- Email: [email protected]
On Stochastic Modelling of Hydrologic Data
224
T . E . 1JNNY l l n i v e r s i t y of Waterloo
INTRODUCTION l i y d r o log i c d a t a pr i iiiar i
and t o o t h e r \ r a r i a l i l e s i n c
l i ~ . d i - 1oo g i c
II>~Llrolog \‘;l
1
c t I\c c n
11
t irnc s c r i i ‘ s . d i ffc\rcnt.
Reprinted from Time Series M e t h o d s in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
@
225
45
24
1 2 3 ( a ) N I G E R R I V t R ( M o n t h l y d a t a ) a t KIANEY:
36
( l i m e P e r i o d 1942.1944)
TIMF IN
I
I
I
tdrN’l‘ll<
TIM
(11) NIGER R I V E R ( Y e a r l y d a t a ) a t \rIj2i.ltY: (Time F c r i o d 1940-1966)
I h Yl..Al?S
F i g . 1 . Group f o r i n a t i o n i n ) . e a r l y mid nionthly t i m e s e r i e s o f f l o ~ I n c o n t r a s t , tlic monthly time s e r i e s a r e marAcd 11)- s e a s o n a l it), ‘i’herc i s a l s o t h e prcx-
a c c o r d i n g t o t h e g e o p h y s i c a l year ( F i g . l a ) .
s e n c e o f c h a r a c t e r i s t i c g r o u p s o f h i g h arid lori v a l u e s aillong t h e !‘cars and a l s o w i t h i n t h e y e a r . The d a i l y time series a r e d i f f e r e n t froin n i o n t h l > ~xiid ).early t inre
series i n t h e s e n s e t h a t t h e s e t i m e s e r i e s a r c c h a r a c t c r i s c d 11). tlic o c c u r r e n c e o f sharp peaks and e x p o n c n t i : i l deca!-.
The c : i u s c - c f f e c t
r e l a t i o n s h i p i n r a i n f a l l - r u n o f f process i s s t r o n g e r i n t h e s h o r t
iii-
t e r v a l time s e r i e s . The p e r i o d i c i t i e s a r c i n d u c e d i n t h e s t r e a n i f l o w time s e r i c s liy
t h e geophysical cycle.
This is reflected i n thc o e c ~ i r ~ c ~ ~ ofi ehigl: c
p r e c i p i t a t i o n and h i g h r u n o f f d u r i n g tlic
nionths, tind lo\c pre-
smiiiicr
c i p i t a t i o n and low r u n o f f d u r i n g t h e w i n t e r iiioriths i n N o r t h c r r i c 1 i mates.
Means a n d v a r i a n c e s o f h y d r o l o g i c
larger i n
sLiinnier
tiiiic
a n d snial ler i n w i n t e r riioriths
scric’s
.
c i t y i n time a l s o o c c u r s i n t h e forin o f trc>nd i n the
:ire
foiiiicl
I ~ ’ u r t ! i c ~ rlion-hciiiiogcri, rl;it:i
;is \ i c ’ 1 1
;is
iii
forin o f g r a d u a l and suddcri v a r i a t i o n s i n t h e stoc1i;ist ic- n:itiiw
226 o f t h e JLita. 'l'he i i i i p l i c a t i o n a r i s i n g from t h e p r e s c n c c o f p e r s i s t e n c c , e s p c c i -
t h a t c o r r e s p o n d i n g t o prolonged wet and d r y p e r i o d s , i s s i g n i -
all!.
f i c a n t from t h e p o i n t o f \.icw of Liater resources . d e s i g n and o p e r a t i o i l s .
Both s h o r t - t e r m p e r s i s t c n
and 1 ong- t crm
p e r s i s t e n c e a r e i i i t c g r a l p a r t s o f h y d r o l o g i c t i m e s e r i e s which t l l e r c b y bccoiiie d i f f e r e n t from those tiine s c r i e s found i n o t h e r d i s c i p 1 i n e s swh
stock-niarket anal>.sis.
;IS
An 1 i i s t o r i c : r l 1). r c c o r d e d s t r c a n i f l o i i tiinc s c r i c s c m lie considcred
t o lie d c r i \ . e i i frorii
;I
s t o c l i a s t i c g e n e r a t i n g nicchanisiri t h a t e v o l v e s
a c c o r d i n g t o c e r t a i n proba1)i l i s t i c l a i c s .
If i t \ccrc possi1)1c t o d c -
c i p h e r t h e s e proIia1)i l i s t i c I a i \ s , t h e n t h e r e r i o ~ i l dI)c a s a t i s f a c t o r ) . iiiodcl for. t h e time s e r i e s .
Ilowcvcr, t h i s procedure i s d i f f i c u l t ,
if
n o t p r a c t i c a l i y i r i i p o s s i l ~ l c . I3ccaiise s t r e a i i i f l o ~ i d a t a r c p r c s c n t oiil]. a s i n g l e t iriie s e r i e s
o f a s t a t ioniir.!.
;I
check
;is
p r o c e s s and a l s o
t o ~ h e t h e rs u c h a series foriris p a r t ;in
crgodi c proccss i s inipossilile.
I n s p i t e o f t h i s , t h e f o l l o w i n g procedure i s o f t e n c a r r i e d o u t i n c o n n e c t i o n Lci t h strcainflo\.; time s e r i e s rriodc11 i n g : Gi\.cn
i)
:in
liistorical1)- recorded d a t a series:
;\ssiiiiic
e r g o d i tit).
i i ) . \ s s ~ i r i i e ;I s t o c l i a s t i c process i i i J ( : a I c u l a t c c.c>rtain p a r a n i e t e r s from t h e r c c o r d c d d a t a arid : i s s m e t h a t tticsc paraiiictcrs a r c t h e s t a t i s t i c s f o r t h c
ii-]
t h a t the s o - d e f i n e d s t o c h a s t i c process i s t h c gc'nerat i 11s iiicchani siii froiii w l i i ch t h e recorded d a t a i s d c r i \ ~ e das ;A s m i p l e .
:\ssiiiiic
'I'hcrcl)~., :ind
iii
t h so
iiian\.
assciiiqit i o n s ,
;i
ered t o lie o h t a i r i c d for. the t i m e s e r i c s .
s t o c l i a s t c iiiodel i s c o n s i d I t s h o u d IIC cniphasiicd
here t h a t t h i s p r o c e d i i r e h e a r s no r e l a t i o n s h i p t o t h e p h y s i c a l p h c n ~ I ~ L ' I I I C on I ~ ; ~\chicti
t h e d a t a has ticen r c ~ c o r d c d . I n a d d i t i o n , t h i s p r o -
c c d u r e i . 0 ~ 1 1 c l o f t c n 1)ccoiric i r r c l c v a n t r\.licn i t s r e s u l t s a r e a p p l ictl
i n connect i o n
iii
t h water r c s o i i r c r s p1anriiiig
t i o i i s c;111 a l s o I)c> r a i s e d view.
froiii
;iiicl
iiianligcnicnt.
Olijec-
l i c u r i s t i c and p h i 1 o s o p h ~ i cp o i n t s o f
A l l t h a t is r e q u i r e d t o coiiiplctc t h c :iIio\,c iiroccdiirc i s
;I
227 feiv p a r a m e t e r s
-
a t t h e most t h r e e o f f o u r
-
d e t e r m i n e d from t h e
d a t a ; o t h e r w i s e t h e whole s e t of d a t a s o l a b o r i o u s l ) . c o l l e c t e d c a n be d i s c a r d e d .
An Iixamp 1c Figure 2 represents
;I
b i v a r i a t e time s e r i e s o f l e n g t h 1 0 0 iini t s .
Soiiie p o i n t s i n t h e s e r i e s a r e shown liy open c i r c l e s mid o t h e r s by
s o l i d ones.
T h i s w i 11 he e x p l a i n e d s u b s e q u e n t l y .
No a t t e m p t has
been made h e r e t o g e n e r a t e an ARblA p r o c e s s o r any o t h e r s t o c h a s t i c
process t o represent thesc time s e r i e s .
I t was not t h e p u r p o s e f o r
They a r e t a k e n from a r e c e n t
Lihich t h e s e s e r i e s Were ' r e c o r d e d ' .
t h e s i s cotiip1etcd a t t h e U n i v e r s i t y o f W a t e r l o o (Flclnncs, 1981)
t%.lJ.
and i t is achnowledgcd h e r e .
The f a c t rem:iins
t h a t , i f t h e d a t a had
h e e n p r i n t e d i n a t a l i u l a r f o r m a t i t would h a v e been eas)' t o e x t r a c t p a r a m e t e r s and d e v e l o p a n ARbN process as a model f o r t h e s e r i e s . This could r e s u l t i n a delusion of t h e r e a l i t y .
t 0
x-
t
I
I
+3
0
N
c
-3 tlmi unit5
F i g . 2 . B i v a r i a t e time s e r i e s a c c o r d i n g t o p r o b a b i l i s t i c laws d e s c r i b e d i n e q u a t i o n s ( 1 2 ) and (1.3) (from r e f : MacTnncs, 1 9 8 0 ) .
228 I n order. t o i l l u s t r a t e t h e n i a i n p o i n t o f this d i s c u s s i o n , con-
s i d e r t h e g c l n e r a t i n g riicchanisiii o f t h e s e r i e s i n 1:ig. 7.
i e s arc’ o h t a i r i c d
;is
saiiiplcs of
L
1
tlio
s e p a r a t e processes.
These s e r -
‘I’he s o l i d
21 I1 d
and
‘r (t-1.)J
: ; [E ( t ) E-
=
[:;.05
:L]
In t h e al)o\.e t* i s the f i n a l time p o i n t a s s o c i a t e d w i t h t h c regime o f t h e e a r 1 i c r d e s c r i b e d p r o c e s s iininediately p r e c e d i n g t h e regiiiic
associated x i t h the random w a l k niodcl.
‘The length o f r u n (nuinlier o f
time p o i n t s ) in each regime is g e n e r a t e d b y a n e q u i l i b r i i i i r i t l i s c r c t e
rencwa 1 process g i\.cn liy: R
=
(.?I
1 + Ii(n,0)
~ h c r cI3 i s
;I
Iiinoniial r a n d o m variahlc d e s c r i h c d by
which i s t h e p r o b a b i l i t y o f exactly 13 o c c u r r e n c e s i n TI i n d e p e n d e n t B e r n o u l 1 1 t r i a l s with t h e proliahi 1 i t y o f a n o c c u r r e n c e i n any one t r i a l liciiig 8. 5
u c h t l l < lt
The \/alucs o f n and 8 a r e 55 and 0 . 2 , rcspcctivcly,
229
Though thcsc s c r i c s
art i f i c i n l crcat ions, t h t y
i1rc
practical rcIcvancc.
110
have
soiiw
i n most hydrologic d;it;r t1ic1.c :ire s e v c ~ r ; i l
gencrating proccsscs a t work otic aftcr the othcr ;it d i ffcrcnt per-
iods i n t imc process. AS
illid ;in
tlicsc cannot be a\cr;igcd i nto :I s i n g I c x gcticr:tt i ng ex:mipIc, ii biviiriatc :lI\Flh p ~ o c c dcrivcxl ~;~ For the
whole s c r i c s i n J:ig. 2
twi11d
lw h;rscd on
;I
v;iluc for tlic f i r s t order
autocorrclation coefficient significant 1y cii ffcrcnt from that for thc process from w h i TJI
t hc $01 i d c i rc- I c?; ;t rc oltta i ricJ.
'I'ltc s o 1 i tl
c i r c l e s ;ind the oi)cn circles rcprcsent d;tt;t s i t t i v a s t 1y tlit'fcxrcitt persistcncc charactcrist ic
iitid
:in)' t intc series m o ~ l c ~th;it l tlocs riot
cons idcr t h i s d i ffcrcnct shou1d hc t rcat cd
;is
itns;:iti st';tctory .
On Pcrs i st cnct I'crs is t c n c t imp 1 i cs
cert :I
ii
iii
dctcrmi n i s t i c rc'1
succcssivc \alms of d a t a i n thc t i m c s t r i c s .
;it
Stich
i onsh i j) I w t \ c w i > ;I
rc1;tt ionsttip
may be diic t o t h c fact t h a t the caitsc r c ~ s i i l t i n gi n t h e cf'f'ect pvr-
sists f o r
succcssivc
:I
sp:in of t inic longcr than one or ntorc iiicrcmcttts I ) c t t \ c c r i d:ttittii.
Thc
ciittsc
twitig of liriti t c d Icngth, thc i ~ ~ i s t -
encc introduced h y tlic c;iiisc i s a l s o o f I imi t c d Icnptli. Siic.rxvtliiig parts of t h e t iinc scrics may csliihit Ji ffcrciit pcrsistcvici.. \ \ h i ~ * his :I I'crsistcnce is oftcii charactcriscd I)\- thc c*or1*i~~ogi-;i111 p l o t of tlic corrclatioii cocffir.icnt ;tg;tinst lag. (hrrc1;tt i o n coe f f i c i e n t : i t each lag is Jcterinincd b\- :I sc;inniiig p r o ~ + ~ ~ J:i~-ross ~irc* t h c whole data. 'l'his hits i i n avcr;tging cfl'cvt. 'I'ftc* f a 1 lacy o!' tisi~:g t h i s indicator t o denote pcrsistciicc is apparent i n tlrc
annual hydro1ogic s c r i c s . efficicrits a t lag 1
:tiid
I n most
;intiii;i
~*;tsic of'
1 s e r i w tlic correl;it i o n
a t higher l a g s itre
foitiid
k-o-
t o Iic irisigtii f i -
cantly diffcrcnt from x r o , ~nciiriingt h i i t tlicy 1 i c v i t h i n the coiifidence bounds of simi lar cocfficicnts f o r
it11
indcpcridcitt s c r i c s .
