r a n g e h i g h e r t h a n ^ , / l ( t A j ) i s n o t a f f e c t e d by t h e d i m i n i s h i n g o f A t . M a t u r e l y

• / i ( 6 0 ) i s a f u n c t i o n o f A t i f x ( t ) c o n t a i n s f r e q u e n c i e s h i g h e r t h a n ~ . I f x ( t ) i s a " w h i t e n o i s e " t h i s dependence i s e x p r e s s e d by t h e f o r m u l a

I n case g ( t ) i s an o b s e r v a t i o n c o h t a i n i n g t h e wave movement, x ( t ) b e h a v e s a p p r o x i m a t e l y l i k e " w h i t e n o i s e " i n t h e f r e q u e n c y - r a n g e i n w h i c h t h e t i d a l c o n s t i t u e n t s o c c u r . The f r e q u e n c i e s o f t h e waves, b e i n g much h i g h e r t h a n -~~ , a r e p r o j e c t e d by r e p e a t e d f o l d i n g somewhere on (O, ~~). I n t h i s c a s e t h e n o i s e level/h{00) can be r e d u c e d by d i m i n i s h i n g A t , by w h i c h t h e v a r i a n c e o f t h e u l t i m a t e r e s u l t i s d e c r e a s e d . U s u a l l y , h o w e v e r , t h e h i g h f r e q u e n c i e s a r e a l r e a d y r e d u c e d by t h e m e a s u r i n g i n s t r u m e n t . E x p e r i e n c e shows t h a t f o r A t - 1 h t h e n o i s e l e v e l i n t h e f r e q u e n c y r a n g e h i g h e r t h a n i s so l o w , t h a t f u r t h e r d i m i n i s h i n g o f A t d o e s n o t a f f e c t t h e n o i s e l i v e l / 1 ( W ) on t h e i n t e r v a l (0, ^ ) . Thus i t f o l l o w s by means o f t h e f o r m u l a e ( 6 . 7 ' ) , ( 6 . 8 ) e t c . t h a t t h e a c c u r a c y o f t h e r e s u l t s w i l l n o t be i n f l u e n c e d by A t i n t h i s c a s e .

c o m p u t e r . The p u n c h i n g and t h e check on i t a r e t h e most t i m e c o n s u m i n g work o f t h o w h o l e t i d a l a n a l y s i s . Because i t i s p r a c t i c a l l y i m p o s s i b l e to c a r r y o u t t h e p u n c h i n g w i t h o u t e r r o r s , a good check i s a b s o l u t e l y n e c e s s a r y . I n t h e c o m p u t e r c e n t r e o f t h e D e l f t T e c h n o l o g i c a l U n i v e r s i t y a d e c e n t p l o t t e r i s a v a i l a b l e . The punched t a p e s a r e r e a d by t h e c o m p u t e r , t h e o u t p u t i s v i a t h e p l o t t e r . On t h e d r a w i n g s o b t a i n e d i n t h i s way t h e e r r o r s a r e d i r e c t l y v i s i b l e and can be c o r r e c t e d . T h i s m e t h o d , t h o u g h somewhat r o u g h , g i v e s v e r y s a t i s f a c t o r y r e s u l t s i n p r a c t i c e . The c o m p u t e r . p r o g r a m i s s t r a i g h t f o r w a r d . A s i m p l i f i e d d i a g r a m o f t h e c o m p u t i n g p r o c e s s i s g i v e n i n f i g u r e 2, A few d e t a i l s a r e d i s c u s s e d w i t h t h e h e l p o f t h i s d i a g r a m . The c o m p u t a t i o n o f t h e r i g h t hand s i d e o f th© n o r m a l e q u a t i o n s t a k e s much t i m e b e c a u s e o f t h e g r e a t number o f t i m e s a c o s i n u s o r s i n u s has t o bo e v a l u a t e d . S u b s t a n t i a l r e d u c t i o n o f c o m p u t e r -t i m e was g a i n e d by -t h e a p p l i c a -t i o n o f an i n -t e r p o l a -t i o n p r o c e d u r e , l i -t I 4 p 39. A t a b l e c o n s i s t i n g o f 256 numbers i s s t o r e d I n t o t h e c o m p u t e r :

0(0)

= ( c o s (e-e) +

2

cos

9

+ cos ( 9 + c ) )

e = ^ , 9 = k e k

- 1 , 2 ,

256.

