PROCEEDINGS OF A N INTERNATIONAL CONFERENCE HELD A T CANADA CENTRE FOR INLAND WATERS, BURLINGTON, ONTARIO, CANADA, OCTO...
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PROCEEDINGS OF A N INTERNATIONAL CONFERENCE HELD A T CANADA CENTRE FOR INLAND WATERS, BURLINGTON, ONTARIO, CANADA, OCTOBER 64,1981
DEVELOPMENTS I N WATER SCIENCE, 17
OTHER TITLES IN THIS SERIES
1
G. BUGLIARELLO AND F. GUNTER
COMPUTER SYSTEMS A N D WATER RESOURCES
2
H.L. GOLTERMAN
PHYSIOL0G:CAL LIMNOLOGY
3
Y.Y. HAIMES, W.A. H AL L AND H.T. FREEDMAN
MULTIOBJECTIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF-METHOD
4
J.J. FRIED
GROUNDWATER POLLUTION
5
N. RAJARATNAM
TURBULENT JETS
6
D. STEPHENSON
PIPELINE DESIGN FOR WATER ENGINEERS
7
v.
HALEK AND J. SVEC
GROUNDWATER HYDRAULICS
8
J.BALEK
HYDROLOGY AND WATER RESOURCES I N TROPICAL AFRICA
9
T.A. McMAHON AND R.G. MElN
RESERVOIR CAPACITY A N D Y I E L D
10 G. KOVACS SEEPAGE HYDRAULICS
1 1 W.H. GRAF AND C.H. MORTIMER (EDITORS) HYDRODYNAMICS OF LAKES: PROCEEDINGS OF A SYMPOSIUM
12-13 OCTOBER, 1978, LAUSANNE, SWITZERLAND
12 W. BACK AND D.A. STEPHENSON (EDITORS) CONTEMPORARY HYDROGEOLOGY: THE GEORGE BURKE M A X E Y MEMORIAL VOLUME
13 M.A. M A R I ~ OAND J.N. LUTHIN SEEPAGE A N D GROUNDWATER
14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE
15 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series)
16
w. BACK AND
R. L ~ T O L L E (EDITORS)
SYMPOSIUM ON GEOCHEMISTRY OF GROUNDWATER
TIME SERIES METHODS PROCEEDINGS OF AN INTERNATIONAL CONFERENCE HELD AT CANADA CENTRE FOR INLAND WATERS, BURLINGTON, ONTARIO, CANADA, OCTOBER 6-8,1981
EDITED BY
A.H. EL- SHAARAWI National Water Research Institute, Burlington, Ont. L7R 4A6 (Canada)
IN COLLABORATION WITH
S.R. ESTERBY National Water Research Institute, Burlington, Ont. L7R 4A6 (Canada)
ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam-Oxford-New York 1982
ELSEVIER SCIENTIFIC PUBLISHING COMPANY Molenwerf 1 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N.Y. 10017
Library of Congress Cataloging in Publication Data
Main entry under title: Time series methods in hydrosciences. (Developments in water science ; 17) Proceedings of the International Conference on Time. Series Methods in Hydrosciences, Canada Centre for Inland Waters, Burlington, Ont., Oct. 6-8, 1981. 1. Hydrology--Mathematicalmodels--Congresses. 2. Time-series analysis--Congresses. I. El-Shaarawi, A H I1 Esterb S. R . 111. IGternat'on 1 Conference or; Time Skries Me%ods in Hydrosciences t1981 : Canada Centre for Inland Waters) IV. Series. GB656.2.M33T55 1982 551.48'0724 82-11378 ISBN 0-444-42102-5 (U.S. )
ISBN 0 4 4 4 4 2 1 0 2 5 (Vol. 17) ISBN 0444-41669-2 (Series)
0 Elsevier Scientific Publishing Company, 1982 All rights reserved. No part o f this publication may be reproduced, stored in a retrieval system or transmitted in any form o r b y any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330,1000 A H Amsterdam, The Netherlands
Printed in The Netherlands
cond.twtu tlze p~oceedingd 0 4 t h e I n t e m a t i u n d M~Ahohi n H q a h o ~ C i t r ~ which c . ~ ~ wm h e l d Conbetence. OM Time at t h e Cazada Certtte dvrr I t h i d IrraCm, B d i n g . t u n , Ont&o, Canada, dctobeh 6, 7 and 8 , 198 1 . The pahtccipantb 0 6 t h e Co,!jetence came 6mm Canada, U S A , UK, H o ~ a n d ,WenZ G e w i y , T M y , Saudi Ambia, A u d t d a , Japan, Tudzey, Nomay and Gneece. InvLted papcM toehe m u ~ n R e dbrj: Tkid
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Financid, L o g h ~ cand m o d hupj3o)Lt wehe phovided by ( 7 ) #le National Watm Raemch ImaXtute, EvlvLtonment Canada, ( 2 ) BaylJidd LabomXotly 604 Manine Science and Swrveyh, V e p m h e n t 0 4 Fi5hetia and Occam, ( 3 1 WcLtcrc Q u u k X t y B m c h , E~zvhonmerd Canada, and ( 4 ) Wate.&Re~aw~cc?.~ Btrancll, OnXahio ILliuLi~ttyolJ Env.i/Lonrner~.t. Foh ad a,U .tltiJ, 1 am indebted t o DL K&h Rodgem, Dhcc;ta~~ National Watt4 Re$eahcl~I r ~ ~ X u t eMa., F . Etdeh, C1ue.d 0 6 Aquatic Pl~ybics and Srj.5terns Divi~iJion ( N W R I ) and A h . 1.D.W. McCUReoch, D.itrcotirn Gerzehd, Bay d i d d LaDotrato/n/ Joa ~ t l a n iScieuzcc? ~ ~ ~ . and Swlve ys
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A.H.
El-Shamaui
VII
CONTENTS
D.R. BRILLINGER, Some C o n t r a s t i n g Examples o f t h e Time and Frequency Domain Approaches t o Time S e r i e s Analysis
1
I A N B. MACNEILL, D e t e c t i o n o f I n t e r v e n t i o n s a t Unknown Times
16
A.M. M A T H A I , D i s t r i b u t i o n o f P a r t i a l Sums w i t h A p p l i c a t i o n s t o Dam Capacity and A c i d Rain
27
R.J. KULPERGER, T e s t i n g f o r Non-Linear S h i f t s i n S t a t i o n a r y @ - M i x i n g Processes
37
M.L. T I K U , A Robust S t a t i s t i c f o r T e s t i n g t h a t two A u t o c o r r e l a t e d Samples come f r o m I d e n t i c a l Populations
45
A.H. EL-SHAARAWI and S . R . ESTERBY, I n f e r e n c e about t h e P o i n t o f Change i n a Regression Model w i t h a S t a t i o n a r y E r r o r Process
55
A.H. EL-SHAARAWI and L . D. DELORME, The Change-Point Problem f o r a Sequence o f Binomial Random V a r i a b l e s
6%
F. ASHKAR, N. EL-JABI, and J . ROUSELLE, E x p l o r a t i o n o f an Extreme Value P a r t i a l T i m e S e r i e s Model i n Hydrosc ience
76
A. A. ABD-ALLA and A. M. ABOUAMMOH, A Comparati ve Study on E s t i m a t i o n o f Parameters o f a Markovian Process - 1
93
U. L. GOURANGA RAO, G e n e r a l i z e d L e a s t Squares Procedure f o r Regression w i t h A u t o c o r r e l a t e d E r r o r s
100
K.W. HIPEL, A . I . MCLEOD and D.J. NOAKES, F i t t i n g Dynamic Models t o H y d r o l o g i c a l T i m e S e r i e s
110
N.T. KOTTEGODA, Some Aspects o f N o n - S t a t i o n a r y Behavi o u r i n Hydrology
130
D.A. CLUIS and P. BOUCHER, P e r s i s t e n c e E s t i m a t i o n f r o m a Time-Series C o n t a i n i n g Occasional M i s s i n g Data
151
VIII Table
06
CoM;t~n,12
CARTWRIGHT, T i d a l A n a l y s i s - A R e t r o s p e c t
170
L.R. M U I R , I d e n t i f i c a t i o n o f I n t e r n a l Tides i n T i d a l C u r r e n t Records f r o m t h e M i d d l e E s t u a r y o f t h e S t . Lawrence
189
D.L. DEWOLFE and R.H. LOUCKS, S i m u l a t i o n o f t h e Low Frequency P o r t i o n o f t h e Sea Level S i g n a l a t Yarmouth, Nova S c o t i a
208
LUNG-FA KU, The Computation o f Tides f r o m I r r e g u l a r l y Sampled Sea Surface H e i g h t Data
21 3
UNNY, On S t o c h a s t i c M o d e l l i n g o f H y d r o l o g i c Data
224
D.E.
T.E.
W. P. BUDGELL, A Dynamic-Stochastic Approach f o r M o d e l l i n g A d v e c t i o n - D i s p e r s i o n Processes i n Open Channels
244
B. DE JONG and A.W. HEEMINK, The Mean and Variance o f Water Currents Induced by I r r e g u l a r Surface Waves
264
L.A. SIEGERSTETTER and W. WAHLIB, Generation o f Weekly Streamflow Data f o r t h e R i v e r Danube-River Main-System
280
VAN-THANH-VAN NGUYEN and JEAN ROUSELLE, P r o b a b i l i s t i c C h a r a c t e r i z a t i o n o f P o i n t and Mean A r e a l R a i n f a l l s
292
M. M I M I K O U and A.R. RAO, A R a i n f a l l - R u n o f f Model f o r D a i l y Flow S y n t h e s i s
29 7
P. VERSACE, M. F I O R E N T I N O and F. ROSSI, A n a l y s i s o f Flood S e r i e s by S t o c h a s t i c Models
31 5
M. BAYAZIT, A Model f o r S i m u l a t i n g Dry and Wet P e r i o d s o f Annual Flow S e r i e s
325
K. MIZUMURA, A Combined S n o w m l t and R a i n f a l l Runoff
341
P.J.W. ROBERTS, A n a l y s i s o f C u r r e n t Meter Data f o r Predi c t i n g Pol 1 u t a n t D i spe r s ion
35 1
A. WILLEN, Should We Search f o r P e r i o d i c i t i e s i n Annual Runoff Again?
36 2
M.G. GOEBEL and T.E. UNNY, Step Ahead Streamflow Forec a s t i n g Using P a t t e r n A n a l y s i s
3 74
Z . SEN, Walsh S o l u t i o n s i n Hydroscience
390
IX
K.W. POTTER and J.F. WALKER, M o d e l l i n g t h e E r r o r i n F l o o d D i s charge M e as u r e ment s
40 5
W. F. CASELTON, I n f o r m a t i o n T h e o r e t i c a l C h a r a c t e r i s t i c s of some S t a t i s t i c a l Models i n t h e Hydrosciences
41 4
D.P. LETTENMAIER and S.J. BURGES, V a l i d a t i o n o f Syn t h e t i c S t reamf 1ow Model s
424
D.P. KRAUEL, F. MILINAZZO, M. PRESS, and W.W. WOLFE, Observation and S i m u l a t i o n o f t h e Sooke Harbour System
445
E. CARONI, F. MANNOCCHI and L. UBERTINI, R a i n f a l l - F l o w R e l a t i o n s h i p i n Some I t a l i a n R i v e r s by M u l t i p l e S t o c h a s t i c Models
455
B.J. NEILSON and B.B. H S I E H , A n a l y s i s o f Water Temperature Records U s i n g a D e t e r m i n i s t i c S t o c h a s t i c Model
465
S.G. RAO and E.W. QUILLAN, S t o c h a s t i c ARIMA Models f o r Mont h 1y S t r e amf 1ow s
4 74
E.H. LLOYD and D. WARREN, The L i n e a r R e s e r v o i r w i t h Seasonal Gamma- D i s t r i b u t e d Markovi an I n f l ows
487
R. M. PHATARFOD, On The Storage Size-Demand-Re1 a b i 1 it y Re 1a t ion s h ip
49 8
J.W. DELLEUR, M. G I N 1 and M. KARAMOUZ, Optimal ARMA Models for the S t a t i s t i c a l Analysis o f Reservoir Operating Rules
51 0
J. STEDINGER and DANIEL P E I , An Annual-Monthly Streamflow Model f o r I n c o r p o r a t i n g Parameter U n c e r t a i n t y i n t o Reservoir Simulation
520
P. BOLZERN, G. FRONZA and G. GUARISO, S t o c h a s t i c Flood P r e d i c t o r s : Experience i n a Small B a s i n
5 30
J.G. SECKLER, Time S e r i e s Mu1t i p l e L i n e a r Regression Models f o r E v a p o r a t i o n f r o m a Free Water Surface
5 3%
R.C. LAZARO, J.W. LABADIE and J.D. SALAS, Optimal Management o f Mu1t i r e s e r v o i r Systems Using Streamflow Fo r e ca s t s
553
X
D.S. GRAHAM and J.M. H I L L , Appropriate Sampling Procedures for Estuarine and Coastal Zone WaterQua1 i ty Measurements
581
S.L. YU and J.F. CRUISE, Time Series Analysis o f Soil Data
600
M. BEHARA and E . KOFLER, Forecasting Under Linear Partial Information
608
1
SOME CONTRASTING EXAKPLSS CF THE TIME AND ?REg'!ENCY
T)OE!sIN
APPRCACHES TC T I F E SSSI ES ANALYSIS
DAVID R. BRILLINGZR U n i v e r s i t y of C a l i f o r n i a , Berkeley ABSTRACT
Two d i s t i n c t approaches t o t h e a n a l y s i s o f time s e r i e s d a t a
-
are i n common use t h e time s i d e and t h e frequency s i d e . The frequency approach i n v o l v e s e s s e n t i a l use of s i n u s o i d s and bands of ( a n g u l a r ) frequency, with F o u r i e r transforms playing an important r o l e . The time approach makes l i t t l e u s e of these. Certain u s efu l techniques are h y b r i d s of t h e s e two approaches. This work proceeding v i a examples, compares and c o n t r a s t s t h e two approaches with r e s p e c t t o modelling, s t a t i s t i c a l i n f e r e n c e and r e s e a r c h e r s ' aims 1
INTRODUCTION
Many, many time series a n a l y s e s have been c a r r i e d o u t a t t h i s
p o i n t i n time. Some of t h e s e a n a l y s e s have been c a r r i e d o u t t o t a l l y i n t h e time domain, some have proceeded e s s e n t i a l l y i n t h e frequency domain, and some have made s u b s t a n t i a l use of b o t h domains. There
are numerous examples i n hydrology of each type of a n a l y s i s . I t seems useful t o examine some time s e r i e s a n a l y s e s t o a t t e m p t t o recogniae t h e s t r e n g t h s and weaknesses of each approach and t o t r y t o d i s c e r n j u s t what l e a d t h e r e s e a r c h e r s involved t o a d o p t the
Particular approach t h a t they d i d . This work p r e s e n t s d e s c r i p t i o n s o f a number of time series a n a l y s e s t h a t t h e a u t h o r has been involved with. Some of t h e s e have been frequency s i d e , some have been time s i d e and some have
been hybrids. Some have been parametric, some have been nonparametReprinted from Time Series Methods in Hydrosciences. by A.H. El-Shaarawi and S.R. Esterby (Editors)
0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
2 r i c e Some have involved l i n e a r systems, some have been concerned with n o n l i n e a r systems. None of t h e s t u d i e s are h y d r o l o g i c a l , however i t i s c l e a r t h a t analagous s i t u a t i o n s do a r i a e i n hydrology.
I t seemed b e s t t o p r e s e n t examples t h a t t h e a u t h o r knew a l l d e t a i l s concerning
2 TIME SERIES A N A L Y S I S hrkey (1978) d e f i n e s our f i e l d of s t u d y a s follows: "Time s e r i e s a n a l y s i s c o n s i s t s o f a l l t h e techniques t h a t , when a p p l i e d t o time s e r i e s d a t a , y i e l d , a t l e a s t sometimes, e i t h e r i n s i g h t o r knowledge, AND e v e r y t h i n g t h a t h e l p s us choose o r understand t h e s e procedures .'I I n t h a t paper he f u r t h e r l i s t s some of t h e aims of time s e r i e s a n a l y s i s . These a r e : 1. discovery of phenomena, 2. modelling,
3 . preparation f o r f u r t h e r i n q u i r y , 4. reaching conclusions, 5. assessment of p r e d i c t a b i l i t y and 6 d e s c r i p t i o n of v a r i a b i l i t y . As one a t t e m p t s t o understand t h e r e l a t i v e merits o f t h e v a r i o u s approaches and techniques of time s e r i e s a n a l y s i s , i t i s worthwhile t o keep t h e above d e f i n i t i o n and aims i n mind. Most r e s e a r c h e r s would seem agreed on what i s a time s i d e a n a l y s i s . There i s u n c e r t a i n t y over j u s t what c o n s t i t u t e s t h e frequency s i d e The f o l l o w i n g v a r i a n t of a s t a t e m e n t i n Bloomfield e t a1 (1981) i s h e l p f u l : frequency s i d e a n a l y s i s i s t h i n k i n g of systems, t h e i r i n p u t s , o u t p u t s and behavior i n s i n u s o i d a l terms.
I t i s e a s i e r t o l i s t techniques t h a t are time s i d e , frequency s i d e o r hybrids. On t h e time s i d e one may l i s t : s t a t e s p a c e , a u t o r e g r e s s i v e moving average (ARMA) and econometric modelling, t r e n d a n a l y s i s , r e g r e s s i o n , p u l s e probing of systems and e m p i r i c a l orthogonal f u n c t i o n s among o t h e r t h i n g s
On t h e frequency s i d e
one may l i s t : s p e c t r a l and c e p s t r a l a n a l y s i s , s e a s o n a l a d j u s t m e n t , harmonic decomposition and s i n u s o i d a l probing of systems
Hybrid
techniques i n c l u d e : complex demodulation, moving spectrum a n a l y s i s and t h e probing of systems by c h i r p s . I n p r a c t i c e i t seems t h a t t h e r e i s u s u a l l y a frequency v e r s i o n of a time s i d e procedure, and
3
vice versa. I t f u r t h e r seems t h a t t h e s e t e c h n i q u e s are g e n e r a l l y a l l i e s , r a t h e r than c o m p e t i t o r s
A number of p r a c t i c a l time s e r i e s a n a l y s e s w i l l now b e d e s c r i b e d and t h e i r type of a n a l y s i s commented on*
3
THE CHANDLER HOBBLE
The p o i n t of i n t e r s e c t i o n of t h e E a r t h ' s a x i s of r o t a t i o n with the p o l a r cap does n o t remain f i x e d , r a t h e r i t wanders a b o u t w i t h i n
a region o f t h e approximate s i z e of a t e n n i s c o u r t - L e t ( X ( t ) , Y ( t ) ) denote t h e c o o r d i n a t e s of t h e p o i n t a t time t , r e l a t i v e t o i t s l o n g
run average p o s i t i o n . S e t Z ( t ) = X ( t )
+
iY(t),
then (from Munk and
lacDonald (1960)) t h e e q u a t i o n s of motion are
w i t h P( t ) t h e e x c i t a t i o n f u n c t i o n whose increments d S ( t ) d e s c r i b e
the change i n t h e E a r t h ' s i n e r t i a t e n s o r i n t h e time i n t e r v a l (t,t+dt)
Supposing t h e p r o c e s s P t o have s t a t i o n a r y increments,
the power spectrum of t h e s e r i e s 2 i s given by
What i s of i n t e r e s t h e r e i s t o d e r i v e an e s t i m a t e of a and t o derive c h a r a c t e r i s t i c s of t h e e x c i t a t i o n p r o c e s s H
I t i s known
t h a t t h e e x c i t a t i o n p r o c e s s c o n t a i n s an annual component, due t o the a l t e r n a t i o n o f s e a s o n s i n t h e southern and n o r t h e r n hemispheres. To b u i l d a s p e c i f i c model, suppose t h a t t h e increments o f s e a s o n a l l y a d j u s t e d QI are white n o i s e with v a r i a n o e a* The spectrum of t h e -2 2 seasonally a d j u s t e d Z i s then lil a1 d /2n The data a v a i l a b l e
-
f o r a n a l y s i s is Z ( t ) p e r t u r b e d by measurement e r r o r f o r t 1p-1
O,..., (In B r i l l i n g e r (1973) i t i s monthly d a t a from 1902 t o 19690) I
Supposing t h e v a r i a n c e of t h e measurement e r r o r s e r i e s t o be p the power spectrum of t h e s e r i e s o f f i r s t d i f f e r e n c e s of t h e seasonally a d j u s t e d d i s c r e t e d a t a i s given by
2
,
4
where a
-
-p + i2f
parameters a,p,x,p,b
Given t h e d a t a one would l i k e e s t i m a t e s of t h e and t o examine t h e v a l i d i t y of t h e model. These
t h i n g s are p o s s i b l e on t h e frequency s i d e . T Let d denote t h e f i n i t e F o u r i e r transform of t h e s e r i e s of
(a)
f i r s t d i f f e r e n c e s of t h e seasonaly c o r r e c t e d data The periodogram T T of t h i s d a t a i s then I (3) The periodogram o r d i n a t e s I (2ns/T), 8
-
1,2,...
b e i n g approximately independent e x p o n e n t i a l v a r i a t e s
with means f (2ns/T), s = 1,2,
respectively, estimating the
parameters by maximizing t h e "Gaussian" l i k e l i h o o d
i s one way t o proceed. ( I n essence t h i s procedure i s suggeeted i n Whittle (1954) .) Estimates d e r i v e d i n t h i s f a s h i o n , and estimates of t h e i r s t a n d a r d e r r o r s a r e p r e s e n t e d i n B r i l l i n g e r (1973). Figure 4 of t h a t paper i s an estimate of t h e power spectrum d e r i v e d by smoothing t h e periodogram t o g e t h e r with t h e e s t i m a t e d above f u n c t i o n a l form. The f i t i s q u i t e good. However t h e nonparametric e s t i m a t e does show a minor peak a t frequency 0154 cycles/month t h a t i s s u s p i c i o u s l y large
This frequency
w a s f u r t h e r i n v e s t i g a t e d by t h e method of complex demodulation. Complex demodulation i s a h y b r i d frequential-temporal
technique
If X(t) denotes t h e s e r i e s of concern, then t h e s t e p s i n v o l v e d are: i. form U ( t )
E
X(t)exp(-iht)
, f o r > the
frequency of i n t e r e s t , ii.
smooth t h e series U ( t ) t o o b t a i n t h e s e r i e s V ( t ) , demodulate a t frequency
A , iii.
t h i s i s t h e complex
graph l V ( t ) I 2 and arg V ( t )
One
of t h e important u s e s of complex demodulation i s t h e d e t e c t i o n of changes with time i n a frequency band o f i n t e r e s t . For t h e frequency
0154 s p e c i a l a c t i v i t y seems t o be p r e s e n t o n l y f o r t h e p e r i o d 1905 to 1914
5
The above a n a l y s i s took p l a c e p r i n c i p a l l y i n t h e frequency domain, b u t p a r t i a l l y i n t h e h y b r i d domain a s well. The advantages of the frequency domain included: a. o p e r a t i o n s on t h e s e r i e s (sampling, s e a s o n a l adjustment
, differencing)
could be handled
d i r e c t l y , b. measurement n o i s e w a s e a s i l y d e a l t with, c. e s t i m a t i o n became a problem of maximizing a n elementary f u n c t i o n , d . s t a n d a r d e r r o r s were a byproduct of t h e e s t i m a t i o n procedure- I t i s f u r t h e r evident t h a t a frequency component p r e s e n t f o r only a r e s t r i c t e d time period could o n l y be discovered by a h y b r i d procedure. This was why complex demodulation was so u s e f u l .
4
FREE OSCILLATIONS OF THE EARTH For a time i n t e r v a l a f t e r a major earthquake t h e E a r t h r i n g s a t
c e r t a i n fundamental f r e q u e n c i e s . This motion i s called i t s f r e e o s c i l l a t i o n s - The f r e q u e n c i e s are c a l l e d i t s e i g e n f r e q u e n c i e s . The estimation o f t h e v a l u e s of t h e e i g e n f r e q u e n c i e s and t h e i r a s s o c i a t e decay r a t e s i s a problem of fundamental importance t o g e o p h y c i s t e building models of t h e Earth. The problem i s t h a t of how t o estimate these parameters given t h e seismogram of a major earthquake. The frequency domain p r o v i d e s an e f f e c t i v e means o f doing t h i s . Complex demodulation provides an e f f e c t i v e meana of checking t h e mechanical model Dynamical c o n s i d e r a t i o n s s u g g e s t t h e f o l l o w i n g model f o r t h e seismogram,
K
+
I
for t
>
0 , with t h e
xk t h e e i g e n f r e q u e n c i e s
of i n t e r e s t , the
6,
t h e i r decay r a t e s , a and $ c o n s t a n t s and E a n o i s e s e r i e s . k k Crude estirr.ates of t h e m a y be d e r i v e d by graphing t h e periodo-
Xk
gram of a data s t r e t c h . The model may be examined by complex demodul a t i n g a t estimated
xk
rn
If t h e smoothing f i l t e r haa a bandwidth
small enough t o exclude o t h e r e i g e n f r e q u e n c i e s , and i f t h e above model h o l d s with t h e n o i s e n o t too s u b s t a n t i a l , then a graph of
6
l o g l V ( t ) l w i l l f a l l o f f i n a l i n e a r f a s h i o n ( e l o p e approximately
-Ok)
and arg V ( t ) w i l l be approximately c o n s t a n t ( i f t h e e s t i m a t e d
frequency i s c l o s e enough t o t h e t r u e one.)
B o l t and B r i l l i n g e r
(1979) p r e s e n t such graphs f o r t h e r e c o r d made a t T r i e s t e of t h e great Chilean earthquake of 1960. The model seem8 confirmed. What
i s needed now are p r e c i s e e s t i m a t e s of t h e unknown parameters and e s t i m a t e s of t h e i r s t a n d a r d e r r o r s . These may be c o n s t r u c t e d a e follows
.
Let a = =
d:(A)
x+ i O ,
b = aexp(id),
T-1 C X ( t ) exp(-ih)
T-1
AT(A) =
and
For
2 in
dT(2)
b
a n i n t e r v a l Ik near
dk , one
Wow i f t h e n o i s e s e r i e s ,
well-separated
E,
T
A ,will
exp(-i%)
T
h a s dX@)
T
bkD
( 3 - %) +
i s s t a t i o n a r y and such t h a t
v a l u e s a r e only weakly dependent, then t h e f i n i t e
F o u r i e r transform v a l u e s d (2xs/T) near
C
t=O
tEO
&
, for
s an i n t e g e r w i t h 2ne/T
be approximately independent complex normal v a r i a t e s
(See B r i l l i n g e r (1981) f o r with mean 0 and v a r i a n c e 2nTf & & ( A ) example a ) The m a x i m u m l i k e l i h o o d estimates of t h e unknown parameters a r e t h u s t h e l e a s t s q u a r e s e s t i m a t e s found by minimizing
where t h e summation i s over f r e q u e n c i e s 2 n s / T i n Ik
Further the
asymptotic d i s t r i b u t i o n of t h e s e e s t i m a t e s may be found d i r e c t l y and s o s t a n d a r d e r r o r s e s t i m a t e d and confidence i n t e r v a l s c o n s t r u c t -
ed. D e t a i l s a r e given i n B o l t and B r i l l i n g e r (1979). Once a g a i n , by going o v e r t o t h e frequency domain a d i r e c t estima t i o n procedure h a s been found. Because e s t i m a t e s of s t a n d a r d e r r o r s a r e p a r t of t h e procedure, e s t i m a t e d e i g e n f r e q u e n c i e s from d i f f e r e n t seismograms may now be combined e f f i c i e n t l y
F u r t h e r t h e approximate
sampling p r o p e r t i e s of t h e e s t i m a t e s a r e c l e a r , b e i n g based on normal v a r i a t e s e A h y b r i d procedure allowed c o n f i r m a t i o n of t h e model.
7
5
TIiE HUMAN PUPILLARY SYSTEM The p u p i l of t h e eye e x h i b i t s a number of n o n l i n e a r c h a r a c t e r -
i s t i c s . When i t i s probed with narrow bandwidth s i n u s o i d a l l i g h t , the motions of i t s diameter d i s p l a y second and p o s s i b l y t h i r d o r d e r harmonics of t h e fundamental f r e q u e n c y - F u r t h e r t h e shape of t h e t r a n s f e r f u n c t i o n e s t i m a t e d by such s i n u s o i d a l probing changes as the amplitude of t h e s t i m u l u s i s v a r i e d and f i n a l l y a dynamic asymmetry i s e x h i b i t e d between responses t o on and o f f s t i m u l i -
I t i s a p p a r e n t t h a t a n o n l i n e a r model needs t o be developed i n o r d e r to d e s c r i b e t h e p u p i l l a r y system9
A u s e f u l model f o r n o n l i n e a r systems i s t h e f o l l o w i n g one d i s c u s s ed by Tick (1961), =
Y(t)
a
+
Ja(t-u)X(u)du
+
Js b(t-u,t-v)X(u)X(v)dudv
+
E(t)
with X , t h e system i n p u t s t a t i o n a r y and Gaussian, with Y t h e system output and with
t
a s t a t i o n a r y n o i s e s e r i e s . L e t A and B denote t h e
l i n e a r and q u a d r a t i c t r a n s f e r f u n c t i o n s of t h i s system,
B(2,p)
=
b(u,v)exp(-du
-ipv)dudv
,
then, i n t h i s case o f Gaussian s t i m u l a t i o n , one h a s t h e r e l a t i o n s h i p s
f XXy
(-2, C‘ I
.
P
Here f X X i s t h e power spectrum o f t h e i n p u t , f y X t h e cross-spectrum of the i n p u t and t h e o u t p u t and fXXy t h e cross-bispectrum
of t h e
input and t h e o u t p u t - ( T h i s l a s t i s t h e F o u r i e r transform o f t h e t h i r d o r d e r cross-moment
function
0 )
These l a s t r e l a t i o n s h i p s a l l o w t h e computation o f e s t i m a t e s of
8
A and B once e s t i m a t e s of t h e s p e c t r a involved have been computed. The s p e c t r a l e s t i m a t e s may be based d i r e c t l y on t h e F o u r i e r transforme o f t h e d a t a s t r e t c h e s a v a i l a b l e . As a f i n a l s t e p a and b may be e s t i m a t e d by back F o u r i e r t r a n s f o r m i n g t h e estimates of A and B, t a k i n g care t o i n s e r t convergence f a c t o r s i n t h e process. Hung e t a1
(1979) p r e s e n t t h e s p e c i f i c computational formulas involved and p r e s e n t an example of t h i s system i d e n t i f i c a t i o n procedure f o r t h e human p u p i l l a r y system- The e s t i m a t e d a and b a r e found t o make
sense p h y s i o l o g i c a l l y and t o be c o n s i s t e n t with c h a r a c t e r i s t i c s noted i n o t h e r t y p e s of experiment with t h e system. The e x t e n t of l i n e a r i t y of t h e system may be measured by t h e ( l i n e a r ) coherence
2
with 1 R I ,< 1 and t h e n e a r e r i t i s t o I, t h e more s t r o n g l y l i n e a r the quadratic t h e system. S e t t i n g U( t ) J]b( t-u, t-v)X(u)X(v)dudv
-
,
coherence i s d e f i n e d as
This too i s bounded by 1, with i t s n e a r n e s s t o 1 i n d i c a t i n g how purely q u a d r a t i c t h e system i s - The s t r e n g t h of l i n e a r p l u s pure 2
I + 1RYW l 2 E s t i m a t e s YX of t h e l i n e a r and q u a d r a t i c coherence f o r t h e human p u p i l a r e
q u a d r a t i c r e l a t i o n s h i p i s measured by IR
p r e s e n t e d i n Hung e t a1 (1979)- The l i n e a r coherence i s l a r g e r , b u t t h e q u a d r a t i c i s i m p o r t a n t as w e l l . The above a n a l y s i s is a frequency domain o n e * Had t h e i n p u t s e r i e s been Gaussian white n o i s e , a and b could have been e s t i m a t e d d i r e c t l y by c r o s s - c o r r e l a t i o n ,
however i n t h e experiments of Hung e t a1 X
could n o t be taken t o be white n o i s e . ( A s i d e remark i s t h a t even i n t h e white n o i s e c a s e , t h e c r o s s - c o r r e l a t i o n s might be b e t t e r computed v i a a ( f a s t ) F o u r i e r transform.)