Thus, i n t h e l i t e r a t u r e , and in water rcsourccs applications :is w l l , annual series a r c oftcn trcatccl as indcpcndent. Ilowcvcr, i t is swit from Fig. 1 t h a t thcrc i l r C \W 1 1 d e f i tied grotip format ions i n :innti:i 1
230 s e r i e s which i s i n d i c a t i v e of s t r o n g p c r s i s t c n c e i n s t r e t c h e s of the series.
I t i s a l s o v a l i d t o note i n t h i s conncction t h e re-
c e n t l y r e c o r d e d examples i n v a r i o u s p a r t s o f t h e world o f 7 bad years, etc. F u r t h c r m o r c , i t i s d i f f i c u l t t o o b t a i n a r c l i a b l c and s t a b l e e s t i m a t e f o r t h e c o r r e l a t i o n c o e f f i c i c n t from t h c d a t a s e r i e s .
With
r e g a r d t o t h e d a t a s c r i e s i n T a b l e 1 of I c n g t h 59, d i f f e r e n t scct i o n s o f t h i s s e r i c s h a v e b c r n used i n c as c s numbcrcd 1 t o 10 f o r TABLE 1 . E x p l a n a t i o n o f t h e Cascs o f Timc S e r i e s Case Xumber 1
Data Values C o n s i d e r e d
R e m a rh s
For T h e Case
1 t o 50
S c r i c s "A" Data P o i n t s . Monthly Discharge i n C.M.S.
2 t o 51
3 2 5 , 228, 2 0 1 , 1 4 3 , 103, 83,
3 t o 52
4 t o 53
l S 1 , 3 0 0 , 251, 6 4 0 , 511, ,708,
5 t o 54
1 2 3 , 1-17, 278, 2 3 2 , 2 6 4 ,
6 t o 55 ?
7
t O
56
8
8 to 5:
3
!I
10
t o 58
248,
2 8 1 , 399, S31, 5 7 2 , 7 0 0 , 4 7 7 , 40!1,
31.3,
2 0 2 , 1 11, 1 4 9 , 12x,
1 4 5 , 3 0 4 , -711, 92,
s:7,
74, 2 3 0 , 1 - 2 ,
389, 162, 1 3 ,
1-15,
231 u n l i k e l y t h a t t h e c o r r e l o g r a m would t e n d t o a c o n s t a n t " p o p u l a t i o n " In e f f e c t , a sample from a s t a -
value with longer length of data.
t i o n a r y and e r g o d i c s t o c h a s t i c p r o c e s s c a n n o t f o r m a model f o r t h e observed time series.
.
0 6 0.4 0.2
0 -0.2
.
0 6
0.4
g
G
-0.2
0 -0.2
Ir
0.6 r'
g
5
0.4
c
w
3.2
2
0
-3.2 fl
3.b
3.4 3.2
0 3.2 7.h
1.4 1.2 0
-u.
2 1
3
h
9
1
2
1
5
'l'hc presciice o f p e r s i s t e n c e i n ! . c : i r l ! ~time s e r i c s , and i n str'cain-
flow time series i n p a r t i c u l a r , \<:is f i r s t h r o i i g l ~ t t o the, : i t t c i i t i o n of c n g i n c c ~ - sI)?
t h e pionceriiig s t u d i e s of (Ilurst, 1951, 1 9 S h ) on l o n g
tcrni s t o r a g e rcqiiireiiicnts i n the> N i l c I h s i n .
232 Through a n e x t e n s i v e a n a l y s i s of g e o p h y s i c a l time s e r i e s (cons i s t i n g o f a n n u a l v a l u e s ) , i n c l u d i n g t h e e x t r e m e l y long time s e r i e s o f t h e a v e r a g e a n n u a l f l o w on t h e r i v e r k i l e , and also o f normal i n d e p e n d e n t s e r i e s g e n e r a t e d by v a r i o u s e x p e r i m e n t s , t l u r s t d e r i v e d the
re 1a t i o n s h i p RK/SN
01
H N
where N i s t h e l e n g t h o f d a t a .
I n the above, R
N
i s t h e ad u s t e d
r a n g e and i t c a n h e e x p r e s s e d f o r an a n n u a l time s e r i e s {x , i = l , 2 , . . . ,
N) o f
and s t a n d a r d d e v i a t i o n N
l e n g t h K y e a r s , w i t h mean x
The term R ./S
i \ , N
[=i N) i s
the adjusted rcscalcd range.
h i
S"
as
c s t iriiatc of
H, d e n o t e d by K l l , was d e f i n e d a s
Hurst observed t h a t t h e value of the exponent I1 i n r e l a t i o n s h i p (0) h a s on the a v e r a g e a v a l u e o f 0.7.3 f o r t h e g e o p h y s i c a l t i m e s e r i e s , and 0 . 5 f o r t h e normal i n d e p e n d e n t s e r i e s .
In hydrologic litcratur-c,
t h e d i s c r e p a n c y i n t h c v a l u e s o f t h e e x p o n e n t i n h y d r o I O g i c a 1 time
s e r i e s and t h a t i n a l l i n d e p e n d e n t s e r i e s has lice11 called t h e llurs-i phenomenon and t h e e x p o n e n t i n r e l a t i o n s h i p Hurst c o e f f i c i e n t .
((I)
i s noii I\rlo\in as t h e
A v a l u e o f t h e ilurst c o e f f i c i c n t g r e a t e r t h a n
0.5 i s c o n s i d e r e d t o i n d i c a t e l o n g - t e r m p c r s i s t e n c c . 1)arnmcter v a l u e s f o r t h e l l u r s t c o e f f i c i e n t d e r i v e d frorii h i s t o r i
c a l r e c o r d i s found s e n s i t i v c t o non-hoiiiogerieities
i n d a t a inc1udi:ig
c h a n g e s i n t h e p r o b a l ~ il i s t i c laws d e f i n i n g t h e g c n e r ; i t iiig of t h e d a t a .
c i e n t g r e a t e r than O.S. to
iiic~c~iiatiisiii
Klemes [ 1 9 7 4 ) h;is sho\\,ri t h a t i n d c ~ p c n d c n t s c r i e s
s i s t e r i c e o f o r d e r zero) w i t h f l u c t w t i n g A r c f e r c ~ n c ciiia!. ~
iiic;ins
-
c s h i I 1 i tcil
a l s o be ~ti:iclc
;it
;in
(~CI.-
Il-coct
this point
Wing (19Sl) who, w h i l e coirimcnting on I l u r s t ' s o r i g i n a l p : i p c > r cx-
pre.;sed doulit a b o u t I l u r s t ' s f i n d i n g s a n d i 1 n p 1 i c d t h a t p e r h a p s
233 d i s c o n t i n u i t i e s i n t h e r e c o r d could have caused an H - c o e f f i c i e n t greater than 0.5. E x t e n s i v e a n a l y s i s of l a r g e assemblage of r e c o r d e d d a t a by ( H u r s t , 1951) encompassing many g e o p h y s i c a l phenomena, e . g . , r a i n f a l l , runo f f , l a k e l e v e l s , t r e e r i n g s and mud v a r v e s , showed t h a t groups of high and low v a l u e s t e n d e d t o o c c u r more f r e q u e n t l y i n n a t u r a l e v e n t s than i n p u r e l y random e v e n t s .
I n a d d i t i o n , H u r s t observed w i t h p a r -
t i c u l a r r e f e r e n c e t o annual streamflow time s e r i e s t h a t groups a s s o c i a t e d w i t h s t r e t c h e s of f l o o d s and d r o u g h t s o c c u r r e d w i t h o u t any r e g u l a r i t y e i t h e r i n t h e i r d u r a t i o n or i n t h e time of o c c u r r e n c e ( F i g . 1 ) . T h i s , t h e n , i s t h e fundamental d i f f e r e n c e between n a t u r a l s t r e a m flow time s e r i e s and o t h e r p u r e l y 'man-made'
s e r i e s such a s those
derived from random p r o c e s s e s , a u t o r e g r e s s i v e p r o c e s s e s and f r a c t i o n a l Gaussian n o i s e s e q u e n c e s . On Some of t h e Commonly Used Models f o r Hydrologic Time S e r i e s C e r t a i n s t o c h a s t i c p r o c e s s e s have been s u g g e s t e d by h y d r o l o g i s t s as models f o r time s e r i e s .
S.pecified s t a t i s t i c s of t h e chosen p r o -
cess a r e a d j u s t e d t o have n u m e r i c a l v a l u e s e q u a l t o t h a t of e q u i v a l e n t p a r a m e t e r s e v a l u a t e d from t h e observed s e r i e s .
The term used
i n h y d r o l o g i c l i t e r a t u r e i s p r e s e r v a t i o n . Thereby i t i s meant t h a t a l l sample f u n c t i o n s o b t a i n e d from t h e p r o c e s s d e l i v e r t h e same p a r a -
meter v a l u e s .
Considered s i g n i f i c a n t i n t h i s c o n n e c t i o n are one o r
more of t h e f o l l o w i n g :
Mean o f t h e s e r i e s , v a r i a n c e , c o r r e l a t i o n
c o e f f i c i e n t s a t l a g 1 and a t h i g h e r l a g s , and Hurst c o e f f i c i e n t . Two commonly used models a r e d i s c u s s e d below. Mandelbrot and van Ness (1968a) d e f i n e f r a c t i o n a l Brownian motion p r o c e s s (fBm) a s : BH(t) =
t
~
fi
1
(t-v)
H-0.5 dB(v) ; 0 < H < 1
(9)
--M
where dB(v) i s t h e d i f f e r e n t i a l of t h e Brownian motion p r o c e s s and H i s a s p e c i f i e d exponent.
T h i s p r o c e s s r e d u c e s t o a Brownian motion
process (Wiener-Levi p r o c e s s ) f o r H = 0.5.
234 The f r a c t i o n a l G a u s s i a n n o i s e p r o c e s s (fGn) i s d e f i n e d a s t h e d e r i v a t i v e o f t h e above p r o c e s s .
The d i s c r e t i s e d v e r s i o n , t h e d i s c r e t e
f r a c t i o n a l G a u s s i a n n o i s e s e q u e n c e (dfCn), i s d e f i n e d by M a n d e l b r o t and van Ncss
(lY68a) as f o l l o b s :
L
1 11-0.5 B [t) = ~___ c (t-v) AB(v+l) 11 v=-m
4G-i
b h e r e t h a s i n t e g e r v a l u e s from
-m
F u r t h e r AB(v) i s t h e
t o present.
f i n i t e d i f f e r e n c e i n t h e Brownian m o t i o n p r o c e s s ~ ~ i At Bh ( v ) = H ( ~ + I + E ) a n d x ( ~ )i s t h e r e a l i z e d v a l u c o f t h e p r o c e s s a t t i m e p o i n t t . t ' h e f a c t t h a t dfGn h a s t h e i s y m p t o t i c p r o p e r t y t h a t i t s a d j u s t e d
B(v+l),
r a n g e I1
N
d e f i n e d i n e q u a t i o n ( 7 ) i s s u c h t h a t 11
N
a
Nl' i s t h e p r i m a r y '
r e a s o n f o r d e v e l o p i n g t h i s p r o c e s s as a model f o r g e o p h y s i c a l t i m e series.
By h e e p i n g ( p r e s e r v i n g ) I 1
= t h e llurst c o e f f i c i e n t derived
from t h e r e c o r d e d time s e r i e s , t h e s a m p l e f u n c t i o n s o f dfCn a r e made t o possess t h e same I I u r s t c o e f f i c i e n t . There i s a b s o l u t e l y no o t h e r s i m i l a r i t y whatever between sample f u n c t i o n s of dfGn and r e c o r d e d time s e r i e s .
The p r o p o n e n t s o f t h e
dfCn models c l a i m t h a t i t i s c a p a b l e o f p r o v i d i n g s a m p l e s w i t h ext r e m e s ( h i g h s and lows) t h a t a r e more s e v e r e t h a n t h a t i n t h e h i s toric series
( M a n d e l b r o t and Wallis, 1 9 6 % ) . T h i s h a s n o t b e e n demon-
s t r a t e d i n m y c o n v i n c i n g manner. There i s n o d o u b t t h a t t h e r e i s a t h e o r e t i c a l b e a u t y
in
the frac-
t i o n a l Brownian m o t i o n p r o c e s s i t s e l f . The t h e o r y and t h e u n d e r l y i n g a s s u m p t i o n s o f b o t h f B m and dfGn a r e p r o v i d e d i n a s e r i e s o f a r t i c l e s 1 1 ~M a n d e l b r o t and van Ness (1YOSa)
and M a n d e l b r o t and W a l l i s (1968b,
1 9 6 9 a , b , c ) . Review a r t i c l e s i n d i c a t i n g t h e i r r e l e v a n c e t o h y d r o l o g y a r e a l s o g i v e n i n C h i , e t a1 ( 1 9 7 3 ) , O'Connel (1974) and Lawrence and Kottegoda
(1977).