V/ith s •= c(0 - ^) cos 0 w i l l be a p p r o x i m a t e d by cos 0 = o(9)-(0-9) 3 ( 0 )

and s i n

0

by s i n

0

- 3(9) -

(9-0)

o( 9 ) ,

© - i e <

0

< 9

+ ^ e .

I n t h i s way a 4 - d e c i m a l a c c u r a c y i s o b t a i n e d . As t h e s t a n d a r d d e v i a t i o n

A j j

o c c u r r i n g i n t h e r e s u l t s i s a.o, d e t e r m i n e d by t h e q u o t i e n t —j^— , see e,g, f o r m u l a

( 6 , 7 ) ,

t h e s o l u t i o n o f t h e n o r m a l e q u a t i o n s i s computed by means o f m a t r i x i n v e r s i o n , A j j , The p r i n c i p a l d i a g o n a l c o n s i s t s i n f a c t o f t h e numbers — ' ' A ' ^ - j , F o r t h e m a t r i x i n v e r s i o n t h e method d e s c r i b e d i n . l i t 8 i s u s e d . T h i s method i s s p e c i a l l y based on t h e i n v e r t i n g o f s y m m e t r i c a l m a t r i c e s . An A l g o l programra u s i n g t h i s m e t h o d i s d e s c r i b e d i n l i t 9 . However t h i s programm d o e s n o t tr'i^e i n t o a c c o u n t t h e symmetry o f t h e m . a t r i x as f a r as t h e r e s e r v a t i o n o f t l i e c o m p u t e r s t o r a g e i s c o n c e r n e d . T h e r e f o r e a m o d i f i c a t i o n was i n t r o d u c e d l i t [ l o l pp 12 13, so t h a t t h e m a t r i x i s s t o r e d as an o n e - d i m e n s i o n a l a r r a y

• 2

t a k i n g ^ m ( m + l ) w o r d s i n s t e a d o f a t w o - d i m e n s i o n a l a r r a y t a k i n g m w o r d s .

The node f a c t o r s f o r two sup.sequent y e a r s f^^' and f j ^ ' a r e s t o r e d , a f t e r w h i c h t h e node f a c t o r f o r t h e m i d d l e t,, o f t h e o b s e r v a t i o n i s o b t a i n e d by l i n e a r i n t e r p o l a t i o n .

r e a d N,m read Wj ^ g ( k A t ) i = 1 , k = - N , - - N f i g u r e 2 compute g = g ' ( t < A t ) = k = L g { k A t ) k=-N g ( k A t ) - g - N, N

compute matrices of normal equations

compute right side of n o r m a l equations 1

compute solution of normal equation by means of m a t r i x inversion

read t i m e of m i d d l e of observation t ^ , ( V + u )j , f.^and f.^ , names of c o n s t i t u e n t s

compute amplitudes of constituents Hj' =

and m o d i f i e d epoch l i t . [ B ] p 7 7 , g. = ( V + u), + c o . t ^ + a r c t g ^

p r i n t names of c o n s t i t u e n t s and Hj ; gj , i compute 'v V I " = E g ( k A t ) -k = -N m N m N -Z Oi Z g(kAt)cos60, k A t - . E hi E j = o 'k=-N J J=1 ' = -^g(kAt) s i n w.kAt c o m p u t e and p r i n t 2 N - m compute residu r ( k A t ) = g ( k A t ) - f ( k A t ) compute V \ / 2 = Z k A t ) k = - N , , -N k=-N p r i n t g { k A t ) and r ( k A t ) k = -N,- ;N p r i n t [ V V I ] a n d [ v V 2 '