I n t h e non-white
case a form of
deconvolution needs t o be c a r r i e d o u t and t h i s i s done e f f e c t i v e l y
9
via frequency domain procedures. Proceeding via t h e frequency domain l e a d t o t h e d e f i n i t i o n of t h e l i n e a r and q u a d r a t i c coherences. These a r e frequency s i d e parameters t h a t prove exceedingly u s e f u l i n p r a c t i c e . There seem t o be no u s e f u l time s i d e analogs.
6
A L I N E A R DESCRIPTION OF NEURON FIRING
I n an important c l a s s of n e u r o p h y s i o l o g i c a l experiments a sequence of c o n s t a n t amplitude e l e c t r i c a l impulses i s taken as i n p u t t o a neuron. The neuron i n t u r n emits a t r a i n of n e a r c o n s t a n t amplitude e l e c t r i c a l impulses
The n e u r o p h y s i o l o g i s t i s i n t e r e s t e d i n d e s c r i -
bing and u n d e r s t a n d i n g t h e p r o c e s s by which an i n p u t t r a i n i s converted t o an o u t p u t t r a i n . To develop a formal d e s c r i p t i o n o f such a p r o c e s s i t i s convenient
t o a s s i m i l a t e t h e i n p u t and o u t p u t p u l s e t r a i n s t o p o i n t p r o c e s s e s
M and N with M(t) t h e number of i n p u t p u l s e s i n t h e time i n t e r v a l ( 0 , t ) and N ( t )
t h e corresponding number of o u t p u t p u l s e s - A l i n e a r
model r e l a t i n g two p o i n t p r o c e s s e s i s d e s c r i b e d by
I
Prob(N p o i n t i n ( t , t + h )
M )
-
[p
+
C a(t
- cj)jh
j
f o r small h , where t h e u . a r e t h e times of i n p u t p u l s e s . I t i s of J i n t e r e s t t o e s t i m a t e t h e f u n c t i o n a and t o c o n s t r u c t a measure o f how a p p r o p r i a t e t h i s model i s i n p r a c t i c a l s i t u a t i o n s . These t h i n g s may be done by means of a frequency s i d e approach. The b a s i c s t a t i s t i c i s once a g a i n a f i n i t e F o u r i e r transform,
.
T
c
exp(-i>g.) O t ) d l ( t ) 0
The periodogram of t h e d a t a i s d e f i n e d as The power spectrum, f
T going t o
o),
(A),
of E I$?A)
.
IMM(A) T
may be d e f i n e d f o r A t
= (2nT)'l
3\ f
l d i o ) I2
0 as t h e l i m i t ,
= 0 i t may b e d e f i n e d by c o n t i n u i t y .
As i n t h e c a s e of o r d i n a r y time s e r i e s , t h e power spectrum may be
10
-
cross-spectrum may be d e f i n e d and e s t i m a t e d i n a similar f a s h i o n . The model l e a d s d i r e c t l y t o t h e r e l a t i o n s h i p fNM(A) with A t h e F o u r i e r transform of a
A(A)f,,(A),
This r e l a t i o n s h i p p r o v i d e s
e s t i m a t e s of A and a i n t u r n . Quite a number of such e s t i m a t e s are given i n B r i l l i n g e r e t a1 (1976) f o r neurons of t h e s e a h a r e . One f a c t o r c a u s i n g t h e forms of A and a t o vary s u b s t a n t i a l l y i s whether t h e synapse i s e x c i t a t o r y ( i n p u t t e n d s t o i n c r e a s e t h e o u t p u t r a t e ) o r i n h i b i t o r y ( i n p u t d e c r e a s e s t h e o u t p u t r a t e ) . The time l a g from i n p u t t o o u t p u t shows up i n t h e estimates a s w e l l , as does t h e r e f r a c t o r y period ( o u t p u t p u l s e s may n o t be spaced a r b i t r a r i l y closely together)
The degree t o which t h e o u t p u t t r a i n may b e determined from t h e i n p u t v i a t h e model p r e s e n t e d i s conveniently measured by t h e
12
= If,,(h) 12/fNN(>)fMM(>),once a g a i n I n t h e examples of B r i l l i n g e r e t a1 (1976) t h i s f u n c t i o n i s found
coherence f u n c t i o n ,
t o vary s u b s t a n t i a l l y with frequency. Generally i t i s much l a r g e r a t t h e lower f r e q u e n c i e s - I t i s s u r p r i s i n g l y l a r g e i n many c a s e s given t h e e s s e n t i a l n o n l i n e a r i t y of t h e system under study. The frequency s i d e approach i s n a t u r a l l y e f f e c t i v e i n d e t e c t i n g p e r i o d i c i t i e s t h a t a r e p r e s e n t and i n one of t h e B r i l l i n g e r e t a1 (1976) examples t h e e s t i m a t e d power spectrum d i s p l a y s a minor peak corresponding t o a p e r i o d i c i t y t h a t r e a l l y could n o t be seen on t h e time s i d e . However, a s t h e above development makes c l e a r , t h e frequency approach f u r t h e r a l l o w s t h e deconvolution of i n p u t from system c h a r a c t e r i s t i c s and l e a d s t o t h e d e f i n i t i o n of a u s e f u l measure of l i n e a r time i n v a r i a n t a s s o c i a t i o n The c i t e d r e f e r e n c e p r e s e n t s a frequency s i d e s o l u t i o n t o an important problem f o r which no o t h e r s o l u t i o n i s p r e s e n t l y known- I t conczrned t h e p h y s i o l o g i c a l connections of t h r e e neurons, L 2 , L 3 and
L 1 0 , of t h e s e a h a r e - The t h r e e neurons were c l e a r l y r e l a t e d ( t h e r e
was s u b s t a n t i a l coherence between a l l p a i r s o f covarying p u l s e trains)
I t was known t h a t L10 was t h e d r i v i n g neuron; however i t was
n o t known i f t h e neurons were i n s e r i e s L l O + L 2 3 L 3
o r L10 --sL2-+
L 3 o r i f L 3 and L2 had no d i r e c t connection, b u t L10 3 L 2 and
11
L10 *L3
only-
P a r t i a l coherence a n a l y s i s i s a u s e f u l t o o l f o r examining such q u e s t i o n s . Denote t h e s p i k e t r a i n s by A , B , C r e s p e c t i v e l y . The p a r t i a l coherence between t r a i n s A and B i s d e f i n e d t o b e t h e coherence between t h e t r a i n s A and B w i t h t h e l i n e a r time i n v a r i a n t e f f e c t s of C removed- I t i s given by t h e modulus-squared
I n t h i s e x p r e s s i o n dependence on
A
of
h a s been suppressed f o r conveni-
ence- I n t h e c a s e r e f e r r e d t o , t h e p a r t i a l coherence of L 3 and L2 with t h e e f f e c t s o f L10 removed w a s n o t s i g n i f i c a n t and t h e presence o f a d i r e c t L 2 t o L3 c o n n e c t a r could be r u l e d o u t e s s e n t i a l l y .
7
THE THRESHOLD MODEL OF NFURON FIRING Suppose t h a t a neuron r e c e i v e s as i n p u t t h e f l u c t u a t i n g e l e c t r i c -
al s i g n a l X ( t ) c P h y s i o l o g i c a l c o n s i d e r a t i o n s s u g g e s t t h e f o l l o w i n g d e s c r i p t i o n of i t s f i r i n g . A membrane p o t e n t i a l
0
i s formed i n t e r n a l l y , where a (
0 )
d e s c r i b e s a summation p r o c e s s and
B ( t ) d e n o t e s t h e time, a t t , s i n c e t h e neuron l a s t f i r e d . The neuron then f i r e s when U ( t ) c r o s s e s a t h r e s h o l d 8
+
E(t),
E
being a noise
process. Given experimental d a t a i t i s o f i n t e r e s t t o v e r i f y and f i t t h i s model
Frequency a n a l y s e s may be c a r r i e d o u t i n t h e manner o f t h e previous s e c t i o n given s t r e t c h e s o f i n p u t and corresponding o u t p u t d a t a -
However g i v e n t h e e s s e n t i a l n o n l i n e a r i t y o f t h e system and t h e feedback from o u t p u t t o i n p u t (due t o t h e p r e s e n c e o f B ( t ) ) t h e s e m a y
n o t be expected t o b e e f f e c t i v e . (As w i l l b e mentioned l a t e r , i n t h e case t h a t X can be t a k e n t o be Gaussian s t a t i o n a r y t h e y a r e o f some we.)
vi ded
I n B r i l l i n g e r and Segundo (1979) a time s i d e s o l u t i o n i s pro-
.
12
L e t Xt,
Ut,
Yt,
t = O,kl,...
denote t h e sampled v e r s i o n s of the
s e r i e s i n v o l v e d - One h a s Yt = 1 i f t h e neuron f i r e d a t time t and
Yt = 0 otherwise. Suppose t h a t t h e n o i s e i s Gaussian w h i t e , then Prob(Yt
I
I
1
Ut)
@(Ut
0
- Q)
with f t h e normal cumulative- F u r t h e r , c o n d i t i o n a l on t h e given i n p u t t h e l i k e l i h o o d f u n c t i o n of t h e data i s
The parameters au and
€3
may now be e s t i m a t e d by maximizing t h i s
l i k e l i h o o d - B r i l l i n g e r and Segundo (1979) p r e s e n t a number of e s t i mates found i n t h i s f a s h i o n f o r t h e neurons R2 and L 5 o f t h e sea
hare. Once t h e s e e s t i m a t e s have been o b t a i n e d , t h e f u n c t i o n Prob(Y = 1
I
u) may b e e s t i m a t e d - This was done- I t was found t o
have t h e sigmoidal shape of 8 I n t h e case t h a t t h e i n p u t X i s Gaussian and t h e feedback e f f e c l i s n o t l a r g e , i t may b e shown t h a t t h e e s t i m a t e d aU d e r i v e d via
cross-spectral
a n a l y s i e a r e , up t o sampling f l u c t u a t i o n s , p r o p o r t i -
onal t o t h e d e s i r e d a
U
(See B r i l l i n g e r (1977)o) Such e s t i m a t e s are
given i n B r i l l i n g e r and Segundo (1979) and good agreement foundFor t h i s problem, a frequency a n a l y s i s could n o t s u f f i c e - The system had a n o n l i n e a r i t y and a feedback was p r e s e n t . By c h o i c e of s p e c i a l i n p u t , (Gaussian), and i f t h e feedback was n o t s t r o n g , t h e frequency a n a l y s i s gave approximate answers; however i t i s b e t t e r t o a d d r e s s t h e system d i r e c t l y .
8
NICHOLSON'S DATA ON SHEEP BLOWFLIES During t h e 1950's t h e A u s t r a l i a n entomologist A - J .
Nicholson
c a r r i e d o u t an e x t e n s i v e series of experiments concerning t h e p o p u l a t i o n v a r i a t i o n of L u c i l i a c u p r i n a ( t h e sheep b l o w f l y ) under vari o u s c o n d i t i o n s - Nicholson maintained p o p u l a t i o n s o f t h e f l i e s on v a r i o u s d i e t s (some c o n s t a n t , some f l u c t u a t i n g ) , e x p e r i e n c i n g
13 d i f f e r e n t forms of competition (between l a r v a e and a d u l t s , f o r egg laying s p a c e , e t c - ) , and o t h e r many o t h e r c o n d i t i o n s . The paper B r i l l i n g e r e t a 1 (1980) r e p o r t s t h e a n a l y s i s of population d a t a f o r a cage maintained under c o n s t a n t c o n d i t i o n s - ?he b a s i c d a t a were t h e numbers of f l i e s emerging and f l i e s d i e i n g i n s u c c e s s i v e two day i n t e r v a l s . From t h e s e s e r i e s , and t h e i n i t i a l c o n d i t i o n s , t h e number
o f a d u l t s a l i v e a t time t could be computed. The amount o f food provided t h e f l i e s was c o n s t a n t and l i m i t e d . This caused t h e population s i z e t o o s c i l l a t e d r a m a t i c a l l y , f o r when many f l i e s were present the females d i d n o t r e c e i v e enough p r o t e i n t o r e a l i z e t h e i r m u -
i m u m f e c u n d i t y . I n consequence many fewer eggs were l a i d and t h e next g e n e r a t i o n smaller
Nicholson r a n t h e experiment f o r approxima-
t e l y 700 d a y s The l i f e c y c l e of a blowfly l a s t s 35 t o 40 days. The a g g r e g a t e numbers d i s p l a y e d an o s c i l l a t i o n with t h i s p e r i o d throughout much of the experiment. O s t e r
(1977) p r e s e n t s t h e F o u r i e r spectrum of t h e
data and a peak does s t a n d o u t - However, while t h e d a t a does have s u b s t a n t i a l s t a t i o n a r y f e a t u r e s , i t a l s o h a s a c h a o t i c appearance i n one s t r e t c h . Spectrum a n a l y s i s does n o t take n o t i c e of a l t e r n a t e behavior i n s e p a r a t e s t r e t c h e s
Complex demodulation w a s n o t espec-
i a l l y i n f o r m a t i v e e i t h e r . A cross-spectrum a n a l y s i s of t h e number of deaths, D t ,
on t h e number of emergences, E t ,
l e d to a p l a u s i b l e
shape f o r t h e impulse response, however t h e coherence was n o t high.
I t seemed t h a t a much b e t t e r d e s c r i p t i o n must be o b t a i n a b l e f o r such f i n e experimental d a t a Considerations of t h e biology involved suggested t h a t t h e probabi l i t y of a blowfly d i e i n g , i n a two day p e r i o d , would depend on i t s age, i , on t h e number, N , i t was competing w i t h , and t h e number, N-,
i t had competed with l a s t time p e r i o d - An e x p r e s s i o n t h a t worked well was
with 8 d e n o t i n g t h e unknown parameter v a l u e s a.
3'
@,
A s t a t e space
14 approach, (Gupta and Mehra
(1974), L i p s t e r and Shiryayev (1978)), was
then taken f o r t h e d e s c r i p t i o n of t h e d a t a . A s t a t e v e c t o r
&
was
defined whose e n t r i e s gave t h e (unobservable except f o r a g e 0 ) memb e r s of each age group. The Kalman-Bucy f i l t e r w a s s e t up f o r
and maximum l i k e l i h o o d e s t i m a t i o n came down t o choosing 0 t o minimize
S p e c i f i c d e t a i l s may be found i n B r i l l i n g e r e t a1 (1980). The model
was found t o provide an e f f e c t i v e descripti*on For t h i s d a t a n o n l i n e a r i t i e s were p r e s e n t
F'urther d i f f e r e n t sub-
groups of t h e population were behaving d i f f e r e n t l y . D e s p i t e t h e presence of understood o s c i l l a t i o n s , a frequency approach was n o t very revealing
9
D I SCITSSI ON This paper h a s d e s c r i b e d a number of time s e r i e s a n a l y s e s proceed-
i n g from a frequency s i d e a n a l y s i s t o a time s i d e a n a l y s i s with some hybrid a n a l y s e s i n between. I n each c a s e no i n i t i a l conimitment was made t o one s i d e or t h e o t h e r , r a t h e r a t some s t a g e one approach became much more r e v e a l i n g than t h e o t h e r Because of space l i m i t a t i o n s some of t h e b a s e s f o r d e c i d i n g on the f i n a l approach w i l l simply be l i s t e d . These a r e : g o a l s and circums t a n c e s , e a s e of ( p h y s i c a l ) i n t e r p r e t a t i o n , s i m p l i c i t y and parsimony, sampling f l u c t u a t i o n s
, computational
diff iculty
, s e n s i t i v i t y , physi-
c a l theory ( v e r s u s b l a c k b o x ) , d a t a q u a l i t y , d a t a q u a n t i t y , e a s e i n d e a l i n g with c o m p l i c a t i o n s , e x p e r t n e s s a v a i l a b l e , r e a l time v e r s u s dead time, e f f i c i e n c y , dangers ( e g - o v e r t i g h t p a r a m e t e r i z a t i o n ) , bandwidth of phenomenon, presence and type of n o n l i n e a r i t i e s , type of n o n s t a t i o n a r i t y .
15 10
REFERENCES
Bloomfield, P O , B r i l l i n g e r , D.R.,
Cleveland, W.S-
and Tukey, J.W.,
1981. The P r a c t i c e of Spectrum Analysis. I n p r e p a r a t i o n , 233 pp* B o l t , B -A and B r i l l i n g e r , D OR , 1979 E s t i m a t i o n of u n c e r t a i n t i e s i n e i g e n s p e c t r a l e s t i m a t e s . Geophys. J. R. a s t r - Soc.,59:593-603*
1973. An e m p i r i c a l i n v e s t i g a t i o n of t h e Chandler wobble. Bull- I n t e r n a t - S t a t i s t . I n s t . , 45: 413-433B r i l l i n g e r , D.R., 1977 The i d e n t i f i c a t i o n o f a p a r t i c u l a r n o n l i n e a r time s e r i e s system. Biometrika, 64: 509-515B r i l l i n g e r , D OR., 1981 Time S e r i e s : Data A n a l y s i s and Theory-
B r i l l i n g e r , D.R.,
Holden-Day,
San Francisco,540 pp-
B r i l l i n g e r , D-R.,
and Segundo, J - P - , 1976. I d e n t i f i c a t -
Bryant, H.L.
i o n of s y n a p t i c i n t e r a c t i o n s . Biol
Cybernetics, 22: 213-2280
B r i l l i n g e r , D-B-, Guckenheimer, J o , Guttorp, Po and O s t e r , G O , 1980. L e c t u r e s on Math. i o L i f e S c i . 1 3 0 A m e r - l a t h . SOC., Providence. B r i l l i n g e r , D.R.
1979
and Segundo, J.P.,
E b p i r i c a l examination of
t h e t h r e s h o l d model of neuron. B i o l . Cybernetics, 35: 213-2200 Cupta, N.K-
1974
and Flehra, R.K.,
likelihood estimation
Computational a s p e c t s of maximum
IEEE Trans
Aut
Control, AC-19 : 771-783
1979 I n t e r p r e t a t i o n of k e r n e l s . 11. Math. B i o s c i e n c e s , 46: 159-1870 L i p s t e r , R.S. and Shiryayev, A.N., 1978. S t a t i s t i c s of Random ProcHung, G o , B r i l l i n g e r , D.R.
and S t a r k , L o ,
esses. S p r i n g e r , New York.
Hunk, W.H.
and MacDonald, G.J.F.,
1960
The Rotation of t h e Earth.
Cambridge P r e s s , Cambridge. Oster,
Go,
1977. Modern Modelling of Continuum Phenomena,
(Editor) Tick, L - J . , systems Tukey, J e w . ,
Amer 1961
Po,
SOC
, Providence
The e s t i m a t i o n o f t r a n s f e r f u n c t i o n s o f q u a d r a t i c
Technometrics, 3: 563-567
1978. Can we p r e d i c t where "time s e r i e s " . I n : D.R.
i n g e r and G.C. Whittle,
Math
R e DiPrima
Brill-
Tiao ( E d i t o r s ) D i r e c t i o n s i n Time S e r i e s . IMS.
19540 Some r e c e n t c o n t r i b u t i o n s t o t h e t h e o r y of s t a t i -
onary p r o c e s s e s - I n : H. Wold, A Study i n t h e Analysis of S t a t i o n a r y Time-Series * Almqvist and Wiksell, Uppsala rn
16
DETECTION OF INTERVENTIONS AT UNKNOWN TIMES I A N 9. MACNEILL S t a t i s t i c a l L a b o r a t o r y , The U n i v e r s i t y o f Western O n t a r i o , London, O n t a r i o , Canada 1
INTRODUCTION Models f o r h y d r o l o g i c a l t i m e s e r i e s a r e c h a r a c t e r i z e d by para-
meters which may s t a y c o n s t a n t o r which may change o v e r t h e course o f time.
While t h e d e t e c t i o n o f changes i n parameters when t h e
t i m e o f change i s s p e c i f i e d i s a r e l a t i v e l y s t a n d a r d s t a t i s t i c a l problem, t h e d e t e c t i o n o f changes when t h e t i m e o f change i s unknown i s a non-standard problem t h a t i s c u r r e n t l y r e c e i v i n g cons iderabl e a t t e n t i o n .
A method o f d e t e c t i n g change o f r e g r e s s i o n parameters a t unknown times i s presented.
A d e r i v a t i o n i s presented o f a l i k e -
l i h o o d r a t i o t y p e s t a t i s t i c f o r d e t e c t i n g changes i n r e g r e s s i o n parameters a t unknown times. s t a t i s t i c a r e discussed.
Distributional properties o f the
The s t a t i s t i c i s t h e n a p p l i e d t o s e v e r a l
periodic series. Models a r e then c o n s i d e r e d f o r i m p r o v i n g t h e s h o r t and i n t e r mediate-term f o r e c a s t i n g c a p a c i t y o f p e r i o d i c models and y e t p r e s e r v i n g b o t h l o n g - t e r m p r e d i c t i v e c a p a c i t y and t h e c l e a r meaning o f t h e model parameters.
The models a r e c o n s t r u c t e d so as t o
have c e r t a i n o f t h e p r o p e r t i e s o f a u t o r e g r e s s i v e schemes b u t y e t t o r e t a i n the basic properties o f p e r i o d i c i t y .
For a u t o r e g r e s -
sions, one uses t h e e n t i r e observed d a t a t o e s t i m a t e t h e parameters. Then one uses these e s t i m a t e s t o e s t i m a t e t h e n e x t o b s e r v a t i o n from s e v e r a l o f those i m m e d i a t e l y p r e c e d i n g i t .
The r e s i d u a l s
formed by t h e d i f f e r e n c e s between t h e a c t u a l and e s t i m a t e d observ a t i o n s may be used t o t e s t g o o d n e s s - o f - f i t .
T h i s regime i s
a p p l i e d t o t h e a m p l i t u d e o f p e r i o d i c components.
The b a s i c a m p l i -
tude i s f i r s t e s t i m a t e d and then i s m o d i f i e d by a d a p t i v e means by t h e a d d i t i o n o f components d e f i n e d by t h e immediately p r e c e d i n g Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
17
observations.
Thus, f o r example, t h e a m p l i t u d e may s h r i n k i f p r e -
s e n t and p r e v i o u s o b s e r v a t i o n s i n d i c a t e t h e presence o f a low amplitude c y c l e .
The e f f e c t o f p r e s e n t o b s e r v a t i o n s on t h e a m p l i -
tude disappears i n t h e long-term; hence t h e b a s i c c y c l e d e f i n e s the long-term p r e d i c t i o n .
2
DOUBLY STOCHASTIC MODELS
A doubly s t o c h a s t i c r e g r e s s i o n model may be d e f i n e d as f o l l o w s . Let { E ( j ) , j
2
11
be a sequence o f independent and i d e n t i c a l l y d i s -
t r i b u t e d e r r o r terms each n o r m a l l y d i s t r i b u t e d w i t h z e r o mean and v a r i a n c e cj2 > 0, and l e t { f i ( t ) , set o f regressor functions.
t > 0, i
=
0, 1,
Also, l e t { B ( j ) , j
quence o f r e g r e s s i o n c o e f f i c i e n t s .
I f p' ( j ) =
...,
p) denote a
11 denote a se( 5 (j ) .. . , 8 p ( j ) ) 2
i s s t o c h a s t i c , t h e n t h e dependent v a r i a b l e s denoted by C Y ( j ) , j
2
13
i n a doubly s t o c h a s t i c r e g r e s s i o n model may be defined as f o l l o w s : Y(j) =
o;i
P fii(j)
fi(j) +
dj), j
2
1.
An approach t o t h e e s t i m a t i o n of v a r i a b l e r e g r e s s i o n parameters i n v o l v i n g r e c u r s i v e l e a s t squares r e g r e s s i o n a n a l y s i s i s g i v e n by Young [1974] who discusses s e v e r a l s t o c h a s t i c models which possess s p e c i f i e d s t r u c t u r e f o r r e g r e s s i o n parameters.
H a r r i s o n and S t e -
vecs [1976] d e f i n e a dynamic l i n e a r model which i s c h a r a c t e r i z e d by s t o c h a s t i c parameters whose e s t i m a t i o n u t i l i z e s t h e r e c u r s i v e procedures o r i g i n a l l y f o r m u l a t e d b y Kalman [1963]; again, t h e s t r u c t u r e f o r t h e s t o c h a s t i c models f o r t h e r e g r e s s i o n parameters
is assumed known.
Brown, D u r b i n and Evans [1975] use r e c u r s i v e r e -
s i d u a l s t o a t t a c k t h e problem of d e t e c t i n g changes o v e r t i m e i n r e g r e s s i o n parameters.
PlacNeill [ 1978a, 1978bl discusses p r o p e r -
t i e s o f raw r e g r e s s i o n r e s i d u a l s t h a t can be used f o r d e t e c t i o n o f changes a t u n s p e c i f i e d times i n r e g r e s s i o n parameters.
For the
same problem, M a c N e i l l [1980] a l s o discusses an a1 t e r n a t i v e s t a t i s tic.
P r i e s t l e y [1965], P r i e s t l e y and Subba Rao [19691, and Subba
Rao and Tong [1974] t r e a t s i m i l a r problems u s i n g a s p e c t r a l approach.
A number o f a u t h o r s , i n c l u d i n g Haggan and Ozaki [1979],
have discussed problems i n v o l v i n g n o n - l i n e a r phenomena and r e l a t e d
18 n o n - l i n e a r models.
A TEST FOR CHANGE
3
OF REGRESSION
AT UNSPECIFIED TIFIE
I n t h e f i r s t i n s t a n c e , i t i s probably a p p r o p r i a t e t o c o n s i d e r t h e r e g r e s s i o n problem t o be s t a t i o n a r y i n t h e sense t h a t
-f!!l) =
$2)
=
*
.
P
= +Ej
and then t e s t t h e n u l l h y p o t h e s i s i m p l i c i t i n t h i s assumption a g a i n s t t h a t o f change a t u n s p e c i f i e d time.
A formulation o f this
problem discussed by !lacNeil1 [1980] i s as f o l l o w s .
...,
I f we l e t
...,
Y' = (Y(l), Y ( n ) ) , E;I = ( ~ ( l ) , ~ ( n ) ) , and Xn be t h e de-n s i g n m a t r i x whose i j t h component i s f . ( t . ) , then, i n s t a n d a r d maJ
1
t r i x form, we may w r i t e t h e r e g r e s s i o n e q u a t i o n as Y = -nx a + E -n -n' h
and t h e Gauss-Plarkov e s t i m a t o r f o r
-i = ( X-n'
x -n
)-I
6, denoted
by
4,
then i s
X'Y -nNn
The s u b s c r i p t s on t h e v e c t o r s and m a t r i c e s a r e o m i t t e d where no confusion results. t o be
1 - v^ where
Pi
s,l
=
N
The v e c t o r o f r e g r e s s i o n r e s i d u a l s i s d e f i n e d t h e ith component o f
v^
is:
f(ti).
The a l t e r n a t i v e h y p o t h e s i s r e q u i r e s changes i n
6 a t unknown times.
To s p e c i f y a l t e r n a t i v e s we l e t $ ( i )
$(i),
=
{Q(i),
..., S p ( i ) i
r e p r e s e n t t h e changes i n t h e v e c t o r of r e g r e s s i o n c o e f f i c i e n t s e f f e c t e d between ith and t h e ( i + l ) t h o b s e r v a t i o n s .
That i s , i f k ( i )
i s t h e v e c t o r of r e g r e s s i o n c o e f f i c i e n t s f o r t h e i t h o b s e r v a t i o n , t h e n j(i+l) = & ( i ) + $i).So t h a t t h e Bayes-type argument i n t r o duced by Chernoff and Sacks [1964] may be used t o e l i m i n a t e n u i s ance parameters, we assume t h a t
g has
a m u l t i v a r i a t e normal d i s t r i -
b u t i o n w i t n z e r o mean and c o v a r i a n c e m a t r i x T21-where -r2 > 0.
We
then l e t d p a r t i c u l a r change sequence be d e f i n e d by: w I
-
= {w
1
where wi
, w 2,
...1
i s 1 i f a change i n ,fjoccurs between t h e i t h and ( i t l ) t h
o b s e r v a t i o n s and i s zero o t h e r w i s e .
Thus, a s i n g l e change through
19 t h e s e r i e s o f o b s e r v a t i o n s would r e q u i r e one component o f
w
t o be
1 and t h e r e s t zero. The a s s i g n a t i o n o f a p r i o r d i s t r i b u t i o n t o t h e c o l l e c t i o n o f a l l p o s s i b l e change sequences, w,, t h e n makes i t p o s s i b l e t o f o r m u l a t e t h e problem i n a way i n t r o d u c e d b y Gardner The nuisance parameters $ ( i ) can t h e n be i n t e g r a t e d o u t
[1969]. and, w i t h
small, t h e l i k e l i h o o d r a t i o s t a t i s t i c f o r t e s t i n g the
T'
n u l l h y p o t h e s i s a g a i n s t change sequences
w, w i t h
a uniform p r i o r
can be shown t o be a p p r o x i m a t e l y p r o p o r t i o n a l t o : c
Xk
where
is
5 with
t h e f i r s t k rows i d e n t i c a l l y equal t o zero.
The a p p r o x i m a t i o n becomes e x a c t as
where, i f Z '
=
{Z , Z , 1
2
T~
vanishes.
..., Z,>, I IZI)' =
Note t h a t :
Z12 + Z 2 + 2
... + Z f .
Associated w i t h t h e sequence o f p a r t i a l sums o f r e g r e s s i o n r e s i d u a l s i s a g e n e r a l i z e d Brownian B r i d g e (see ElacNeill 1978b) which we s h a l l denote by { b f ( t ) ,
t
E
[0,11>.
The s t o c h a s t i c
i nt e g r a 1
1'0
!3F(t)dt
i s t h e n r e l a t e d t o a Crdmer-von Mises t y p e s t a t i s t i c d e f i n e d upon t h e sequence o f p a r t i a l sums o f r e g r e s s i o n r e s i d u a l s ; some examp l e s a r e c o n s i d e r e d by HacNei11 [1978al.
L e t uf and
mean and v a r i a n c e o f t h e s t o c h a s t i c i n t e g r a l .
OF
denote t h e
Then i t may be
shown t h a t : E(Q)
21
02 'If
n i$
and
(i-1)(3i-57) n
Var(Q) where ($X.)
"J
y
Zda2f
n
i g 2 jgz
[min{(i-l),
Ei
(j-l)}]
i s t h e i t h row o f t h e d e s i g n m a t r i x ' P P = c x c f Q ( t .1) f Q ( t j ) . Q=O i Q jQ = R=O
(Xi-Xj)2 5, and
x
We n e x t d i s c u s s d i s t r i b u t i o n t h e o r y f o r Qn which i s a q u a d r a t i c form i n independent normal v a r i a t e s .
The m a t r i x o f t h e q u a d r a t i c
.Ol .025 .05 .10 -50 .90 .95 -975 .99
12
1.062 1.796 1.a51 2.366 1.884 2.992 2.537 3.930 9.023 11.887 3 0 . 4 ~ 1 44.764 41.803 61.499 53.669 78.909 69.8al 102. 6d8
10 2.790 3.476 4.323 5.589 16.480 61.527 84.890 1Q8.909 141 -641
14 3.765 4.773 5.870 7.512 21.504 81 3 9 9 112.996 143.672 186.824
16 a.993 6.255 7.632 9.609 27.869 ic)4.r)8i 142.83R 183.199 238.198
19 6.369 7.922 9.608 12.145 34.648 129.28~ 177.391 227.486 295.766
29
Q
u
21
form i s of the form
p
where
p
Assume t h a t {X.}?
tion matrix.
J J=1
i s the usual regression projeci s the s e t o f eigenvalues of the
matrix PJ p ordered from l a r g e s t t o smallest. e r i s t i c function f o r Q 4s: n
If w e l e t D(2is) = @ - ' ( s ) ,
X
Then the charact-
= 2 i s , and assume t h a t t h e r e
a r e 2n
o b s e r v a t i o n s , t h e n , p r o v i d e d t h e e i g e n v a l u e s a r e d i s t i n c t , it c a n be shown t h a t :
r2: 1 -
P[QZn
5 a] =
1
1 - -
n
c ( J=1
T r .
- 1 p
-A
e
-P dX.
X(-D(X))l'z
IT----
2j-1
If some of the eigenvalues are n o t d i s t i n c t - then one may: compute ( 1 ) with these einenvalues removed; compute the X 2 d i s t r i b u tion associated with the repeated eiqenvalues; and convolve the resultincl d i s t r i b u t i o n s t o obtain the d i s t r i b u t i o n of 1Je l e t 11 be the matrix whose ( i , j ) t h component i s (Xi*X.)min[(i-l),(j-l)]y let
Qn
(I - X(X'X)-'Xl), 1 , = 7gn P M E gn. p
=
-
-1
a n d consider
0
I t i s then a straightforward numerical nroblem t o comoute usino a Dackaoe such as EISPPK tlle eiaenvalues o f p 11 P a n d t o obtain the N
N
c h a r a c t e r i s t i c function which we may then invert. for a sinqle sinusoid aonear in Table 1 .