? h e dfGn i n v o l v e s summation from i n f i n i t e p a s t to t h e p r e s e n t . I n o t h e r w o r d s , what happened i n t h e d i s t a n t p a s t is c o n s i d e r e d as an i n f l u e n c i n g f a c t o r i n t h e p r e s e n t occurrence. This concept i s t h e a n t i t h e s i s o f t h a t o f t h e blarhov p r o c e s s e s and Marhov c h a i n s . I t
235 s h o u l d b c c o n s i d e r e d i n t h i s c o n n c c t i o n t h a t Markov c h a i n s h a v e n o t o n l y t h c o r c t i c a l c l e g a n c c b u t t h e y also h a v e f o u n d wide a p p l i c a t i o n s
i n h y d r a u l i c s and h y d r o l o g y .
These i n c l u d e a p p l i c a t i o n s i n s t o r a g e
t h c o r y (hloran, 1951; I ' r a l ~ h u , 1 9 6 7 ; I,10>~d, 1967; Klcmcs, 1981; S o a r e s e t a l , 1 9 7 i ) , and i n e s t i m a t i o n t h e o r y and f o r c c a s t i n g
(Jazwinski,
For a comprehensive s e t of a r t i c l e s with a p p l i c a t i o n s i n
1970).
hydrology s e e C h i u (1978) and a l s o Unny ( 1 9 7 7 ) . As a result
af t h e d i f f i c u l t i e s i n v o l v e d i n t h e i n f i n i t c summation
a p p r o x i m a t i o n s h a v e becn d c v c l o p e d for t h e dfGn.
These i n c l u d c t h e
Types I a n d J I a p p r o x i m a t i o n s o f b l a n d e l b r o t and Wallis ( 1 9 6 9 c ) , t h e f a s t f r a c t i o n a l Giiiissian n o i s e a p p r o x i m a t i o n ( f f G n ) o f M a n d e l b r o t , ( 1 9 7 1 a ) , and t h e f i l t e r e d fGn of
M a t a l a s and W a l l i s ( 1 9 7 1 b ) .
The a u t o c o v a r i a n c e f u n c t i o n o f dfGn i s f o u n d t o t e n d t o z e r o v e r y I t i s primarily t h e r e s u l t o f n o n - s t a t i o n n r i t y i n t h c dfGn.
slowly.
111 f a c t , n o n - d e c a y i n g c o r r e l o g r a m s arc c o n s i d e r e d t o i n d i c a t e non-
s t a t i o n a r i t y i n t h e d a t a according t o t h e procedure adopted i n t h e ARISlA n i o d c l l i n g o f t h e time s e r i e s (Box and J e n k i n s , 1 9 7 0 ) .
In t h c s c
i n s t a n c e s . t h e d a t a i n t h e s e r i e s arc s u c c e s s i v c l v d i f f e r e n c e d . i f r v d t i m e s , u n t i l a d e c a y i n g c o r r e l o g r a m i s o b t a i n e d . Elodell-
i n g t h e n i n v o l v e s f i t t i n g an ARMA model o f o r d e r ( p , q ) o f t h e form O(B) ( x
t
-x)
= 0(B)a
(11)
t
to the differences data.
In t h e above,
0 ( B ) = ( 1 - @ , B - O 2 B 2.. .dpBp)
and
8 ( B ) = ( 1 - 0 1 B - e , B 2 . ..8 B q ) q
(12)
w i t h B 'is t h e backward s h i f t o p e r a t o r and t h e @Is and 0 ' s a r c s p e c i fied coefficients.
Further, a
t
i s normal i n d e p e n d e n t l y d i s t r i b u t e ?
x
i s t h e mean o f random \ a r i a b l c h i t h z e r o mean and v a r i a n c e u a' and t h e s e r i c s . The s t a t i s t i c s o f t h e ARMA p r o c e s s are f u n c t i o n s o f t h e
c o e f f i c i e n t s , a s well as t h e s p e c i f i e d v a l u e s f o r
x and u a .
Assunling
e r g o d i c i t y , t h e s t a t i s t i c s a r c e v a l u a t e d from t h e r e c o r d e d series, thus enahling t h e determination o f t h e c o e f f i c i e n t s i n t h e A I W I (p,q) process.
17ic s a m p l c f u n c t i o n s p r e s e r v e t h e m c l n , t h e v a r i a n c e and
t h e c o r r e l o g r a n i . A p a r t from t h i s p r e s e r v a t i o n , t h e r e i s n o s i m i l a r i t y
236 w h a t e v e r between sampel f u n c t i o n s of t h e ARMA p r o c e s s and t h e recorded s e r i e s . S i n c e t h e p u b l i c a t i o n of t h e book by Box and J e n k i n s ( 1 9 7 0 ) , t h e r e h a s b e e n a f l o o d of a r t i c l e s on t h e ARMA ( p , q ) models o r , e q u i v a l e n t l y , on ARIMA ( p , d , q ) models i n t h e h y d r o l o g i c c o n t e x t . S e a s o n a l and n o n - s e a s o n a l models and many o t h e r i n f i n i t e v a r i a t i o n s of t h e s e models have b e e n r e p o r t e d .
"Best" models h a v e b e e n de-
t e r m i n e d f o r a g i v e n t i m e s e r i e s u s i n g c r i t e r i a s u c h as t h e Akaike i n f o r m a t i o n c r i t e r i a (Akaike, 1 9 7 4 ) .
I t i s s u r p r i s i n g t h a t much of
t h e developments i n t h e ARIMA m o d e l l i n g d u r i n g t h e l a s t d e c a J e h a s t a k e n p l a c e w i t h o u t any c o n c e r n b e i n g e x p r e s s e d as t o t h e o b j e c t i v e s of m o d e l l i n g and t h e a p p l i c a t i o n made of t h e s e models.
The scanrled
p a r a m e t e r s employed i n t h e development of ARIMA models f o r g i v e n s t r e a m f l o w t i m e s e r i e s are of q u e s t i o n a b l e r e l e v a n c e b e c a u s e of t h e f a c t t h a t t h e s e n a t u r a l g e o p h y s i c a l t i m e s e r i e s do n o t e v o l v e according t o simple p r o b a b i l i s t i c l a w s .
Despite these shortcomings,
t h e ease t h a t accompanies t h e u s e of preprogrammed l o g i c h a s s t i m u l a t e d a n a c c e p t a n c e of t h e s e models.
I n many c a s e s model d e v e l o p -
ment f o r a g i v e n s e r i e s h a s b e e n r e d u c e d t o t h e l e v e l of a mecha n i s t i c p r o c e d u r e c a r r i e d o u t on t h e machine.
Inference about
s t r e a m f l o w phenomena a r e b e i n g made w i t h o u t any r e f e r e n c e t o t h e p h y s i c a l n a t u r e of t h e problem a n d , i n e x t r e m e cases, w i t h o u t any consideration other than the data sheet.
The o n l y p r e r e q u i s i t e t o
p r o v i d i n g a n i n f e r e n c e h a s become a c a p a c i t y t o program; i n f a c t much less b e c a u s e t h e programs a r e a l r e a d y a v a i l a b l e on t h e s y s t e m . On t h e Requirements of Models f o r T i m e S e r i e s i n Hydrology Models f o r h i s t o r i c a l l y r e c o r d e d t i m e s e r i e s i n t h e h y d r o l o g i c c o n t e x t a r e r e q u i r e d so t h a t s u c h models c a n b e used f o r e x t r a p o l a t i o n of d a t a i n t o f u t t i r e times beyond t h e p r e s e n t .
I t h a s be-
come a n a c c e p t e d p r a c t i c e i n t h e l a s t two d e c a d e s o r s o t o c o n s i d e r t h e s e e x t r a p o l a t e d d a t a i n t h e d e s i g n and p l a n n i n g of water resources systems.
T h i s i s b a s e d on t h e u n d e r s t a n d i n g t h a t t h e p a s t
r e p r e s e n t e d by t h e h i s t o r i c a l d a t a w i l l n e v e r b e r e p e a t e d and t h a t
237 data series employed in water resources applications should be such that they are likely to occur in probabilistic terms in the performance time horizon of the system which lies in the future. Data extrapolation is required in various formats.
Specifical-
ly four different formats are discussed below: a) Generation of unbiased equiprobable samples for use in long term planning and design.
The purpose is to pro-
vide several and various scenarios on which the efficacy of the proposed design can be tested.
b) Generation of biased equiprobable samples for use in planning and operation of the system in the short term in the immediate future.
The purpose now is to obtain differ-
ent scenarios biased to the present time. c) Forecasting on a stochastic basis data for several periods ahead.
Forecasting involves the determination of the ex-
pected value and the probability distribution of the future event on a period by period basis.
Such forecasted samples
are required as an aid to decision making on the operation of the system for the next few time periods.
d) Deterministic or stochastic forecasting of a single datum on a single step ahead basis.
This forecasted value is used
in the actual scheduling of the real-time operation of the system. The general purpose of the extrapolation of data is to provide an understanding at the present time of future events so tilat certain decisions can be taken based on this understanding. This purpose includes the successful exaggeration of extremes in the historical data as well as generation of extrapolated data with increased information content derived from a priori sources. pose does
However, the pur-
not involve prediction with any specified "Degree of
Accuracy" the real-time events into the future. A l s o , then, there is no such thing as a correct model or a "best" model; however, there are appropriate models; and the only justification for the validity of a model is that based on an investigation whether the
238 p u r p o s e f o r which t h e model h a s b e e n d e v e l o p e d i s s e r v e d by i t s use.
T h i s a l s o l e a d s t o t h e c o n c l u s i o n t h a t , f o r any g i v e n h i s -
t o r i c a l d a t a r e c o r d e d up t o t h e p r e s e n t t i m e , i t i s n e c e s s a r y t o have s e p a r a t e models
f o r e x t r a p o l a t i o n of d a t a n o t e d i n f o r m a t s
a t o d above. C o n s i d e r t h e case of d a t a e x t r a p o l a t i o n , f o r m a t a , w i t h t h e p u r p o s e of g e n e r a t i n g e q u i p r o b a b l e s a m p l e s . t o as d a t a s y n t h e s i s . tioned.
This is o f t e n r e f e r r e d
A s a n a p p l i c a t i o n t h e f o l l o w i n g c a n b e men-
The several s a m p l e s o f i n f l o w a r e r o u t e d t h r o u g h a reser-
v o i r system and, u s i n g a n o p t i m i z a t i o n procedure, samples of o p t i This is t h e i m p l i c i t stochas-
m a l r e l e a s e p o l i c i e s are d e t e r m i n e d . tic optimization.
T h i s p r o c e d u r e r e s u l t s i n t h e development of l o n g
t e r m r u l e curves i n system operation.
It is c l e a r l y seen, then,
t h a t a model f o r d a t a e x t r a p o l a t i o n s h o u l d be s u c h t h a t i t s h o u l d be c a p a b l e of p r o v i d i n g s a m p l e s w i t h e x t r e m e s of f l o o d and d r o u g h t s e q u e n c e s , s o t h a t t h e v a l i d i t y of t h e development of r u l e c u r v e s could be investigated with regard t o these samples.
A model f o r
f o r m a t "a" c a n be c o n s i d e r e d t o g e n e r a t e e q u i p r o b a b l e s c e n a r i o s i n t o t h e f u t u r e i f t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e s a t i s f i e d by t h e samples d e r i v e d from t h e model: ( a ) t h e samples e x h i b i t e x t r e m e f l o o d s e q u e n c e s w i t h i n a r a n g e l y i n g on b o t h s i d e s o f t h a t found i n t h e h i s t o r -
ical sample; (b) t h e s a m p l e s p r o v i d e e x t r e m e d r o u g h t s e q u e n c e s w i t h i n a r a n g e l y i n g on b o t h s i d e s of t h a t found i n t h e h i s t o r i c a l sample; ( c ) t h e s a m p l e s p r o v i d e d i s t r i b u t i o n of d a t a i n t h e s t a t e s p a c e
similar t o t h a t embedded i n t h e h i s t o r i c a l d a t a . S a t i s f a c t i o n of t h e above t h r e e c o n d i t i o n s s h o u l d b e t h e c r i t e r i a i n j u s t i f y i n g a model.
These c o n d i t i o n s a r e s p e c i f i c and q u a n t i -
f iable. C o n s i d e r t h e f o r m a t "d" c o n n e c t e d w i t h t h e f o r e c a s t i n g of a s i n g l e datum on a s t e p ahead b a s i s .
This i s required i n real-time
o p e r a t i o n which i s t h e s c h e d u l i n g of t h e s y s t e m o p e r a t i o n f o r t h e
239 next time period.
At the completion of the time period when the
actual measured value is available, it is used to update the system states and the so updated states form the initial conditions for decisions on real-time operation for the succeeding time period based on a new forecasted value. Again, for emphasis, it should be stated that a comparison of the forecasted value on a step ahead basis with the value occurring in real-time is excluded as a purpose. criteria for validating the model. the system in real-time.