I n l i t p 234 a c o n t r o l e i s g i v e n c h e c k i n g t h e a c c u r a c y o f t h e c o m p u t a t i o n s . Por t h a t p u r p o s e VV1 Z g ( k A t ) - E a. r g ( k A t ) c o s 60. k A t - E hj s g ( k A t ) s i n to. k A t . ' l< = -N J = 1 k=-N and VV2 t r ^ ( k A t ) a r e c o m p u t e d , r ( k A t ) = g ( k A t ) - f ( k A t ) . VV1 = ^VV2 i f t h e e x a c t s o l u t i o n o f t h e n o r m a l e q u a t i o n s has been o b t a i n e d . Thus i f an i n a c c u r a t e s o l u t i o n has been o b t a i n e d , e.g. i n consequence o f an i l l - c o n d i t i o n e d m a t r i x , t h e n t h i s f a c t v d l l become o b v i o u s by c o m p a r i s o n o f [ w i ] and r v V 2 l . The 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 u l t s can r o u g h l y be c a l c u l a t e d by means o f t h e power s p e c t r u m o f r ( k A t ) , as i n d i c a t e d i n p a r a g r a p h 6. M o r e o v e r , i t c a n be seen w h e t h e r t h e r e a r e o m i t t e d c o n s t i t u e n t s t h e a m p l i t u d e s o f w h i c h c a n n o t be n e g l e c t e d . 8. R e c a p i t u l a t i o n o f f o r m u l a e o f s t o c h a s t i c p r o c e s s e s . An e x t e n s i v e d e s c r i p t i o n o f s t o c h a s t i c p r o c e s s e s can be f o u n d i n l i t 2 . A s t o c h a s t i c p r o c e s s i n a f a m i l y o f s t o c h a s t i c v a r i a b l e s x ( t ) , i n w h i c h t b e l o n g s t o a s e t T l i t 2 p 46. T h i s s e t may be a c o n t i n u o u s s e t as w e l l as a d i s c r e t e o n e , i t may ba f i n i t e o r i n f i n i t e . I n t h e p r e s e n t c a s e , i n w h i c h x ( t ) s h o u l d r e p r e s e n t t h e e r r a t i c phenomena i n t h e o b s e r v a t i o n g ( t ) , T i s c o n t i n u o u s and r e p r e s e n t s a p e r i o d , t b e i n g t h e t i m e . So a s t o c h a s t i c v a r i a b l e x ( t ) b e l o n g s t o e v e r y t e T. These x ( t ) may be d e p e n d e n t among

each o t h e r o r n o t . A r e a l i s a t i o n o f a l l t h e s e s t o c h a s t i c v a r i a b l e s c o n s t i t u t e s a r e a l i s a t i o n o f t h e s t o c h a s t i c p r o c e s s ,

A c l a s s i c e x a m p l e o f a s t o c h a s t i c p r o c e s s may be g i v e n by a s e t o f i d e n t i c a l n o i s e g e n e r a t o r s . Each o f t h e s e g e n e r a t o r s e m i t s a s i g n a l . These s i g n a l s a r e i d e n t i c a l i n t h e i r s t a t i s t i c a l b e h a v i o u r , b u t a t e v e r y a r b i t r a r y moment t h e v o l t a g e o f each o f t h e g e n e r a t o r s w i l l be an" o t h e r o n e . The s i g n a l

e m i t t e d b y e a c h o f t h e g e n e r a t o r s i s a r e a l i s a t i o n o f a s t o c h a s t i c p r o c e s s . The p r o c e s s i t s e l f d e s c r i b e s t h e b e h a v i o u r o f t h e s e t o f a l l g e n e r a t o r s . I f t h e s t a t i s t i c a l p r o p e r t i e s o f t h e p r o c e s s a r e t i m e - i n d e p e n d e n t , t h e p r o c e s s i s a s o - c a l l e d s t a t i o n a r y p r o c e s s . O n l y s t a t i o n a r y p r o c e s s e s w i t h z e r o a v e r a g e w i l l be c o n s i d e r e d .

The c o v a r i a n c e f u n c t i o n and t h e power s p e c t r u m a r e most i m p o r t a n t f o r t h e i n s i g h t i n t o t h e b e h a v i o u r o f t h e s t o c h a s t i c p r o c e s s .

Tho c o v a r i a n c e f u n c t i o n p ( i ) i o d e f i n e d by P(T ) = E f X ( t ) / ( t + a ) ] ( 8 . 1 ) I f x _ ( t ) i s a r e a l p r o c e o s t h e n n a t u r a l y x * ( t ) « x( t ) For a w i d e c l a s s o f p r o c e s s e s ( e r g o d i c p r o c e s s e s j h o l d s :

T

-o-oo

T

The i n t e g r a l i s t a k e n o v e r a r e a l i s a t i o n o f t h o s t o c h a s t i c p r o c e s s . P(T) = / e'''*'^ d B(6o ) h o l d s g o o d . J-in I f P(T) i s an a b s o l u t e l y i n t e g r a b l e f u n c t i o n on (-00 , 00 ) - / i > { t ^ ) e x i s t s , so t h a t P ( T ) - J o " ' " * ^ ( t O ) d W and / 3 ( ^ ) » Ö7 / o '•'"^ p ( x ) d T l i t

p

518-522.