4
Some r e s u l t s
ESTIVATION OF REGRESSION PAPAPIETER PROCESSES
'*\hen the t e s t for chancre of recrression parameters a t u n k n o w n time r e j e c t s the null hyoothesis, one focuses attention on the parameter orocess. The method o f estimation of {,f(t), t > 11 used below i s t h a t of recursive rearession whereby '(+' i s estimated b y l e a s t souares usinn a senment o f lennth k of the observations centred a r o u n d t . This i s the method o f "moving reoression" re-
22 The l e a s t s q u a r e s e s t i m a t o r f o r
f e r r e d t o b y Brown e t a2 [ 1 9 7 5 ] .
c ( t ) , i s Tiven by the following enuation:
-' @ ( t , k ) = (XI ( t , k )
X(t,k))-'X'(t,k) Y(t,k) where t h e d e s i g n m a t r i x and v e c t o r o f dependent v a r i a b l e s u t i l i z e denote k,t r e s p e c t i v e l y , and x ' ( t ) = I f Pk,t
o n l y t h e k o b s e r v a t i o n s c e n t r e d a r o u n d t. (X'(t,k)
&(t,k))-'
(f7(t),fl(t)
and X ' ( t , k ) Y ( t , k ) , N
,..., f P ( t ) ) ,
N
and C
t h e n one may use t h e f o l l o w i n o r e c u r s i v e
r e l a t i o n s t o compute e s t i m a t e s o f t h e r e g r e s s i o n p r o c e s s : -Pk + l , t
=
-Fk , t
- -Pk , t
-X(t+k+l)(l+Z'(t+k+l)F'k,t x' ( t + k + l
N
-k, P t+l= P-k+l C -k,t+l
=
X(t+k+l))-'
) Pk,t'
,t +P-k+l ,t-X ( t ) ( l - X ' ( t ) P k + l N
C +Y ( t + k + l ) X ( t + k + l ) - Y ( t -k,t N
,tE(t))-'X'(t)Ek+l
,t' and
)X( t ) .
T h i s i s t h e a l g o r i t h m o f P l a c k e t t [1950] t h e use o f w h i c h i n a t i m e s e r i e s c o n t e x t i s d i s c u s s e d b y Young [ 1 9 7 4 ] ,
i n a regression
c o n t e x t b y Brown e t a2 [ 1 9 7 5 ] , and i n a s h o r t - t e r m f o r e c a s t i n p c o n t e x t b y H a r r i s o n and S t e v e n s [19761. Eacl. component o f t h e e s t i m a t o r , f ( t , k ) , o f a l l components o f c e n t r e d a b o u t t.
-
? ( a )
i s a l i n e a r combination
f o r a l l k values o f t h e t i m e narameter
-
More p r e c i s e l y , i f B ( t , k )
i s a (p+l)xl vector
whose Lth component i s
then
p( t + k / 2 , N
k) =
(3' ( t ,k ) X ( t, k ) ) - ' X ( t ,
k ) g ( t ,k ) + ( X ( t, k ) X ( t, k ) )-'B( t , k ) .
Thus t h e r e l a t i o n hetween t ( t ) and [ ( t , k )
i s comPlicated by t h e
p r e s e n c e o f m o v i n g a v e r a g e s i n t h e n o i s e p r o c e s s and b y t h e p r e sence o f c o r r e l a t i o n between t h e components o f t h e e s t i m a t i o n v e c t o r induced by t h e e s t i m a t i o n Procedure. !le p r o c e e d t o e x p l o r e t h e e m p i r i c a l p r o p e r t i e s o f t h e e s t i m a t o r s o f t h e s t o c h a s t i c process o f r e q r e s s i o n c o e f f i c i e n t s
by f i t t i n g
A R I V A models ( s e e Sox and J e n k i n s [ 1 9 7 0 ] ) t o t h e v a r i o u s components o f t h i s process.
lde f i r s t c o n s i d e r t h e I J o l f e r s u n s p o t s e r i e s f o r
Sunspot series Fitted model
/
Estirmated mean value
50
Figure 1 .
100 150 Square root of yearly su n sp o t numbers: 1701 1900
-
200 years
N
w
-
O.O.,
-a/2
.(
*
b
I
50
F i g u r e 2.
150
100
-
Estimated phase angle of yearly nunspot seriee for 1701-1900 with w 0.561.
200 yearo
25
the period 1700-1960. We f i t t h e following p e r i o d i c model; w = 0.561: Y ( t ) = p0 + f 1 cos w t + p 2 s i n w t + & ( t ) =
f 0 + y sin(wt +
dj)
+
E(t).
The r e l e v a n t Q s t a t i s t i c s a r e as f o l l o w s :
E ( q ) - 2.05
Q = 27.09
x 105 ,
l o 5 , and JVar(0) = 0.75 x l o 5 . Hence the hypothesis of no chanqe of r e q r e s s i o n parameters i s r e j e c t e d . The r e c u r s i v e r e o r e s s i o n procedure i s apD1ied f o r v a r i o u s values o f k . As one m i q h t expect, t ? e process C p ( t , k ) , t > 03 i s raaoed f o r small values o f k and smooth f o r l a r n e v a l u e s . Finure 1 c o n t a i n s Dlots of t h e y e a r l y sunspot s e r i e s f o r the period 17011900. Firlure 2 c o n t a i n s a p l o t o f t h e e s t i m a t e d ohase a n n l e f o r t h e s e r i e s . 'ale make the assunption t h a t each comnonent orocess has a mean value and f i t ARIW, models t o t h e d e v i a t i o n s from t h i s mean. The f o r e c a s t f u n c t i o n s r e a r e s s t o t h e s e means so one does n o t l o s e t h e lona-term n r e d i c t i v e p r o o e r t i e s o f a simple sunusoid. I n t h e s h o r t - t e r m , t h model i s a d a p t i v e . 5
x
CQNCLUS I O N
An a d a p t i v e harmon c r e g r e s s i o n model o f a douhly s t o c h a s t i c nature f i t t e d t o data i n d i c a t e s t h a t such models a r e capable of improving botll f i t s t o the d a t a and f o r e c a s t s o f f u t u r e o b s e r v a t i o n s . REFERENCES
Box, G . E . P . and J e n k i n s , G.V., 1970. Time S e r i e s Analysis: Forec a s t i n n and Control. Yolden-Day, San Francisco. Brown, R . L . , D u r b i n , J . and Evans, J . P . , 1975. Techniclues f o r t e s t i n a the constancy o f r e n r e s s i o n r e l a t i o n s h i m over time. J . Roy. S t a t i s t . SOC. S e r . R 37: 149-192. Chernoff, H . and Zacks, S . , 1964. Estimatino t h e c u r r e n t mean o f a normal d i s t r i b u t i o n which i s sub o Ev i den l y t h i s c a s e i n c l u d e s i n d e p e n d e n t e x p o n e n t i a l i n p u t s w i t h d f f e r e n t mean v a l u e s i n w h i c h c a s e a = . . . = a I a r e n o t a1 I t h e same. I f some o f t h e A . ' s J -'xP of (2.5) e v i d e n t l y t h e number of a r g u m e n t s i n
2
= I and A I' P a r e equal then i s reduced.
In
o r d e r t o make t h e c o n v e r g e n c e of t h e m u l t i p l e s e r i e s i n ( 2 . 4 ) f a s t e r one can use t h e f o l l o w i n g t e c h n i q u e . R e p l a c e y b y y / B so t h a t @
and choose B
B>O,
2 converges f a s t .
t h a t f o r s m a l l va ues o f t h e a r g u m e n t s T h i s r e p I acement
02
I t i s we1 I-known
approximates t o u n i t y .
s e q u i v a l e n t t o c o n s i d e r i n g t h e d e n s i t y of B Y .
I n s t e a d o f s m p l i f y i n g t h e v a r i o u s f a c t o r s o f TI(I-1.t) J i n terms o f ( I - X ) one c a n a l s o p r o c e e d a s f o l l o w s . C o n s i d e r I the identity
(1-X.t) I
= (l-yt)(A./y)(l-(l-y/Ai)/(l-yt)) I
f o r some y>O. Then p r o c e e d i n g as b e f o r e o n e can i n v e r t t h e moment generating f u n c t i o n f o r t < I/Y, d e n s i t y i n t h e f o l l o w i n g form.
l/Ai,
i = l , ...,p
to g e t t h e
32
g? ( y ) = A 7 1
y>O,
a
. .. 'pp r ( a I
If
I
a
I
+. ..+a - I
.-Yh
P
P
f12
i n which case t h e
...,p.
)-
+...+a
i s of p v a r i a b l e s
(y-'-Xy')y,
i=l,
y i s chosen such t h a t
parameters a
I
+ . . . + u and
and choose B
f12
part t o
then the c o e f f i c i e n t
P
so t h a t
g2
y.
i s a gamma d e n s i t y w i t h t h e
Now i f y i s r e p l a c e d by y/B,
p>O
i s approximated t o u n i t y then t h e
d e n s i t y o f BY i s a p p r o x i m a t e d t o a gamma d e n s i t y w i t h t h e p a r a met-ers u + . . . f a I P
3.
and yB.
ANOTHER R E P R E S E N T A T I O N FOR THE DENS I T Y WITH GAMMA I NPUTS
w i t h X I , ... %P ? b e i n g i n d e p e n d e n t gamma v a r i a b l e s w i t h t h e d e n s i t i e s g i v e n i n A g a i n c o n s i d e r t h e c a s e where Y=X
+...+ X
I
( 2 . 1 ) . C o n s i d e r t h e c a s e when al,...,a a r e a l l e q u a l t o a . I n P
t h i s case
P
(3.I )
II( I - X . t ) - " j = I L - t B I - " j=I J where I i s a n i d e n t i t y m a t r i x ,
B i s a symmetric p o s i t i v e d e f i n i t e
' I ,
m a t r i x w i t h t h e eigen values J,.,
-1
determinant o f L-tB.
1 %I - t B I - "
,..., p
l/A.,i=l I
.. .
and IL-t31 d e n o t e s t h e
But
= lBl-ayclp( I - y t ) - a p
for t < I/y,
j = l , ...,p
.
lL-(L-yB-')/(
where K=(k
.
I)*.
I-yt)
.,k
p
I-'
),
k
> k > I - 2 -
a r e nonnegative integers > k > 0, k = k +. .+k k l ,. .,k - P I P' P = and CK d e n o t e s a z o n a l p o l y n o m i a l o f o r d e r k and D
.
Il ( a - ( i - l ) / 2 )
i=l
F o r d e t a i Is r e g a r d i n g z o n a l p o l y n o m a l s s e e
ki
M a t h a i & Saxena ( 1 9 7 8 ) and t h e r e f e r e n c e s t h e r e i n . By
n v e r t i ng
t h e above moment g e n e r a t i n g f u n c t i o n one g e t s t h e dens t y a s f o l lows.
33
CK(l-yB-')/(k! ( a p ) k ) ,
(3.3)
y > 0
This i s another representation f o r t h e density of
Y
d e n s i t y of
Y. S i n c e t h e
i s u n i q u e , b y c o m p a r i n g ( 3 . 3 ) and (2 6 ) we g e t t h e
fol lowing m
Theorem I .
k=O 1
K(a)K
= !J2(a, ..,a;
for y > 0
( ~ / y C ) K~( i - y B - l ) / ( k ! ! a p ) k
up;(y
y > 0, X i > 0,
i = l , ...,p.
S i m i a r r e s u t s can b e o b t a i n e d when a l , . . . , a a r e a l l P i n t e g e r s n w h i c h c a s e one c a n look upon A. b e i n g r e p e a t e d a I
times,
i=l,
. ..
9 P .
If the inputs XI,
...,X
P
r e p r e s e n t t h e i n p u t s of sedimer,
S
i n t o a dam o v e r a p e r i o d of p o c c a s i o n s t h e n one c a n compute t h e p r o b a b i l i t y t h a t t h e t o t a l sedimentation i s less than a prea s s i g n e d number 6,
t h a t is,
P{Y 56; by u s i n g any one
of t h e
e x p l i c i t c o m p u t a b l e r e p r e s e n t a t i o n s f o r g ( y ) g i v e n above. T h a t P IS,
P { Y 26} =
1;
gp(Y)dY
where t e r m b y t e r m i n t e g r a t i o n i s v a l i d . 4.
AMOUNT
OF
ACID RAIN
C o n s i d e r t h e s i t u a t i o n o f s u l p h u r d i o x i d e o r o t h e r such p o l l u t a n t s d i s t r i b u t e d i n a c e r t a i n r e g i o n i n t h e atmosphere a c c o r d i n g t o a 3 - v a r i a t e normal d i s t r i b u t i o n w i t h t h e c e n t r e a t t h e p o i n t i ~ '=
where p i = E ( X i ) ,
i=1,2,3,
w i t h Xi
d e n o t i n g t h e i t h c o - o r d i n a t e o r t h e i t h v a r i a b l e w i t h t h e conc e n t r a t i o n of t h e p o l l u t a n t s p r o p o r t i o n a l t o t h e probabi I i t y
34 c o n t e n t . C o n s i d e r a c l o u d p a t c h i n t h e r e g i o n i n t h e shape o f an e l l i p s o i d w i t h a c e n t r e and axes o f symmetry o f i t s own. The amount o f a c i d v a p o u r c a r r i e d by t h e c l o u d i s p r o p o r t i o n a l t o t h e p r o b a b i l i t y c o n t e n t o f t h i s e l l i p s o i d . The amount o f a c i d r a i n t h a t can b e e x p e c t e d o u t o f t h i s c l o u d i s p r o p o r t i o n a l l o t h e p r o b a b i l i t y o f t h i s e l l i p s o i d . Thus i f t h e e l l i p s o i d i s d e s c r i b e d by (X-a)'C(X-a)
i s g i v e n as
o
y a r c t h e same q u a n t i t i e s a p p c s r i n q i n (2.6).
36 REFERENCES
G i I I i land, D.C., and Hansen,E.R. 1974. A n o t e on some s e r i e s r e p r e s e n t a t i o n s o f t h e i n t e g r a l o f a b i v a r i a t e normal d i s t r i b u t i o n o v e r an o f f s e t c i r c l e . Naval Research L o g i s t i c s Q u a r t e r l y , 21,No.I: 207-21 I . Helstrom,C.1978. Approximate e v a l u a t i o n o f d e t e c t i o n probabil i t i e s i n r a d a r and o p t i c a l c o m m u n i c a t i o n s . l E E E T r a n s . 14: 630-640. Aerospace E l e c t r o . S y s t e m s . , A E S M a c N e i l l , l . B . 1 9 7 4 . T e s t s f o r change o f p a r a m e t e r a t unknown t i m e s and d i s t r i b u ~ t i o n so f some r e l a t e d f u n c t i o n a l s o n B r o w n i a n m o t i o n . A n n . S t a t i s t . , 2 : 950-962. Mathai,A.M.1980. Moments o f t h e t r a c e o f a n o n c e n t r a l W i s h a r t m a t r i x . ,Cornmiin. S t a t i s t . ( T h e o r . M e t h . ) A 9 ( 8 ): 795-80 I Mathai,A.M. and Saxena,R.K.1978. The H - f u n c t i o n w i t h A p p l i c a t i o n s i n S t a t i s t i c s and 0 - t h e r D i s c i p l i n e s . W i l e y H a l s t e d , New Y o r k . Rice,S.O. 1981. D i s t r i b u t i o n o f q u a d r a t i c f o r m s i n normal random v a r i a b l e s - E v a l u a t i o n by numerical i n t e g r a t i o n . S I A M J . S c i . S t a t . Comput.l,No.4: 438-448. Ruben,H. 1962. P r o b a b i I i t y c o n t e n t o f r e g i o n s u n d e r s p h e r i c a l normal d i s t r i b u t i o n I V : The d i s t r i b u t i o n o f homogeneous and nonhomogeneous q u a d r a t i c f u n c t i o n s o f normal v a r i a b l e s , Ann. M a t h . S t a t i s t . , 33: 542-570.
.
37
TESTING FOR NON-LINEAR SHIFTS IN STATIONARY @-MIXING PROCESSES
R. J. KULPERGER
Department o f Mathematical Sciences, McMaster U n i v e r s i t y , Hamilton, Ontario, Canada, L8S 4K1
INTRODUCTION
-
-
; n 2 11 and Y = {Y ; n 2 11 be two independent n n stationary processes, and suppose we observe data {X ..., Xm } and 1' It is of interest to know if the processes X and Y {Yl, . . . , Yn}.
Let
X = {X
-
differ in a specified manner.
Suppose
X
and Y
-
have continuous
1 1 distribution functions (df's) F and G respectively.
One type of
change is a generalized shift, so that
A(x)
+ A(X 1) have the 1 In the iid case, Doksum (1974) considered this
is a shift function, and (1.1) says Y1 and X
same distribution.
and obtained an estimate conditions,where N = n
+
AN (x) of A(x), m.
weakly to a Gaussian process.
under some identifiability
(x) - A(x) ) converged N Weak convergence is essentially conver-
Doksum showed &(A
gence in distribution for processes.
See Billingsley (1968) for more
details. Now suppose X and Y are @-mixing processes (see Billingsley (1968)
-
for definitions). dence.
-.,
Mixing in general is a type of asymptotic indepen-
Here the X process is mixing means that the random variables
.. .,
X ) and f (X . s 2 k k 2 s' independent for large n , where f
fl (X1,
but otherwise arbitrary.
+
n) are approximately statistically
and f a are measureable functions, 1 In other words, the random variables far
apart in time behave approximately independently, but random variables near by in time may be dependent.
Many time series models should have
Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
38 this type of property.
A special type of mixing is @-mixing.
Kulperger (1981) showed 6 ( A (x) - A(x)) converges weakly to a N Gaussian process A(x). This result is given in section 2. In the iid case, A(x) is a scaled version of a Brownian bridge; but not so in the @-mixing case.
In section 3, we carry out the estimation pro-
cedure on some simulated data. structure of A(x). B(t) is given by
This requires estimating the covariance
The covariance structure of a Brownian bridge
Cov(B(t), B(s))
=
t(l - s), 0
5
t
5
s
I, which
5
simplifies the calculations in the independent data case. A possible type of application is in a time series two sample problem.
If a pollutant, or some other type of intervention, is made
or added into a system, by taking segments far apart in time before and after this incident, one can estimate @,(x). This happens if, for example, a factory is built and dumps sewage or some other pollutant into the system, for example a river system.
Suppose the underlying
series is somehow connected to temperature. Then A(x) has the physical interpretation of being the reaction to this intervention.
It may be
that A(x) is near zero for low temperatures, but has a large effect for higher temperatures. Looking only for constant shifts or average changes may not show any change, when the above case may be happening. The two sample procedure presented here also avoids many parametric assumptions.
THE ESTIMATES The identifiability condition on A(x) is that A(x) decreasing. From (1.1) we obtain n(x) -1 a df H, H (x) = inf(y: H(y) 2 x).
=
-1
G
+ x is non-
(F(x)) - x , where for
From the data, we can obtain empirical distribution functions (edf's m
F (x) m
1 =I m
1
I
(X.), (-m,x) I
and
1 G (x) = n n
n
c
I
(Y , ) (-m,x) 3
I
where
I is the indicator function of the set A. These estimate A F and G respectively. Thus a natural estimate of A(x) is
AN (x) =
-1 Gn (F,(x))
(2.1)
- X.
From Billingsley (1968),under certain conditions, as m
3
39
where U . I
F(X.) and h (x) = I C o , t l ( ~ -) t. I t
=
Similarly as
6 (Gn (x) -
n
3
G(x))
m +
V(G(x))
weakly,
(2.4)
where V is a continuous Gaussian process on [ O , l ] ,
where 2 , 7
=
Theorem 2.1
G (Y . ) 3
independent of W,
.
(Kulperger (1981))
Suppose G has a positive density g. Then under conditions for m which (2.2) and (2.4) hold, and if - + E ( o , l ) , then n &(AN(x) - A(x)) + A ( x ) weakly, where
In the iid case, V and W are independent Brownian bridges.
In
the $-mixing case they are not Brownian bridges, since the covariances (2.3) and (2.5) depend on F and G respectively.
In general, V and
W are not even deterministically time changed Brownian bridges.
The covariance functions of V and W must be estimated, as well as
g, the density function of G.
Kulperger (1981 , Theorem 4.1) obtained
consistent estimates of the limiting covariance functions of the V and W
processes.
To illustrate these computations, as well as to give some indication of how well they perform in some sense in a specific case, we simulate
-
-
one realization of such an X and Y process.
This is described in
40 the next section.
A S I M U L A T I O N EXAMPLE AND NUMERICAL COMPUTATIONS For this simulation, we consider a 10-dependent process.
*
*
- . . , Xm+9
X1,
9
be iid N ( 0 , l ) r.v.'s and X .
=
1
Marginally X . is a N ( 0 , l O )
*
r.v..
Let Y1,
*
1: xj+i , i = l , j=O
...,
N ( 0 , l ) ' s are generated by the Box-Muller (1958) method.
200, so that
A
=
m.
*
-
=
...,
Y be iid N ( 0 , l ) T.V. s, m+ 9
In particular, notice that the Y process is not Gaussian.
n = m
Let
The iid
We take
-. 2 I
Using Theorem 2.1, we obtain approximate or asymptotic marginal confidence intervals for f~(x),for x
=
-5, -4,
...,
5.
To do some we
rnust compute consistent estimates of the corresponding Var(A(x)) terms. Suppose we wish to construct a confidence interval at x 0' -1 -1 9 (G (F (xo)) by gn (Gn (Fm(xO)1 ) , where
Estimate
1
n
-a
h = fin . We take f? n estimate of g(x).
and
=
1, cx
1 so that (3.1) is a consistent 5'
= -
From ( 2 . 3 ) , for each t, var(W(t)) f (x)(y) t,t
1
=
=
2 ~ ~ f '( 0~1 ) , where trt
1 eiycov(h (U1), ht(Uk)) is the spectral density of the t
k
time series {h (U ) } evaluated at frequency y, and i = J-1. We now t k estimate this spectral density at frequency 0 from Kulperger (1981). Let
rm
2n. =
{Aj:
j
-
m
greatest integer function.
j
=
1,
* ]
is the
Let h = h (F (X,)). Define the trmrj,X t m J
estimated finite Fourier transforms for y m
m } ,where [
. .. , [ B E
rm
by
41 m
1 e -iyJ
Notice that
=
o
for y
rm.
E
Let
1
(3.
where
M m
=
[ p &] and the sum is over y , & Tm.
Under the assumption that
3
a3
with
t
,
t,t
var (V(t))
=
2Tf)'( (0) trt
where ft,t (y) is the spectral density of {h t (G(Yl )
y.
frequency
&
var(V(t)) by
n
and
il
and
t
=
) >
evaluated at
Define the estimated finite Fourier transform
27T
y
2n;i(L(O),
F (x ) . n O
=
Similarly observe that
where
(XI
(0) in +-,t
2 7 ~ ; ' ~(0) ) + 2Ff
We then estimate var (W(F(x,) ) ) by
m.
.45.
=
Law ({U , } j2l) is absolutely continuous with
respect to Lebesque measure on [O,l]
probability as m +
We take
I'
n
=
I
3
by
,
3 {y.: y , = ----, j
G (x).
=
-
1/2
n
Fm (x), s
=
1,
. .. ,
[B&]}.
Estimate
(0), the analogue of ( 3 . 2 ) , with
2i;:yL
t
n
=
m
replaced
Estimate Var ( A ( x ) ) by
G (XI.
n
Figure 1 contains the true and estimated
A(*).
A(*) and AN(.), the
Figure 2 contains il with the estimated N marginal 95% confidence intervals. The lengths of these confidence solid line being
( a )
intervals are of the right size for these processes with these sample sizes, in the sense that they are close to the intervals wc would hav
42
[;LURE
17.9
1
14.2
10.4
6.6
2.9 0.0 -.8
-4.5
-7.13
-5.21
-3.28
-1.36
-.56
2.49
- -
4.41
6.34
8.26
FTGURE 2
17.9
14.2
t
10.4
6.6 t
t
2.9 0.0 -.8
-4.5
-7.13
-5.21
-3.28
-1.36
.56
2.49
x I- -
4.41
6.34
8.26
43 obtained if var(A(x)) were really known. G(x
+
A(x))
=
F(x)
with density $.
=
For our simulation
@(x/JlO), where 0 is the standard normal d.f.,
Therefore g (x
+ n(x) ) (1 + A' (x))
=
($ (L)
410
/fi.
= g(x + ~cx)). q(~-l(~(x)))
Notice that
The sum in (2.3) cannot easily be obtained in closed form, since it
X process, X I , F(XI1)
where
...,
1(ax)4
(x)dx. However observations of the X allows us to obtain U = F(Xl), . , Ull 1 11
involved terms of the form
.
and hence compute the random variables
t. 1
=
o(x./V%),
x
i
=
i - 6, i
=
I,
...,
11.
Simulating many replications of this allows us to estimate (2.4) by the Law of Large Numbers. The ratios of the lengths of the estimated to the tru asymptotic confidence intervals are thus obtained, and recorded in Table 1. These ratios are correct to one decimal place, since the Monte-Carlo estimates of Var(W(t)) and Var(V(t)) are based on sample sizes 200. The ratios seem to be quite reasonable in this example.
ACKNOWLEDGEMENTS I wish to thank N. Sorokowsky for programming the simulation.
also wish to thank Jackie Collin for typing this paper.
I
This work
was partially supported by grant number A5176 from the Natural Science and Engineering Research Council of Canada.
44
TABLE 1 x
A(X)
-5 -4 -3 -2 -1 0 1 2 3 4 5
.0669 .1799 .4743 1.1920 2.6894 5.000 7.3106 8.8080 9.5257 9.8201 9.9331
2 ci
q x )
1.008 1.7914 1.4446 1.5561 2.1666 4.0825 6.2555 8.6029 8.4722 8.8200 9.0295
0
2
2
(x)/u
(x)
1.53 0.72 1.09 .88 .32 .63 1.35 1.77 1.57 .37 1.09
Estimated 95% marginal confidence interval -2.88, -2.10, -2.01, -2.77, -1.29, -1.54, - .79 2.97, 4.37, 6.91, 5.35,
4.90 5.68 4.90 5.88 5.62 9.71 13.30 14.24 12.57 10.73 12.70
( x ) is estimated from 200 Monte-Carlo runs. The ratios
G L ( x ) to o L ( x ) are accurate to 1 decimal place. RE FERENCE S
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. Box, G. and M.E. Muller (1958). A note on the generation of random normal deviates. Ann. Math. Statist., 29, 610. Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two sample case. Ann. Statist., 2, 267-277. Kulperger, R. (1981). Estimating non-linear shifts in stationary #-mixing processes in the two sample case. Preprint.
45
A ROBUST STATISTIC FOR TESTING THAT TWO AUTOCORRELATED SAMPLES COME FROM IDENTICAL POPULATIONS
M . L . TIKU
Department of Mathematical Sciences, McMaster University, Hami 1 t o n , Canada
ABSTRACT
Testing t h a t two independent samples come from identical POPUlations i s a common s t a t i s t i c a l problem. If the random var ables within the samples a r e i i d (independently a n d i d e n t i c a l l y d i s t r i b u t e d ) , Tiku (1980) gives a robust s t a t i s t i c ( t h a t i s a s t a t i s t i c whose null d i s t r i b u t i o n i s f a i r l y insensitive t o underlying populations) which i s a l s o remarkably powerful. I n t h i s paper, we give an analogous s t a t i s t i c which can be used i f the random variables a r e moderately autocorrelated; t h i s s t a t i s t i c i s shown t o be robust and powerful. 1.
INTRODUCTION
Testing t h a t two independent samples come from identical populations i s a common s t a t i s t i c a l problem. If the underlying populations are normal and i f one has simple random samples, one would naturally employ the Student's t s t a t i s t i c t = ( i , - ~,)/[si't(l/n,)+(l/n,)~l. This s t a t i s t i c , however, i s n o t robust t o most nonnormal populations prevalent in practice; see Tiku ( 1 9 7 1 ) a n d Subrahmaniam e t a l . (1975), f o r example. A s t a t i s t i c (based on Tiku's modified maximum likelihood e s t i m a t o r s ) , analogous t o t , which i s robust and i s a l s o remarkably powerful was given by Tiku (1980); see a l s o Tiku a n d Singh (1981). I n t h i s paper, we propose a s t a t i s t i c which can be used i f the random variables are moderately autocorrelated t h r o u g h a f i r s t - o r d e r stationary stochastic process. The proposed s t a t i s t i c Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
46
i s shown t o be robust t o b o t h symmetric a n d skew populations and remarkably powerful against location s h i f t s . THE TEST STATISTIC
2.
Let =
J’1 , i a n d Y2,i
where u
LJ1
+
i
= 1,2,.
u 2 + u ~ , ~i , =
=
~
U1,iY
=
..,nl+l
1,2,
...,n 2 + 1 ,
,8, ~u ~ , +~ e- l t~,
W e assume t h a t e l t and e2t are i i d with mean E(e) = 0 and variance V(e)=02. Note t h a t V ( Y , , ~ ) = u 2 / ( 1 - 0 , 2) and V ( Y , , ~ ) = ~ ~ / ( 12- 0 ~ ) ; 0
< lgil
Ho:
hlHo): nl=20,n2=10
0 s
-0.5 0.0
5 1
.054 .045 .042 .035 .031 .012 .009 .007 .005 .004
5 1
.051 .046 .048 .038 .040 -011 ,007 .009 .006 .008
B -0.5 0.0 0.1 0.5 Normal .049 .042 .052 .052 .009 .008 .010 .013 Double-Exponential .050 .054 .050 .058 .012 .012 .011 .013
5 1
5 1
5 1
5 1
0.1
0.5
0.7
nl=n2=20 P,
0.7
-0.5 0.0
.050 .010
.050 .047 .043 .044 .029 .008 .008 .008 .004 .002
.063 .018
.057 .056 .053 .050 .039 .010 .014 .011 .009 .004
.053 -044 .047 .037 .034 .014 .007 .008 .005 .005
Student's t, d f 4 .046 .046 ,049 .046 .055 .010 .009 .012 .012 .016
.047 .051 .052 .040 .039 .010 .006 .008 .005 .005
.050 .047 .045 .040 .036 .012 .008 .008 .006 .006
Student's t, df 3 .050 .053 .053 .053 .062 .010 .011 .011 .012 .014
.050 .052 .058 .041 .043 .012 .009 .009 .007 .005
.053 -047 .044 .040 .042 .012 .008 .008 .009 .010
Student's* t ,df 2 .057 .053 .055 .060 .059 .OlO .Oll .010 .015 .019
.059 .057 .054 .053 .046 .012 .009 .Oll .009 .007
.053 .045 .042 .035 .036 .008 .008 .O. .007
0.90N(OY1)-t 0.10N(0,3) .050 .049 .051 .052 .050 .011 .010 .013 .013 .013
.048 .045 .047 .046 .032 .012 .009 .006 .006 .005
.012
0.1
0.5
0.7
0
r.
0
m
0
7
0
0
. .
0 0
m o
"
ww
99
d - 0
r-b
. .
G - 0 0 0
99
WCO G - 0
. .
0 0
N m m o
wm
99
m o
r.r-
. .
b0 0 0
99
d-I-
c o o
mI-
.
z
- I
I
I I . .
m o
m m 0 0
C O N G-I0 0
. .
m m
G - 0 0 0
. .
. .
. .
0 0
e m m o
d-m
. .
0 0
m o
m w
. .
0 0
m o
. .
b0 0 0
m a
. .
wm d - 0 0 0
. .
0 0
m m m o
d-m 0 0
..
d - 0
m m
. .
b0 0 0
0 0
. .
m o
r m
O m m o 0 0
0 0
m 0
0 0 . .
a m
m-
wm
99
r.w
d - 0
99 m o
N
0 0 . .
b c o
m m
m o
0 0
. . O *
G-co m o
. .
r-d-
m o
0 0
. .
00
x
NN
I
m o 0 0 . .
m 0
ah
m o
0 0
a m
. .
m o
00 . .
- . w m
N O 0 0
mrn
0 0 . .
m o
0 0 . .
m o
0 0 . .
m o
G-m
. .
ar.