The following can form
Perform physical operation of
Simulation on the computer of the real-
time physical operation is an alternative procedure.
After having
completed the operation for a reasonable horizon of time, evaluate the results.
Justification of the model can now be based on any
criteria that is an appropriate function of these results.
For
example, the following questions are valid on a post operation basis. Was the operation, so far carried out, optimal? Was there any failure (withdrawal below targeted or required level) involved in the operation? Could the operation have been improved if a different model had been used for step ahead forecasting? Some Further Thoughts on Modelling The severe shortcomings of ARIMA models and dfGn models €or data synthesis have been noted previously.
Primarily these models ne-
glect the consideration of the distinguishing characteristics of well defined groups in the data record. The existence of groups as postulated by Hurst is evident from Fig. 1. Even a visual examination will indicate the extreme interrelationship between succeeding datum values in each group.
There is, then, a need for in-
vestigations pertaining to these groups so that the intrarelationship between identifiable groups as well as the interrelationship within each groun could be properly considered in time series modelling. For example, consider the data record in Fig. 2.
It is obvious
that the open circles represent data that have less variation or
240 p e r t u r b a t i o n from one a n o t h e r , w h i l e t h e s o l i d c i r c l e s show d a t a t h a t have a m o d e r a t e l y l a r g e o s c i l l a t o r y b e h a v i o u r , s t r o n g i n t e r dependence and a s m a l l n e g a t i v e c o r r e l a t i o n between x i and x 2 , p e r hpas, with a lag.
C l e a r l y , t h e open and s o l i d c i r c l e s , as s e e n
from t h e d a t a , r e p r e s e n t two e a s i l y d i s c e r n i b l e random b e h a v i o u r types.
I n many cases, i n most h y d r o l o g i c cases, t h i s u n d e r s t a n d i n g
c a n be enhanced by a p r i o r i i n f o r m a t i o n c o n c e r n i n g t h e d a t a s e t .
Is t h e r e any p r o c e d u r e , t h e n , t h a t would e n a b l e u s t o d i v i d e t h e data into several separate classes? Models of t i m e s e r i e s s h o u l d be b a s e d on a n a n a l y s i s of d a t a and i t s s y n t h e s i s .
A n a l y s i s i s t h e p r o c e s s o f d e t e r m i n i n g t h e fun-
d a m e n t a l components, g r o u p s , e t c . , embodied i n t h e d a t a by s e p a r a t i o n and i s o l a t i o n .
I t s p u r p o s e i s f o r c l o s e s c r u t i n y and examin-
a t i o n of t h e c o n s t i t u e n t components, as w e l l as f o r a c c u r a t e r e s o l u t i o n of a n o v e r a l l s t r u c t u r e o r t h e n a t u r e of t h e whole o r p a r t s of t h e d a t a s e t . A n a l y s i s of e m p i r i c a l i n f o r m a t i o n o r d a t a from t h e p h y s i c a l world i s t h e r e s u l t of mapping of t h i s i n f o r m a t i o n from one form t o an-
other.
The mapping s h o u l d b e b a s e d on t r a i n i n g s e t of d a t a and
supervised learning procedures.
The meaning of t h i s l a t t e r t e r m
o f t e n used i n c o n n e c t i o n w i t h p a t t e r n r e c o g n i t i o n and p a t t e r n analysis
i s q u i t e obvious.
It involves t h e inclusion a t the a n a l y s i s
s t a g e of any e x p e r i e n c e b a s e d u n d e r s t a n d i n g of t h e a n a l y s t , a s w e l l as his
knowledge of t h e c a u s a t i v e f o r c e s t h a t c r e a t e t h e p r o g r e s s -
i o n of d a t a i n t i m e (Unny e t a l , 1 9 8 1 ) . S y n t h e s i s r e p r e s e n t s t h e a c t i o n of combining v a r i o u s p a r t s o r compo:,ents h a v i n g d i f f e r e n t c h a r a c t e r i s t i c s i n t o one c o h e r e n t , cons i s t e n t whole.
I t i s t h e r e s u l t of remapping i n t h e o r i g i n a l f o r m a t
of t h e d a t a c o n f i g u r a t i o n r e c o g n i z e d i n t h e l e a r n i n g p h a s e .
It i s
q u i t e o b v i o u s t h a t a n a l y s i s and s y n t h e s i s i n t e r a c t w i t h e a c h o t h e r . B r e a k i n g up i n t o components i s n o t p o s s i b l e w i t h o u t s p e c i f y i n g t h e manner i n which t h e components c o u l d be p u t t o g e t h e r . The two s t e p mapping p r o c e d u r e l e a d i n g t o a n a l y s i s and s y n t h e s i s c a n be r e p r e a t e d a number of t i m e s w i t h c o n t i n u e d improvements i n
241 t h e l e a r n i n g p r o c e d u r e , p r o v i d e d a b a s i s e x i s t s f o r s u c h improvements.
T h i s b a s i s i s t h e u n d e r s t a n d i n g b u i l t on t h e i n t e r a c t i o n
of t h e a n a l y s t w i t h t h e p h y s i c a l w o r l d . R e c e n t l y a s e r i e s o f a r t i c l e s have a p p e a r e d t h a t employ c o n c e p t s of
p a t t e r n r e c o g n i t i o n f o r d a t a s y n t h e s i s (Panu and Unny, 1 9 8 0 a , b , c
and d ; Unny e t a l , 1 9 8 1 ) .
A p a t t e r n i s a shape r e p r e s e n t a t i o n of
s e c t i o n s of t h e p h y s i c a l w o r l d , f o r example, s e c t i o n s of streamf l o w t i m e wave form c o r r e s p o n d i n g t o g e o p h y s i c a l s e a s o n s .
Patterns
r e s u l t from i n n u m e r a b l e c a u s e s and a s t u d y of p a t t e r n s i s a s t u d y of all t h e s e c a u s e s .
A c h r o n o l o g i c a l r e f l e c t i o n of t h e c a u s a t i v e
mechanism i s c o n t a i n e d i n a s e r i e s of s u c h p a t t e r n s .
What i s
a t t e m p t e d i n t h e a r t i c l e s n o t e d above i s t h e development of a t e c h nique t h a t provides f l e x i b i l i t y i n data processing, t h a t accepts i n p u t from t h e a n a l y s t and t h a t i s a d a p t i v e t o t h e r e q u i r e m e n t s s a t i s f y i n g v a r i o u s o b j e c t i v e s of m o d e l l i n g .
The a p p r o a c h o c c u p i e s
a n i n t e r m e d i a t e p o s i t i o n between a p u r e l y s u b j e c t i v e e x p e r i e n c e b a s e d f o r m u l a t i o n and a t o t a l l y machine d e r i v e d a l t e r n a t i v e .
The
m o t i v a t i o n h a s been t o a v o i d i r r e l e v a n t r e s u l t s a r i s i n g o u t of t h e u s e of p r e c o n c e i v e d models and preprogrammed l o g i c t h a t impose a n e x t e r n a l s t r u c t u r e upon t h e o t h e r w i s e u n i q u e b e h a v i o u r o f t h e t i m e
series. CONCLUDING REMARKS
The main p o i n t o f d i s c u s s i o n c o n t a i n e d i n t h i s p a p e r c a n b e summarized as f o l l o w s :
The l a s t d e c a d e h a s s e e n v a r i o u s a t t e m p t s
a t r e f i n i n g some of t h e p r e v i o u s l y p r o p o s e d models f o r t i m e s e r i e s synthesis.
Much of t h e developments i n t h i s r e g a r d h a s t a k e n p l a c e
w i t h o u t due r e g a r d g i v e n t o t h e u n i q u e n a t u r e of t h e p h y s i c a l problem and w i t h o u t c o n s i d e r a t i o n of t h e o r i g i n of d a t a ( s t o c k m a r k e t data versus streamflow d a t a ) .
Perhaps i t is t h e a p p r o p r i a t e t i m e
t o take a f r e s h look a t t i m e series modelling procedures i n t h e hydrologic context.
242
REFERENCES Akaike, H., 1974. A New Look at the Statistical Model Identification, IEEE Trans, Automatic Control, 19(6): 716-723. Box, G.E.P. and Jenkins, G.M., 1973. Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, California. Chi, M., Neal, E. and Young, G.K., 1973. Practical Application of Fractional Brownian Motion and Noise to Synthetic Hydrology, Water Resour. Res., 9: 1569-1582. Chiu, C.L., (Editor), 1978. Applications of Kalman Filter to Hydrology, Hydraulic and Water Resources, Proc. A.G.U., Chapman Conference, University of Pittsburgh. Hurst, H.E., 1951. Long-term Storage Capacity of Reservoirs, Trans. A.S.C.E., 116: 770-808. Hurst, H.E., 1956. Methods of Using Long-term Storage in Reservoirs, Proc. Instn. Civil Engrs., 1: 519-543. Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory, Academic Press, New York. Klemes, V., 1974. The Hurst Phenomena - A Puzzle? Water Resour. Res., lO(4): 675-688. Klemes, V., 1981. Applied Stochastic Theory of Storage in Evolution, Advances in Hydrosciences, 12: 79-141. Lawrance, A.J. and Kottegoda, N.T., 1977. Stochastic Modelling of Riverflow Time Series, Jour. Royal Statist. SOC. Series A, 140(1) : 1-47. Lloyd, E.H., 1967. Stochastic Reservoir Theory, Advances in Hydrosciences, 4: 281-339. Mandelbrot, B.B., 1971. A Fast Fractional Gaussian Noise Generator, Water Resour. Res. 76(3): 543-553. Mandelbrot, B.B. and vanNess, J.W., 1968a. Fractional Brownian Motions, Fractional Noises and Applications, SOC. Ind. Appl. Math. Rev., l O ( 4 ) : 422-437. Mandelbrot, B.B. and Wallis, J.R., 1968b. Noah, Joseph and Operational Hydrology, Water Resour. Res., 4(5): 909-918. Mandelbrot, B.B. and Wallis, J.R., 1969a. Computer Experiments with Fractional Gaussian Noises. Part 1 - Averages and Variances; Part 2 - Rescaled Ranges and Spectra; and Part 3 - Mathematical Appendix, Water Resour. Res., 5(1): 228-267. Mandelbrot, B.B. and Wallis, J.R., 1969b. Some Long Run Properties of Geophysical Records, Water Resour. Res. 5(2): 321-340. Mandelbrot, B.B. and Willis, J.R., 1969c. Robustness of the Rescaled Range R/S in the Measurement of Non-cyclic Long-run Statistical Dependence, Water Resour. Res., 5 ( 5 ) : 967-988.
243 Matalas, N.C. and Wallis, J.R., 1971. Statistical Properties of Multi-variate Fractional Gaussian Noise Processes, Water Resources Research, 7(6): 1460-1668. MacInnes, C.D., 1981. Multiple Time Series Data Extrapolation in Water Resources Engineering Applications using Pattern Recognition Techniques, Doctoral Dissertation, Department of Civil Engineering, University of Waterloo, Ontario, Canada. Moran, P.A.P., 1954. A Probability Theory of Dams and Storage Systems, Australian Journal of Applied Science, 5: 116-124. O’Connell, P.E., 1974. Stochastic Modelling of Long-term Persistence in Streamflow Sequences,” Ph.D. Thesis, Univ. of London, England. Panu, U.S., 1978. Stochastic Synthesis of Monthly Streamflows Based on Pattern Recognition, Doctoral Dissertation, Department of Civil Engineering, University of Waterloo, Ontario. Panu, U.S. and Unny, T.E., 1980a. Extension and Application of Feature Prediction Model for Synthesis of Hydrologic Records, Water Resources Research, 16(1): 77-96. Panu, U.S. and Unny, T.E., 1980b. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition I. General Methodology of the Approach, Journal of Hydrology, 46: 5-34. Panu, U . S . and Unny, T.E., 1980c. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition 11. Application of Natural Watersheds, Journal o f Hydrology, 46: 197-217. Panu, U.S. and Unny, T.E., 1980d. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition 111. Performance Evaluation of the Methodology, Journal of Hydrology, 46: 219-237. Prabhu, N.U., 1964. Time Dependent Results in Storage Theory. J. Applied Probability, Vol 1: 1-46. Soares, E.F., Unny, T.E. and Lennox, W.C., 1977. On a Stochastic Sediment Storage Model for Reservoirs, in Stochastic Processes in Water Resources Engineering. (Eds.) L. Gottschalk, G. Lindh and L. de Marie, Water Resources Publications, Fort Collins, Colorado, 141-166.
.
Unny, T.E., Panu, U.S., MacInnes, C.D. and Wong, A.K.C., 19 Pattern Analysis and Synthesis of Time Dependent Hydrologic Data. Advances in Hydrosciences, Vol 12: 195-295. Unny, T.E. 1977. Transient and non-stationary Random Processes, in Stochastic Processes in Water Resources Engineering, (Eds.) L. Gottschalk, G. Lindh and L. de Marie, Water Resources Publications, Fort Collins, Colorado. Wing, S.P., 1951. Discussion on Long-term Storage Capacity of Reserviors, by Hurst, H.E., Trans. A.S.C.E., 116: 807-808.