/ 3 ( (A) ) b e i n g t h e power s p e c t r u m o f t h e p r o c e s s x ^ ( t ) .

I f x ( t ) i s a r e a l p r o c e s s p{i) i s a r e a l one and even f u n c t i o n , a s f o l l o w s d i r e c t l y f r o m t h e d e f i n i t i o n . A l s o i s a r e a l and even f u n c t i o n . I n t h i s case w i l l h o l d : ) c o 8 WT d w =

2

/ ^ ( w ) c o s 60 T dtO 1 1

A"

/ i C ' ^ ) 2n I P ( t ) c o 9 6 0 T d i : « — / p ( T) c 0 3 6 0 T d - t . / i ( 6 o ) g i v e s t h e d i s t r i b u t i o n o f t h e e n e r g y i n f _ ( t ) o v e r t h e f r e q u e n c i e s , l i t 3 p 7-8, 84-88, i n such a way t h a t ƒ] ^ ( ^ > ) d ^ o - p ( 0 ) = J^"^^ l ƒ x 2 ( t ) d t . put So //3((/0)d/>0 i s t h e t o t a l e n e r g y i n x ( t ) . I n t h o m a t h e m a t i c a l sense /S ( w ) i a d e f i n e d b o t h on t h e p o s i t i v e a s w e l l a s on t h e n e g a t i v e f r e q u e n c i e s . P h y s i c a l l y s p e a k i n g , h o w e v e r , n e g a t i v e f r e q u e n c i e s have no m e a n i n g , t h e r e f o r e v e r y o f t e n t h e s p e c t r u m

on p o s i t i v e f r e q u o n c i o s i s c o n s i d e r e d o n l y , t a k i n g y j ' l c o ) 2y2>{oo) as power specti-um. ( 8 . 2 ) .

ƒ ^3'(w)dtO i s t h o e n e r g y i n t h o f r e q u e n c y r a n g e ( 6 0 ^ , i^.^)

I n t h e f o l l o w i n g t h e p r i m e w i l l bo o m i t t e d .

I n p a r a g r a p h 5 was sho'ffn i n w h a t manner by t h e s a m p l i n g no f r e q u e n c i e s h i g h e r t h a n - ~ c o u l d be d i s t i n g u i s h e d f r o m f r e q u e n c i e s o u t o f t h e i n t e r v a l

( ° .

i f )

-By means o f ( 5 . 1 ) i t can be soen t h a t

/3 ( 6 0 ) b e i n g t h e power s p e c t r u m o f t h e sampled f u n c t i o n . I n t h i s c a s e a l l e n e r g y o f x ( t ) iv. i n t h e f r e q u e n c y r a n g e {- ^ , ^ M o r \^ ^ ^x' > 2k 7T I n e a c h o f t h e i n t e g r a l s = <^ ^ i s i n t r o d u c e d . Henco: n / i ( w ) d w E ^ * y d ( ^ ) d w = / / 3( t o ) d ^ J , 2kn As s t a t e d e a r l i e r by means o f ( 8 . 3 ) i t f o l l o w s a g a i n t h a t f u r t h e r d i m i n i s h i n g o f A t d o e s n o t i n f l u e n c e / ^ ( ^ ) anyhow, as once A t i s so s m a l l t h a t o u t s i d e t h e i n t e r v a l ( - ^ » ^ ) e n e r g y i s p r e s e n t , ( o r o u t s i d e (O, ~) i f o n l y p o s i t i v e f r e q u e n c i e s a r e c o n s i d e r e d ) .

Por e v e r y s t a t i o n a r y p r o c e s s a h a r m o n i c a n a l y s i s can be g i v e n by moans of'. ( 8 , 4 ) x ( t ) - f e'"^ d z ( ^ ) l i t [ 2 ] p 527 i s a s t o c h a s t i c p r o c e s s such t h a t E f ( 2 ( - z t ) ) ( 2( ^ 0 ^ ) - z ( ^.,) ) * I - 0 i f ( 60 , W ) and ( ^0 , ^ ) a r e d i s j u n c t i n t e r v a l s . 4 3 2 1