0 0
-a
G-m
m a
rn-
0 0
. .
d - 7
h-
. .
G - 0 0 0
. .
r c o
Kro
-4-0 0 0
. .
Nr.
. .
r-h
0 0
r.0
coo
rn-
. .
G-r0 0
. .
G - 0
G--
mr-
. .
G-r0 0
mr-
. .
00
. . Om rnr-
m-
99
0 0
G - 0 0 0
00
-4-0 0 0
G-co m o
. .
0 0
- 0
0
. .
r m m o
m I
0 0
0
0
.r 0
m 0
7
0
0 0
rn I
0
.r 0
m 0
w
o r -
h
3
z w
I-
z o .
0 0
0 v
7 L n
w o 1 1 m I - Q
4
*r
h
x m
n 0
W
-0
fa 7
W L
n
m .r
.r
X
2 m
E
-0 S
L
fa
c, S
a
u
W m 0 -c S
0
c,
.r
L
a >
W
m
n
aJ
0 S
0
.*
-x
49
50 E x p e c t i n g t h e small sample n u l l d i s t r i b u t i o n o f T t o be a p p r o x i m a t e l y S t u d e n t ' s t w i t h A +A 3 degrees o f freedom, we s i m u l a t e d 1 2; t h e mean, v a r i a n c e , skewness 8 , = p 2 / p 3 and k u r t o s i s 6; = i-I / v 2 3 2 4 2 o f T f o r numerous u n d e r l y i n g p o p u l a t i o n s ( b o t h symmetric and skew) and, t o o u r s u r p r i s e , found them v e r y c l o s e t o 0, 1, 0 and 3, r e s p e c t i v e l y , r a t h e r t h a n c l o s e t o t h e c o r r e s p o n d i n g values o f t h e
B < 0.5, i n d i c a t i n g 0.5, t h e n u l l d i s t r i b u t i o n o f T
S t u d e n t ' s t d i s t r i b u t i o n , w i t h i n t h e range -1.0 < approximate n o r m a l i t y of T; f o r f~>
t u r n e d o u t t o be i n t r a c t a b l e w i t h values o f t h e v a r i a n c e and 6; much d i f f e r e n t t h a n 1 and 3, r e s p e c t i v e l y ( s e e a l s o L j u n g and Box, 1980). However, i t i s t h e range -0.5
: a
j=l tj
8. f o r t=1,2
4 u
t
=
Yt
>: b t j fjj
f o r t=nl+l,n1+2
- j=l
a2 =
a2 i s
hood estimate of
1 and
{pj
},
can be o b t a i n e d by t h e method of l e a s t P
-
{8j
given
n
,...,n ,
h
,...,n 1 and
and t h e maximum l i k e l i -
1 ( u , - ~ ~ - ~ ) ~ / The n . h
j o i n t maximum
t =1
l i k e l i h o o d f u n c t i o n f o r n1 and p i s o b t a i n e d from t h e j o i n t l i k e l i h o o d f u n c t i o n of and
CJ
2
{Oj},
{pj},
a2, p and "1 by r e p l a c i n g
by t h e i r maximum
r e p l a c i n g ut and
Lt
by
CJ*
li':elihp-J.
and
;' i n
{ej},
{fjj}
estimates, o r e q u i v a l e n t l y by f(ul,u2,
l i k e l i h o o d estimate of p i s t h e s o l u t i o n t o
...,u n ) . a
The maximum
l o g f l a p = 0.
For u o =
0 , t h i s is n..
n
h
t=I w h i l e f o r uO-N(O,
(2.4) CJ
2
/ I - p 2 ),
^p
i s t h e r o o t of t h e c u b i c e q u a t i o n pf
t=2 f o r which
I PI
t-1
t
n . . u u = 0 t = 2 t-1 t
1
A
1.
Denote t h e maximum l i k e l i h o o d estimate of p a r t i c u l a r v a l u e s of n 1 and p, by G2(nl, p ) .
CJ
2
,
calculated for
The j o i n t maximum
l i k e l i h o o d f u n c t i o n f o r n l and p, assuming uo - N ( 0 , u 2 / 1 - p 2 ) , L e ( n l , p> =
(2.5)
^2 (0
(nl, p)}
-1112
(1-p')'
and assuming uo = 0, is
1
L a ( n l , p> = { a 2 ( n l , p> -n/2 w i t h t h e r e l a t i v e maximum l i k e l i h o o d f u n c t i o n g i v e n by n
-
R(n 1, p) = L ( n 1, p ) / L ( n 1, p)
is
(2. 6)
59 and i n e a c h case
-2 0
( n l , p) is c a l c u l a t e d a c c o r d i n g t o t h e a p p r o p r i a t e
p r o c e d u r e d e s c r i b e d above.
Maximum l i k e l i h o o d f u n c t i o n f o r nL
2.2
A c o n d i t i o n a l p r o c e d u r e used t o e l i m i n a t e t h e dependence between t h e
random v a r i a b l e s ( P l a c k e t t , 1960) can be used w i t h model (1.1).
For
t h e number of o b s e r v a t i o n s odd, n=2k+l, t h e c o n d i t i o n a l d i s t r i b u t i o n of ~2
= (u2,u4
,...,U2k)
g i v e n 21 = ( u 1 , u 3
,...
is multivariate
normal w i t h mean, v a r i a n c e and c o v a r i a n c e s = )p(l+p2)-1(u2i-l E ( u ~ ~
+ u 2i+l 1
v a r (u2i> = a2( 1 - p 2 ) / ( l + p 2 ) and cov ( u 2 i , u z S ) = 0 f o r i R e p l a c i n g ut by y t
-
(2.8)
4
P
1 atjej
for t
nl+l,n1+2
,...,n
j =I
q
Y t -j_Clbtj B j
or t
=
,...,k. 1 , 2 , ...,n l and
s , where i = 1,2 =
by
and w r i t i n g A f o r p / ( 1 + p 2 ) ,
t h e mean becomes, f o r even v a l u e s of n l , P
E(Y ) 2t for t
=
=
P
J=I
1,2
a
2 t , j ej
+ A(Y2t-1 + Y 2 t + l> - A j c= (l a 2 t - 1 , j +a 2 t + l , j. ) eJ. ’
n ,...,+ -I,
60
for t
...,-n 1-12
1,2,
=
'
q
> =j=lI bnl+l,j Bj + E(Yn 1+1
1
P
A(Y
+Y
n1 n1+2
>--A(
1
a 8 + b f3.) j=l n1,j j j = 1 nl,j J
and
(2.10) a
D
j=1
for t
2t,j
J
-$-,..., n +3 n,
=
>-Ai(b + b j=l 2t-1,j 2t+l,j with var (y
,...,p)
Ylj for hej (j=1,2
2t
) = a2(1-p2)/(l+p2).
and y2j for Af3j (j=l,2
By writing
,...q),
then
(2.9) and (2.10) are in the form of the usual linear model when n1 is Let {Gj}, {Ylj}, {yzj}, {ij 1,
known.
?,
and
^a2
be the usual
least squares estimates for the parameters of the model.
Substituting
these estimates in the conditional distribution of u2 given u1 produces a likelihood function for "1 alone. Inferences about "1 can then be made by examining the relative maximum likelihood function which is
where f
"1
(uz,..
.,~2k(u1,..., ~ 2 k + ~is) the conditional (u2,. ..,U2k) given (ul,. ..,~2k+l) with the
distribution of parameters {Oj}, {Ylj}, {Q},
(2.11)
{Yq}, u2 , and A replaced by
their estimates. 2.3
Maximum likelihood function for n1 assuming ~0 Under the assumption of independence, substitution of the maximum
likelihood estimates of function for {8j }, function for "1,
{& },
{ej },
{Pj } and
o2 into the joint likelihood
u2 and n 1 gives the maximum likelihood
61
Under t h e assumption of p=O, t h e m a r g i n a l o r c o n d i t i o n a l l i k e l i h o o d f u n c t i o n might be p r e f e r r e d ( E s t e r b y and El-Shaarawi,
1981b).
However,
s i n c e maximum l i k e l i h o o d f u n c t i o n s have been used above, t h e maximum l i k e l i h o o d f u n c t i o n w i l l be used h e r e t o make r e s u l t s more comparable.
2.4
Computational c o n s i d e r a t i o n s I n t h e j o i n t l i k e l i h o o d f u n c t i o n s f o r n 1 and p, t h e maximum
l i k e l i h o o d e s t i m a t e s of
{ej }, {pj }
2 are o b t a i n e d u s i n g t h e
and
0
method of l e a s t s q u a r e s when p and n 1 are given.
This involves
m i n i m i z a t i o n of t h e t e r m i n t h e exponent i n b o t h ( 2 . 2 )
and ( 2 . 3 ) .
W r i t e model (1.1) i n t h e form
+
yt = a ' a -t where
art =
and a'
=
u
t
(atl, a t 2 ,
(01,0 2 , . .
- -.
.,BP,
atp, b t l y bt2y.. (31, (32
,...,Bq).
-
btq>
Thus ut-put-l
can be
w r i t t e n as (Y t - -t'a)
- P(Yt-l -a' -t-1- a)
yt-pyt-l
and
=
-
a't-pa't-l
- are o b t a i n e d as
Z t
=
and t h u s t h e maximum l i k e l i h o o d
A
estimates
Let
= (Y~-PY~-~)
=
(X'X)- 1X'z, - where z'
and X i s t h e m a t r i x w i t h t h e tth row e q u a l t o
=
(21,22,
...,zn)
?It.
The j o i n t l i k e l i h o o d f u n c t i o n s f o r n1 and p c a n be o b t a i n e d by an i t e r a t i v e procedure. 1.
Assume an i n i t i a l v a l u e f o r p.
2.
For t h e p a r t i c u l a r v a l u e of p, form
3.
Let nl= 1 , 2
-a^ and
^a2
=
,...,n-1 (2'5-
z.
and f o r e a c h v a l u e of "1, form X , o b t a i n
X'z)/n. h
4.
Choose t h e maximum l i k e l i h o o d estimate of "1,
n
1, as t h e
v a l u e of n 1 which maximizes t h e l i k e l i h o o d f u n c t i o n , e i t h e r L e ( n l , p) o r L a ( n l , p ) *
62 5.
For nl-1, calculate
^p
u s e 3 o b t a i n e d assuming t h i s v a l u e of "1 and
{Gt}
From
where Gt=yt-fift&
it
o r (2.5),
u s i n g e i t h e r e q u a t i o n (2.4)
obtain
depending upon which
likelihood function is being calculated.
6.
I f t h i s i s t h e f i r s t i t e r a t i o n , assume p=p and r e t u r n t o s t e p 2.
If it is not the f i r s t i t e r a t i o n , test i f the
d i f f e r e n c e between t h e p r e v i o u s and p r e s e n t estimate of p i s less t h a n some s p e c i f i e d v a l u e and r e t u r n t o 2 i f i t i s
not.
7.
Using t h e f i n a l v a l u e of
c,
c a l c u l a t e R(n1, p) u s i n g t h e
a p p r o p r i a t e e x p r e s s i o n f o r L ( n 1 , p),
e i t h e r (2.6)
o r (2.7).
The l i k e l i h o o d f u n c t i o n s f o r n 1 a l o n e do n o t i n v o l v e i t e r a t i o n .
For
each a l l o w a b l e v a l u e of n l , t h e p a r a m e t e r s of t h e model a r e e s t i m a t e d
,.
by l e a s t s q u a r e s and a 2 ( n 1 ) o b t a i n e d .
Next, f i l i s chosen as t h e v a l u e
2 of n l f o r which CJ ( n l ) i s t h e minimum.
procedure, a vector
I n t h e case of t h e c o n d i t i o n a l
= ( y 2 , y 4 , . . . , y 2 k ) i s formed and f o r e a c h v a l u e
of n1 a m a t r i x X must be formed i n a manner similar t o t h a t used f o r the j o i n t likelihood functions.
3.
MONTE CARL0 STUDY The b e h a v i o u r of t h e f o u r l i k e l i h o o d f u n c t i o n s d e s c r i b e d above w a s
s t u d i e d by g e n e r a t i n g d a t a based upon a s i m p l e v e r s i o n of model (1.1). I t was assumed t h a t t h e change c o n s i s t e d of a change from one mean l e v e l t o a n o t h e r , i.e.
...,n 1 ) n1+1, ...,n ) .
Y t = 01
+ Ut
( t = 1,2,
Y t = B1
+ U t
(t =
The v a l u e s of t h e p a r a m e t e r s were t a k e n as 81 = 1098,
2
=
-0.25,
0, 0.25,
1981a), n=100, n l = 5 0 and p - 0 . 9 0 ,
0.50,
0.75,
0.90.
hundred samples were g e n e r a t e d w i t h u o and
For e a c h v a l u e of
-0.75, p,
two
{ e t } o b t a i n e d as pseudo-
random normal d e v i a t e s g e n e r a t e d by t h e p o l a r method. of
850,
128* ( t h e e s t i m a t e s obtained f o r t h e Nile River discharge d a t a ,
E s t e r b y and El-Shaarawi, -0.50,
=
I n i t i a l values
p=O were used i n t h e c o m p u t a t i o n of both t h e e x a c t and a p p r o x i m a t e
j o i n t likelihood functions.
F o r each of t h e two hundred s a m p l e s , o n l y
63 one r e a l r o o t s a t i s f y i n g
I pI c l
was found i n t h e c o m p u t a t i o n of t h e
exact j o i n t l i k e l i h o o d f u n c t i o n s . tional considerations.
See s t e p 5 i n t h e s e c t i o n Computa-
T h i s w a s a s c e r t a i n e d by a p r o v i s i o n i n t h e
computer program t o p r i n t a message i f i n f a c t more t h a n one s u c h r e a l r o o t w a s found.
4
The mean of G I , t h e number of t i m e s o u t of 200 f o r which
6 are
where n 1 = 50 w a s used t o g e n e r a t e t h e s a m p l e s , and t h e mean of given f o r a l l t h e l i k e l i h o o d f u n c t i o n s i n Table 1.
50,
All l i k e l i h o o d
TABLE 1 Comparison of t h e estimates of n 1 and p from t h e f o u r l i k e l i h o o d f u n c t i o n s based on 200 samples Value of p Method
Mean
-0.90
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
0.90
exact
50
50
50
50
50
50
49
49
51
approximate assume P=o conditional
50
50
50
50
50
50
49
49
47
50
50
50
50
50
50
50
50
50
50
50
50
51
50
49
49
49
47
exact
43
38
52
62
77
91
104
133
152
approximate assume
44
38
52
62
77
91
106
136
157
81
71
66
66
76
92
118
148
171
141
143
153
149
167
184
189
187
185
exact
-0.89
-0.74
-0.51
-0.26
-0.02
0.21 0.45
0.68
0.83
approximate
-0.89
-0.74
-0.51
-0.26
-0.02
0.21
0.69
0.84
of n1
-Number with Glfnl
P=o conditional
Mean of P
--
0.45
.----.-_I
_ _ I -
f u n c t i o n s gave b e t t e r estimates of n 1 f o r samples g e n e r a t e d w i t h p and, o v e r a l l , t h e j o i n t l i k e l i h o o d f u n c t i o n s , e x a c t and a p p r o x i m a t e , performed a b o u t t h e same but b e t t e r t h a n t h e o t h e r two l i k e l i h o o d functions.
The l i k e l i h o o d f u n c t i o n and t h e € r e q u e n c y d i s t r i b u t i o n of
C
64
TABLE 2 Average of t h e e x a c t r e l a t i v e maximum l i k e l i h o o d f u n c t i o n Re(nl, p) based on t h e 200 samples Value of p "1
-.90
-.75
-.50
-.25
.OO
.25
.50
.75
.90
1
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
.001 .007
.(
1 .003 .017 0 .003 .001 .OOO .OOO
.038 .011 .004 .001
.001
.001 .001
.ooo .ooo .c 0 .003
.024
.057 .035 .022 ,016 .016 .016 .011 .012
1.1 f -tt
nl=63 nl=72
.OOO
.OOO
.046 .073 .064 ,105 .133 .192 .661 ,167 .140 .094 .075 .064 .070
1
1.
.loo .103 .087 .482
-31 5
.102
.076
4
1
[ .02,
1
.08
] [ .03,
-06 ]
*OiO
[ .005, . 030]
1
1
i
nl=75
T
C . 001,.0071
99
-~
4
1
i n d i c a t e s t h a t t h e v a l u e remains c o n s t a n t a t t h e l a s t r e c o r d e d number. i n d i c a t e s t h a t v a l u e s are w i t h i n t h e c l o s e d i n t e r v a l g i v e n by t h e p a i r of numbers i n t h e b r a c k e t s [ ,
3.
65
1
10
,I
I
1525-
Point of Change Parameter N t 20 30 40 50 60 70 80
,
L
I
i
. I
i
90
99
p--o75
i;m
fi--ora
-
0, lo85
6,-
Y,
Fig. 1 .
P
859
6 = 133
Plots of the generated d a t a , f i t t e d means and R m ( n , , i ) f o r samples from Monte Carlo study.
66 are examined i n more d e t a i l f o r t h e e x a c t l i k e l i h o o d f u n c t i o n i n T a b l e s 2 and 3 .
Both t h e s e t a b l e s show t h a t as p i n c r e a s e s t h e number
of v a l u e s of n1 which are p l a u s i b l e a l s o i n c r e a s e s .
O r i n terms of the
f r e q u e n c y d i s t r i b u t i o n , g i v e n i n Table 3 , a l l estimates of n 1 w e r e l a r g e r t h a n 45 but no l a r g e r t h a n 52 when p = -0.90
w a s used t o
g e n e r a t e t h e samples, but 12% of t h e samples had estimates < I 0 o r 390 f o r p=0.90. Examples of t h e samples g e n e r a t e d and t h e r e l a t i v e maximum l i k e l i hood f u n c t i o n of n l a t
p';,
Re(nl,i),
are g i v e n i n F i g u r e 1.
The
g e n e r a t e d d a t a , t h e e s t i m a t e d c o n s t a n t l i n e s and t h e r e l a t i v e l i k e l i hood f u n c t i o n are shown on each p l o t and t h e v a l u e of p used t o gener a t e t h e sample and t h e estimates of t h e p a r a m e t e r s are g i v e n b e s i d e the plot.
The f i r s t two p l o t s are examples f o r which t h e e s t i m a t e s of
TABLE 3 Frequency d i s t r i b u t i o n of 61 o b t a i n e d from t h e e x a c t l i k e l i h o o d f u n c t i o n and t h e 200 samples h
n1 6 c1
R e l a t i v e f r e q u e n c y of n 1
c 1 o r n 1 6 c2 f o r p =
or ~2 G
6 G
6 G
6 =
%
10 20 30 40 45 48 50 52 55
5 3 60 3 70 >, 80 >/ 90
-0.90
0 0 0 0 0
-0.75
0 0 0 0
-0.50
0 0 0 0 0 0.02
0.01
0 0
0.79 0.01
0.81 0
0.74
0 0
0 0 0 0
0 0 0 0
0
0
0 0 0
0.01
-0.25
0.00
0.25
0.50
0.75
0.90
0 0 0 0 0 0.06 0.69 0.04
0 0 0 0 0.02 0.08 0.62 0.09
0 0 0
0.02 0.02 0.02 0.07
0.04
0.06
0.11
0.25
0.22
0.45 0.34 0.31
0 0 0 0 0
0 0 0 0 0
0.01 0.02 0.13 0.55
0.12 0.03
0.01 0 0 0
0.48
0.18 0.11 0.05 0.03 0 0
0.08
0.10
0.13 0.19
0.19 0.26 0.31 0.36 0.24 0.38 0.35 0.28 0.19 0.11 0.06
0.26 0.20 0.10 0.05 0.03
t h e p a r a m e t e r s were c l o s e t o t h e v a l u e s used t o g e n e r a t e t h e d a t a .
The
t h i r d p l o t i l l u s t r a t e s t h e t y p e of sample which can be g e n e r a t e d w i t h
67 positive
and which r e s u l t s i n estimates of "1 and p f a r from t h e
values used t o g e n e r a t e t h e d a t a . I n summary, t h e Monte C a r l o s t u d y shows t h a t f o r n e g a t i v e p o r
moderate p o s i t i v e p t h e j o i n t l i k e l i h o o d f u n c t i o n s , exact and a p p r o x i mate, and t h e l i k e l i h o o d f u n c t i o n assuming p=O c a n be u s e d t o e s t i m a t e 31.
The l i k e l i h o o d f u n c t i o n f o r "1 based upon c o n d i t i o n i n g d i d n o t
p e r f o r m w e l l e v e n though t h e sample s i z e u s e d w a s 100.
F u r t h e r work i s
r e q u i r e d t o e x p l a i n t h e p r o p e r t i e s of t h e l i k e l i h o o d f u n c t i o n s s u g g e s t e d by t h e Monte C a r l o s t u d y .
REFERENCES 1975. T e c h n i q u e s f o r Brown, R.L., D u r b i n , J. and Evans, J . M . , t e s t i n g t h e c o n s t a n c y of r e g r e s s i o n r e l a t i o n s h i p s o v e r t i m e (with Discussion). J . R . S t a t i s t . SOC. B, 37:149-192. Cobb, G.W., 1978. The problem of t h e N i l e : conditional solution B i o m e t r i k a , 65(2):243-251. t o a c h a n g e p o i n t problem. Cox, D.R. and H i n k l e y , D.V., 1974. T h e o r e t i c a l s t a t i s t i c s . Chapman and H a l l , London, 511 pp. E s t e r b y , S.R. and El-Shaarawi, A.H., 1981a. L i k e l i h o o d i n f e r e n c e a b o u t t h e p o i n t of change i n a r e g r e s s i o n regime. J. Hydrology, 53 :17-30. 1981b. I n f e r e n c e a b o u t t h e E s t e r b y , S.R. and El-Shaarawi, A.H., Appl. S t a t i s t . , p o i n t of change i n a r e g r e s s i o n model. 30 :277-285. 1970. I n f e r e n c e a b o u t t h e c h a n g e - p o i n t i n a H i n k l e y , D.V., s e q u e n c e of random v a r i a b l e s . B i o m e t r i k a , 57(1):1-17. H i n k l e y , D.V., 1971. I n f e r e n c e a b o u t t h e change-point from c u m u l a t i v e sum t e s t s . B i o m e t r i k a , 5 8 ( 3 ) : 509-523. 1956. Graphs of c u m u l a t i v e r e s i d u a l s . Q.J.R. Kraus, E.B., M e t e o r o l . SOC., 82:96-98. P e t t i t t , A.N., 1979. A n o n - p a r a m e t r i c a p p r o a c h t o t h e change-point problem. Appl. S t a t i s t . , 28(2):126-135. 1960. P r i n c i p l e s of r e g r e s s i o n a n a l y s i s . Oxford: P l a c k e t t , R.L., C l a r e n d o n Press, London, 173 pp.
68
THE CHANGE-POINT PROBLEM FOR A SEQUENCE OF BINOMIAL RANDOM VARIABLES A.H.
EL-SHAARAWI AND L.D.
DELORME
N a t i o n a l Water Research I n s t i t u t e
ABSTRACT Three s t a t i s t i c s a r e p r e s e n t e d f o r d e t e c t i n g a change i n a sequence o f ordered b i n o m i a l random v a r i a b l e s .
The f i r s t two a r e based on t h e
c o n d i t i o n a l d i s t r i b u t i o n o f t h e random v a r i a b l e s g i v e n t h e i r sum, w h i l e t h e t h i r d s t a t i s t i c i s based on t h e e m p i r c a l l o g i s t i c t r a n s f o r m approach.
The use o f these s t a t i s t i c s i s i l l u s t r a t e d u s i n g d a t a f r o m a
lake-sediment c o r e . INTRODUCTION The change-point
roblem has r e c e i v cl c o n s i d e r a b l e a t t e n t i o n r e c e n t l y
i n the f i e l d o f s t a t i s t i c s .
Roughly speaking, t h i s problem i s concerned
w i t h d e v e l o p i n g procedures f o r t e s t i n g t h e h y p o t h e s i s t h a t t h e parameters o f t h e d i s t r i b u t i o n o f a sequence o f o r d e r e d random v a r i a b l e s are n o t constant.
I f t h e h y p o t h e s i s o f change i s accepted t h e n t h e
problem i s how t o make i n f e r e n c e s about b o t h t h e p o s i t i o n o f change and t h e magnitude o f change.
Quandt (1 958,1960,1972),
Quandt and Ramsey
(1978), H i n k l e y (1969), Brown et a1 (1975), Feder (1975a,b),
Ferreira
(1975) and E s t e r b y and El-Shaarawi (1981a) have d i s c u s s e d t h e changep o i n t problem f o r l i n e a r r e g r e s s i o n models.
The case o f b i n a r y random
v a r i a b l e s was c o n s i d e r e d by P e t t i t t (1979), H i n k l e y (1970), M c G i l c h r i s t and Woodyer (1975) and Page (1955,1957). I n w a t e r q u a l i t y s t u d i e s , t h e change-point problem has many a p p l i c a tions.
F o r example, i n t h e s t u d y o f l a k e sediment c o r e s one m i g h t be
i n t e r e s t e d i n d e t e r m i n i n g i f a change has o c c u r r e d i n t h e r e l a t i v e o r a b s o l u t e abundance o f a p a r t i c u l a r b i o l o g i c a l i n d i c a t o r and o f d e t e r m i n i n g t h e p a t t e r n and t h e depth a t which t h e change has o c c u r r e d . Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors)
o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
69
A p p l i c a t i o n s t o w a t e r q u a l i t y problems have been c o n s i d e r e d by E s t e r b y and El-Shaarawi (1981a, 1981b). I n t h i s paper, a number o f approaches a r e p r e s e n t e d t h a t can be used f o r t h e a n a l y s i s o f t h e change-point problem i n a sequence o f ordered b i n o m i a l random v a r i a b l e s .
These t e c h n i q u e s a r e t h e n a p p l i e d
t o t h e s t u d y o f a d a t a s e t f r o m a lake-sediment c o r e . THE MODEL Suppose t h a t we have a sequence o f n o r d e r e d independent b i n o m i a l random v a r i a b l e s xl...,xn.
Assume t h a t xi
( f o r i = 1,2,...,k)
has t h e
distribution m.-x. 1
('1) x.
o2x i
1
, and x ~ (+f o ~r i
= 1,2,
...,n - k )
has t h e d i s t r i b u t i o n
m.-x (Lo2) i i
The i n t e g e r k ( t h e change-point)
i s unknown.
The problem i s t o t e s t
any o f t h e h y p o t h e s i s
:'H H: 0 f 0 1 2'
a1
02.
I f any o f t h e s e h y p o t h e s i s i s r e j e c t e d , t h e problem i s t h e n how t o e s t i m a t e k and t o make i n f e r e n c e s about i t s v a l u e s . I n t h e b i n a r y case, where mi = 1 ( i = 1,2
,..., n )
and xi
( i = 1,2,...,
n ) t a k e s o n l y t h e values 0 and 1, M c G i l c h r i s t and Woodyer (1975) have developed s t a t i s t i c s f o r t e s t i n g H,
+
H , H- when n i s even.
These
s t a t i s t i c s have been g e n e r a l i z e d b y P e t t i t t (1979) f o r a l l values o f n.
Fol 1ow ing Pet t it t t h e s t a t is t ic s u = Max j u k I ln yn U
+
=
Max
UkYn
1< O n U- =
-Min U k,n ln
70 c o u l d be used f o r t e s t i n g H , H S k - k x and
and H- r e s p e c t i v e l y , where U
=
k,n The e x a c t s i g n i f i c a n c e l e v e l s
...
+
Sk =
+
+ xk. and H- can be o b t a i n e d u s i n g t h e Kolmogorov-
a s s o c i a t e d w i t h H, H
Smirnov two sample s t a t i s t i c s ( P e t t i t t ( 1 9 7 9 ) ) . I n t h e n o n - b i n a r y case, t h e s i g n i f i c a n c e l e v e l s a s s o c i a t e d w i t h t h e above s t a t i s t i c s a r e n o t e x a c t l y known, however, a c o n s e r v a t i v e v a l u e For t h e
f o r t h e s i g n i f i c a n c e l e v e l i s g i v e n by P e t t i t t ( 1 9 7 9 ) . b i n o m i a l case, U ‘k,n
takes the form
k,n
S k - TPk,
=
where Pk
(ml +
=
I f t h e mi’s
...
+ mk)/M, M
=
+
ml
...
+ M n and T
=
nx.
a r e m o d e r a t e l y l a r g e , t h e n i t can be shown t h a t t h e
x
given under t h e c o n d i t i o n a l d i s t r i b u t i o n o f Ul,n...,... ” ( n - 1 ) ,n a s s u m p t i o n o1 = o2 c o n v e r g e s i n d i s t r i b u t i o n t o t h e m u l t i v a r i a t e normal
0 and v a r i a n c e c o v a r i a n c e m a t r i x V, when t h e c o v a r i a n c e w i t h mean T(M-T) between Si and S . ( f o r i f j ) i s - ~ Pipj, lt h e v a r i a n c e o f Si 3
is
An a p p r o x i m a t e s i g n i f i c a n c e l e v e l f o r U can be T(M-T) Pi ( l - P i ) . M- 1 obtained u s i n g Sidak (1968) i n e q u a l i t y
Y
*.-
..X,xlCL
n',Y
-x,-
~ > \ . j n L T L ~ , % Y L k L e.Lfrercrr-c y frulx,
When xt c a u s e s y t i n s t a n t a n e o u s l y a s i g n i f i c a n t v a l u e of t h e
sample CCF w i l l e x i s t a t l a y z e r o .
A d e s c r i p t i o n of t h e various
t y p e s o f c a u s a l r e l a t i o n s h i p s i s p r o v i d e d by a u t h o r s such a s P i e r e and Hauqh (1977) and H i p e l e t a l .
(1981).
To i d e n t i f y t h e form o f t h e dynamic and n o i s e components when
i n t e r v e n t i o n s a r e n ' t p r e s e n t , Haugh and B o x (1977) p r o v i d e a t e c h nique which i s b a s e d upon t h e r e s i d u a l CCF.
By u t i l i z i n g a p h y s i c a l
u n d e r s t a n d i n g o f t h e problem p l u s s t a t i s t i c a l p r o c e d u r e s , H i p e l e t
116 al.
(1981) s u g g e s t a n e m p i r i c a l approach f o r i d e n t i f y i n g t r a n s f e r
functions plus the noise t e r m .
The e m p i r i c a l a p p r o a c h f o r i d e n t i f y i n g
the noise t e r m i s described i n the next section. 2.3
IDENTIFYING THE N O I S E COMPONENT A f t e r t h e dynamic component h a s b e e n d e s i g n e d , t h e n o i s e component
i s t e n t a t i v e l y assumed t o b e w h i t e n o i s e and c o n s e q u e n t l y t h e t r a n s f e r f u n c t i o n - n o i s e model i n (1) h a s t h e form
In practice, the noise t e r m is usually correlated. obtaining the estimated r e s i d u a l series,
at '
Therefore, a f t e r
f o r t h e model i n (9),
t h e t y p e o f ARMA model t o f i t t o t h i s s e r i e s c a n b e a s c e r t a i n e d by f o l l o w i n g t h e u s u a l t h r e e s t a g e s o f model c o n s t r u c t i o n f o r ARMA models.
S u b s e q u e n t l y , t h e i d e n t i f i e d form o f t h e ARMA model c a n b e
i n (1) and MLE's f o r a l l t h e model p a r a m e t e r s c a n be t simultaneously estimated. D i a g n o s t i c c h e c k s c a n t h e n b e employed t o
used f o r N
i n s u r e t h a t t h e r e s i d u a l assumptions a r e s a t i s f i e d .
I f more t h a n
one dynamic model p a s s e s d i a g n o s t i c t e s t s , t h e A I C c a n be employed t o a s s i s t i n s e l e c t i n g t h e b e s t model. 2.4
ESTIMATION O F MISSING DATA The model u s e d f o r e s t i m a t i n g m i s s i n g d a t a i s a s p e c i a l case o f
t h e g e n e r a l i n t e r v e n t i o n t r a n s f e r f u n c t i o n - n o i s e model g i v e n i n (1) (Baracos e t a l . ,
1 9 8 1 ; D ' A s t o u s and H i p e l , 1 9 7 9 ) .