T . E . 1JNNY l l n i v e r s i t y of Waterloo
INTRODUCTION l i y d r o log i c d a t a pr i iiiar i
and t o o t h e r \ r a r i a l i l e s i n c
l i ~ . d i - 1oo g i c
II>~Llrolog \‘;l
1
c t I\c c n
11
t irnc s c r i i ‘ s . d i ffc\rcnt.
Reprinted from Time Series M e t h o d s in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
@
225
45
24
1 2 3 ( a ) N I G E R R I V t R ( M o n t h l y d a t a ) a t KIANEY:
36
( l i m e P e r i o d 1942.1944)
TIMF IN
I
I
I
tdrN’l‘ll<
TIM
(11) NIGER R I V E R ( Y e a r l y d a t a ) a t \rIj2i.ltY: (Time F c r i o d 1940-1966)
I h Yl..Al?S
F i g . 1 . Group f o r i n a t i o n i n ) . e a r l y mid nionthly t i m e s e r i e s o f f l o ~ I n c o n t r a s t , tlic monthly time s e r i e s a r e marAcd 11)- s e a s o n a l it), ‘i’herc i s a l s o t h e prcx-
a c c o r d i n g t o t h e g e o p h y s i c a l year ( F i g . l a ) .
s e n c e o f c h a r a c t e r i s t i c g r o u p s o f h i g h arid lori v a l u e s aillong t h e !‘cars and a l s o w i t h i n t h e y e a r . The d a i l y time series a r e d i f f e r e n t froin n i o n t h l > ~xiid ).early t inre
series i n t h e s e n s e t h a t t h e s e t i m e s e r i e s a r c c h a r a c t c r i s c d 11). tlic o c c u r r e n c e o f sharp peaks and e x p o n c n t i : i l deca!-.
The c : i u s c - c f f e c t
r e l a t i o n s h i p i n r a i n f a l l - r u n o f f process i s s t r o n g e r i n t h e s h o r t
iii-
t e r v a l time s e r i e s . The p e r i o d i c i t i e s a r c i n d u c e d i n t h e s t r e a n i f l o w time s e r i c s liy
t h e geophysical cycle.
This is reflected i n thc o e c ~ i r ~ c ~ ~ ofi ehigl: c
p r e c i p i t a t i o n and h i g h r u n o f f d u r i n g tlic
nionths, tind lo\c pre-
smiiiicr
c i p i t a t i o n and low r u n o f f d u r i n g t h e w i n t e r iiioriths i n N o r t h c r r i c 1 i mates.
Means a n d v a r i a n c e s o f h y d r o l o g i c
larger i n
sLiinnier
tiiiic
a n d snial ler i n w i n t e r riioriths
scric’s
.
c i t y i n time a l s o o c c u r s i n t h e forin o f trc>nd i n the
:ire
foiiiicl
I ~ ’ u r t ! i c ~ rlion-hciiiiogcri, rl;it:i
;is \ i c ’ 1 1
;is
iii
forin o f g r a d u a l and suddcri v a r i a t i o n s i n t h e stoc1i;ist ic- n:itiiw
226 o f t h e JLita. 'l'he i i i i p l i c a t i o n a r i s i n g from t h e p r e s c n c c o f p e r s i s t e n c c , e s p c c i -
t h a t c o r r e s p o n d i n g t o prolonged wet and d r y p e r i o d s , i s s i g n i -
all!.
f i c a n t from t h e p o i n t o f \.icw of Liater resources . d e s i g n and o p e r a t i o i l s .
Both s h o r t - t e r m p e r s i s t c n
and 1 ong- t crm
p e r s i s t e n c e a r e i i i t c g r a l p a r t s o f h y d r o l o g i c t i m e s e r i e s which t l l e r c b y bccoiiie d i f f e r e n t from those tiine s c r i e s found i n o t h e r d i s c i p 1 i n e s swh
stock-niarket anal>.sis.
;IS
An 1 i i s t o r i c : r l 1). r c c o r d e d s t r c a n i f l o i i tiinc s c r i c s c m lie considcred
t o lie d c r i \ . e i i frorii
;I
s t o c l i a s t i c g e n e r a t i n g nicchanisiri t h a t e v o l v e s
a c c o r d i n g t o c e r t a i n proba1)i l i s t i c l a i c s .
If i t \ccrc possi1)1c t o d c -
c i p h e r t h e s e proIia1)i l i s t i c I a i \ s , t h e n t h e r e r i o ~ i l dI)c a s a t i s f a c t o r ) . iiiodcl for. t h e time s e r i e s .
Ilowcvcr, t h i s procedure i s d i f f i c u l t ,
if
n o t p r a c t i c a l i y i r i i p o s s i l ~ l c . I3ccaiise s t r e a i i i f l o ~ i d a t a r c p r c s c n t oiil]. a s i n g l e t iriie s e r i e s
o f a s t a t ioniir.!.
;I
check
;is
p r o c e s s and a l s o
t o ~ h e t h e rs u c h a series foriris p a r t ;in
crgodi c proccss i s inipossilile.
I n s p i t e o f t h i s , t h e f o l l o w i n g procedure i s o f t e n c a r r i e d o u t i n c o n n e c t i o n Lci t h strcainflo\.; time s e r i e s rriodc11 i n g : Gi\.cn
i)
:in
liistorical1)- recorded d a t a series:
;\ssiiiiic
e r g o d i tit).
i i ) . \ s s ~ i r i i e ;I s t o c l i a s t i c process i i i J ( : a I c u l a t c c.c>rtain p a r a n i e t e r s from t h e r c c o r d c d d a t a arid : i s s m e t h a t tticsc paraiiictcrs a r c t h e s t a t i s t i c s f o r t h c
ii-]
t h a t the s o - d e f i n e d s t o c h a s t i c process i s t h c gc'nerat i 11s iiicchani siii froiii w l i i ch t h e recorded d a t a i s d c r i \ ~ e das ;A s m i p l e .
:\ssiiiiic
'I'hcrcl)~., :ind
iii
t h so
iiian\.
assciiiqit i o n s ,
;i
ered t o lie o h t a i r i c d for. the t i m e s e r i c s .
s t o c l i a s t c iiiodel i s c o n s i d I t s h o u d IIC cniphasiicd
here t h a t t h i s p r o c e d i i r e h e a r s no r e l a t i o n s h i p t o t h e p h y s i c a l p h c n ~ I ~ L ' I I I C on I ~ ; ~\chicti
t h e d a t a has ticen r c ~ c o r d c d . I n a d d i t i o n , t h i s p r o -
c c d u r e i . 0 ~ 1 1 c l o f t c n 1)ccoiric i r r c l c v a n t r\.licn i t s r e s u l t s a r e a p p l ictl
i n connect i o n
iii
t h water r c s o i i r c r s p1anriiiig
t i o i i s c;111 a l s o I)c> r a i s e d view.
froiii
;iiicl
iiianligcnicnt.
Olijec-
l i c u r i s t i c and p h i 1 o s o p h ~ i cp o i n t s o f
A l l t h a t is r e q u i r e d t o coiiiplctc t h c :iIio\,c iiroccdiirc i s
;I
227 feiv p a r a m e t e r s
-
a t t h e most t h r e e o f f o u r
-
d e t e r m i n e d from t h e
d a t a ; o t h e r w i s e t h e whole s e t of d a t a s o l a b o r i o u s l ) . c o l l e c t e d c a n be d i s c a r d e d .
An Iixamp 1c Figure 2 represents
;I
b i v a r i a t e time s e r i e s o f l e n g t h 1 0 0 iini t s .
Soiiie p o i n t s i n t h e s e r i e s a r e shown liy open c i r c l e s mid o t h e r s by
s o l i d ones.
T h i s w i 11 he e x p l a i n e d s u b s e q u e n t l y .
No a t t e m p t has
been made h e r e t o g e n e r a t e an ARblA p r o c e s s o r any o t h e r s t o c h a s t i c
process t o represent thesc time s e r i e s .
I t was not t h e p u r p o s e f o r
They a r e t a k e n from a r e c e n t
Lihich t h e s e s e r i e s Were ' r e c o r d e d ' .
t h e s i s cotiip1etcd a t t h e U n i v e r s i t y o f W a t e r l o o (Flclnncs, 1981)
t%.lJ.
and i t is achnowledgcd h e r e .
The f a c t rem:iins
t h a t , i f t h e d a t a had
h e e n p r i n t e d i n a t a l i u l a r f o r m a t i t would h a v e been eas)' t o e x t r a c t p a r a m e t e r s and d e v e l o p a n ARbN process as a model f o r t h e s e r i e s . This could r e s u l t i n a delusion of t h e r e a l i t y .
t 0
x-
t
I
I
+3
0
N
c
-3 tlmi unit5
F i g . 2 . B i v a r i a t e time s e r i e s a c c o r d i n g t o p r o b a b i l i s t i c laws d e s c r i b e d i n e q u a t i o n s ( 1 2 ) and (1.3) (from r e f : MacTnncs, 1 9 8 0 ) .
228 I n order. t o i l l u s t r a t e t h e n i a i n p o i n t o f this d i s c u s s i o n , con-
s i d e r t h e g c l n e r a t i n g riicchanisiii o f t h e s e r i e s i n 1:ig. 7.
i e s arc’ o h t a i r i c d
;is
saiiiplcs of
L
1
tlio
s e p a r a t e processes.
These s e r -
‘I’he s o l i d
21 I1 d
and
‘r (t-1.)J
: ; [E ( t ) E-
=
[:;.05
:L]
In t h e al)o\.e t* i s the f i n a l time p o i n t a s s o c i a t e d w i t h t h c regime o f t h e e a r 1 i c r d e s c r i b e d p r o c e s s iininediately p r e c e d i n g t h e regiiiic
associated x i t h the random w a l k niodcl.
‘The length o f r u n (nuinlier o f
time p o i n t s ) in each regime is g e n e r a t e d b y a n e q u i l i b r i i i i r i t l i s c r c t e
rencwa 1 process g i\.cn liy: R
=
(.?I
1 + Ii(n,0)
~ h c r cI3 i s
;I
Iiinoniial r a n d o m variahlc d e s c r i h c d by
which i s t h e p r o b a b i l i t y o f exactly 13 o c c u r r e n c e s i n TI i n d e p e n d e n t B e r n o u l 1 1 t r i a l s with t h e proliahi 1 i t y o f a n o c c u r r e n c e i n any one t r i a l liciiig 8. 5
u c h t l l < lt
The \/alucs o f n and 8 a r e 55 and 0 . 2 , rcspcctivcly,
229
Though thcsc s c r i c s
art i f i c i n l crcat ions, t h t y
i1rc
practical rcIcvancc.
110
have
soiiw
i n most hydrologic d;it;r t1ic1.c :ire s e v c ~ r ; i l
gencrating proccsscs a t work otic aftcr the othcr ;it d i ffcrcnt per-
iods i n t imc process. AS
illid ;in
tlicsc cannot be a\cr;igcd i nto :I s i n g I c x gcticr:tt i ng ex:mipIc, ii biviiriatc :lI\Flh p ~ o c c dcrivcxl ~;~ For the
whole s c r i c s i n J:ig. 2
twi11d
lw h;rscd on
;I
v;iluc for tlic f i r s t order
autocorrclation coefficient significant 1y cii ffcrcnt from that for thc process from w h i TJI
t hc $01 i d c i rc- I c?; ;t rc oltta i ricJ.
'I'ltc s o 1 i tl
c i r c l e s ;ind the oi)cn circles rcprcsent d;tt;t s i t t i v a s t 1y tlit'fcxrcitt persistcncc charactcrist ic
iitid
:in)' t intc series m o ~ l c ~th;it l tlocs riot
cons idcr t h i s d i ffcrcnct shou1d hc t rcat cd
;is
itns;:iti st';tctory .
On Pcrs i st cnct I'crs is t c n c t imp 1 i cs
cert :I
ii
iii
dctcrmi n i s t i c rc'1
succcssivc \alms of d a t a i n thc t i m c s t r i c s .
;it
Stich
i onsh i j) I w t \ c w i > ;I
rc1;tt ionsttip
may be diic t o t h c fact t h a t the caitsc r c ~ s i i l t i n gi n t h e cf'f'ect pvr-
sists f o r
succcssivc
:I
sp:in of t inic longcr than one or ntorc iiicrcmcttts I ) c t t \ c c r i d:ttittii.
Thc
ciittsc
twitig of liriti t c d Icngth, thc i ~ ~ i s t -
encc introduced h y tlic c;iiisc i s a l s o o f I imi t c d Icnptli. Siic.rxvtliiig parts of t h e t iinc scrics may csliihit Ji ffcrciit pcrsistcvici.. \ \ h i ~ * his :I I'crsistcnce is oftcii charactcriscd I)\- thc c*or1*i~~ogi-;i111 p l o t of tlic corrclatioii cocffir.icnt ;tg;tinst lag. (hrrc1;tt i o n coe f f i c i e n t : i t each lag is Jcterinincd b\- :I sc;inniiig p r o ~ + ~ ~ J:i~-ross ~irc* t h c whole data. 'l'his hits i i n avcr;tging cfl'cvt. 'I'ftc* f a 1 lacy o!' tisi~:g t h i s indicator t o denote pcrsistciicc is apparent i n tlrc
annual hydro1ogic s c r i c s . efficicrits a t lag 1
:tiid
I n most
;intiii;i
~*;tsic of'
1 s e r i w tlic correl;it i o n
a t higher l a g s itre
foitiid
k-o-
t o Iic irisigtii f i -
cantly diffcrcnt from x r o , ~nciiriingt h i i t tlicy 1 i c v i t h i n the coiifidence bounds of simi lar cocfficicnts f o r
it11
indcpcridcitt s c r i c s .
Thus, i n t h e l i t e r a t u r e , and in water rcsourccs applications :is w l l , annual series a r c oftcn trcatccl as indcpcndent. Ilowcvcr, i t is swit from Fig. 1 t h a t thcrc i l r C \W 1 1 d e f i tied grotip format ions i n :innti:i 1
230 s e r i e s which i s i n d i c a t i v e of s t r o n g p c r s i s t c n c e i n s t r e t c h e s of the series.
I t i s a l s o v a l i d t o note i n t h i s conncction t h e re-
c e n t l y r e c o r d e d examples i n v a r i o u s p a r t s o f t h e world o f 7 bad years, etc. F u r t h c r m o r c , i t i s d i f f i c u l t t o o b t a i n a r c l i a b l c and s t a b l e e s t i m a t e f o r t h e c o r r e l a t i o n c o e f f i c i c n t from t h c d a t a s e r i e s .
With
r e g a r d t o t h e d a t a s c r i e s i n T a b l e 1 of I c n g t h 59, d i f f e r e n t scct i o n s o f t h i s s e r i c s h a v e b c r n used i n c as c s numbcrcd 1 t o 10 f o r TABLE 1 . E x p l a n a t i o n o f t h e Cascs o f Timc S e r i e s Case Xumber 1
Data Values C o n s i d e r e d
R e m a rh s
For T h e Case
1 t o 50
S c r i c s "A" Data P o i n t s . Monthly Discharge i n C.M.S.
2 t o 51
3 2 5 , 228, 2 0 1 , 1 4 3 , 103, 83,
3 t o 52
4 t o 53
l S 1 , 3 0 0 , 251, 6 4 0 , 511, ,708,
5 t o 54
1 2 3 , 1-17, 278, 2 3 2 , 2 6 4 ,
6 t o 55 ?
7
t O
56
8
8 to 5:
3
!I
10
t o 58
248,
2 8 1 , 399, S31, 5 7 2 , 7 0 0 , 4 7 7 , 40!1,
31.3,
2 0 2 , 1 11, 1 4 9 , 12x,
1 4 5 , 3 0 4 , -711, 92,
s:7,
74, 2 3 0 , 1 - 2 ,
389, 162, 1 3 ,
1-15,
231 u n l i k e l y t h a t t h e c o r r e l o g r a m would t e n d t o a c o n s t a n t " p o p u l a t i o n " In e f f e c t , a sample from a s t a -
value with longer length of data.
t i o n a r y and e r g o d i c s t o c h a s t i c p r o c e s s c a n n o t f o r m a model f o r t h e observed time series.
.
0 6 0.4 0.2
0 -0.2
.
0 6
0.4
g
G
-0.2
0 -0.2
Ir
0.6 r'
g
5
0.4
c
w
3.2
2
0
-3.2 fl
3.b
3.4 3.2
0 3.2 7.h
1.4 1.2 0
-u.
2 1
3
h
9
1
2
1
5
'l'hc presciice o f p e r s i s t e n c e i n ! . c : i r l ! ~time s e r i c s , and i n str'cain-
flow time series i n p a r t i c u l a r , \<:is f i r s t h r o i i g l ~ t t o the, : i t t c i i t i o n of c n g i n c c ~ - sI)?
t h e pionceriiig s t u d i e s of (Ilurst, 1951, 1 9 S h ) on l o n g
tcrni s t o r a g e rcqiiireiiicnts i n the> N i l c I h s i n .
232 Through a n e x t e n s i v e a n a l y s i s of g e o p h y s i c a l time s e r i e s (cons i s t i n g o f a n n u a l v a l u e s ) , i n c l u d i n g t h e e x t r e m e l y long time s e r i e s o f t h e a v e r a g e a n n u a l f l o w on t h e r i v e r k i l e , and also o f normal i n d e p e n d e n t s e r i e s g e n e r a t e d by v a r i o u s e x p e r i m e n t s , t l u r s t d e r i v e d the
re 1a t i o n s h i p RK/SN
01
H N
where N i s t h e l e n g t h o f d a t a .
I n the above, R
N
i s t h e ad u s t e d
r a n g e and i t c a n h e e x p r e s s e d f o r an a n n u a l time s e r i e s {x , i = l , 2 , . . . ,
N) o f
and s t a n d a r d d e v i a t i o n N
l e n g t h K y e a r s , w i t h mean x
The term R ./S
i \ , N
[=i N) i s
the adjusted rcscalcd range.
h i
S"
as
c s t iriiatc of
H, d e n o t e d by K l l , was d e f i n e d a s
Hurst observed t h a t t h e value of the exponent I1 i n r e l a t i o n s h i p (0) h a s on the a v e r a g e a v a l u e o f 0.7.3 f o r t h e g e o p h y s i c a l t i m e s e r i e s , and 0 . 5 f o r t h e normal i n d e p e n d e n t s e r i e s .
In hydrologic litcratur-c,
t h e d i s c r e p a n c y i n t h c v a l u e s o f t h e e x p o n e n t i n h y d r o I O g i c a 1 time
s e r i e s and t h a t i n a l l i n d e p e n d e n t s e r i e s has lice11 called t h e llurs-i phenomenon and t h e e x p o n e n t i n r e l a t i o n s h i p Hurst c o e f f i c i e n t .
((I)
i s noii I\rlo\in as t h e
A v a l u e o f t h e ilurst c o e f f i c i c n t g r e a t e r t h a n
0.5 i s c o n s i d e r e d t o i n d i c a t e l o n g - t e r m p c r s i s t e n c c . 1)arnmcter v a l u e s f o r t h e l l u r s t c o e f f i c i e n t d e r i v e d frorii h i s t o r i
c a l r e c o r d i s found s e n s i t i v c t o non-hoiiiogerieities
i n d a t a inc1udi:ig
c h a n g e s i n t h e p r o b a l ~ il i s t i c laws d e f i n i n g t h e g c n e r ; i t iiig of t h e d a t a .
c i e n t g r e a t e r than O.S. to
iiic~c~iiatiisiii
Klemes [ 1 9 7 4 ) h;is sho\\,ri t h a t i n d c ~ p c n d c n t s c r i e s
s i s t e r i c e o f o r d e r zero) w i t h f l u c t w t i n g A r c f e r c ~ n c ciiia!. ~
iiic;ins
-
c s h i I 1 i tcil
a l s o be ~ti:iclc
;it
;in
(~CI.-
Il-coct
this point
Wing (19Sl) who, w h i l e coirimcnting on I l u r s t ' s o r i g i n a l p : i p c > r cx-
pre.;sed doulit a b o u t I l u r s t ' s f i n d i n g s a n d i 1 n p 1 i c d t h a t p e r h a p s
233 d i s c o n t i n u i t i e s i n t h e r e c o r d could have caused an H - c o e f f i c i e n t greater than 0.5. E x t e n s i v e a n a l y s i s of l a r g e assemblage of r e c o r d e d d a t a by ( H u r s t , 1951) encompassing many g e o p h y s i c a l phenomena, e . g . , r a i n f a l l , runo f f , l a k e l e v e l s , t r e e r i n g s and mud v a r v e s , showed t h a t groups of high and low v a l u e s t e n d e d t o o c c u r more f r e q u e n t l y i n n a t u r a l e v e n t s than i n p u r e l y random e v e n t s .
I n a d d i t i o n , H u r s t observed w i t h p a r -
t i c u l a r r e f e r e n c e t o annual streamflow time s e r i e s t h a t groups a s s o c i a t e d w i t h s t r e t c h e s of f l o o d s and d r o u g h t s o c c u r r e d w i t h o u t any r e g u l a r i t y e i t h e r i n t h e i r d u r a t i o n or i n t h e time of o c c u r r e n c e ( F i g . 1 ) . T h i s , t h e n , i s t h e fundamental d i f f e r e n c e between n a t u r a l s t r e a m flow time s e r i e s and o t h e r p u r e l y 'man-made'
s e r i e s such a s those
derived from random p r o c e s s e s , a u t o r e g r e s s i v e p r o c e s s e s and f r a c t i o n a l Gaussian n o i s e s e q u e n c e s . On Some of t h e Commonly Used Models f o r Hydrologic Time S e r i e s C e r t a i n s t o c h a s t i c p r o c e s s e s have been s u g g e s t e d by h y d r o l o g i s t s as models f o r time s e r i e s .
S.pecified s t a t i s t i c s of t h e chosen p r o -
cess a r e a d j u s t e d t o have n u m e r i c a l v a l u e s e q u a l t o t h a t of e q u i v a l e n t p a r a m e t e r s e v a l u a t e d from t h e observed s e r i e s .
The term used
i n h y d r o l o g i c l i t e r a t u r e i s p r e s e r v a t i o n . Thereby i t i s meant t h a t a l l sample f u n c t i o n s o b t a i n e d from t h e p r o c e s s d e l i v e r t h e same p a r a -
meter v a l u e s .
Considered s i g n i f i c a n t i n t h i s c o n n e c t i o n are one o r
more of t h e f o l l o w i n g :
Mean o f t h e s e r i e s , v a r i a n c e , c o r r e l a t i o n
c o e f f i c i e n t s a t l a g 1 and a t h i g h e r l a g s , and Hurst c o e f f i c i e n t . Two commonly used models a r e d i s c u s s e d below. Mandelbrot and van Ness (1968a) d e f i n e f r a c t i o n a l Brownian motion p r o c e s s (fBm) a s : BH(t) =
t
~
fi
1
(t-v)
H-0.5 dB(v) ; 0 < H < 1
(9)
--M
where dB(v) i s t h e d i f f e r e n t i a l of t h e Brownian motion p r o c e s s and H i s a s p e c i f i e d exponent.
T h i s p r o c e s s r e d u c e s t o a Brownian motion
process (Wiener-Levi p r o c e s s ) f o r H = 0.5.
234 The f r a c t i o n a l G a u s s i a n n o i s e p r o c e s s (fGn) i s d e f i n e d a s t h e d e r i v a t i v e o f t h e above p r o c e s s .
The d i s c r e t i s e d v e r s i o n , t h e d i s c r e t e
f r a c t i o n a l G a u s s i a n n o i s e s e q u e n c e (dfCn), i s d e f i n e d by M a n d e l b r o t and van Ncss
(lY68a) as f o l l o b s :
L
1 11-0.5 B [t) = ~___ c (t-v) AB(v+l) 11 v=-m
4G-i
b h e r e t h a s i n t e g e r v a l u e s from
-m
F u r t h e r AB(v) i s t h e
t o present.
f i n i t e d i f f e r e n c e i n t h e Brownian m o t i o n p r o c e s s ~ ~ i At Bh ( v ) = H ( ~ + I + E ) a n d x ( ~ )i s t h e r e a l i z e d v a l u c o f t h e p r o c e s s a t t i m e p o i n t t . t ' h e f a c t t h a t dfGn h a s t h e i s y m p t o t i c p r o p e r t y t h a t i t s a d j u s t e d
B(v+l),
r a n g e I1
N
d e f i n e d i n e q u a t i o n ( 7 ) i s s u c h t h a t 11
N
a
Nl' i s t h e p r i m a r y '
r e a s o n f o r d e v e l o p i n g t h i s p r o c e s s as a model f o r g e o p h y s i c a l t i m e series.
By h e e p i n g ( p r e s e r v i n g ) I 1
= t h e llurst c o e f f i c i e n t derived
from t h e r e c o r d e d time s e r i e s , t h e s a m p l e f u n c t i o n s o f dfCn a r e made t o possess t h e same I I u r s t c o e f f i c i e n t . There i s a b s o l u t e l y no o t h e r s i m i l a r i t y whatever between sample f u n c t i o n s of dfGn and r e c o r d e d time s e r i e s .
The p r o p o n e n t s o f t h e
dfCn models c l a i m t h a t i t i s c a p a b l e o f p r o v i d i n g s a m p l e s w i t h ext r e m e s ( h i g h s and lows) t h a t a r e more s e v e r e t h a n t h a t i n t h e h i s toric series
( M a n d e l b r o t and Wallis, 1 9 6 % ) . T h i s h a s n o t b e e n demon-
s t r a t e d i n m y c o n v i n c i n g manner. There i s n o d o u b t t h a t t h e r e i s a t h e o r e t i c a l b e a u t y
in
the frac-
t i o n a l Brownian m o t i o n p r o c e s s i t s e l f . The t h e o r y and t h e u n d e r l y i n g a s s u m p t i o n s o f b o t h f B m and dfGn a r e p r o v i d e d i n a s e r i e s o f a r t i c l e s 1 1 ~M a n d e l b r o t and van Ness (1YOSa)
and M a n d e l b r o t and W a l l i s (1968b,
1 9 6 9 a , b , c ) . Review a r t i c l e s i n d i c a t i n g t h e i r r e l e v a n c e t o h y d r o l o g y a r e a l s o g i v e n i n C h i , e t a1 ( 1 9 7 3 ) , O'Connel (1974) and Lawrence and Kottegoda
(1977).