When d e a l i n g w i t h
monthly d a t a t h e f i r s t s t e p i n t h e e s t i m a t i o n p r o c e d u r e i s t o subs t i t u t e t h e a p p r o p r i a t e m o n t h l y means f o r a l l o f t h e m i s s i n g d a t a . This s u b s t i t u t i o n i s necessary t o achieve a zero value f o r each m i s s i n g d a t a p o i n t when t h e s e r i e s i s d e s e a s o n a l i z e d by s u b t r a c t i n g o u t t h e e s t i m a t e d m o n t h l y means a n d d i v i d i n g by t h e e s t i m a t e d m o n t h l y s t a n d a r d d e v i a t i o n f o r each o b s e r v a t i o n .
T o d e m o n s t r a t e how a n
i n t e r v e n t i o n model c a n be employed t o e s t i m a t e a s i n g l e m i s s i n g o b s e r v a t i o n , c o n s i d e r t h e case where t h e r e i s a s i n g l e m i s s i n g d a t a point a t t i m e t
=
T.
The model u s e d t o e s t i m a t e t h e m i s s i n g
117 o b s e r v a t i o n f o r a s i n g l e s e r i e s such a s y
t
is
i s s e t t o z e r o when t h e s e r i e s i s d e s e a s o n a l i z e d ; and t i s a p u l s e i n t e r v e n t i o n d e f i n e d by
where y
it
~
(0
,
50, V may be assumed t o be n o r m a l l y d i s t r i b u t e d w i t h mean 2N(N-1) and v a r i a n c e 4 ( N - Z ) / ( N - l ) [ .
A g a i n as shown i n T a b l e 1,
n e a r l y a l l s t a n d a r d i s e d v a l u e s o f V a r e w i t h i n t h e l i m i t s o f -11.96 and hence non-randomness i s n o t i n d i c a t e d : Rescaled Range T e s t
. . ,xN)
( x l ,x2,.
** r
The a d j u s t e d r e s c a l e d ra-nge o f t h e o b s e r v a t i o n s
i s obtained from
max ( x 1+X 2+...+ x i - i i ) h r o > r c > l
and
0 > ri > ro > rc > -1.
This almost unbiased estimator r i s very easy t o apply, b u t presents a drawback for values of ri greater t h a t 0.9, as the singular point around r i = 1 and the sampling errors make the definition of rc less accurate. Bias related t o the estimator rD Compressing the tirne-series by simply neglecting the occasional missing values gives way t o the estimator rD; doing so, the length of the sample i s reduced t o (N-n) and each missing d a t a leads in + XimXi,l equation (1) t o the suppression of two products XimXi+l a t the numerator and a t the same time a new product X i - l X i + l is created; a t the denominator, one of the squares Xim2 disappears. If we suppose an exponential law for the decrease of the autocorrelation with the lag, a case of pure persistence fairly common in in hydrometeorology and which includes the often-used first-order Markov model, the equation (1) becomes:
158 TAtlLt 1:
ro
1
Values o f rc f o r d i f f e r e n t values o f ro and ri -0.9
-1
-0.8
-0.1
-0.6
-0.5
-0.3
-0.4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1
0.8
0.9
1-
‘1
-1
-1.m
-0.9
-0.90
-0.8
-0.80
-0.1
-0.70
-0.6
-0.81 -0.60
-0.5
-0.11
-0.4
-0.83 - 0 . S -0.40
-0.3
-0.96 -0.61 -0.45 -0.30
-0.50
-0.2
-0.76 -0.52 -0.14 -0.20
-0.1
-0.81 -0.59 -0.38 -0.22 -0.10
0
-1.00 -0.67 -0.43 -0.25 -0.11
0.00
0.1
- 0 . 1 1 - 0 . 4 9 -0.28 -0.12
0.00
0.10
0.2
-0.90 -0.56 -0.12 -0.14
0.m
0.11 0.20
0.3
-0.63 -0.31 -0.16
0.00
0.12 0.22 0.30
0.4
-0.80 -0.43 -0.18
0.00
0.14 0.24 0.31 0.40
0.5
-0.53 -0.21
0.00
0.15 0.21 0.36 0.44 0.50
0.6
-0.70 -0.21
0.00
0.18 0.11 0.41 0.49 0.55 0.60
0.1
-0.36
0.00
0.22 0.37 0.41 0.55 0.61 0.66 0.70
0.8
-0.60
0.00
0.28 0.45 0.56 0.63 0.69 0.14 0.11 0.80
0.00
0.43 0.61 0.10 0.11 0.81 0.84 0.81 0.88 0.90
0.9
1
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.oc
TABLE 2: T h e o r e t i c a l b i a s e s
p-T and sampling v a r i a n c e s v a r r f o r s h o r t samples d e r i v e d from Markovian p a r e n t p o p u l a t i o n s
N
100
P
P - r
0
80
-
-60
40
-
20
0 - r -
var r
P - r
var r
-
0 - r
var r
0.013
0.017
0.017
0.025
0.025
0.050
0.050
0.018
0.012
0.023
0.017
0.035
0.025
0.070
0.049
0.009
0.023
0.012
0.030
0.016
0.045
0.024
0.090
0.048
0.022
0.009
0.028
0.011
0.037
0.015
0.055
0.023
0.110
0.045
0.4
0.026
0.008
0.033
0.010
0.043
0.014
0.065
0.021
0.130
0.042
0.5
0.030
0.007
0.038
0.009
0.050
0.013
0.075
0.019
0.150
0.038
0.6
0.034
0.006
0.043
0.008
0.057
0.011
0.085
0.016
0.170
0.032
0.7
0.038
0.005
0.048
0.006
0.063
0.008
0.095
0.01 3
0.210
0.018
0.8
0.042
0.004
0.053
0.004
0.070
0.006
0.105
0.009
0.210
0.018
0.9
0.046
0.003
0.058
0.002
0.077
0.003
0.115
0.005
0.230
0.010
1
0.050
0
0.063
0
0.083
0
0.125
0 -
0.250
0
var r
p - r
var
0.010
n.oio
0.013
0.1
0.014
0.010
0.2
0.018
0.3
r
159 +I
c
FIGURE 1 :
:
'I
FIGLIFE 2:
I
Theoretical bias corrections related to various estimators r^ in the large sample case.
Nomograph relating r to r0 and C r i'
FTGITRE 3:
Compared efficiencies in the bias corrections (N= 6n, n= 9) based on 2090 entries.
160 N(N-2n-1) r + nNr2 (N-n) (N-1) E(rD) = which can be i n v e r t e d as:
- +
-(I2ntl ) r =
[1
-
N
-
(2”” ) I 2 + 4
(1- n ) (1-
) E[~D]
N
N
’
(15)
2 “ N T h i s expression can be expanded t o o r d e r N-2: (1 +
r = E [ r D ]( 1 - -)2 n + l N
n( 3-E[ r ~)+I ] t
N
One can v e r i f y t h a t i f n =0, E ( r D ) = r, i f E [ r D ] = 0 , r = O i n d e pendently o f n and N, and i f E [ r ] = 1, r = l t o t h e o r d e r N-2. The replacement o f E [ r D ] by i t s samp?e e s t i m a t e r y i e l d s an adequate b i as c o r r e c t ion :
r
-
rD= rD[1 ( l - r D ) N
+
2 N2
~2
( 2 - r ~ ) + 2n ( l - r D ) ]
(16)
I f we su pose a s u f f i c i e n t number o f m i s s i n g data ( n a 1) and by p u t t i n g = a G 1, t h e n t h e preceeding e q u a t i o n s reduce t o : N
1
One can note t h a t f o r a l l values o f rD, t h e b i a s i s always p o s i t i v e ; t h e e s t i m a t o r rDunderestimates t h e t r u e v a l u e o f r as ro did. Comparison o f t h e b i a s e s The biases o f ro, ri and rD, g i v e n by e q u a t i o n s ( 6 ) , (11) and (18) are compared on t h e F i g u r e 2, f o r a = 0.05 and 0.20. We suggest t h e use o f ro f o r sampled values l o w e r t h a n 0.5 and o f ri f o r sampled values h i g h e r than 0.5, which minimizes t h e i n f l u e n c e o f t h e sampling e r r o r on t h e e s t i m a t o r ’ s b i a s c o r r e c t i o n s . Even i f t h e b i a s on rDa r e s m a l l e r , we p r e f e r ro and ri as no hypothes i s i s made on t h e g e n e r a t i n g process o f t h e p a r e n t p o p u l a t i o n .
161
The biases of rc being much smaller cannot be represented on t h i s f i gure. 5. The short sample case
The complete sample (no missing value) More interesting t h a t an unbiased evaluation of the autocorrelation r of a sample i s the estimation of the autocorrelation p of the parent population; SOPER et a l , (1918), ORCUTT (1948) and SASTRY (1951) worked on the re1 ationships between autocorrel ations estimated from a few traces of relatively short ( N G 100) samples and t h a t of t h e i r parent population; in a review of previous works, MARIOTT and POPE (1954) have shown t h a t bias may a r i s e from two sources: i f the mean value of the parent population i s known, autocorrel ations estimated from short samples are generally biased toward zero; i f the mean of the parent population i s not known and has t o be estimated from the sample, i t induces another bias (except for p = 0 ) , which i s always negative for a long series. The combinaison of the two biases can e i t h e r compensate or reinforce each other; as they are not independent, they cannot be investigated separately. For t h a t reason, WALLIS and O'CONNELL (1972) have used the simulation technique t o evaluate the biases resulting from samples of various lengths drawn from parent population f o r known auto-correlation p ; they have shown t h a t i f the parent population i s generated by a Markovian process of order one, the theoretical bias correction due t o KENDALL (1954) and derived from a circular series assumption i s equally valid for the case of classical oDen series. This length correction can be written as: p - r = -
1 (1 + N
4p) =
4E[r] t 1 N-4
For the same type of parent population, BOX and JENKINS (1970) have devel opped the sampling variance for short samples: var(r) =
1-02 N
As we are going t o use t h i s case as a reference, the theoretic a l biases and sampling variances for complete samples of length N = 100, 80, 60, 40 and 20 and parent populations of autocorrelation p = O,(O.l),l have been tabulated in the Table 2. The incomplete sample (occasional missing values) Given the short sample bias correction, equation (19), and the estimator's bias corrections, equations ( 3 ) , (8) and (15), we questioned whether successive application of those two corrections could improve significantly the bias in the autocorrelation in the case of short samples with occasional missing values. Bearing in
162
mind t h a t the sampl i n g variance i s general l y increased by a bias correction and t h a t one generally disposes of a single sample, we should verify t h a t the f i n a l sampling variance was kept i n reasonable limits t o really benefit from an almost unbiased estimation; i n this regard, one can note t h a t the equations (19) and ( 3 ) being linear, the variance amplification factor i s independent of the order in which the corrections are carried o u t ; this is not the case for the equations (8) and (15), so t h a t anterior and posterior length corrections yield different sampl i n g variances. The a n a l y t i c a l expressions derived from the combinations of the theoretical corrections as well as their numerical values for the cases considered here are given in CLUIS and BOUCHER (1981). SIMULATIONS AND RESULTS
To test the efficiencies of the proposed theoretical bias corrections pertaining t o the various estimators, we used a Monte Carlo technique by generating 2000 synthetic sequences of length 100, 80, 60, 40 and 20 from a Markovian parent population of known parameter p ; those series were created using the recurring formula:
I n . this expression,
are NIP ( O , l ) , provided by the a l g o r i t h m A t f i r s t , the missing values were i n troduced in appropriate number by suppressing a t random some el ements of the sequence i n each of the 2000 samples; as found also by KNOKE (1979) i n a study on some s t a t i s t i c s of the lag-one a u t o correlation distribution, the results were remarkably unaffected by the location of the missing d a t a when the number of missing d a t a d i d not exceed 20% of the sample length: the means and the so, variances calculated w i t h 2000 series were stable w i t h i n i t was decided t o give missing values fixed positions i n each of the 2000 samples, making them approximately equispaced. E~
o f BOX and MULLER (1958).
Five series of results obtained by simulation and concerning residual biases and sampl i ng variances w i 1 1 be discussed here : series 1: on complete samples (no missing value) t o serve as reference for the efficiencies of the estimators; series 2: applied;
on incomplete samples f o r w h i c h no correction has been
series 3: on incomplete samp es for w h i c h only the length correct i o n has been applied; series 4: on incomplete samp es for which only the estimator's correction has been applied;
163 s e r i e s 5: on i n c o m p l e te samples f o r which b o t h c o r r e c t i o n s have been a p p l i e d . The r e s u l t s o f t h e s e r i e s 1 shown on t h e Ta b l e 2 p r o v e t h e h i g h e f f i c i e n c y o f t h e KENDALL l e n g t h b i a s c o r r e c t i o n and e x h i b i t a 1arge i n c r e a s e o f t h e sampling v a r i a n c e s f o r v e r y s h o r t samples (N=40, N=20), b u t a l s o v e ry s i m i l a r r e s u l t s t o t h e t h e o r e t i c a l values o f t h e T a b l e 1 f o r l a r g e r samples (N=100, N=80), e s p e c i a l l y ¶ i f p i s c l o s e t o 1. F o r t h e s e r i e s 2, 3, 4 and 5, t y p i c a l r e s u l t s c o n c e r n i n g t h e cases N=100, n=10, N=60, n=15 and N=20, n=4 a r e p r e s e n t e d on t h e Tables 4, 5, 6, 7; t h e T a b l e 4 d i s p l a y s t h e magnitude o f t h e b i a ses w i t h o u t any c o r r e c t i o n a p p l i e d t o t h e e s t i m a t o r s ; a l l t h e e s t i m a t o r s y i e l d poor r e s u l t s , b u t rc and rDa r e r e l a t i v e l y more e f f i c i e n t f o r t h e l a r g e r s e r i e s ; t h e Tables 5 and 6 p r e s e n t s t h e r e s u l t s o f t h e s e r i e s 3 and 4 when o n l y one c o r r e c t i o n has been a p p l i e d ; t h e r e s i d u a l b i a s e s a r e s t i l l v e r y i m p o r t a n t , and e x c e p t i n one case, always p o s i t i v e ; one can n o t e t h a t i f one has t o a pply o n l y one c o r r e c t i o n , t h e l e n g t h c o r r e c t i o n i s g e n e r a l l y more e f f i c i e n t than the estimator's correction. For t h e study o f the s e r i e s 5, t h e two c o r r e c t i o n s can be a p p l i e d i n two p o s s i b l e p e r m u t a t i o n s i n t h e case o f ro, ri and rDand i n f i v e p o s s i b l e permut a t i o n s i n t h e case o r rc; t h e b e t t e r r e s u l t s b o t h f o r t h e c o r r e c t i o n o f t h e b i a s e s and t h e r e d u c t i o n o f t h e sampling v a r i a n c e were c o n s i s t e n t l y o b t a i n e d when t h e l e n g t h c o r r e c t i o n i s a p p l i e d f i r s t and a r e d i s p l a y e d on t h e T a b l e 7; t h e e s t i m a t o r ro proves t o be t h e worst f o r l a r g e v a l u e s o f p and t h e b e s t f o r small v a l u e s o f t h e e s t i m a t o r ri i s t h e b e s t e s t i m a t o r f o r small v a l u e s o f p ; b o t h rD and rc a r e v e ry e f f i c i e n t i n t h e whole range of values o f p ; t h e r e i s an advantage i n u s i n g r c as i t s t h e o r e t i c a l b i a s c o r p;
r e c t i o n makes no h y p o th e s i s about t h e g e n e r a t i n g process o f t h e p arent p o p u l a t i o n which i s n o t t h e case f o r rD;t h e r e i s a l s o a disadvantage r e l a t e d t o i t s poor d e f i n i t i o n around p = 1 which can be seen f o r v e ry s h o r t samples (N = 20). The F i g u r e 3 shows t h e p ro g re s s to w a r d unbiasedness when n o t u s i n g o r when u s i n g one o r two c o r r e c t i o n s i n t h e case o f a sample o f l e n g t h 60 c o n t a i n i n g 9 m i s s i n g values. I n parallel t o the bias c o r r e c t i o n s , t h e sampling v a r i a n c e s i n c r e a s e d s l i g h t l y , w i t h no noteworthy d i f f e r e n c e between t h e e s t i m a t o r s , t h e sampling v a r i a n ces correspond i ng t o t h e u l t i m a t e b i as c o r r e c t ion b e i ng mu1t i p l ied by a f a c t o r 2 o r 3 compared w i t h t h e ones o b t a i n e d on t h e T a b l e 3 f o r complete samples.
COMPARISON WITH THE FISHER'S TRANSFORMATION When two p o p u l a t i o n s a r e c o r r e l a t e d , t h e d i s t r i b u t i o n o f t h e i r c o r r e l a t i o n c o e f f i c i e n t s i s n e i t h e r Gaussian n o r even s y m e t r i c a l ,
164 TABLE 3: R e s i d u a l b i a s e s and sampling v a r i a n c e s f o r l e n g t h - c o r r e c t e d a u t o c o r r e l a t i o n s o f complete samples
_ .
N
80
100
-
-
60
20
40
P
P - r
var r
p - r
var r
-
P - r
var r
0 - r
var r
P - r
var r
0
0.003
n.oio
0.003
0.013
0.003
0.018
0.005
0.028
0.000
0.068
0.1
-0.001
0.011
0.001
0.014
0.003
0.018
0.005
0.028
.O.
006
0.069
0.2
-0.003
0.010
.0.004
0.013
0.003
0.018
-0.003
0.028
.0.003
0.064
0.3
-0.000
0.010
0.002
0.012
0.001
0.017
-0.002
0.028
.0.003
0.066
0.4
0.003
0.009
0.005
0.012
0.005
0.017
0.006
0.027
0.003
0.067
-0.000 0.008
0.001
0.011
o.noi
0.015
-0.001
0.025
0.005
0.065
0.5 0.6
0.005
0.008
0.003
0.010
0.004
0.014
0.005
0.022
0.009
0.060
0.7
0.000
0.006
0.000
0.008
0.002
0.012
0.007
0.021
0.020
0.057
0.8
0.004
0.005
0.007
0.007
0.008
0.010
0.017
0.018
0.027
0.054
0.9
0.006
0.004
0.009
0.005
0.016
0.008
0.024
0.015
0.044
0.046
0.9!
0.008
0.003
0.011
0.004
0.018
0.006
0.032
0.013
0.066
0.044
Rased on 2000 e n t r i e s
TABLE 4 :
O r i g i n a l b i a s e s induced by t h e e s t i m a t o r s (no c o r r e c t i o n a p p l i e d )
P
r0
N=60
n.10
N=100
-
‘i
‘c
--0
N=20
n=9
ro
ri
n=4 rD
rc
0.011 -0.09; 0.013 0.010 0.021 -0.136 0.02: 0.020 0.051 -0.157 0.065 0.050
0.1 0.022 -0.077
0.044 0.084 -0.119 0.097 0.082
0.2 0.03P -0.061
0.063 0.135 -0.070 0.141 0.133
I / / / 1
0.3 0.057 -0.042 0.048 0.063 0.092 -0.062 0.08( 0.092 0.178 -0.028 0.174 0.174
I
0.4 0.072 -0.028 0.056 0.077 0.119 -0.036 0.09C 0.116 0.233 0.019 0.213 0.225 0.5 0.087 -0.01E 0.061 0.08C 0.13E -0.022 0.095 0.125 0.279 0.054 0.242 0.263 0.6 0.106 0.7
0.001 0.06e o.09e 0.167 0.002 0.llC 0.144 0.330
0.12c 0.01c 0.065 0.097 0.196 0.020 0.111 0.152 0.388 0.133 0.290 0.347
0.8 0.140 0.025 n. 068 0.099 0.231 0.9
0.092 0.265 0.304
0.17C
0.044 0.111 0.162 0.463 0.174 0.314 0.401
0.04C 0.068 0.096 0.287 0.069 0.121 0.174 0.569 0.218 0.341 0.486
0.95 0.205 0.047 0.061 0.097 0.356
-- ----
0.080 0.121 0.193 0.697 0.250 0.366 0.629
165 TABLE 5: Residuals biases after the application
I I
N-100
I
n.10
N.60
n=9
‘i
r~
ro
rc
r~
of the lenth correction only
I rc
N=20 ro
ri
n=4 r~
rc
0.001 -0.106 1.002 0.000 0.005 -0.164 0.006 0.003 0.001 -0.259 0.002 -0.000
0 0.1
0.009 -0.095 1.009 0.013 0.021 -0.142 0.020 0.022 0.017 -0.236 0.013
0.015
0.2
0.021 -0.082 1.016 0.027 0.033 -0.129 0.023 0.035 0.056 -0.200 0.038
0.054
0.3
0.036 -0.067 1.025 0.043 0.059 -0.106 0.040 0.059 0.084 -0.173 0.048
0.080
0.4
0.048 -0.056 1.029 0.053 0.082 -0.085 0.049 0.078 0.129 -0.139 0.068
0.119
0.5
0.059 -0.048 7.029 0.058 0.092 -0.077 0.043 0.080 0.161 -0.120 0.072
0.141
0.6
0.075 -0.034 3.032 0.067 0.119 -0.058 0.047 0.093 0 . 2 0 ~ -0.098 0.071
0.167
0.7
0.085 -0.030 0.024 0.062 0.142 -0.046 0.040 0.095 0.247 -0.071 0.070
0.197
0.8
0.102 - 0 . 0 1 ~ 0.023 0.059 0.173 -0.028 0.038 0.098 0.3if
0.239
0.9
0.129 -0.007 0.017
-0.045 0.069
0.052 0.226 -0.008 0.034 0.104 0.42: -0.015 0,071 0.32C
0.95 0.163 -0.001 0.015 0.052 0.295
0.000 0.029 0.121 0.57:
0.012 0.088 0.487
TABLE 6: Residuals biases after the application of the estimator’s correction only
N=~OO
P
-
‘0
‘i
N=60
n=10 ‘0
‘C
--
n=9
N-20
n=4
r0
.__
0
0.01: 0.106 0.01t 0.014 0.02f 0.027 0.037 o . 0 ~ 0.070 0.069 0.134 0.094
0.1
0.01: 0.022 0.017 0.01: 0.031 10.046 0.040 0.03: 0.078 0.096 0.137 n.098
0.2
0.01E 0.032 0.02c
0.3
O.O2€
0.4
0.031 0.05C 0.031 0.03E 0.05j ‘0.087 0.059 0.061 0.170 0.203 0.190 0.194
0.5
0.033 0.052 0.034 0.042 0.054 ln.o~70.056 0.065 0.194 0.223 0.203 0.218
0.6
0.043 0.055
0.7
0.046 0.056 0.04C 0.052 0.08: 0.097 0.070 0.092 0.269 0.270 0.334 0.287
0.8
0.055 0.06C 0.047 0.059 0.105 0.103 0.081 0.10~ 0.334 0.293 0.255 0.339
0.9
0.076 0.061 0.052 0.065 0.151 0.110 0.092 0.12: 0.442 0.319 0.282 0.430
I
0.021 0.03C i0.054 0.036 0.037 0.110 0.136 0.154 0.130
0.044 0.027 0.031 0.04: ~0.073 0.050 0.051
0.131 0.167 0.164 0.154
0.041 0.05; 0.071 0.097 0.007 0.085 0.227 0.245 0.215 0.249
0.95 0.109 0.062 0.056 0.072 0.222 0.112 0.097 0.151 0.600 0.345 0.307 0.608
-
--
based on 2000 entries
166 TABLE 7:
U l t i m a t e biases (both c o r r e c t i o n s a p p l l e d ) N=100
P
0
ro
n.10
‘1
‘0
0.002 -0.001
rc
N.60
n=9
ri
‘0
ro
0.004 0.003
0.ooz i -0.004
N=20
rc
0.014 0.011
n=4
ri
ro
ro
rc
0.001 -0.054 0.056 0.044
0.1
-0.002
0.004
0.2
-0.002
0.009 -0.001 0.002 -0.004 I 0.010 -0.001.0.002
0.3
0.002
0.017
0.003 0.007
0. 005i
0.022
0.007 0.013
0.002 -0.016
0.4
0.003
0.019
0.002 0.010
0.011
0.030
0.010 0.020
0.026 -0.000 0.036 0.039
0.5
0.003
0.016
0.002 0.009
0.001
0.021
0.000 0.012
0.031 -0.004
0.030 0.032
0.6
0.008
0.019
0.005 0.015
0.011
0.023
0.005 0.019
0.047 -0.007
0.023 0.033
0.001 0.000 -0.004I 0.009
0.010 0.008 -0.014
-0.048
0.037 0.013
0.001 -0.026 0.038 0.021 0.028 0.022
0.7
0.007
0.012
0.000 0.010
0.011 ’
0.015
0.003 0.017
0.075 -0.006
0.025 0.039
0.8
0.013
0.010
0.004 0.011
0.033 I 0.014
0.008 0.020
0.131 -0.009
0.031 0.047
0.9
0.030
0.007
0.006 0.012
0.075 I 0.012
0.014 0.026
0.241 -0.007
0.044 0.155
0.951 0.062
0.006
0.008 0.013
0.145 I
0.017 0.033
0.426 -0.011 0.064 0.303
0.009
based on 2000 e n t r i e s
TABLE 8 :
R e s i d u a l b i a s e s and s a p l i n g v a r i a n c e s u s i n g F i s h e r ’ s t r a n s f o n a t ion ( N= 100)
no l e n g t h c o r r e c t i o n n.10
11x5 var rf
0
-
with length correction
n-5
7 var
rf
P-Tf
n-10
var r f
var r f
0
0.067
0.009
0.135
0.009
n.nni
n.1119
0.000
0.064
0.1
O.ORR
0.009
0.171
o.no8
-o.nii
0.019
-0.070
0.058
0.2
0.106
o.nio
0.210
n.oo9
-0.02~
0.m
-0.119
0.058
0.3
0.130
0.009
0.255
0.009
-0.039
n.nm
-0.143
0.048
0.4
0.156
n.ow
0.303
o.on9
-0.048
0.021
-0.159
0.037
0.5
0.175
0.009
0.347
o.on8
-0.069
0.019
-0.165
0.026
0.6
0.207
0.008
0.401
0.009
-0.068
0.017
-0.135
0.021
0.7
n.231
o.nm
0.448
o.no8
-0.060
o.niz
-0.102
0,012
0 .R
0.271
0.008
0.510
o.no8
-0.035
n.oo9
-0.n4n
0.009
0.9
0.311
0.006
0.571
0.007
-0.013
0.nn5
0.027
n.no6
0.95
0.336
n.nofi
0.607
0.007
-0.1-145
n.003
0.005
n.005
hased on Zflflfl e n t r i e s .
167 which makes i m p o s s i b l e t h e t e s t s o f hypotheses; FISHER (1921) gave a s o l u t i o n t o t h i s problem by t r a n s f o r m i n g sampled v a l u e s o f r i n t o a q u a n t i t y Z almost n o r m a l l y d i s t r i b u t e d w i t h a s t a n d a r d d e v i a t i o n a z p r a c t i c a l l y independent o f t h e c o r r e l a t i o n l e v e l : Z =
t [
Ln(1 + rf)
-
Ln ( l - r f ) ]= t a n h - l r f
or
(21)
rf = tanh Z D e a l i n g w i t h a u t o c o r r e l a t i o n s , we a r e i n t h e case o f " i n t r a c l a s s " o r " f r a t e r n a l " c o r r e l a t i o n w i t h r e g a r d t o t h e p a r e n t popul a t i o n ; t e tandard d e v i a t i o n o f t h e transformed v a r i a t e i s then !I)-? where m i s t h e a Z = (m- l e n g t h o f t h e sequence f o r which 2 t h e t r a n s f o r m e d v a l u e Z i s c a l c u l a t e d . Making use o f t h i s p r o p e r ty, we d e r i v e d an e s t i m a t o r r f i n t h e case o f o c c a s i o n a l m i s s i n g values; on each sequence k o f u n i n t e r u p t e d data, an a u t o c o r r e l a t i o n c o e f f i c i e n t rk i s c a l c u l a t e d and e v e n t u a l l y c o r r e c t e d f o r l e n g t h ; t h e c o r r e s p o n d i n g t r a n s f o r m e d v a l u e s Zk a r e t h e n compounded i n t o a g l o b a l v a l u e Z:
rf = ta n h Z One shoul d n o t e t h a t t h e a p p l i c a t i o n o f t h e F i s h e r ' s t r a n s f o r mat ion i s l e s s r e s t r i c t i v e t h a n t h e p r e v i o u s l y designed e s t i m a t o r s as i t does n o t r e q u i r e f o r t h e m i s s i n g v a l u e s t o be o c c a s i o n a l ; t h e r e s i d u a l b i a s i s always n e g a t i v e and s m a l l ; f o r e q u i d i s t a n t m i s s i n g data, t h e expansion o f - t h e p r e v i o u s e x p r e s s i o n p e r m i t s t o f i g u r e i t s va ue:
r-rf
=
-
rf ( 1 - r f P 2N ( n + 1)
(I+-) 2N
I n our case t l i s e x p r e s s i o n was k e p t below 10-3.
To t e s t t h e e f f i c i e n c y o f such an e s t i m a t o r , we used t h e same s i m u l a t i o n program as p r e v i o u s l y ; 2000 samples o f l e n g t h 100 were generated, c o n t a i n i n g 5 and 10 m i s s i n g v a l u e s a p p r o x i m a t e l y e q u i s Fo r such paced, which c r e a t e d sequences o f about 16 and 8 data. s h o r t samples, i t seems necessary t o a p p l y t h e l e n g t h c o r r e c t i o n
168 g i v e n by e q u a t i o n ( 9 ) , b u t we found i n t h e l i t e r a t u r e no r e p o r t o f such a use w i t h t h e F i s h e r ' s t r a n s f o r m a t i o n ; t h i s c r e a t e s a d i f f i c u l t y as, f o r very s h o r t samples, some c o r r e c t e d values o f rkbecome l a r g e r than one, e s p e c i a l l y f o r l a r g e values o f p ; so we const r a i n e d rk between - 0.98 and + 0.98. The r e s u l t s a r e shown on t h e Table 8; t h e improvement, due t o t h e l e n g t h c o r r e c t i o n i s q u i t e v i s i b l e , b u t even w i t h it, t h e e f f i c i e n c y o f t h e e s t i m a t o r d e r i v e d f r o m t h e F i s h e r ' s t r a n s f o r m a t i o n i s l a r g e l y worse t h a n o t h e r e s t i m a t o r e f f i c i e n c i e s g i v e n by t h e Table 7. One can a l s o suggest t h a t t h e values presented i n Table 8 a r e p o s s i b l y t o o o p t i m i s t i c , due t o t h e c o n s t r a i n t s imposed upon t h e values o f rk. CONCLUSIONS Various e s t i m a t o r s have been t e s t e d i n o r d e r t o e s t i m a t e t h e 1 ag-one a u t o c o r r e l a t i o n c o e f f i c i e n t o f a t i m e - s e r i e s c o n t a i n i n g occasional m i s s i n g values; t h e corresponding b i a s c o r r e c t i o n s have been f i r s t e s t a b l i s h e d t h e o r e t i c a l l y i n t h e case o f l a r g e samples, then combined w i t h a l e n g t h c o r r e c t i o n t o y i e l d a procedure v a l i d f o r s h o r t samples; i t has been demonstrated t h a t t h e compression o f a t i m e - s e r i e s , a common p r a c t i c e i n t h i s s i t u a t i o n , should n o t be performed w i t h o u t a p p l y i n g t h e proper c o r r e c t i o n s . The r e s u l t s t e s t e d by a Monte-Carlo t e c h n i q u e w i t h Markovian s e r i e s seemed t o be more e f f e c t i v e t h a n t h e c l a s s i c a l Z - t r a n s f o r m a t i o n and should be u s e f u l f o r a whole range o f r e a l - l i f e h y d r o m e t e o r o l o g i c a l t i m e s e r i e s whose p e r s i s t e n c e s t r u c t u r e i s a p p r o x i m a t e l y Markovian. REFERENCES
ANDERSON, R.L. (1942). D i s t r i b u t i o n o f t h e s e r i a l c o r r e l a t i o n c o e f f i c i e n t . Ann. Math. Stat., B: 1-13. BOX, J.P. and G.M. J E N K I N S (1970). Time s e r i e s a n a l y s i s , f o r e c a s t i n g and c o n t r o l . Holden-Day, San Francisco, C a l i f . , 553 p. BOX, G.E.P. and M.E. MULLER (1958). A n o t e on t h e g e n e r a t i o n o f random normal deviates. Ann. Math. S t a t i s t . , 29: 610-611. BRUBACHER, S.R. and G.T. WILSON (1976). Interpolating time series with applications t o the estimation o f h o l i d a y e f f e c t s on e l e c t r i c i t y demand. J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , London, England, S e r i e s C ( A p p l i e d S t a t i s t i c s ) , 25 ( 2 ) : 107-116. CLUIS, D. and P. BOUCHER (1981). E s t i m a t i o n de l ' a u t o c o r r e l a t i o n d ' o r d r e 1 d ' u n k h a n t i l l o n c o u r t comportant des Val eurs manquantes o c c a s i o n n e l l es. Technical Report No 138. INRS-Eau, U n i v e r s i t c du Quebec, Que., Canada. D'ASTOUS, F. and K.W. HIPEL (1979). A n a l y s i n g environmental t i m e s e r i e s . ASCE, Env. Eng. Div., 105 (EE5) : 979-992.
169 FISHER, R.A. (1921). On t h e p r o b a b l e e r r o r o f a c o e f f i c i e n t o f c o r r e l a t i o n deduced from a small sample. Metron., l ( 4 ) : 3-32. JENKINS, M.G and D.G. WATTS (1968). S p e c t r a l a n a l y s i s and i t s a p p l i c a t i o n s . Holden-Day, San Fr a n c i s c o , C a l i f . , 525 p. KENDALL, M. G. ( 1954). Note on b i a s i n t h e e s t i m a t i o n o f a u t o c o r r e l a t i o n . Biometrika,
41: 403-404. KNOKE, J.D. (1979). Normal appro x i ma ti o n s f o r s e r i a l c o r r e l a t i o n s t a t i s t i c s . Biometries, 35: 491-495. MARRIOTT, F.H.C. and J.A. POPE (1954). Bias i n t h e e s t i m a t i o n o f a u t o c o r r e l a t i o n , B i o m e t r i k a , 42: 390-
402. ORCUTT, G.H. (1948). A st udy o f t h e a u t o r e g r e s s i v e n a t u r e o f t h e t i m e s e r i e s used f o r T i n b e r g e n ' s model o f t h e economic system o f t h e U n i t e d S t a t e s , 1919-193.2. J.R. S t a t i s t . SOC., B. 10: 1-54. PARZEN, E. (1964). An approach t o e m p i r i c a l t i m e s e r i e s a n a l y s i s . Radio-Science,
6 8 ( 9 ) : 937-951. PREECE, D.A. (1971). I t e r a t i v e procedures f o r m i s s i n g v a l u e s i n experiments. 'Technom e t r i c s , 13 (4):743-753. RODRIGUEZ-ITURBE, I. (1971). S t r u c t u r a l a n a l y s i s o f hydro1 o g i c a l sequences. Proceeding Warsaw Symposium. I n : Mathematical models i n h y d r o l o g y , 3: 1157-
1165. SASTRY, A.S.R. (1951). Bias i n estimation o f s e r i a l c o e f f i c i e n t s .
Sankhya, 11: 281-
296. SOPER, H.E., A.W. YOUNG, B.M. CAVE, A. LEE and K. PEARSON (1916). On t h e d i s t r i b u t i o n o f t h e c o r r e l a t i o n c o e f f i c i e n t i n small samples - a c o o p e r a t i v e study. B i o m e t r i k a , 11: 328-413. WALLIS, J.R. and N.C. MATALAS (1971). Correloqram a n a l y s i s ' r e v i s i t e d . Wat. Resour. Res., 7 ( 6 ) : 1448-
1459.
WALLIS, J.R. and P.E. O'CONNELL (1972). Small sample e s t i m a t i o n o f pl. Water Resour. Res.,
8 ( 3 ) : 707-
712. WILKINSON, G.N. (1958). E s t i m a t i o n o f m i s s i n g values f o r t h e a n a l y s i s o f incompl e t e data. B i o m e t r i k a , 14 ( 2 ) : 257-286.
17@ TIDAL ANALYSIS - A RETROSPECT
D. E . CARTWRI GHT I n s t i t u t e o f Oceanographic S c i e n c e s , B i d s t o n , UK
LIMITATIONS
TO PROGRESS
I f i r s t became i n v o l v e d w i t h t i d a l a n a l y s i s - as an o c e a n o g r a p h e r as d i s t i n c t f r o m one c o m m i t t e d t o p r o d u c i n g t i d e - t a b l e s - i n t h e l a t e 1 9 5 0 ' s when p r i m i t i v e e l e c t r o n i c d i g i t a l c o m p u t e r s f i r s t became a c c e s s i b l e t o t h e g e n e r a l s c i e n t i s t .
A t t h a t time, t i d e -
t a b l e s were s t i l l h a n d w r i t t e n b y o p e r a t o r s s i t t i n g i n f r o n t o f t i d e - p r e d i c t i n g machines, and t i d a l d a t a were a n a l y s e d w i t h t h e a i d o f s h e e t s o f s q u a r e d p a p e r , s t e n c i l s and m e c h a n i c a l d e s k - a d d e r s . Twenty-odd Years l a t e r , such methods may appear l a u g h a b l e , b u t i n f a c t t h e i r t i d e p r e d i c t i o n s were v e r y good, t h e methods h a v i n g been e v o l v e d b y e x p e r t m a t h e m a t i c i a n s who r e g a r d e d them as h a v i n g reached a p l a t e a u o f p r a c t i c a l p e r f e c t i o n .
T h e r e f o r e , a1 t h o u g h
modern computers opened up a new r a n g e o f a n a l y t i c a l t e c h n i q u e s w h i c h were p r e v i o u s l y u n t h i n k a b l e i n t e r m s o f l a b o u r , t h e i r improvement i n a c c u r a c y o f p r e d i c t i o n was a t b e s t o n l y m a r g i n a l .
To be more e x p l i c i t , a d a t a s e r i e s o f sea s u r f a c e e l e v a t i o n z ( t ) may be e x p r e s s e d as z(t)
=
r;(t)
+
R(t)
where t i s t h e t i m e ( p r e f e r a b l y ' U n i v e r s a l ' o r Greenwich Mean T i m e ) , 5 i s t h e p a r t o f t h e s i g n a l which i s d i r e c t l y r e l a t e d t o t h e t i d e -
g e n e r a t i n g p o t e n t i a l o f t h e Moon and Sun, and R i s t h e r e s i d u a l w h i c h i s u n c o r r e l a t e d w i t h r, and w h i c h depends on m e t e o r o l o g i c a l and o t h e r n o n - t i d a l i n f l u e n c e s .
A t a typical estuarine s i t e i n
t h e UK t h e t o t a l v a r i a n c e o f z f o r a c e r t a i n p e r i o d was 2 . 4 2 0 ~ 1 ~ . O f t h i s , 2.361m2 was w i t h i n t h e p o s s i b l e s p e c t r a l bands o f t h e
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) @ 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
171 t i d e s , making 2.361 an upper bound t o t h e v a r i a n c e o f 5 and 0.059 a l o w e r bound f o r t h e v a r i a n c e o f R. Comparison o f t h e same d a t a w i t h t h r e e d i f f e r e n t t i d a l syntheses ( t h e o r e t i c a l a p p r o x i m a t i o n s t o c ) gave t h e f o l l o w i n g r e s i d u a l v a r i a n c e s : Elementary harmonic method
(60
'Extended' harmonic method
(114 terms)
0.114m2
Advanced response method
(47
0.088m2
terms) terms)
0.125~1~
I t i s c l e a r t h a t t h e most advanced methods o f t i d a l a n a l y s i s
developed d u r i n g t h e computer age can o n l y improve p r e d i c t i o n variance ( i . e .
reduce t h e r e s i d u a l ) by a v e r y small f r a c t i o n o f
the t o t a l variance.
To reduce i t more e f f e c t i v e l y one has t o
i n v e s t i g a g e methods o f p r e d i c t i n g R( t ) t h r o u g h w e a t h e r - f o r e c a s t models, and t h i s i s o u t s i d e t h e scope o f t h i s survey.
I n terms o f
r e p r e s e n t a t i o n o f c ( t ) however t h e modern methods a r e v e r y good indeed, a l t h o u g h p o o r e r cases a r e found i n r e g i o n s s t r o n g l y a f f e c t e d b y s h a l l o w w a t e r and v a r i a b l e r i v e r r u n - o f f . i n t e r e s t e d i n the t i d e s ' p e r se'
, there
For those
have been c o n s i d e r a b l e
advances i n u n d e r s t a n d i n g t h e i r p h y s i c a l n a t u r e i n r e l a t i o n t o t h e i r g e n e r a t i n g f o r c e s and t h e i r p r o p a g a t i o n i n t h e ocean b a s i n s , ( C a r t w r i g h t , 1977).
The p r i n c i p a l t e c h n i q u e s i n t i m e s e r i e s
a n a l y s i s which have made t h i s p o s s i b l e have been t h e use o f s p e c t r a l a n a l y s i s f o r e x p l o r i n g t h e n a t u r e o f b o t h r ; ( t ) and R ( t ) , and t h e use o f d i r e c t l y computed t i m e s e r i e s o f t h e t i d e - g e n e r a t i n g potential. I n t h e f o l l o w i n g s e c t i o n s I s h a l l r e v i e w b r i e f l y some o f t h e uses which have been made o f these and o t h e r techniques, and a l s o p o i n t o u t some areas where t h e r e i s s t i l l need f o r improvement. INITIAL DATA SCANNING R i s i n g a n a l y t i c a l p r e c i s i o n has emphasised t h e need t o check d a t a s e r i e s c a r e f u l l y b e f o r e any s e r i o u s a n a l y s i s .
Digital tide
gauges and automated c h a r t readers have reduced t h e frequency o f
172 some t y p e s o f e r r o r , b u t many sea l e v e l r e c o r d s a r e s t i l l r e c o r d e d b y moving pen and d i g i t i s e d b y a human r e a d e r , and t h e s e may c o n t a i n a host of e r r o r s o f various o r i g i n .
The s i m p l e s t e r r o r d e t e c t o r s
w h i c h have been w i d e l y used a r e t h e W i e n e r p r e d i c t o r
( 1 ) and t h e
i n t e r p o l a t o r (2),
k ~ ' ( t )=
c 1
Z(t-kb)
W' k
k z"(t) =
c
1
W"
k
[~(t-ks)]
d e s i g n e d so t h a t z ' - z o r z " - z has a v a r i a n c e v e r y much l e s s t h a n that of z itself.
(6 i s t h e t i m e i n t e r v a l o f t h e s e r i e s ) .
On
s c a n n i n g i n t , any v a l u e s z ' o r z " d i f f e r i n g f r o m z b y more t h a n a p r e - a s s i g n e d q u a n t i t y i s f l a g g e d , and t h e d a t a checked, i f p o s s i b l e back t o i t s o r i g i n a l s o u r c e .
Some f a v o u r t h e Wiener
p r e d i c t o r ( 1 ) because i f I z ' ( t ) - z ( t ) l i s l a r g e b u t a l l p r e v i o u s k values are small i t i s a f a i r i n d i c a t i o n t h a t z ( t ) i t s e l f i s s u s p e c t , whereas ( 2 ) g i v e s a s y m m e t r i c a l p a t t e r n o f anomalous d i f f e r e n c e s b e f o r e and a f t e r each t r u e e r r o r . allows a lower threshold o f d e t e c t a b i l i t y .
However, ( 2 ) a l w a y s
Several forms o f ( 2 )
a r e d i s c u s s e d b y K a r u n a r a t n e ( 1 9 8 0 ) , i n c l u d i n g t h e use o f a 25h interval. A n o t h e r e r r o r - d e t e c t i n g method w h i c h we have f o u n d v e r y e f f e c t i v e a t B i d s t o n i n r e c e n t y e a r s i s t h e automated p l o t t i n g of the residual series a f t e r subtraction o f a f a i r t i d a l synthesis.
A l l forms o f e r r o r a r e r e v e a l e d a t a g l a n c e and t h e method i s e s p e c i a l l y e f f e c t i v e i n i d e n t i f y i n g a1 1-too-common t i m i n g e r r o r s (Pugh & V a s s i e , 1978: G r a f f & K a r u n a r a t n e , 1 9 8 0 ) . SPECTRAL ANALYSIS Spectral a n a l y s i s i s a v i t a l t o o l f o r understanding t h e nature o f t i d a l data.
I t has t r a n s f o r m e d t h i n k i n g f r o m t h e o l d e r
173
textbooks which t r e a t t i d e s as i f they were an isolated l i n e spectral process, t o the modern concept o f a weakly nonlinear signal highly correlated with i t s source function embedded in a continuous noise background. The source function i s of course the tide-generating potential , or more usefully, the time-varying p a r t of i t s leading spherical harmonics which divide naturally i n t o the tidal 'species' 0,1,2,3.. . . representing long-period, d i u r n a l , semi-diurnal t i d e s e t c . I t i s important t o preserve spectral phase. Spectral analysis t h r o u g h the standard method employing cross-correlation with the source function i s n o t in my experience very f r u i t f u l , because of the very high power density in the M2 l i n e and others, which s p i l l over into neighbouring f i l t e r b a n d s , masking the f i n e r d e t a i l one seeks there. The normal approach i s t h r o u g h fourier analysis of synodic periods of data, t h a t i s periods f o r which most of the major l i n e s come near t o an integral number of cycles, important examples being 29,59,355,738 days Unfortunately, each of these contains a large prime f a c t o r , so the ' f a s t f o u r i e r transform' technique i s inapplicable in i t s deal form involving powers of 2, or i f adapted t o take large pr me factors i t loses much of i t s advantage in speed. Besides t h i s , one does n o t usually require the complete transform. Franco & Rock (1971), however, have adapted the f a s t f o u r i e r transform t o the 'harmonic method' of t i d a l analysis. I have a personal preference f o r a basic spectral unit consisting of a f o u r i e r transform of a 59d span of data with a ' h a n n i n g ' window: C s ( t ) = 2N-I
N/ 2 2
z(t+rs)(l+cos2rr/N) exp (2risr/N)
(3)
r=-N/2 where f o r 6 = 1 h o u r , N = 59x24 = 1416, a n d s takes a l l integral values from 0 t o about 60P where P i s the highest t i d a l species of i n t e r e s t . The transform i s effected by a common algorithm such
174 as ' W a t t - I t e r a t i o n ' , adapted t o make use o f t h e f a c t t h a t t i s u s u a l l y stepped s e q u e n t i a l l y i n steps o f T days where T i s t y p i c a l l y i n t h e range 5-15.
(The ' h a n n i n g ' , r e p r e s e n t e d by t h e
m i d d l e b r a c k e t , i s o m i t t e d f r o m t h e f i r s t stage o f computation and b r o u g h t i n a t t h e end by a p p l y i n g (&,l,k)
smoothing t o t h e
sequence o f harmon ics ) . The s p e c t r a l f i l t e r ( 3 ) n e a t l y separates t h e t i d a l 'group c e n t r e d on p c y c l e s l l u n a r day + q c y c l e s l m o n t h (OGpdl2, -4GqG4)
,
i n t o t h e s p e c t r a l elements s = 57p + Zq, w h i l e t h e Overspill i s
so small t h a t o u t s i d e t h e species-bands t h e elements C s may be f a i r l y taken as measures o f t h e n o n - t i d a l continuum. A s e t o f C s f o r a s i n g l e value o f t has some l i m i t e d use, b u t t h e most u s e f u l a p p l i c a t i o n s stem f r o m a sequence o f t e x t e n d i n g o v e r say one o r several y e a r s . (i)
These a p p l i c a t i o n s a r e o f t h r e e s o r t s :
The complex sequences C s ( t ) f o r s e l e c t e d values o f s o f t i d a l i n t e r e s t may be f u r t h e r f o u r i e r analysed a t h i g h e r r e s o l u t i o n t o r e v e a l t h e t i d a l s t r u c t u r e w i t h i n each 'group'.
(ii)
A p a r a l l e l sequence C i ( t ) may be generated f r o m a r e l a t e d t i m e s e r i e s such as t h e t i d a l p o t e n t i a l o r some m e t e o r o l o g i c a l f u n c t i o n ( f o r t h e n o n - t i d a l values o f s ) and t h e mean t r a n s f e r f u n c t i o n and coherence e v a l u a t e d a t each frequency ( s l 5 9 c y c l e s d - ' ) f r o m t h e r e l a t i o n s ( c f Munk & C a r t w r i g h t , 1966)
zs - < c ; c y c ; c;*> where
*
2
q-( I C s' c*s I > )
2
/< c;I
denote t h e complex c o n j u g a t e and
o v e r a l l values o f t .
c
2
>4CSI
2
>
(4)
> an ensemble average
175
( i i i ) For t i d a l groups s possibly containing two d i s t i n c t t i d a l elements , (eg linear and nonlinear, gravitational and radiational) , cross spectra can be made w i t h two parallel s e r i e s f o r each s , and the d i s t i n c t elements separated by invert i n g a cross -corre 1a t i on mat r i x . Examples of ( i ) are the harmonic development of the t i d e potentials from spectral analysis of 18y sequences (Cartwright & Taylor, 1971; Cartwright & Edden, 1973), the resolution of an unexpected term a t exactly 1 cyclellunar day (Cartwright , 1975, 1976) , a detailed examination of the s t r u c t u r e of the non-gravitational solar t i d e s (Cartwright & Edden, 1977), and several other examinations of oceanic t i d a l records. Procedure ( i i ) was applied a t cy-' resolution (N6 = 710d) i n the analyses of long t i d a l s e r i e s by Munk & Cartwright (1966), and, as presented here, t o an a l a l y s i s of surges and surge-tide interaction round Britain by Cartwright (1968) and in numerous cases where an admittance a t cm-' resolution or a variance spectrum covering t i d a l and non-tidal frequencies are required. Procedure ( i i i ) has had some appl ications t o separating admittances t o t i d a l e f f e c t s with close o r overlapping frequency structure mentioned under ( i ) , and f o r a quick examination of the different c h a r a c t e r i s t i c s of say S2 and K2 (radiational e f f e c t s ) or 2N2 and u 2 (nonlinear e f f e c t s ) . THE NOISE CONTINUUM I t i s worth commenting on the main features o f the continuum
because although well known t o s p e c i a l i s t s they are l i t t l e used by practitioners. By d e f i n i t i o n , the continuum represents a random variation. I t may be partly related t o weather parameters b u t i t must usually be regarded as unpredictable noise, possibly with seasonal variations in general l e v e l . I t s spectrum, l i k e t h a t o f most geophsical variables, r i s e s monotonical ly towards the lowest measurable frequencies, and presents a threshold f o r the
176 d e t e c t a b i l i t y o f t i d a l components i n a r e c o r d o f g i v e n d u r a t i o n . Because o f t h i s frequency-dependence,
the diurnal t i d e s i n places
l i k e t h e A t l a n t i c Ocean where t h e y a r e weak a r e u s u a l l y l e s s r e l i a b l y e s t i m a t e d t h a n t h e v e r y weak t e r - d i u r n a l t i d e s w i t h f r e q u e n c i e s n e a r 2 . 9 cd-’ o f magnitude l o w e r .
where t h e c o n t i n u u m i s t y p i c a l l y an o r d e r
The m o s t n o t a b l e c a s u a l t i e s a r e o f c o u r s e t h e
t i d e s o f long period (species 0).
The c o n t i n u u m d e n s i t y a t t h e
m o n t h l y f r e q u e n c y Mm, say, i s t y p i c a l l y 3cm
2
(cycle/year)-l
so t h e
noise variance o f a s p e c t r a l element d e r i v e d from N y e a r s ’ data i s a b o u t 3/N cm
2
.
A t y p i c a l a m p l i t u d e o f t h e Mm t i d e i s e x p e c t e d t o
be 1 cm o r l e s s , w i t h a v a r i a n c e o f a t most 0.5 cm
2
.
Ifwe t a k e as
c r i t e r i o n f o r r e l i a b i l i t y o f an e s t i m a t e , t h a t i t s s i g n a l v a r i a n c e s h o u l d be t e n t i m e s t h e n o i s e v a r i a n c e , t h e n Mm w o u l d t y p i c a l l y require N
=
60 y e a r s f o r a p r o p e r e s t i m a t i o n .
( D e s p i t e t h i s , one
s t i l l sees l i s t s o f h a r m o n i c c o n s t a n t s f r o m l y a n a l y s e s w h i c h r e l i g i o u s l y q u o t e a m p l i t u d e s and phases f o r Mm and as i f t h e y had some p r e d i c t i v e v a l u e . ) Some improvement i n t h e l o w f r e q u e n c y c o n t i n u u m l e v e l can be e f f e c t e d b y s u b t r a c t i n g a c o n v o l u t i o n o f t h e atmospheric pressure f i e l d ( C a r t w r i g h t , p.45,
1968), b u t i n general t h e o n l y l o n g - p e r i o d
t i d e s w h i c h can be r e l i a b l y e s t i m a t e d f r o m a f e w y e a r s ’ d a t a a r e t h e s e a s o n a l y e a r l y (Sa) and p o s s i b l y t h e ha1 f - y e a r l y ( S s a ) components. ?he most a p p r o p r i a t e d a t a s e t s f o r t h e s e a r e t h e l o n g s e r i e s o f m o n t h l y sea l e v e l s a v a i l a b l e f r o m t h e I n t e r n a t i o n a l Permanent S e r v i c e f o r Mean Sea L e v e l . A n o t h e r awkward p r o p e r t y o f t h e c o n t i n u u m i s i t s i n v a r i a b l e t e n d e n c y t o r i s e i n t h e n e i g h b o u r h o o d o f t h e s t r o n g t i d a l bands, even c o n v e r g i n g i n c u s p - l i k e f a s h i o n on i n d i v i d u a l s t r o n g l i n e s l i k e M%. T h i s p r o p e r t y was f i r s t d i s c o v e r e d b y Munk, Z e t l e r & Groves ( 1 9 6 5 ) , b u t i t i s a l s o examined i n Munk & C a r t w r i g h t , (1968).
( 1 9 6 6 ) and C a r t w r i g h t
I t s cause has been v a g u e l y a s c r i b e d t o i n t e r n a l t i d e s o r
t o some m o d u l a t i o n o f t h e t i d e s b y t h e w e a t h e r c o n t i n u u m , b u t i n my e x p e r i e n c e o f d e a l i n g w i t h common t i d e gauge r e c o r d s t h e c u s p - l i k e
177 r i s e i s i n p r a c t i c e more l i k e l y t o be due t o m e d i o c r e i n s t r u m e n t a l maintenance, n o t a b l y i n t i m e - k e e p i n g . T i d a l cusps make i t i m p o s s i b l e t o e s t i m a t e t h e r e l i a b i l i t y o f t i d a l constants from the inter-species spectral noise l e v e l , otherw i s e f a i r l y e a s y t o compute.
I t w o u l d be m o s t u s e f u l i f a l l
e s t i m a t e d s e t s o f t i d a l c o n s t a n t s were accompanied b y s p e c i e s - b a n d v a r i a n c e s o f t h e o r i g i n a l d a t a and o f t h e r e s i d u a l s a s s o c i a t e d w i t h the l i s t e d constants.
The l a t t e r g i v e s a r e l i a b l e measure o f t h e
t r u e c u s p - v a r i a n c e w i t h i n t h e s p e c i e s concerned, a g a i n s t w h i c h t h e l i k e l y v a r i a t i o n o f t h e i n d i v i d u a l t e r m s may be assessed.
The t o t a l
r e s i d u a l v a r i a n c e i s n o t s u f f i c i e n t because i t i n c l u d e s much l o w f r e q u e n c y and i n t e r - s p e c i e s n o i s e . EXTENDED
HARMONIC METHODS
F o r i t s compromise between a c c u r a c y and s i m p l i c i t y t h e ' h a r m o n i c method' r e m a i n s as i t was a c e n t u r y ago, t h e b e s t p r i n c i p l e f o r r o u t i n e p r a c t i c a l d e a l i n g w i t h weakly n o n l i n e a r t i d a l systems. i s well-known,
As
i t e x p r e s s e s t h e t i d a l p a r t o f sea l e v e l a t a g i v e n
place i n t h e form
where sn i s a s e t o f known f r e q u e n c i e s , n o t d i f f e r i n g b y l e s s t h a n about 1 c y - l , and x n ( o ) a r e known i n i t i a l phases a t t h e c o n t e m p o r a r y epoch t = O ;
fn,un a r e s l o w m o d u l a t i n g f u n c t i o n s , m o s t l y w i t h t h e
p e r i o d 1 8 . 6 ~o f t h e l u n a r node, d e r i v e d f r o m t h e c o r r e s p o n d i n g t e r m s i n t h e t i d e - g e n e r a t i n g p o t e n t i a l , and Hn,Gn
are a r b i t r a r y constants
t o be a s s i g n e d t o t h e p l a c e i n q u e s t i o n . The i n g e n i o u s b u t n o i s e - l e a k i n g f i l t e r s w i t h i n t e g r a l m u l t i p l i e r s used t o e x t r a c t Hn,Gn
i n t h e o l d hand c o m p u t a t i o n s (most n e a t l y
summarised i n t h e c o n t e x t o f more r e c e n t t e c h n i q u e s b y Godin ( 1 9 7 2 ) ) , have l o n g s i n c e been superseded b y t h e s u p e r i o r t e c h n i q u e s made
178 p o s s i b l e by a u t o m a t i c computers such as ' l e a s t - s q u a r e s ' and FFT (Franc0 & Rock 1971 )
(Horn, 1960)
.
One o f t h e vaguest aspects o f ' h a r m o n i c ' p r a c t i c e i s how t o a s s i g n t h e frequencies sn and t h e i r t o t a l number n ' o u t o f t h e many hundred p o s s i b l e terms appearing i n t h e t i d a l p o t e n t i a l and i t s cross-products.
The o l d procedures were l i m i t e d by t h e number o f
terms which t h e l a r g e s t t i d e - p r e d i c t i n g machines c o u l d handle, i n t h e range 45-60.
By s t u d y i n g t h e power s p e c t r a o f r e s i d u a l s , Z e t l e r
& Cummings (1967) and R o s s i t e r & Lennon (1968) i n d e p e n d e n t l y a r r i v e d a t n ' = 114 as a s u i t a b l e number, a l t h o u g h t h e i r c h o i c e o f i n d i v i d u a l terms d i f f e r e d .
Some o f t h e new terms chosen r e q u i r e d
c o n s i d e r a b l e c o n t o r t i o n o f t h e o r i g i n a l harmonic concept.
For
example, MNKZS, i m p l y i n g a q u i n t i c i n t e r a c t i o n between t h e p r i m a r y c o n s t i t u e n t s M,
N2, S,,
and K 2 , t o produce a new t e r m o f species 2,
was found t o have s i g n i f i c a n t a m p l i t u d e a t t h e t h r e e s h a l l o w w a t e r s t a t i o n s considered.
T h i s and o t h e r odd-order i n t e r a c t i o n s i d e n t i f i e d
a r e almost c e r t a i n l y symptoms o f s t r o n g f r i c t i o n , as d i s t i n c t f r o m even-order i n t e r a c t i o n s which a r i s e f r o m a d v e c t i v e terms i n t h e dynamics o f s h a l l o w w a t e r p r o p a g a t i o n . I d e n t i f i c a t i o n o f a c e r t a i n t y p e o f i n t e r a c t i v e t e r m because i t stands a l o n e i n frequency, d y a m i c a l l y i m p l i e s t h e e x i s t e n c e o f s i m i l a r and perhaps s t r o n g e r i n t e r a c t i o n s between a l l s i m i l a r terms. However, many o f these a r e hidden by t h e f a c t t h a t t h e i r combined f r e q u e n c i e s c o i n c i d e w i t h t h e frequencies o f p r i m a r y l i n e a r terms o r w i t h l o w e r - o r d e r i n t e r a c t i o n s which a r e a l r e a d y t a k e n i n t o account. For example, t h e presence o f 2SM2 must i m p l y t h e presence o f and 2MN,,
ZMS,
b u t because t h e l a t t e r c o i n c i d e i n t h e i r c e n t r a l frequenc es
w i t h p2 and L2 r e s p e c t i v e l y , t h e y a r e i g n o r e d as i n d i v i d u a l e f f e c t s I n p r a c t i c e , t h i s works up t o a p o i n t , b u t i n v o l v e s i n a c c u r a c y when (f,u)
f a c t o r s a r e assigned, because these a r e q u i t e d i f f e r e n t i n t h
two cases. Amin (1976) s e r i o u s l y t a c k l e d t h i s d i f f i c u l t y by h a r m o n i c a l l y a n a l y s i n g n e a r l y 19 y e a r s o f d a t a f r o m Southend.
T h i s process
r e s o l v e d t h e l i n e s t r u c t u r e o f t h e spectrum t o ' n o d a l s p l i t t i n g '
,
179 l e v e l , so t h a t t h e t r u e ( f , u ) m o d u l a t i o n of each c o n s t i t u e n t c o u l d be e s t i m a t e d d i r e c t l y w i t h o u t r e c o u r s e t o t h e known m o d u l a t i o n o f the p o t e n t i a l . Amin was t h e r e b y a b l e t o assess t h e r e s p e c t i v e c o n t r i b u t i o n s o f l i n e a r and i n t e r a c t i v e terms ( p r o v i d e d no more t h a n two i n f l u e n c e s
were p r e s e n t ) and a s s i g n a more a c c u r a t e harmonic f o r m u l a t i o n t h a n i s usually possible.
Amin a l s o i d e n t i f i e d t h e seasonal m o d u l a t i o n s
t o M2, f i r s t s t u d i e d by Corkan (1934) who named them Ma2 and MA2*, s i n c e r e c o g n i s e d as widespread i n t h e N o r t h Sea and p r o b a b l y elsewherl (e.g. Pugh & Vassie, 1976).
A 19-year a n a l y s i s o f t h e t i d e s a t t h e
l o n g - e s t a b l i s h e d s t a t i o n a t B r e s t , France i s p r e s e n t e d by Simon (1980 t h i s i n c l u d e s t h e c u r i o s i t y o f a d i s t i n c t harmonic t e r m generated by a f a u l t i n t h e tide-gauge mechanism, i d e n t i f i e d by DesnoFs (1977). I f t h e harmonic method i s t o be s t r e t c h e d t o i t s t h e o r e t i c a l l i m i t , one should do away w i t h t h e ( f , u )
f a c t o r s i n ( 5 ) , except i n
cases o f simple oceanic t i d e s , and r e p r e s e n t t h e f u l l s e t o f p u r e harmonic c o n s t a n t s down t o n o d a l - s p l i t t i n g , w i t h s i x parameters t o denote t h e frequency.
Amin (1976) r e c o r d s 326 terms up t o species 6 ?
w i t h o u t i n c l u d i n g t h e 20 terms ( w i t h o u t n o d a l - s p l i t t i n g ) b e l o n g i n g t o h i g h e r species l i s t e d by R o s s i t e r & Lennon (1968) f o r t h e same place.
No doubt, more terms would be r e q u i r e d f o r a p o r t w i t h
stronger d i u r n a l i n e q u a l i t y .
However, t h e mere number o f terms
presents no d e t e r r e n t t o a modern computer, and t h e y may be e a s i e r t o deal w i t h t h a n t h e r a t h e r awkward ( f , u )
factors.
Analysis o f
course would r e q u i r e 18-19 y e a r s o f d a t a , and i t would be a l l t h e more i m p o r t a n t t o s p e c i f y t h e t h r e s h o l d o f t h e n o i s e continuum.
*
Because of t h e f r e q u e n t need f o r computerised t y p e s e t t i n g , I have recommended t h e n o t a t i o n MB2 f o r t h e h i g h e r frequency seasonal m o d u l a t i o n .
180 RESPONSE METHODS
The ' response' method f t i d a l analysis was i troduced by Munk & Cartwright (1966) as a research tool rather than as a means of bettering the accuracy of predictions. A l t h o u g h comparisons of i t s prediction accuracy with t h a t of harmonic methods (e.g. Z e t l e r , C a r t w r i g h t & Berkman , 1979) has always shown the ' response' predictions t o be s l i g h t l y b e t t e r , the margin of improvement i s n o t enough t o j u s t i f y replacing familiar routine procedures by the rather bulky and unfamiliar programs involved i n the 'response' formalism. There i s no space here f o r a f u l l discussion of the method, b u t I should l i k e t o summarise points of improvement which have been made since the publication of the original paper. The aim i s t o escape from the enslavement t o a multitude of independent time-harmonic terms by expressing i n a few parameters the response functions of the measured t i d e t o the leading spherical harmonics of the tide-generating potential. ( I n the e a r l i e s t work, the response functions were referred t o the 'equilibrium t i d e ' a t the place of measurement, equivalent t o using phase lags in the 'Kappa' notation, b u t t h i s was soon found t o be an i r r e l e v a n t complication, so a l l response functions are now referred t o the same time-variable part of the potential a t the Greenwich meridian, equivalent t o phase lags i n the G-notation.) The gravitational potential V ( e , x , t ) due t o the Moon and the Sun on a sphere w i t h the Earth's equatorial radius can be computed a t time t in the form g-lV(e,A,t)
=
3
n
2
2
[a;(t)U:(
m=o is north c o l a t i t u d e ,
e,A)
+ b:(t)V:(
e,~)l
n=2
where
0
x
i s e a s t longitude, a n d
(6)
181
i s t h e s p h e r i c a l harmonic o f o r d e r m degree n i n a s t a n d a r d normalisation.
m i s identical with the t i d a l 'species'.