? h e dfGn i n v o l v e s summation from i n f i n i t e p a s t to t h e p r e s e n t . I n o t h e r w o r d s , what happened i n t h e d i s t a n t p a s t is c o n s i d e r e d as an i n f l u e n c i n g f a c t o r i n t h e p r e s e n t occurrence. This concept i s t h e a n t i t h e s i s o f t h a t o f t h e blarhov p r o c e s s e s and Marhov c h a i n s . I t
235 s h o u l d b c c o n s i d e r e d i n t h i s c o n n c c t i o n t h a t Markov c h a i n s h a v e n o t o n l y t h c o r c t i c a l c l e g a n c c b u t t h e y also h a v e f o u n d wide a p p l i c a t i o n s
i n h y d r a u l i c s and h y d r o l o g y .
These i n c l u d e a p p l i c a t i o n s i n s t o r a g e
t h c o r y (hloran, 1951; I ' r a l ~ h u , 1 9 6 7 ; I,10>~d, 1967; Klcmcs, 1981; S o a r e s e t a l , 1 9 7 i ) , and i n e s t i m a t i o n t h e o r y and f o r c c a s t i n g
(Jazwinski,
For a comprehensive s e t of a r t i c l e s with a p p l i c a t i o n s i n
1970).
hydrology s e e C h i u (1978) and a l s o Unny ( 1 9 7 7 ) . As a result
af t h e d i f f i c u l t i e s i n v o l v e d i n t h e i n f i n i t c summation
a p p r o x i m a t i o n s h a v e becn d c v c l o p e d for t h e dfGn.
These i n c l u d c t h e
Types I a n d J I a p p r o x i m a t i o n s o f b l a n d e l b r o t and Wallis ( 1 9 6 9 c ) , t h e f a s t f r a c t i o n a l Giiiissian n o i s e a p p r o x i m a t i o n ( f f G n ) o f M a n d e l b r o t , ( 1 9 7 1 a ) , and t h e f i l t e r e d fGn of
M a t a l a s and W a l l i s ( 1 9 7 1 b ) .
The a u t o c o v a r i a n c e f u n c t i o n o f dfGn i s f o u n d t o t e n d t o z e r o v e r y I t i s primarily t h e r e s u l t o f n o n - s t a t i o n n r i t y i n t h c dfGn.
slowly.
111 f a c t , n o n - d e c a y i n g c o r r e l o g r a m s arc c o n s i d e r e d t o i n d i c a t e non-
s t a t i o n a r i t y i n t h e d a t a according t o t h e procedure adopted i n t h e ARISlA n i o d c l l i n g o f t h e time s e r i e s (Box and J e n k i n s , 1 9 7 0 ) .
In t h c s c
i n s t a n c e s . t h e d a t a i n t h e s e r i e s arc s u c c e s s i v c l v d i f f e r e n c e d . i f r v d t i m e s , u n t i l a d e c a y i n g c o r r e l o g r a m i s o b t a i n e d . Elodell-
i n g t h e n i n v o l v e s f i t t i n g an ARMA model o f o r d e r ( p , q ) o f t h e form O(B) ( x
t
-x)
= 0(B)a
(11)
t
to the differences data.
In t h e above,
0 ( B ) = ( 1 - @ , B - O 2 B 2.. .dpBp)
and
8 ( B ) = ( 1 - 0 1 B - e , B 2 . ..8 B q ) q
(12)
w i t h B 'is t h e backward s h i f t o p e r a t o r and t h e @Is and 0 ' s a r c s p e c i fied coefficients.
Further, a
t
i s normal i n d e p e n d e n t l y d i s t r i b u t e ?
x
i s t h e mean o f random \ a r i a b l c h i t h z e r o mean and v a r i a n c e u a' and t h e s e r i c s . The s t a t i s t i c s o f t h e ARMA p r o c e s s are f u n c t i o n s o f t h e
c o e f f i c i e n t s , a s well as t h e s p e c i f i e d v a l u e s f o r
x and u a .
Assunling
e r g o d i c i t y , t h e s t a t i s t i c s a r c e v a l u a t e d from t h e r e c o r d e d series, thus enahling t h e determination o f t h e c o e f f i c i e n t s i n t h e A I W I (p,q) process.
17ic s a m p l c f u n c t i o n s p r e s e r v e t h e m c l n , t h e v a r i a n c e and
t h e c o r r e l o g r a n i . A p a r t from t h i s p r e s e r v a t i o n , t h e r e i s n o s i m i l a r i t y
236 w h a t e v e r between sampel f u n c t i o n s of t h e ARMA p r o c e s s and t h e recorded s e r i e s . S i n c e t h e p u b l i c a t i o n of t h e book by Box and J e n k i n s ( 1 9 7 0 ) , t h e r e h a s b e e n a f l o o d of a r t i c l e s on t h e ARMA ( p , q ) models o r , e q u i v a l e n t l y , on ARIMA ( p , d , q ) models i n t h e h y d r o l o g i c c o n t e x t . S e a s o n a l and n o n - s e a s o n a l models and many o t h e r i n f i n i t e v a r i a t i o n s of t h e s e models have b e e n r e p o r t e d .
"Best" models h a v e b e e n de-
t e r m i n e d f o r a g i v e n t i m e s e r i e s u s i n g c r i t e r i a s u c h as t h e Akaike i n f o r m a t i o n c r i t e r i a (Akaike, 1 9 7 4 ) .
I t i s s u r p r i s i n g t h a t much of
t h e developments i n t h e ARIMA m o d e l l i n g d u r i n g t h e l a s t d e c a J e h a s t a k e n p l a c e w i t h o u t any c o n c e r n b e i n g e x p r e s s e d as t o t h e o b j e c t i v e s of m o d e l l i n g and t h e a p p l i c a t i o n made of t h e s e models.
The scanrled
p a r a m e t e r s employed i n t h e development of ARIMA models f o r g i v e n s t r e a m f l o w t i m e s e r i e s are of q u e s t i o n a b l e r e l e v a n c e b e c a u s e of t h e f a c t t h a t t h e s e n a t u r a l g e o p h y s i c a l t i m e s e r i e s do n o t e v o l v e according t o simple p r o b a b i l i s t i c l a w s .
Despite these shortcomings,
t h e ease t h a t accompanies t h e u s e of preprogrammed l o g i c h a s s t i m u l a t e d a n a c c e p t a n c e of t h e s e models.
I n many c a s e s model d e v e l o p -
ment f o r a g i v e n s e r i e s h a s b e e n r e d u c e d t o t h e l e v e l of a mecha n i s t i c p r o c e d u r e c a r r i e d o u t on t h e machine.
Inference about
s t r e a m f l o w phenomena a r e b e i n g made w i t h o u t any r e f e r e n c e t o t h e p h y s i c a l n a t u r e of t h e problem a n d , i n e x t r e m e cases, w i t h o u t any consideration other than the data sheet.
The o n l y p r e r e q u i s i t e t o
p r o v i d i n g a n i n f e r e n c e h a s become a c a p a c i t y t o program; i n f a c t much less b e c a u s e t h e programs a r e a l r e a d y a v a i l a b l e on t h e s y s t e m . On t h e Requirements of Models f o r T i m e S e r i e s i n Hydrology Models f o r h i s t o r i c a l l y r e c o r d e d t i m e s e r i e s i n t h e h y d r o l o g i c c o n t e x t a r e r e q u i r e d so t h a t s u c h models c a n b e used f o r e x t r a p o l a t i o n of d a t a i n t o f u t t i r e times beyond t h e p r e s e n t .
I t h a s be-
come a n a c c e p t e d p r a c t i c e i n t h e l a s t two d e c a d e s o r s o t o c o n s i d e r t h e s e e x t r a p o l a t e d d a t a i n t h e d e s i g n and p l a n n i n g of water resources systems.
T h i s i s b a s e d on t h e u n d e r s t a n d i n g t h a t t h e p a s t
r e p r e s e n t e d by t h e h i s t o r i c a l d a t a w i l l n e v e r b e r e p e a t e d and t h a t
237 data series employed in water resources applications should be such that they are likely to occur in probabilistic terms in the performance time horizon of the system which lies in the future. Data extrapolation is required in various formats.
Specifical-
ly four different formats are discussed below: a) Generation of unbiased equiprobable samples for use in long term planning and design.
The purpose is to pro-
vide several and various scenarios on which the efficacy of the proposed design can be tested.
b) Generation of biased equiprobable samples for use in planning and operation of the system in the short term in the immediate future.
The purpose now is to obtain differ-
ent scenarios biased to the present time. c) Forecasting on a stochastic basis data for several periods ahead.
Forecasting involves the determination of the ex-
pected value and the probability distribution of the future event on a period by period basis.
Such forecasted samples
are required as an aid to decision making on the operation of the system for the next few time periods.
d) Deterministic or stochastic forecasting of a single datum on a single step ahead basis.
This forecasted value is used
in the actual scheduling of the real-time operation of the system. The general purpose of the extrapolation of data is to provide an understanding at the present time of future events so tilat certain decisions can be taken based on this understanding. This purpose includes the successful exaggeration of extremes in the historical data as well as generation of extrapolated data with increased information content derived from a priori sources. pose does
However, the pur-
not involve prediction with any specified "Degree of
Accuracy" the real-time events into the future. A l s o , then, there is no such thing as a correct model or a "best" model; however, there are appropriate models; and the only justification for the validity of a model is that based on an investigation whether the
238 p u r p o s e f o r which t h e model h a s b e e n d e v e l o p e d i s s e r v e d by i t s use.
T h i s a l s o l e a d s t o t h e c o n c l u s i o n t h a t , f o r any g i v e n h i s -
t o r i c a l d a t a r e c o r d e d up t o t h e p r e s e n t t i m e , i t i s n e c e s s a r y t o have s e p a r a t e models
f o r e x t r a p o l a t i o n of d a t a n o t e d i n f o r m a t s
a t o d above. C o n s i d e r t h e case of d a t a e x t r a p o l a t i o n , f o r m a t a , w i t h t h e p u r p o s e of g e n e r a t i n g e q u i p r o b a b l e s a m p l e s . t o as d a t a s y n t h e s i s . tioned.
This is o f t e n r e f e r r e d
A s a n a p p l i c a t i o n t h e f o l l o w i n g c a n b e men-
The several s a m p l e s o f i n f l o w a r e r o u t e d t h r o u g h a reser-
v o i r system and, u s i n g a n o p t i m i z a t i o n procedure, samples of o p t i This is t h e i m p l i c i t stochas-
m a l r e l e a s e p o l i c i e s are d e t e r m i n e d . tic optimization.
T h i s p r o c e d u r e r e s u l t s i n t h e development of l o n g
t e r m r u l e curves i n system operation.
It is c l e a r l y seen, then,
t h a t a model f o r d a t a e x t r a p o l a t i o n s h o u l d be s u c h t h a t i t s h o u l d be c a p a b l e of p r o v i d i n g s a m p l e s w i t h e x t r e m e s of f l o o d and d r o u g h t s e q u e n c e s , s o t h a t t h e v a l i d i t y of t h e development of r u l e c u r v e s could be investigated with regard t o these samples.
A model f o r
f o r m a t "a" c a n be c o n s i d e r e d t o g e n e r a t e e q u i p r o b a b l e s c e n a r i o s i n t o t h e f u t u r e i f t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e s a t i s f i e d by t h e samples d e r i v e d from t h e model: ( a ) t h e samples e x h i b i t e x t r e m e f l o o d s e q u e n c e s w i t h i n a r a n g e l y i n g on b o t h s i d e s o f t h a t found i n t h e h i s t o r -
ical sample; (b) t h e s a m p l e s p r o v i d e e x t r e m e d r o u g h t s e q u e n c e s w i t h i n a r a n g e l y i n g on b o t h s i d e s of t h a t found i n t h e h i s t o r i c a l sample; ( c ) t h e s a m p l e s p r o v i d e d i s t r i b u t i o n of d a t a i n t h e s t a t e s p a c e
similar t o t h a t embedded i n t h e h i s t o r i c a l d a t a . S a t i s f a c t i o n of t h e above t h r e e c o n d i t i o n s s h o u l d b e t h e c r i t e r i a i n j u s t i f y i n g a model.
These c o n d i t i o n s a r e s p e c i f i c and q u a n t i -
f iable. C o n s i d e r t h e f o r m a t "d" c o n n e c t e d w i t h t h e f o r e c a s t i n g of a s i n g l e datum on a s t e p ahead b a s i s .
This i s required i n real-time
o p e r a t i o n which i s t h e s c h e d u l i n g of t h e s y s t e m o p e r a t i o n f o r t h e
239 next time period.
At the completion of the time period when the
actual measured value is available, it is used to update the system states and the so updated states form the initial conditions for decisions on real-time operation for the succeeding time period based on a new forecasted value. Again, for emphasis, it should be stated that a comparison of the forecasted value on a step ahead basis with the value occurring in real-time is excluded as a purpose. criteria for validating the model. the system in real-time.