The f o u r
terms w i t h degree 3 a r e much s m a l l e r t h a n t h e t h r e e t e r m s o f degree 2 b u t t h e y p r o d u c e t i d a l e f f e c t s w h i c h a r e d e t e c t a b l e . Terms w i t h d e g r e e h i g h e r t h a n 3 can be e n t i r e l y n e g l e c t e d . The scheme i s t o e x p r e s s t h a t p a r t o f t h e t i d a l sea l e v e l w h i c h i s r e l a t e d t o t h e h a r m o n i c m,n by a r e l a t i o n o f t h e t y p e ( d r o p p i n g t h e s u f f i c e s m,n f o r c o n v e n i e n c e ) :
s c:(t)
= 2
[u,a(t-sAt)
+ vSb(t-s3t)]
(8)
s=-s where ws = u s + i v s i s a s e t o f a r b i t r a r y ' r e s p o n s e w e i g h t s ' f o r Munk & C a r t w r i g h t ( 1 9 6 6 ) j u s t i f y an i n v a r i a b l e c h o i c e
t h e system.
o f 2 days f o r t h e i n c r e m e n t a l t i m e l a g A t , and t h e use o f n e g a t i v e as w e l l as p o s i t i v e l a g s w h i c h a p p e a r s t o v i o l a t e p h y s i c a l l a w s . They a l s o s u g g e s t e d t h a t S=3 i s a s u i t a b l e maximum l i m i t t o t h e summation, b u t l a t e r i n v e s t i g a t i o n s ( C a r t w r i g h t ,
(1968) showed
t h a t S=3 t e n d s t o ' o v e r f i t ' t h e d a t a , and I have f o u n d i n g e n e r a l t h a t S=2 ( 5 complex r e s p o n s e w e i g h t s ) g i v e s as good a r e p r e s e n t a t i o n as n o i s e l e v e l s w i l l p e r m i t .
F o r t h e weaker h a r m o n i c s w i t h d e g r e e
n=3, s = l o r even 0 i s a d e q u a t e .
See a l s o Z e t l e r & Munk, ( 1 9 7 5 ) .
I n d i v i d u a l l y , t h e r e s p o n s e w e i g h t s ws mean v e r y l i t t l e , b u t t a k e n as a c o m p l e t e g r o u p t h e y d e f i n e t h e a d m i t t a n c e o f t h e r e s p o n s e system a t f r e q u e n c y f,
Z(f)
=
s c
ws exp ( - 2 n i f n t )
(9)
s=-s which i s t h e most p h y s i c a l l y m e a n i n g f u l q u a n t i t y w h i c h emerges f r o m the a n a l y s i s .
I n c o n j u n c t i o n w i t h t h e known t i m e - h a r m o n i c d e v e l o p m e n t
o f the p o t e n t i a l c o e f f i c i e n t s a t ( t ) , (Cartwright & Tayler,
1971;
C a r t w r i g h t & Edden, 1 9 7 3 ) , Z ( f ) may be u s e d t o d e r i v e any h a r m o n i c amplitude o f
:,
and i t s phase Lag,
182
G = nm - A r g [Z(o/2n)l, (10) e q u i v a l e n t l y t o and more a c c u r a t e l y t h a n t h e d i r e c t ' h a r m o n i c ' method. The e s s e n t i a l d i f f e r e n c e between t h e t w o methods i s i n f a c t t h a t t h e r e s p o n s e method assumes t h e e x i s t e n c e o f a r e a s o n a b l y smooth admittance f u n c t i o n Z ( f )
, whereas t h e h a r m o n i c method makes no use
o f t h e p h y s i c a l r e a l i t y o f t h e system.
Munk & C a r t w r i g h t ' s " c r e d o o f
smoothness'' ( o f t h e a d m i t t a n c e ) has been w e l l v i n d i c a t e d i n numerous t e s t i n g examples. Groves & Reynolds ( 1 9 7 6 ) p o i n t e d o u t an i n h e r e n t c l u m s i n e s s o f t h e r e s p o n s e r e p r e s e n t a t i o n ( 8 ) -in t h a t , w h i l e a ( t ) and b ( t ) a r e m u t u a l l y o r t h o g o n a l f u n c t i o n s , and so i n t h e l o n g t e r m a r e t h e c o r r e s p o n d i n g f u n c t i o n s w i t h d i f f e r i n g -(m,n) with d i f f e r e n t lags
SAt
, t h e e l e m e n t s o f (8)
are rather strongly correlated, r e s u l t i n g
i n l a c k o f convergence and i n s t a b i l i t y i n t h e w e i g h t s w s .
I n order
t o remedy t h i s , t h e y r e p l a c e d t h e f u n c t i o n s a,b i n (8') b y l i n e a r c o m b i n a t i o n s o f a,b w h i c h t h e y computed t o f o r m a c o m p l e t e l y orthogonal s e t o f f u n c t i o n s which t h e y c a l l e d ' o r t h o t i d e s ' .
Use o f
t h e s e o r t h o t i d e s does i n d e e d r e s t o r e c o n v e r g e n c e and s t a b i l i t y t o t h e i r r e s p o n s e w e i g h t s , b u t s i n c e t h e s e a r e l i n e a r l y r e l a t e d t o ws and t h e o r t h o g o n a l p r o p e r t i e s o f t h e o r t h o t i d e s t h e m s e l v e s r e q u i r e 18.6 y e a r averages i n t h e i r c r o s s - p r o d u c t s ,
t h e i r use does n o t add
any c o m p u t a t i o n a l a d v a n t a g e s f o r t h e f o r m a l i s m 8, ( A l c o c k & C a r t w r i g h t , 1978). C o m p l i c a t i o n s a r e r e q u i r e d t o d e a l w i t h t h e s l i g h t l y anomalous b e h a v i o u r o f t h e s o l a r t i d e s and w i t h n o n l i n e a r t e r m s .
The f o r m e r
i s allowed f o r b y t h e a d d i t i o n o f a ' r a d i a t i o n a l ' p o t e n t i a l a n a l o g o u s t o ( 6 ) , d e r i v e d f r o m t h e S u n ' s p o s i t i o n and r e q u i r i n g a d d i t i o n a l response weights, unlagged i n t i m e .
The t i m e - h a r m o n i c s
o f t h e r a d i a t i o n a l p o t e n t i a l are l i s t e d i n t h e Table 6 o f C a r t w r i g h t & T a y l e r (1971).
t o year.
They a r e e f f e c t i v e ,
but r a t h e r variable from year
D e t a i l e d a n a l y s i s has s u g g e s t e d t h a t t h e a n o m a l i e s i n
s p e c i e s 2 a r e more c l o s e l y r e l a t e d t o t h e a t m o s p h e r i c t i d e t h a n t o the radiational p o t e n t i a l , but the subtle differences are close t o
183 t h e t i d a l n o i s e l e v e l ( C a r t w r i g h t & Edden 1977). Treatment o f n o n l i n e a r e f f e c t s as
' response' processes has proved
t o be t h e l e a s t s a t i s f a c t o r y aspect o f t h e scheme, a l t h o u g h adequate as an a p p r o x i m a t i o n . further 'potential p o t e n t i a l s ,a: c:(t)
The p r e s e n t procedure i s t o add
' f u n c t i o n s derived from products o f the primary I f t h e s e a r e expressed as complex v a r i a b l e s ,
b:.
+ ib:(t),
= a:(t)
(11)
m m' then cn cn i s a s u i t a b l e r e f e r e n c e f u n c t i o n c o n t a i n i n g terms o n l y w i t h t h e sum o f t h e frequencies o f t h e p r i m a r y terms o f species m,m';
t h a t i s , i t d e f i n e s a new f u n c t i o n o f species (mtm')
of n o n l i n e a r o r i g i n . o f species (m-m').
S i m i l a r l y , cm c m l * d e f i n e s a new f u n c t i o n n n I n p r a c t i c e , however, i t i s reasonable t o
suppose t h a t n o n l i n e a r terms, b e i n g l o c a l l y generated, a r e more c l o s e l y r e l a t e d t o t h e p r o d u c t s o f t h e l o c a l t i d e i t s e l f than t o p r o d u c t s of t h e p o t e n t i a l .
A c c o r d i n g l y , we compute l i n e a r terms
s i m i l a r t o ( 8 ) , a p p r o x i m a t i n g t o t h e observed t i d e s o f species 1 and 2, and form p r o d u c t s analogous t o c:
c t ' f r o m them.
Response
weights a r e t h e n assigned t o these p r o d u c t s (which i n c l u d e t r i p l e and h i g h e r o r d e r i n t e r a c t i o n s ) a l o n g w i t h t h e l i n e a r terms f o r t h e g r a v i t a t i o n a l and r a d i a t i o n a l p o t e n t i a l s i n a l e a s t - s q u a r e s e v a l uat i o n process. Where t h e n o n l i n e a r terms a r e weak t h e r e a r e no problems and t h e d e s c r i b e d n o n l i n e a r f o r m a l i s m s i g n i f i c a n t l y improves t h e p r e d i c t a b l e variance.
However, I have experienced some cases where even t h e
r e l a t i v e l y s i m p l e (2+2) i n t e r a c t i o n does n o t s a t i s f a c t o r i l y account f o r a l l t h e observed species 4 v a r i a n c e , even w i t h t h e a d d i t i o n o f some time-lagged terms.
I n one case a t l e a s t i t appears t h a t t h e
assumption t h a t t h e n o n l i n e a r i t y i s generated l o c a l l y i s o n l y p a r t i a l l y Val i d .
Again, t h e i m p o r t a n t t r i p l e i n t e r a c t i o n ( 2 + 2 - 2 ) ,
which embodies t h e m a j o r f r i c t i o n e f f e c t , seems t o r e q u i r e more subtlety o f definition.
Probably, a c l o s e r m o d e l l i n g o f t h e
r e l e v a n t dynamical process i s needed.
184 I n a c u r r e n t v e r s i o n o f t h e ' r e s p o n s e ' a n a l y s i s package, p r o v i s i o n i s made f o r up t o 71 complex r e s p o n s e w e i g h t s , i n c l u d i n g 1 0 r a d i a t i o n a l t e r m s , some annual m o d u l a t i o n s and n o n l i n e a r f o r m s up t o o r d e r 5.
I t i s h i g h l y i m p r o b a b l e t h a t a c a s e c o u l d a r i s e where
a l l t h e s e t e r m s a r e needed s i m u l t a n e o u s l y .
Many v a r i a b l e s a r e
m u t u a l l y c o r r e l a t e d , r e s u l t i n g i n an u n s t a b l e n o r m a l m a t r i x and 1a r g e , u n r e a l is t ic r e s p o n s e w e i gh t s .
Cons id e r a b l e p r e - s e l e c t ion
i s r e q u i r e d , and t h i s can o n l y be done as a r e s u l t o f e x p e r i e n c e and p r e l i m i n a r y s p e c t r a l a n a l y s i s .
I p r e f e r t o r e g a r d a response
a n a l y s i s as a f i n a l means o f o p t i m i s i n g t h e d e f i n i t i o n o f a s e t o f a d m i t t a n c e o f t e r m s whose p r e s e n c e and a p p r o x i m a t e m a g n i t u d e a r e a l r e a d y known b y o t h e r t e c h n i q u e s o r b y knowledge o f t h e c h a r a c t e r i s t i c s o f t h e l o c a l sea a r e a . COMPUTING THE TIDAL POTENTIAL F i n a l l y , I s h o u l d l i k e t o comment on methods o f c o m p u t i n g t h e t i d e - g e n e r a t i n g p o t e n t i a l , on w h i c h much o f t h e modern r e s e a r c h i s based.
A c c u r a c y depends on t h e c o m p u t a t i o n o f t h e l u n a r and s o l a r
p o s i t i o n s , and t h e s t a n d a r d s used b y g e o p h y s i c i s t s and a s t r o n o m e r s v a r y enormously.
The l u n a r f o r m u l a e used i n Munk & C a r t w r i g h t
(1966) do n o t compare w e l l w i t h a s t r o n o m i c a l ephemerides a1 t h o u g h they e v i d e n t l y g i v e passable r e s u l t s a t t h e l e v e l o f accuracy r e q u i r e d by t i d a l a n a l y s i s .
My own programs use a s e l e c t i o n o f
some 280 o f t h e ' B r o w n ' t e r m s and o t h e r r e f i n e m e n t s used i n modern ephemerides t o m a i n t a i n a c o n s i s t e n t a c c u r a c y o f 2" i n l a t i t u d e and l o n g i t u d e and
i n p a r a l l a x ( C a r t w r i g h t & Tayler, 1971).
This
i s c e r t a i n l y e x c e s s i v e , b u t was done w i t h a v i e w t o c h e c k i n g t h e c o m p u t a t i o n s a g a i n s t t h e p u b l i s h e d ephemerides and r e m o v i n g a l l p o s s i b l e doubt about e r r o r s f r o m t h i s source. Some compromise between o r b i t a l p r e c i s i o n and b u l k o f c o m p u t a t i o n i s d e s i r a b l e f o r t h e p u r p o s e s o f good q u a l i t y t i d a l r e s e a r c h , and I o u t l i n e below a complete s e t o f formulae which achieves t h e i d e a l balance.
185 APPEND1 X - POTENTIAL FORMULAE L e t t be (Ephemeris) t i m e i n days counted f r o m 1900 January 0.5 ( i . e . December 31 noon) and T=t/36525. The mean l o n g i t u d e s o f t h e f o l l o w i n g q u a n t i t i e s a r e g i v e n i n 0
t h e f o r m L = Lo
Name
f
L1 T
+ L2TL i n r e v o l u t i o n s
Symbol
Moon M. Perigee Node Sun S. Perigee
n= 1 where B1 = s-p,
S
P n h P'
B = h-p', 2
L1
L2
1336.855231 11.302872 -5.37261 7 100.0021 36 0.004775
-0.000003 -0.000029 0.000006 0.000001 0.000001
LO
0.751206 0.928693 0.71 9954 0.776935 0.781169
B
=
s-n,
3
are as i n t h e f o l l o w i n g t a b l e , where
B 4
=
s-h, and Ai
f
=
3422".540
and kn( i 1
186
AMPLITUDES A i Longitude
Parallax ( E )
i
Latitude ( 6 )
rad
x
x l o m 5 rad
0 -61 1149 -200 -324 - 80 93 10976 -2224 -53 -100 72 373 -1 03
100000 -29 82 4 0 -12 56 90 5450 1003 -28 42 34 297 -9
0 0 0 0 0 0 0 1 0 0 0 2 0 0 2 0 0 1 0 0 0 1 0-2 1 0 0 2 1 0 0 0 1 0 0-2 1 1 0 0 1 1 0-2 1-1 0 0 2 0 0 0 2 0 0-2 0 0 1 0 0 0 1 2 0 0-1 2 1 0 1 0 1 0-1 0 1 0 1-2 1 0-1-2
(p)
8950 51 306 491 48 1 -78 -99
For both Sun and Moon, Right Ascension R and D e c l i n a t i o n D a r e given by the formulae sin D
cos R cos 0 sin R cos D
= = =
cos 6
cos
S
sin cos sin
p
sin
E
+ sin
S
cos
E
p,
-
sin 6 sin E , and f i n a l l y , the r e q u i r e d t i m e - v a r i a b l e s i n ( 6 , 11) a r e given by cos 6
p
cos
E
t h e sum o f t h e s o l a r and l u n a r c o n t r i b u t i o n s t o a2
Ci 2
c2
etc.
( ~ / 5 ) ’ h 2 ( ~ / f ( )3 ~s i n 2 D - 1 ) , = -(Sn/5)’ h 2 (c/-
q= 1 G(m) c o s (@(m)
+
-}q T t m
(3)
xt)]
where w ( t ) i s t h e i m p u l s e r e s p o n s e a t t i m e t ,
G( f ) i s t h e g a i n a t f r e q u e n c y f , Q ( f ) i s t h e phase d i f f e r e n c e a t f r e q u e n c y f , 1
and
i s t h e bandwidth of t h e d i s c r e t e g a i n and phase estimates.
2iii
S t a r t i n g w i t h t h e d a t a i n Table 1 and u t i l i z i n g e q u a t i o n ( 3 ) , t h e c a l c u l a t e d v a l u e s f o r t h e i m p u l s e r e s p o n s e f u n c t i o n e x t e n d i n g away from t h e c e n t r a l v a l u e a t 6-hour
i n t e r v a l s are shown i n Table 2.
The f i l t e r e d and d e c i m a t e d p o r t i o n of t h e r e c o r d a t S a i n t John w a s t h e n f i l t e r e d w i t h t h e i m p u l s e r e s p o n s e w e i g h t s shown i n T a b l e 2, y i e l d i n g a s i m u l a t e d r e c o r d a t Yarmouth. The s i m u l a t e d s i g n a l a t Yarmouth w a s t h e n compared w i t h t h e ( s u i t a b l y l a g g e d ) s i g n a l a t Yarmouth f o r t h e p e r i o d of t h e t e s t d a t a (1112 y e a r s ) .
This
comparison
showed
that
the
maximum d i f f e r e n c e
between t h e r e a l s i g n a l and s i m u l a t e d s i g n a l w a s 5 cm which m e t o u r acceptability criterion. The h o u r l y d a t a a t S a i n t John w a s p e r i o d 1960-1976 and was
t h e n made c o n t i n u o u s f o r
the
by f i l l i n g i n t h e s m a l l gaps w i t h p r e d i c t e d t i d e s
t h e n low-passed,
decimated
and
f i l t e r e d with
the
impulse
212 Table 2.
Values or weights for the impulse response function, Saint John to Yarmouth. Time
Weight
Time
Weight
Time
Weight
-16 -15 -14 -13 -12 -1 1 -10
-0.033 -0.022 -0.013 -0.030 -0.027 -0.030 -0.005 0.023 0.002 -0.060 -0.002
-
-0.020 -0.020 0.040 -0.025 0.239 0.641 0.155 0.006 -0.017 -0.030 -0.003
6 7
-0.013 -0.032 -0.025 -0.013 0.005 -0.043
- 9
- 8
- 7 - 6 response weights.
5 4 3 2
- 1 0 1 2 3
4 5
8 9 10 11 12 13 14 15 16
-0.000
-0.013 -0.012 0.043 -0.033
The Yarmouth hourly data was low-passed, filtered
and lagged to provide alignment with the simulated Yarmouth data. The two signals were then spliced together, yielding a continuous Yarmouth signal from 1960 to 1978. DISCUSSION
Noting that the maximum difference between the actual and simulated signals at Yarmouth was only 5 cm over a period of a year and a half, it would appear that this method of simulating one signal from a related one is sound.
The resulting Yarmouth signal, mostly
observed but partly simulated was then, after further filtering and decimation to
60-hour
data, used for exploring
the relationship
between herring year-class strength and the oceanographic environment.
Although far from complete, this exploration done by others
indicates very high correlations in the order of 0.9
between the
herring year-class strength and the sea level at Yarmouth. REFERENCES
Holloway, J.L.,
Jr.
and space fields. Metuzals, K.,
1959.
Smoothing and filtering of time series
Advances in Geophysics, pp. 351-389.
Sinclair, M. and Sutcliffe, W.
analysis of recruitment variability
1978.
A preliminary
in 4WK herring.
paper, Bedford Institute of Oceanography, Dartmouth, N . S . ,
Working Canada.
213
THE COMPUTATION OF TIDES FROM IRREGULARLY SAMPLED SEA SURFACE HEIGHT DATA
LUNG-FA KU, CANADIAN HYDROGRAPHIC SERVICE,
I
OTTAWA
INTRODUCTION S e v e r a l i n v e s t i g a t o r s have a t t e m p t e d t o o b t a i n t h e
g e o m e t r i c a l d i s t r i b u t i o n o f t i d e s i n a r e q i o n u s i n q t h e sea s u r f a c e h e i g h t d a t a o b t a i n e d f r o m GEOS-3 w i t h o u t any success
(Won and M i l l e r , 1979; Brown and H u t c h i n s o n , 1980; Maul and Yanaway,
1977; Parke, 1980; and Ku, 1 9 8 2 ) .
A n o t h e r approach
i s t o d i v i d e t h e ocean i n t o s m a l l areas where t h e s p a t i a l d i s t r i b u t i o n o f t i d e s can be n e q l e c t e d .
The p r o b l e m i s t h e n
reduced t o m e r e l y t h a t o f a t i m e s e r i e s a n a l y s i s .
T h i s paper
d i s c u s s e s p r o b l e m s encounteed i n t h e a n a l y s i s due t o t h e i r r e g u l a r i t y o f t h e s a m p l i n g i n t e r v a l and t h e b i a s caused b y t h e q e o i d a l h e i g h t r e m a i n e d i n t h e sea s u r f a c e h e i q h t . I1
HARMONIC ANALYSIS OF TIDES Assuminq Ak i s t h e complex a m p l i t u d e o f t h e k t h
t i d a l c o n s t i t u e n t w i t h t h e frequency r k s u r f a c e h e i g h t sampled a t t n
,
,
hn i s t h e n t h sea
then t h e l e a s t squares f i t
s o l u t i o n o f Ak can be e x p r e s s e d i n m a t r i x n o t a t o n as
The element o f [ R ]
and [ Y ]
are
Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
214
r n k = exp ( i C k t n )
(4)
w h i c h i s b a s i c a l l y N t i m e s t h e F o u r i e r t r a n s f o r m o f hn a t t h e angular frequency C k . S i n c e t h e sampled sea s u r f a c e h e i g h t hn can be e x p r e s s e d as t h e p r o d u c t o f t h e h e i g h t h ( t ) and a d a t a s a m p l e r
s,
Yk can t h e r e f o r e be r e p r e s e n t e d b y t h e c o n v o l u t i o n between
t h e F o u r i e r t r a n s f o r m s o f t h e sea s u r f a c e h e i g h t
H
(G) and t h e
d a t a s a m p l e r S (U). The d a t a sampler i s u s u a l l y chosen t o reduce t h e a l i a s i n q i n yk.
I11
DATA SAMPLER The s a m p l i n q t i m e i n t e r v a l o f t h e s e a s u r f a c e
h e i g h t d a t a o b t a i n e d f r o m GEOS-3 i s p r i m a r i l y d e t e r m i n e d b y the period o f the s a t e l l i t e . does n o t f o r m a c l o s e d c u r v e . offset
The s a t e l l i t e ' s t r a c k on e a r t h It i s u s u a l l y described b y t h e F o r GEOS-3,
of the equatorial crossinq o f the track.
t h e e q u a t o r i a l c r o s s i n q moves westward 25.32" f o r each revolution.
Dependinq on t h e l o n q i t u d i n a l w i d t h o f t h e s t u d y
area, t h e d a t a c o l l e c t e d may have come i n a b u r s t o f s e v e r a l passes,followed
b y a gap o f s e v e r a l passes.
To i n v e s t i g a t e
t h e e f f e c t o f b u r s t i n g i n t h e d a t a sampler on t h e h a r m o n i c a n a l y s i s o f t i d e s , l e t us assume t h a t t h e sampler t a k e s t h e f o r m shown i n F i g u r e 1.
The p e r i o d o f t h e s a t e l l i t e i s T, t h e
d u r a t i o n o f t h e b u r s t i s KT, and t h e b u r s t r e p e a t s a t an i n t e r v a l o f LT.
The t o t a l number o f b u r s t s i s
N.
215
m el , -TT\
(N
I
,
I
Figure 1.
)LT
ASCENDING NODE DESCENDING NODE
Simplified data sampler (T 1s the period of the satellltel.
The F o u r i e r t r a n s f o r m o f t h i s sampler a t a frequency Q
is
K
S(V) = ( 5
k=l
N e x p ( i C k t ) ) ( Z exp ( - i C L T n ) ) n =1
(6)
The t e r m i n t h e f i r s t b r a c k e t r e p r e s e n t s t h e F o u r i e r t r a n s f o r m o f t h e sampler w i t h i n a b u r s t , and t h e second one represents t h a t o f t h e sampler b u r s t .
Equation ( 6 ) can a l s o
be expressed as
S(C) =
functions
S i n (G'KT/2) Sin ( G T / ~ )
Sin ( NLT/2) exp ( - i c T ( K t l - ( N + l ) L ) / 2 ) S i n ( aL T / 2 )
Therefore,
S ( G ) i s t h e product o f t w o d i f f r a c t i o n
t h e f i r s t one i s due t o t h e sampler w i t h i n a
b u r s t and t h e second one i s due t o t h e r e p e t i t i o n o f t h e burst.
S nce t h e second d i f f r a c t i o n f u n c t i o n f l u c t u a t e s
f a s t e r t h a n t h e f i r s t , t h e f i r s t f u n c t i o n has t h e e f f e c t o f m o d u l a t i n q t h e a m p l i t u d e o f t h e second f u n c t i o n . o f S ( 0) w i l l r e p e a t a t 1/LT, l / k t , i t s f i r s t zero c r o s s i n q a t 1/NT.
The f e a t u r e
and 1/T i n t e r v a l s , w i t h
21 6 Near t h e e q u a t o r ,
an a r e a w i t h a l o n g i t u d i n a l w i d t h
l e s s t h a n 25", t h e b u r s t w i l l d i s a p p e a r , b e i n g r e p l a c e d b y one pass about e v e r y 7T.
I n r e a l i t y , d a t a m i g h t n o t be
c o l l e c t e d a t e v e r y pass o v e r t h e area, o r some d a t a must be d i s c a r d e d because o f i t s p o o r qua1 it y .
Consequent1 y,
a l t i m e t e r d a t a would possess many gaps. Table 1 l i s t s t h e samplinq t i m e o f t h e d a t a s u p p l i e d b y NASA i n N.E.
P a c i f i c Ocean.
Some d a t a f r o m o t h e r
passes a r e n o t i n c l u d e d due t o t h e p o s s i b l e l a r g e e r r o r indicated i n the analysis of the heiqht difference at c r o s s i n q p o i n t s between t w o s a t e l l i t e t r a c k s .
I t shows t h a t
most o f them a r e t h e m u l t i p l e o f 14T, where T = 101.8 m i n u t e s i s t h e p e r i o d o f t h e GEOS-3 s a t e l l i t e .
C o n s e q u e n t l y , we
d e f i n e 14T as t h e median o f t h e s a m p l i n g i n t e r v a l .
& 02
F i g u r e 2.
04
06
08
.
.
10 cyCIe/hr
Spectrum o f t h e d a t a s a m p l e r .
The s p e c t r u m o f t h i s d a t a sampler i s shown i n F i g u r e 2.
I t has a s i m i l a r f e a t u r e as t h e s p e c t r u m o f a
f i x e d i n t e r v a l d a t a sampler.
The m a j o r l o b e r e p e a t s a t an
i n t e r v a l o f .0414 c y c l e / h o u r which i s e q u a l t o 1/14T. T h e r e f o r e , t h e median o f t h e s a m p l i n q i n t e r v a l i s e q u i v a l e n t
t o t h e sampling i n t e r v a l f o r a f i x e d i n t e r v a l d a t a sampler. The m a j o r l o b e and s e v e r a l n e i g h b o r i n g m i n o r l o b e s , however, does n o t d e c r e a s e t o zero, t h e y r e a c h a minimum i n s t e a d .
217 Therefore, t h e r e s o l u t i o n o f t h e spectrum o f t h e i r r e q u l a r l y sampled d a t a w i l l be p o o r e r t h a n t h a t sampled r e g u l a r l y .
The
f i r s t minimum w i t h a m a q n i t u d e o f -08 occurs a t about
1.2 x 10-4 c y c l e l h r which i s about t w i c e 1/Td, where Td i s A l l m i n o r l o b e s which a r e
the duration o f the observation.
n o t a d j a c e n t t o t h e m a j o r l o b e s m a i n t a i n a m a q n i t u d e o f about
.05. IV
THE B I A S I N THE ANALYSIS By i q n o r i n q t h e e f f e c t o f t h e m i n o r l o b e s i n S ( a ) ,
e q u a t i o n ( 5 ) can be a p p r o x i m a t e d as
Where
r si s
t h e anqular samplinq frequency.
This i s a well
known p r o b l e m w h i c h i s c a l l e d a l i a s i n q . For most d a t a , t h i s p r o b l e m can be overcome b y low-pass f i l t e r i n q t h e d a t a p r o v i d e d t h a t Gk < .5 Cs, A s i m i l a r remedy i s n o t a v a i l a b l e f o r d a t a sampled a t i r r e g u l a r i n t e r v a l s shown i n T a b l e 1.
Therefore,
the covariance vector
c o u l d he b i a s e d by t h e s p e c t r a l e n e r q y o f t h e r e a l sea s u r f a c e heiqht at other frequencies.
S i n c e t h e sea s u r f a c e h e i g h t i s
t h e d i f f e r e n c e between t h e a l t i m e t e r h e i g h t and t h e s a t e l l i t e h e i q h t , t h e s p e c t r u m o f t h e sea s u r f a c e h e i g h t c o n t a i n s t h e spectrum o f t h e s a t e l l i t e h e i g h t which has n o t been a c c o u n t e d f o r i n t h e o r b i t a l computation.
This effect,
however, i s
n e q l i g i b l e ( K u , 1982). F i q u r e 3 p l o t s t h e t r a c k o f t h e s a t e l l i t e passes used i n t h i s study.
I t c o v e r s an a r e a w i t h a l a r q e change i n t h e
q e o i d as shown i n t h e same f i g u r e .
To r e d u c e t h i s e f f e c t i n
I
1
F i g u r e 3. S a t e l l i t e t r a c k s and t h e
4
I
1
g r a v i m e t r i c qeoi d '01
(meter). I
i
I
i
!
J
.
l D c - - T - - T - ~ II0
1
'40
.
. .
.
r---7---0 Long
t h e computation, t h e q r a v i m e t r i c qeoidal h e i q h t s supplied b y
NASA a r e s u b t r a c t e d f r o m t h e sea s u r f a c e h e i q h t s .
The averaqe
sea s u r f a c e h e i q h t i s t h e n computed f o r each p a s s .
After
r e m o v i n q t h e mean o f a l l t h e averaqe sea s u r f a c e h e i q h t s f r o m t h e s e h e i q h t s , t h e i r s p e c t r u m i s computed and p l o t t e d i n F i q u r e 4.
The s i x most i m p o r t a n t t i d a l components i n t h i s
a r e a a r e 01, P 1 and K 1 i n t h e d i u r n a l t i d a l f r e q u e n c y band, and N2, M2 and S 2 i n t h e s e m i - d i u r n a l
t i d a l f r e q u e n c y band.
T h e i r f r e q u e n c i e s have been i n d i c a t e d i n t h e same f i q u r e . s p e c t r u m shows a l a r g e v a r i a t i o n a t a l l f r e q u e n c i e s ,
The
and t h e r e
i s no s i g n i f i c a n t l y l a r q e peak a t any o f t h e s i x t i d a l frequencies.
The s p e c t r u m i s c l e a r l y a l i a s e d as shown b y t h e
numberinq o f some o f t h e peaks between 0. and -0414 c y c l e / h r . c m2/c ph
F i g u r e 4.
Spectrum
o f t h e a v e r a g e sea .UP0
.a40
.060
.o*o
.#OD r"C,./n,
surface heights.
219 Since t h e qravimetric qeoidal h e i q h t provided b y NASA m i q h t n o t be i d e n t i c a l w i t h t h e a c t u a l q e o i d i n t h i s
area, i t i s p o s s i b l e t h a t some o f t h e v a r i a n c e s i n t h e sea surface heiqht data o r i q i n a t e d from t h e residues i n t h e aeoidal heiqhts.
Ku ( 1 9 8 2 ) i n d i c a t e s t h a t t h e r e s i d u e s c o u l d
be s t r o n q e r a l o n q t h e n o r t h w e s t d i r e c t i o n .