The following can form
Perform physical operation of
Simulation on the computer of the real-
time physical operation is an alternative procedure.
After having
completed the operation for a reasonable horizon of time, evaluate the results.
Justification of the model can now be based on any
criteria that is an appropriate function of these results.
For
example, the following questions are valid on a post operation basis. Was the operation, so far carried out, optimal? Was there any failure (withdrawal below targeted or required level) involved in the operation? Could the operation have been improved if a different model had been used for step ahead forecasting? Some Further Thoughts on Modelling The severe shortcomings of ARIMA models and dfGn models €or data synthesis have been noted previously.
Primarily these models ne-
glect the consideration of the distinguishing characteristics of well defined groups in the data record. The existence of groups as postulated by Hurst is evident from Fig. 1. Even a visual examination will indicate the extreme interrelationship between succeeding datum values in each group.
There is, then, a need for in-
vestigations pertaining to these groups so that the intrarelationship between identifiable groups as well as the interrelationship within each groun could be properly considered in time series modelling. For example, consider the data record in Fig. 2.
It is obvious
that the open circles represent data that have less variation or
240 p e r t u r b a t i o n from one a n o t h e r , w h i l e t h e s o l i d c i r c l e s show d a t a t h a t have a m o d e r a t e l y l a r g e o s c i l l a t o r y b e h a v i o u r , s t r o n g i n t e r dependence and a s m a l l n e g a t i v e c o r r e l a t i o n between x i and x 2 , p e r hpas, with a lag.
C l e a r l y , t h e open and s o l i d c i r c l e s , as s e e n
from t h e d a t a , r e p r e s e n t two e a s i l y d i s c e r n i b l e random b e h a v i o u r types.
I n many cases, i n most h y d r o l o g i c cases, t h i s u n d e r s t a n d i n g
c a n be enhanced by a p r i o r i i n f o r m a t i o n c o n c e r n i n g t h e d a t a s e t .
Is t h e r e any p r o c e d u r e , t h e n , t h a t would e n a b l e u s t o d i v i d e t h e data into several separate classes? Models of t i m e s e r i e s s h o u l d be b a s e d on a n a n a l y s i s of d a t a and i t s s y n t h e s i s .
A n a l y s i s i s t h e p r o c e s s o f d e t e r m i n i n g t h e fun-
d a m e n t a l components, g r o u p s , e t c . , embodied i n t h e d a t a by s e p a r a t i o n and i s o l a t i o n .
I t s p u r p o s e i s f o r c l o s e s c r u t i n y and examin-
a t i o n of t h e c o n s t i t u e n t components, as w e l l as f o r a c c u r a t e r e s o l u t i o n of a n o v e r a l l s t r u c t u r e o r t h e n a t u r e of t h e whole o r p a r t s of t h e d a t a s e t . A n a l y s i s of e m p i r i c a l i n f o r m a t i o n o r d a t a from t h e p h y s i c a l world i s t h e r e s u l t of mapping of t h i s i n f o r m a t i o n from one form t o an-
other.
The mapping s h o u l d b e b a s e d on t r a i n i n g s e t of d a t a and
supervised learning procedures.
The meaning of t h i s l a t t e r t e r m
o f t e n used i n c o n n e c t i o n w i t h p a t t e r n r e c o g n i t i o n and p a t t e r n analysis
i s q u i t e obvious.
It involves t h e inclusion a t the a n a l y s i s
s t a g e of any e x p e r i e n c e b a s e d u n d e r s t a n d i n g of t h e a n a l y s t , a s w e l l as his
knowledge of t h e c a u s a t i v e f o r c e s t h a t c r e a t e t h e p r o g r e s s -
i o n of d a t a i n t i m e (Unny e t a l , 1 9 8 1 ) . S y n t h e s i s r e p r e s e n t s t h e a c t i o n of combining v a r i o u s p a r t s o r compo:,ents h a v i n g d i f f e r e n t c h a r a c t e r i s t i c s i n t o one c o h e r e n t , cons i s t e n t whole.
I t i s t h e r e s u l t of remapping i n t h e o r i g i n a l f o r m a t
of t h e d a t a c o n f i g u r a t i o n r e c o g n i z e d i n t h e l e a r n i n g p h a s e .
It i s
q u i t e o b v i o u s t h a t a n a l y s i s and s y n t h e s i s i n t e r a c t w i t h e a c h o t h e r . B r e a k i n g up i n t o components i s n o t p o s s i b l e w i t h o u t s p e c i f y i n g t h e manner i n which t h e components c o u l d be p u t t o g e t h e r . The two s t e p mapping p r o c e d u r e l e a d i n g t o a n a l y s i s and s y n t h e s i s c a n be r e p r e a t e d a number of t i m e s w i t h c o n t i n u e d improvements i n
241 t h e l e a r n i n g p r o c e d u r e , p r o v i d e d a b a s i s e x i s t s f o r s u c h improvements.
T h i s b a s i s i s t h e u n d e r s t a n d i n g b u i l t on t h e i n t e r a c t i o n
of t h e a n a l y s t w i t h t h e p h y s i c a l w o r l d . R e c e n t l y a s e r i e s o f a r t i c l e s have a p p e a r e d t h a t employ c o n c e p t s of
p a t t e r n r e c o g n i t i o n f o r d a t a s y n t h e s i s (Panu and Unny, 1 9 8 0 a , b , c
and d ; Unny e t a l , 1 9 8 1 ) .
A p a t t e r n i s a shape r e p r e s e n t a t i o n of
s e c t i o n s of t h e p h y s i c a l w o r l d , f o r example, s e c t i o n s of streamf l o w t i m e wave form c o r r e s p o n d i n g t o g e o p h y s i c a l s e a s o n s .
Patterns
r e s u l t from i n n u m e r a b l e c a u s e s and a s t u d y of p a t t e r n s i s a s t u d y of all t h e s e c a u s e s .
A c h r o n o l o g i c a l r e f l e c t i o n of t h e c a u s a t i v e
mechanism i s c o n t a i n e d i n a s e r i e s of s u c h p a t t e r n s .
What i s
a t t e m p t e d i n t h e a r t i c l e s n o t e d above i s t h e development of a t e c h nique t h a t provides f l e x i b i l i t y i n data processing, t h a t accepts i n p u t from t h e a n a l y s t and t h a t i s a d a p t i v e t o t h e r e q u i r e m e n t s s a t i s f y i n g v a r i o u s o b j e c t i v e s of m o d e l l i n g .
The a p p r o a c h o c c u p i e s
a n i n t e r m e d i a t e p o s i t i o n between a p u r e l y s u b j e c t i v e e x p e r i e n c e b a s e d f o r m u l a t i o n and a t o t a l l y machine d e r i v e d a l t e r n a t i v e .
The
m o t i v a t i o n h a s been t o a v o i d i r r e l e v a n t r e s u l t s a r i s i n g o u t of t h e u s e of p r e c o n c e i v e d models and preprogrammed l o g i c t h a t impose a n e x t e r n a l s t r u c t u r e upon t h e o t h e r w i s e u n i q u e b e h a v i o u r o f t h e t i m e
series. CONCLUDING REMARKS
The main p o i n t o f d i s c u s s i o n c o n t a i n e d i n t h i s p a p e r c a n b e summarized as f o l l o w s :
The l a s t d e c a d e h a s s e e n v a r i o u s a t t e m p t s
a t r e f i n i n g some of t h e p r e v i o u s l y p r o p o s e d models f o r t i m e s e r i e s synthesis.
Much of t h e developments i n t h i s r e g a r d h a s t a k e n p l a c e
w i t h o u t due r e g a r d g i v e n t o t h e u n i q u e n a t u r e of t h e p h y s i c a l problem and w i t h o u t c o n s i d e r a t i o n of t h e o r i g i n of d a t a ( s t o c k m a r k e t data versus streamflow d a t a ) .
Perhaps i t is t h e a p p r o p r i a t e t i m e
t o take a f r e s h look a t t i m e series modelling procedures i n t h e hydrologic context.
242
REFERENCES Akaike, H., 1974. A New Look at the Statistical Model Identification, IEEE Trans, Automatic Control, 19(6): 716-723. Box, G.E.P. and Jenkins, G.M., 1973. Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, California. Chi, M., Neal, E. and Young, G.K., 1973. Practical Application of Fractional Brownian Motion and Noise to Synthetic Hydrology, Water Resour. Res., 9: 1569-1582. Chiu, C.L., (Editor), 1978. Applications of Kalman Filter to Hydrology, Hydraulic and Water Resources, Proc. A.G.U., Chapman Conference, University of Pittsburgh. Hurst, H.E., 1951. Long-term Storage Capacity of Reservoirs, Trans. A.S.C.E., 116: 770-808. Hurst, H.E., 1956. Methods of Using Long-term Storage in Reservoirs, Proc. Instn. Civil Engrs., 1: 519-543. Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory, Academic Press, New York. Klemes, V., 1974. The Hurst Phenomena - A Puzzle? Water Resour. Res., lO(4): 675-688. Klemes, V., 1981. Applied Stochastic Theory of Storage in Evolution, Advances in Hydrosciences, 12: 79-141. Lawrance, A.J. and Kottegoda, N.T., 1977. Stochastic Modelling of Riverflow Time Series, Jour. Royal Statist. SOC. Series A, 140(1) : 1-47. Lloyd, E.H., 1967. Stochastic Reservoir Theory, Advances in Hydrosciences, 4: 281-339. Mandelbrot, B.B., 1971. A Fast Fractional Gaussian Noise Generator, Water Resour. Res. 76(3): 543-553. Mandelbrot, B.B. and vanNess, J.W., 1968a. Fractional Brownian Motions, Fractional Noises and Applications, SOC. Ind. Appl. Math. Rev., l O ( 4 ) : 422-437. Mandelbrot, B.B. and Wallis, J.R., 1968b. Noah, Joseph and Operational Hydrology, Water Resour. Res., 4(5): 909-918. Mandelbrot, B.B. and Wallis, J.R., 1969a. Computer Experiments with Fractional Gaussian Noises. Part 1 - Averages and Variances; Part 2 - Rescaled Ranges and Spectra; and Part 3 - Mathematical Appendix, Water Resour. Res., 5(1): 228-267. Mandelbrot, B.B. and Wallis, J.R., 1969b. Some Long Run Properties of Geophysical Records, Water Resour. Res. 5(2): 321-340. Mandelbrot, B.B. and Willis, J.R., 1969c. Robustness of the Rescaled Range R/S in the Measurement of Non-cyclic Long-run Statistical Dependence, Water Resour. Res., 5 ( 5 ) : 967-988.
243 Matalas, N.C. and Wallis, J.R., 1971. Statistical Properties of Multi-variate Fractional Gaussian Noise Processes, Water Resources Research, 7(6): 1460-1668. MacInnes, C.D., 1981. Multiple Time Series Data Extrapolation in Water Resources Engineering Applications using Pattern Recognition Techniques, Doctoral Dissertation, Department of Civil Engineering, University of Waterloo, Ontario, Canada. Moran, P.A.P., 1954. A Probability Theory of Dams and Storage Systems, Australian Journal of Applied Science, 5: 116-124. O’Connell, P.E., 1974. Stochastic Modelling of Long-term Persistence in Streamflow Sequences,” Ph.D. Thesis, Univ. of London, England. Panu, U.S., 1978. Stochastic Synthesis of Monthly Streamflows Based on Pattern Recognition, Doctoral Dissertation, Department of Civil Engineering, University of Waterloo, Ontario. Panu, U.S. and Unny, T.E., 1980a. Extension and Application of Feature Prediction Model for Synthesis of Hydrologic Records, Water Resources Research, 16(1): 77-96. Panu, U.S. and Unny, T.E., 1980b. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition I. General Methodology of the Approach, Journal of Hydrology, 46: 5-34. Panu, U . S . and Unny, T.E., 1980c. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition 11. Application of Natural Watersheds, Journal o f Hydrology, 46: 197-217. Panu, U.S. and Unny, T.E., 1980d. Stochastic Synthesis of Hydrologic Data Based on Concepts of Pattern Recognition 111. Performance Evaluation of the Methodology, Journal of Hydrology, 46: 219-237. Prabhu, N.U., 1964. Time Dependent Results in Storage Theory. J. Applied Probability, Vol 1: 1-46. Soares, E.F., Unny, T.E. and Lennox, W.C., 1977. On a Stochastic Sediment Storage Model for Reservoirs, in Stochastic Processes in Water Resources Engineering. (Eds.) L. Gottschalk, G. Lindh and L. de Marie, Water Resources Publications, Fort Collins, Colorado, 141-166.
.
Unny, T.E., Panu, U.S., MacInnes, C.D. and Wong, A.K.C., 19 Pattern Analysis and Synthesis of Time Dependent Hydrologic Data. Advances in Hydrosciences, Vol 12: 195-295. Unny, T.E. 1977. Transient and non-stationary Random Processes, in Stochastic Processes in Water Resources Engineering, (Eds.) L. Gottschalk, G. Lindh and L. de Marie, Water Resources Publications, Fort Collins, Colorado. Wing, S.P., 1951. Discussion on Long-term Storage Capacity of Reserviors, by Hurst, H.E., Trans. A.S.C.E., 116: 807-808.