To s t u d y i t s
e f f e c t i n t h e s p e c t r u m o f t h e sea s u r f a c e h e i g h t , we assume t h a t t h e r e m a i n d e r o f t h e q e o i d a l h e i q h t i n t h e sea s u r f a c e
Table I Sarnplinq Time I n t e r v a l s n
kn
0 300 1 357 2 599 3 855 4 1267 5 1438 6 *1527 7 1537 8 1949 9 2220 10 2234 11 2318 12 2319 13 2347 14 2361 15 2390 16 2404 17 2419 18 2447 19 2461 20 2503 21 2504 22 2560 23 2617 24 2631
*
kn
kn/14
57 242 256 412 171 89 10 412 271 14 84 1 28 14 29 14 15 28 14 42 1 56 57 14
4.1 17.3 18.3 29.4 12.2 6.4 .7 29.4 19.4 1.0 6.0 .1 2.0 1.0 2.1 1.0 1.1 2.0 1.0 3.0
.1 4.0 4.1 1.0
i n d i c a t e s a s c e n d i n q pass
kn
n
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
kn
2844 2859 2972 3043 3072 3087 3115 3229 3342 *3745 *4669 492 1 5049 5248 5319 *5323 5333 5361 5390 5405 5916 5986 * 7882 *7910
kn = kn
-
213 15 113 71 29 15 28 114 113 403 924 252 128 199 71 4 10 28 29 14 512 70 1896 28
kn-1
kn/14 15.2
1.1 8.1 5.1 2.1 1.1 2 .o 8.1 8.1 24.8 66.0 18.0 9.1 14.2 5.1 .3 .7 2.0 2.1 1.0 36.6 5 .O 135.4 14.0
220
heiqht data i s a l i n e a r function o f the l a t i t u d e
Q and t h e
l o n q i t u d e A as hq = 9
-
50 - ( A -220)
(PI
A s y n t h e t i c time s e r i e s i s then qenerated by t a k i n q t h e sample o f hq a t t h e t i m e o f t h e s a t e l l i t e passes shown i n T a b l e 1 and a t t h e c o r r e s p o n d i n q c e n t r e s o f t h e t r a c k shown i n F i q u r e 3. F i q u r e 5.
The s p e c t r u m o f t h i s t i m e s e r i e s i s shown i n
Some o f t h e peaks i n t h e f i q u r e have been numbered
t o indicate the effect of aliasinq.
.02
.06
.04
.10
.08
c y cIe/ hour
F i g u r e 5.
Spectrum o f t h e s i m l a t e d
geoldal neignt resiaues.
V
THE HARMONIC ANALYSIS OF THE SEA SURFACE HEIGHT The s i q n i f i c a n t t i d a l p e r i o d i c components i n t h i s
a r e a a r e 01, P 1 , K 1 , N2, M2 and S2.
Accordinq t o t h e q l o b a l
ocean t i d a l c o m p u t a t i o n c a r r i e d o u t b y S c h w i d e r s k i (1979), and t h e t i d a l measurement r e p o r t e d b y Rapatz and H u q q e t t ( 1 9 7 2 ) , t h e e x p e c t e d v a l u e s o f t h e a m p l i t u d e s and phases of t h e s e components a r e q i v e n T a b l e 2.
C o n v e n t i o n a l t i d e qauqes
measure t h e t i d e as a chanqe i n t h e sea s u r f a c e h e i q h t w i t h r e s p e c t t o t h e sea b o t t o m , w h i l e t h e t i d e d e r i v e d f r o m t h e a l t i m e t e r h e i g h t and t h e s a t e l l i t e h e i g h t i s t h e change o f t h e
221
Table 2 Mean t i d e s computed f r o m t h e harmonic a n a l y s i s o f sea s u r f a c e h e i q h t s (crn and d e q r e e )
3 0
A,
=
-
-
e
H
A
A
G
G
R
65( 43)
- 146
71
25
118
240
86
l o % (5 7 )
-106
108
15
244
250
93
92( 5 4 )
-49
97
40
3 29
260
91
50( 46)
-39
49
15
255
250
34
55(45)
10
54
80
296
265
44
35 ( 5 0 )
-46
35
25
314
300
12
J, = 134
154
N
=
49
a m p l i t u d e and Greenwich phase l a q o b t a i n e d f r o m
=
t h e harmonic a n a l y s i s ( )
=
standard e r r o r
A,G
=
a m p l i t u d e and phase a f t e r c o r r e c t i n q a s t r o n o r n i c a modulation
bar
4
=
,~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
1 2 3
24
( 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
~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
If i t \ccrc possi1)1c t o d c -
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 .
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 ) . Ilowcvcr, t h i s procedure i s d i f f i c u l t ,
iiiodcl for. t h e time s e r i e s .
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 . Lihich t h e s e s e r i e s Were ' r e c o r d e d ' .
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
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.. I'crsistcnce is oftcii charactcriscd I)\- thc c*or1*i~~ogi-;i111 \ \ h i ~ * his :I 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 Data Values C o n s i d e r e d Case Xumber 1
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 -0.2
g G
0 -0.2
Ir
0.6 r'
g
0.4
c
5
3.2
w
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 , \ 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 : t
BH(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.
244
A DYNAMIC-STOCHASTIC APPROACH FOR MODELLING ADVECTION-DISPERSION PROCESSES IN OPEN CHANNELS W.P.
BUDGELL
Bayfield Laboratory f o r Marine Science and Surveys, Department of Fisheries and Oceans, Canada Centre f o r Inland Waters, Burlington, Ontario, Canada
ABSTRACT
A combined stochastic-deterministic model has been developed t o describe the temporal and spatial distribution of conservative substances in open channel flows. The model consists of a f i n i t e difference approximation t o the one-dimensional advection-dispersion equation embedded within a stochastic f i l t e r . The time and measurement updates of the estimated concentrations and t h e i r covariance are carried out t h r o u g h the use of a factored form of the covariance matrix. The resulting f i l t e r i n g algorithm i s more computationally stable t h a n the standard Kalman f i l t e r approach. The dynamicstochastic model i s shown t o perform well when i t i s applied t o simulated observations of s a l i n i t y i n a n Arctic estuary. I t i s shown t h a t t h i s type of modelling approach can be used as a t o o l in planning field experiments.
1. INTRODUCTION The time and space distribution of a conservative substance in rivers and estuaries can often be described using the one-dimensional t ime-dependent advect i on-di spersi on equation (Harl eman, 1971 ; Hann and Young, 1972; Hinwood and Wallace, 1975). If the cross-sectional area, velocity and dispersion coefficient are constants, an analytical sol _tion t o the equation can be obtained (Harleman, 1971). For r e a l i s t i c channel geometry and flow conditions, i t i s necessary t o make use of numerical techniques (Roache, 1 9 7 2 ) . Although numerical advectiondispersion models have often provided r e l i a b l e r e s u l t s , i t should be noted t h a t these models are crude approximations t o the actual
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
245
transport processes t a k i n g place in open channels (Fischer, 1973, 1976). Errors in the specified cross-sectional area, flow f i e l d , dispersion coefficients and boundary conditions and the numerical discretization of the original partial d i f f e r e n t i a l equation introduce uncertai nty or n o i se into the mdell i ng process. This model 1 i ng error , or system noise, i s propagated t h r o u g h time and space by the deterministic numerical model. Because of the dynamic nature of the problem, the variance of the errors in predicted concentrations w i l l increase exponentially with time for the case of constant coefficients i n the o r i g i n a l equation. I f time s e r i e s observations of concentration are available from the river or estuary under considerztion, the modelling error a t each time step can be estimated and the model r e s u l t s can be corrected. By upd a t i n g the computed concentrations using observations, l e s s e r r o r i s propagated t h r o u g h the model. A major d i f f i c u l t y associated with t h i s procedure i s t h a t the observations will also contain a certain degree of error o r measurement noise. Thus, the actual corrections t o be applied t o the computed concentrations wiil n o t be the difference between the observed and computed values, b u t rather some portion of t h a t difference. The magnitude of the correction will depend upon t h e r e l i a b i l i t y of the observations r e l a t i v e t o t h a t of the model. A means of computing the optimal corrections t o be applied t o the computed values a t each time step i s t h r o u g h the use of the Kalman f i l t e r (Kalman, 1960; Kalman and Bucy, 1961). Kalman f i l t e r theory has been a p p l ied t o sate1 1 i t e tracking (Jazwinski , 1970) , a i r pol 1 u t i o n monitori n g (Desalu, Gould and Schweppe, 1974; Bankoff and Hanzevak, 1975; Koda and Seinfeld, 1978; Fronza, S p i r i t 0 and Toniell i , 1 9 7 9 ) , water resources problems (Chiu, 1978) and the estimation of water levels a n d velocities in tidal estuaries (Budge11 , 1981). DeGuida, Connor and Pearce ( 1 9 7 7 ) have used a Kalman f i l t e r t o combine observations of concentration with a f i n i t e element numerical model of the time-dependent two-dimensional h o r i z o n t a l distribution of estuarine pollution. The model includes advection, dispersion and
246
s o u r c e - s i n k terms.
However, Koda and S e i n f e l d (1978) have n o t e d t h a t
such a s t r a i g h t f o r w a r d a p p l i c a t i o n o f t h e Kalman and Bucy (1961) f i l t e r i n g a l g o r i t h m t o l a r g e s c a l e d i s t r i b u t e d p a r a m e t e r systems (systems w i t h b o t h t i m e and space dependence) can l e a d t o f i l t e r d i v e r g e n c e and c o m p u t a t i o n a l i n s t a b i l i t y .
These c o m p u t a t i o n a l problems
are a t t r i b u t a b l e t o t h e covariance m a t r i x associated w i t h t h e estimated c o n c e n t r a t i o n s becomi n g n o n - p o s i t i ve d e f i n i t e . I n t h i s paper, s t a b i l i t y problems a r e a v o i d e d by i m p l e m e n t i n g a square r o o t f o r m o f t h e Kalman f i l t e r i n t h e e s t i m a t i o n o f t h e c r o s s - s e c t i o n a l l y averaged c o n c e n t r a t i o n o f a c o n s e r v a t i v e substance i n r i v e r s and e s t u a r i e s .
The f i l t e r i s c o n s t r u c t e d a r o u n d an i m p l i c i t
f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n o f t h e time-dependent one-dimensional advection-dispersion equation.
A square r o o t f o r m u l a t i o n f o r t h e
f i l t e r e n s u r e s t h a t t h e c o v a r i a n c e m a t r i x r e m a i n s p o s i t i v e d e f i n i t e and r e d u c e s t h e c o m p u t a t i o n a l b u r d e n f r o m t h a t imposed by t h e c o n v e n t i o n a l Kalman f i l t e r a l g o r i t h m .
2. THE DETERMINISTIC MODEL The n u m e r i c a l model combined w i t h t h e s t o c h a s t i c f i l t e r i s r e f e r r e d t o as a d y n a m i c - s t o c h a s t i c model.
The d e t e r m i n i s t i c component o f t h e
d y n a m i c - s t o c h a s t i c model i s governed by t h e o n e - d i m e n s i o n a l advection-di spersion equation d e s c r i b i n g t h e d i s t r i b u t i o n o f a c o n s e r v a t i v e c o n s t i t u e n t i n open c h a n n e l s (Harleman, 1971) :
Boundary c o n d i t i o n s must be s p e c i f i e d a t t h e upper and l o w e r ends of' t h e channel. f o l l ows :
The u p s t r e a m boundary c o n d i t i o n s a r e s p e c i f i e d as
247 c(0,t) a2c
-
I
ax2
= c o ( t ) f o r Q(O,t)> 0
(2)
f o r Q(O,t)< 0
= 0
x=o
and t h e downstream boundary c o n d i t i o n s a r e as f o l l o w s : c(L,t)
= cL(t) f o r Q(L,t)-< 0
--I a2c ax2
= o
(3)
f o r Q(L,t) > 0
x=L
where c ( 0 , t )
and c ( L , t )
a r e t h e c o n c e n t r a t i o n s a t t h e upstream and
downstream ends, r e s p e c t i v e l y , o f an open channel and c o ( t ) and c L ( t ) are s p e c i f i e d c o n c e n t r a t i o n s a t t h e upstream and downstream ends.
These boundary c o n d i t i o n s are d e s c r i b e d i n g r e a t e r d e t a i l by
Thatcher and H a r l eman ( 1972). Since A, Q and E can be t i m e - and space-dependent parameters, necessary t o s o l v e ( 1 ) t o ( 3 ) u s i n g numerical approximations.
it i s
In this
study t h e Stone and B r i a n (1963) s i x - p o i n t f i n i t e d i f f e r e n c e scheme has been used t o approximate t h e t i m e d e r i v a t i v e and a d v e c t i v e f l u x terms i n (1).
The d i s p e r s i o n t e r m i s modelled u s i n g t h e Crank-Nicholson
(1947) scheme.
The r e s u l t i n g f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n
possesses second o r d e r accuracy i n space and time,
produces no
numerical d i s p e r s i o n and i s s t a b l e f o r c e l l P e c l e t numbers l e s s t h a n 20 (Lam, 1977).
There i s no s t a b i l i t y r e s t r i c t i o n on t h e t i m e step.
When t h e f i n i t e d i f f e r e n c e e q u a t i o n s are a p p l i e d t o t h e N-2 i n t e r i o r g r i d p o i n t s o f t h e d i s c r e t i z e d open channel and t h e f i n i t e d i f f e r e n c e approximations o f boundary c o n d i t i o n e q u a t i o n s ( 2 ) and ( 3 ) a r e imposed, t h e r e s u l t i s a set o f
N l i n e a r equations i n N unknowns.
I n matrix
form t h i s may be expressed as: A(n,n+l)
c(n+l)
= B(n,n+l)
c(n) + G(n,n+l) u(n+l)
(4)
I f f l o w i s i n t o a boundary from t h e i n t e r i o r o f t h e computational
r e g i o n , t h e s p e c i f i e d c o n c e n t r a t i o n f o r t h a t boundary c o n d i t i o n i s
248
not. used i n t h e c o m p u t a t i o n o f c(n+l). G(n,n+l) t h e c o r r e s p o n d i n g column o f A(n,n+l) Since -
T h i s i s a c c o m p l i s h e d by s e t t i n g
t o zero.
i s a t r i - d i a g o n a l m a t r i x and t h e r i g h t - h a n d s i d e o f
( 4 ) c o n s t i t u t e s a known v e c t o r i f t h e i n i t i a l c o n d i t i o n s a r e s u p p l i e d ,
c ( n + l ) can be o b t a i n e d i n an e f f i c i e n t manner u s i n g t h e w e l l known Thomas a l g o r i t h m (Roache, 1972, p.349).
E q u a t i o n ( 4 ) can be e x p r e s s e d
i n an a l t e r n a t e manner as: c(n+l) = O(n,n+l)
c(n) + g(n,n+l)
Although the matrices O(n,n+l)
u(n+l)
(5)
and g ( n , n + l )
a r e n e i t h e r computed n o r
s t o r e d , t h e y s e r v e t o r e p r e s e n t t h e sequence o f l i n e a r o p e r a t i o n s c ( n + l ) g i v e n c ( n ) and u ( n + l ) . performed t o o b t a i n -
3. THE STOCHASTIC FILTER The p r o c e s s d e s c r i b e d by ( 5 ) i s p u r e l y d e t e r m i n i s t i c .
Given t h e
c o r r e c t values o f c ( n ) and u(n+l), c ( n + l ) w i l l be known w i t h certainty.
Unfortunately, the d i s t r i b u t i o n of a conservative solute i n
dynamic open channel f l o w s i s f a r f r o m p e r f e c t l y d e s c r i b e d by (1) t o (4).
Errors associated w i t h the cross-sectional
o r i g i n a l three-dimensional
integration o f the
mass t r a n s p o r t e q u a t i o n , t h e n u m e r i c a l
approximation o f a continuous p a r t i a l d i f f e r e n t i a l equation, t h e s p e c i f i c a t i o n o f t h e t i m e - and space-dependent p a r a m e t e r s Q and E , and t h e s p e c i f i c a t i o n o f boundary c o n d i t i o n s a l l r e s u l t i n c o n s i d e r a b l e u n c e r t a i n t y b e i n g a s s o c i a t e d w i t h t h e computed c o n c e n t r a t i o n v e c t o r , c(n+l).
T h i s u n c e r t a i n t y may be c o n s i d e r e d t o r e s u l t f r o m n o i s e , o r e r r o r , caused by i m p e r f e c t m o d e l l i n g o f t h e p r o c e s s under c o n s i d e r a t i o n . t h e e f f e c t s o f system n o i s e , or m o d e l l i n g e r r o r , a r e i n c l u d e d , t h e system model may be d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n : c(n+l)
-
= o(n,n+l)
c(n) + n(n,n+l)
-u ( n + l )
-
t w(nt1)
If
2 49
where w ( n + l ) i s a vector of length N containing system noise, or sources of uncertainty in the modelling process. Thus, each grid point of the model has noise associated w i t h i t . The system noise i s assumed t o be Gaussian and uncorrelated with zero mean and covariance Q(n). Taking the expected value o f ( 6 ) yields the time update equation for concentration : i(n+l) =
Q(n,n+l) i ( n ) + -R ( n , n + l ) u(n+l)
(7)
where c ( n + l ) i s the one step ahead prediction, o r the expected value of c ( n t 1 ) conditioned on information up t o time n A t , and c ( n ) i s the f i l t e r e d estimate, o r the expected value of c(n) conditioned on information up t o time nAt. Subtracting ( 7 ) from ( 6 ) , squaring and taking the expected value gives the covariance time update equation:
P(n+l)
=
o(n,n+l) P ( n ) 2 T ( n , n + l ) + Q(n+l)
(8)
where P ( n + l ) and P ( n ) are the covariances associated with i ( n + l ) and C ( n ) , respectively. If observations are available, they can be used t o improve the accuracy of the estimates o f c( n ) . Measurements have error associated with them. If i t can be assumed t h a t measurement error g(n) i s additive noise then the observations z ( n ) are related t o the s t a t e , or concentration, vector i n the following manner:
The measurement noise i s assumed t o be uncorrelated and Gaussian with zero mean and covariance R(n). If observations are included in the estimation process, the following measurement updates, or f i l t e r estimates, can be obtained for the s t a t e vector and covariance matrix (Jazwinski, 1970);
250
c(nt1)
= i(n+l) t K(n+l)
P(n+l)
= ^P(n+l)
[z(n+l)
-
fic(n+l)]
-K(n+l) H^P(n+l)
where
i s the Kalman gain matrix.
j(a)) (c(n), P(n)).
Prediction for t = % A t , % > n (;(&),
i s accomplished using ( 7 ) and ( 8 ) with i n i t i a l condition Equations ( 7 ) t h r o u g h ( 1 2 ) constitute the d i s c r e t e form of the Kalman-Bucy f i l t e r (Kalman, 1960; Kalman and Bucy, 1961). From (11) i t can be seen that the measurement update of the covariance matrix, P ( n + l ) , i s computed by subtracting the positive d e f i n i t e matrix K ( n + l ) H- P ( n + l ) from the positive d e f i n i t e matrix P ( n + l ) . Because of round-off e r r o r s , the resulting matrix may become non-positive d e f i n i t e or weakly positive d e f i n i t e ultimately causing severe computational i n s t a b i l i t y when the matrix inverse in ( 1 2 ) i s computed (Koda and Sei nfel d , 1978). One means of avoiding these d i f f i c u l t i e s i s the application o f square root f i l t e r i n g theory. Desalu, Gould, and Schweppe (1974) ,Koda and Seinfeld (1978), and Budge11 (1981) have found t h a t applying square
root f i l t e r i n g t o distributed parameter s t a t e estimation problems results in stable a1 gorithms. The covariance square root f i l t e r used here i s an algorithm based upon t r i a n g u l a r square r o o t factorization of the estimation e r r o r covariance matrix. The f i l t e r covariance m a t r i x may be factored as follows: i(n)
=
i(n)
i(n)
h(n) j T ( n ) i s a unit upper triangular matrix
where
-
and
D ( n ) i s a diagonal matrix.
-
Similarly :
251
The matrix U ( n + l ) i s computed in ( 1 6 ) by applying the tri-diagonal algorithm required t o solve ( 4 ) t o the factor matrix U ( n ) . I t should be noted t h a t O( n , n + l ) merely represents the sequence of operations carried out by the equation solver. The matrix O ( n , n + l ) i s never actually computed. The factor matrices U ( n t 1 ) and D ( n + l ) are computed from ( 1 7 ) by a p p l y i n g a modified weighted Gram-Schmidt (MWGS) o r t h o g o n a l ization procedure (Bierman, 1977) which i s reputed t o have accuracy comparable t o the Householder a l g o r i t h m . Unlike the classical procedure, the modified algorithm produces almost orthogonal vectors and pivoting i s unnecessary The s t a t e and covariance measurement updates are accompl i shed t h r o u g h the equations: I
.
-
m
1 [ki(ntl)-1h .
;(n+l)
o(n+l)
iT(n+l)](19)
-
i =1 where h i i s a column vector specifying the location of the i-th measurement sensor such t h a t HT = [- - h l ,h] ;
,...
-1 k.
( n + l ) i s a column vector specifying the gain associated with the
252 i - t h measurement a t t me ( n + l ) A t such t h a t
K(n+l)
=
[kl(n+l)
,...
and z i ( n + l ) i s t h e measurement f r o m t h e i - t h sensor a t t i m e ( n + l ) A t . The g a i n v e c t o r s k i ( n ) a r e o b t a i n e d one a t a t i m e u s i n g t h e UDUT estimate-covariance
u p d a t i n g a l g o r i t h m o f Bierman (1977).
This
updating a l g o r i t h m i s n u m e r i c a l l y s t a b l e since numerical d i f f e r e n c i n g i s avoided i n t h e computation o f J .
Using t h e f a c t o r e d form o f t h e
c o v a r i a n c e m a t r i c e s f o r t h e t i m e and measurement u p d a t e s r e s u l t s i n a c o m p u t a t i o n a l l y s t a b l e a l g o r i t h m t h a t r e q u i r e s fewer a r i t h m e t i c o p e r a t i o n s t h a n t h e c o n v e n t i o n a l Kalman f i l t e r (Bierman, 1977).
n
BAKER ,LAKE
d
CHESTERFIELD INLET
1 0 0
F i g . 1.
-
-
,
50km
Model s c h e m a t i z a t i o n o f C h e s t e r f i e l d I n l e t .
4. A TEST SIMULATION The c h a r a c t e r i s t i c s o f t h e combined d y n a m i c - s t o c h a s t i c model as r e p r e s e n t e d by ( 1 2 ) t o ( 1 9 ) a r e i l l u s t r a t e d u s i n g s i m u l a t e d
253
observations of s a l i n i t y concentration from Chesterfield I n l e t , situated on the Northwest coast of Hudson Bay i n the Canadian Arctic. As shown in Figure 1, the estuary i s discretized into 48 g r i d points with a Ax of 5000 m. The channel depth and t o p w i d t h were obtained from a study by Budgell (1976). Dispersion coefficients were obtained from Roff et a1 (1980) and varied from 500 t o 5000 m2/sec. The time- and space-dependent flow rates and water levels were obtained using a numerical tidal model developed previously (Budgell , 1 9 7 6 ) . The boundary conditions consisted of no flow t h r o u g h the upper end of Baker Lake and predicted tidal water surface elevations a t the m o u t h of the estuary. The tidal water level predictions were obtained for the m o n t h of September, 1978 using tidal harmonic constituents (Godin, 1972). The predominant tidal constituent i n the water level and channel flow time s e r i e s i s the lunar semi-diurnal with a period of 12.42 hours. Typical amp1 itudes of the cross-sectionally averaged velocity are 0.3 t o 1.0 m/sec. The simulated t e s t data were generated using ( 6 ) and ( 9 ) on a time step of 20 minutes. The true s t a t e vector of s a l i n i t y concentrations a t time ( n + l ) A t was obtained from ( 6 ) . Uncorrelated Gaussian noise w ( n t 1 ) was added t o the concentrations computed by the numerical model in ( 4 ) t o produce the t r u e s t a t e vector c ( n + l ) . The variance of the system noise was specified as being 10 percent o f the variance a t t r i b u t a b l e t o tidal fluctuations in s a l i n i t y as computed with the deterministic numerical model ( 4 ) . The system noise variance varied from zero t o 0.5 p p t 2 t h r o u g h o u t the estuary. I n essence, then, s a l i n i t i e s are computed by running the deterministic model for one time step. A small quantity of Gaussian uncorrelated system noise i s added t o these computed values t o produce the true s a l i n i t y concentrations. These s a l i n i t i e s then constitute the i n i t i a l condition for the next time level. The numerical model i s then run for another time step. As before, a vector of system noise values i s added t o the computed concentrations t o create t r u e s a l i n i t i e s a t
254 t h e next t i m e l e v e l .
T h i s process i s repeated t o o b t a i n s t a t e
( s a l i n i t y ) v e c t o r s f o r t h e d e s i r e d l e n g t h o f record.
I n t h i s manner,
system n o i s e added t o t h e numerical model a t each t i m e s t e p propagates through space and t i m e a1 t e r i n g f u t u r e s a l i n i t y values throughout t h e estuary
.
The boundary c o n d i t i o n s used i n t h e c r e a t i o n o f t h e data a r e a sal i n i t y o f zero p p t a t t h e upstream end a t ebb t i d e and a sal i n i t y o f
32 ppt p l u s a random n o i s e component a t t h e downstream end d u r i n g f l o o d tide.
Otherwise,
boundaries.
a c o n d i t i o n o f a2c/a2x
=
0 was s p e c i f i e d a t t h e
The random p e r t u r b a t i o n a p p l i e d t o t h e downstream (ocean)
end o f t h e e s t u a r y a t f l o o d t i d e has a v a r i a n c e o f 0.04 ppt'.
This
p e r t u r b a t i o n s i m u l a t e s e r r o r i n t h e s p e c i f i c a t i o n o f t h e boundary condition. I n o r d e r t o o b t a i n i n i t i a l c o n d i t i o n s , t h e d e t e r m i n i s t i c numerical model was used t o compute s a l i n i t i e s f o r a 1 5 day period.
The
s a l i n i t i e s averaged over t h e f i n a l t i d a l c y c l e were used as t h e i n i t i a l values i n t h e c r e a t i o n o f t h e t e s t data. Measurements were s i m u l a t e d by adding u n c o r r e l ated Gaussian n o i s e w i t h a v a r i a n c e o f 0.04 ppt2 t o t h e " t r u e " s a l i n i t y values a t s p e c i f i e d measurement sensor l o c a t i o n s .
The measurement noise, b e i n g a d d i t i v e ,
was n o t propagated t h r o u g h t h e numerical model and does n o t have any e f f e c t on t h e t r u e s a l i n i t i e s .
Measurement l o c a t i o n s were spread a t
equal i n t e r v a l s throughout t h e estuary. When t h e d y n a m i c - s t o c h a s t i c model was a p p l i e d t o t h e data set, i t was found t o perform w e l l .
Shown i n F i g u r e 2 are t y p i c a l r e s u l t s from
one o f 5 measurement l o c a t i o n s (m=5).
I t can be seen t h a t t h e
estimated concentration ( f i l t e r estimate) c l o s e l y tracks the actual concentration ( t r u e state).
However, when t h e d e t e r m i n i s t i c numerical
model (numerical model o n l y ) i s appl i e d t o t h e same s i t u a t i o n , t h e agreement i s c o n s i d e r a b l y worse.
The l a c k o f a measurement u p d a t i n g
c a p a b i l i t y i n t h e numerical model means t h i s model cannot t r a c k t h e t r u e state.
The pronounced s i n u s o i d a l c o r r e l a t i o n s t r u c t u r e i n t h e
e r r o r s o f t h e numerical model values are a t t r i b u t a b l e t o t h e s t r o n g
255 m
S E C T I O N 25 FILTER ESTIMRTE . _ _N_ U M_ ERICRL MODEL ONLY __ T R U E S T R T E - .-. - __
10
15
20
25
30
T I M E IHRSI
35
40
45
50
Fig. 2 Stochastic-dynamic model and deterministic numerical model estimates v s . the true state at a measurement location.
S E C T I O N 29 _-_-. --...
__ n-,
., .
FILTER ESTIMATE NUMERICAL MODEL ONLY TRUE STRTE
I
Fig. 3 Stochastic-dynamic model and deterministic numerical model estimates v s . the true state between measurement locations.
256 t i d a l f o r c i n g i n t h e advective f l u x term a(Qc)/ax.
A l t h o u g h t h e system
n o i s e i n p u t a t each t i m e s t e p i s u n c o r r e l a t e d , t h e n o i s e i s p r o p a g a t e d t h r o u g h space and t i m e , c o n d i t i o n e d by t h e a d v e c t i o n and d i s p e r s i o n t e r m s i n t h e mass b a l a n c e e q u a t i o n .
The a d v e c t i o n t e r m w i l l t e n d t o
produce a harmonic c o r r e l a t i o n s t r u c t u r e ,
whereas t h e d i s p e r s i o n t e r m
w i l l t e n d t o f i l t e r o u t h i g h f r e q u e n c y and h i g h wave number c o n t r i b u tions t o the correlation structure. The a c c u r a c y o f t h e d y n a m i c - s t o c h a s t i c model e s t i m a t e s d e t e r i o r a t e s w i t h i n c r e a s i n g d i s t a n c e f r o m measurement l o c a t i o n s .
Data from g r i d
p o i n t 29, s i t u a t e d h a l f way between two s e n s o r s , i s shown i n F i g u r e 3. The f i l t e r e s t i m a t e s do n o t f o l l o w t h e t r u e s t a t e n e a r l y as c l o s e l y as a t t h e measurement l o c a t i o n as shown i n F i g u r e 2. e r r o r s have a s t r o n g s i n u s o i d a l component.
Furthermore, t h e
T h i s i s because system
n o i s e g e n e r a t e d from 3 g r i d p o i n t s on e i t h e r s i d e o f t h e l o c a t i o n i n q u e s t i o n i s b e i n g a d v e c t e d and d i s p e r s e d t h r o u g h it.
The p e r f o r m a n c e
o f t h e d y n a m i c - s t o c h a s t i c model i s s t i l l s u p e r i o r t o t h a t o f t h e d e t e r m i n i s t i c model because o f t h e measurement u p d a t e s c a r r i e d o u t a t g r i d p o i n t s 25 and 33.
As one moves f a r t h e r f r o m t h e l o c a t i o n o f a measure-
ment u p d a t e t h e f i l t e r p e r f o r m a n c e degrades because o f t h e c u m u l a t i v e e f f e c t o f random system n o i s e i n p u t .
I n order t o determine t h e l e v e l o f u n c e r t a i n t y associated w i t h t h e s t a t e estimates, vector.
i t i s n e c e s s a r y t o examine t h e c o v a r i a n c e o f t h e s t a t e
Shown i n F i g u r e 4 i s t h e l o n g i t u d i n a l d i s t r i b u t i o n o f t h e mean
square e r r o r (MSE) o f t h e s t a t e e s t i m a t e s and t h e f i l t e r v a r i a n c e as e s t i m a t e d by t h e d y n a m i c - s t o c h a s t i c model f o r a t e s t case i n w h i c h "observations"
a r e a v a i l a b l e from a s i n g l e sensor s i t u a t e d a t t h e mid-
p o i n t of t h e e s t u a r y .
B o t h t h e mean square e r r o r and e s t i m a t e d
v a r i a n c e s have been averaged o v e r 4 t i d a l c y c l e s (150 t i m e s t e p s ) .
The
d i s t a n c e i s r e l a t i v e t o t h e l o c a t i o n o f t h e g r i d p o i n t number 1 i n F i g u r e 1.
It can be seen f r o m F i g u r e 4 t h a t a t most l o c a t i o n s t h e mean
s q u a r e e r r o r i s l a r g e r t h a n t h e v a r i a n c e e s t i m a t e d by t h e dynamics t o c h a s t i c model.
The l a r g e s t v a l u e s f o r t h e e s t i m a t e d v a r i a n c e and
2 57
4
-
e-
MERN SQURRE ERROR ------ - MODEL COMPUTED V R R I R N C E
.oo
Fig. 4
L o n g i t u d i n a l d i s t r i b u t i o n of t h e mean s q u a r e e r r o r and model-computed
-
-
Tf-
cu7
v a r i a n r e w i t h one m e a s u r e m e n t s e n s o r .
.- - - - - - -
+
MEAN SQUARE ERROR MODEL COMPUTED V A R I R N C E MEASUREMENT L O C R T I O N
+-
Fig. 5
L o n g i t u d i n a l d i s t r i b u t i o n o f t h e mean s q u a r e e r r o r and model-computed
v a r i a i c e w i t h 9 me,isuremeiit scnsors.
258
MSE o c c u r a t c o n s t r i c t i o n s i n t h e channel w h i l e t h e minimum v a l u e s
o c c u r a t embayments and a t t h e s i n g l e measurement l o c a t i o n . C o n s t r i c t i o n s t e n d t o a m p l i f y t h e a d v e c t i v e f l u x and t h u s t h e n o i s e field.
T h i s t e n d e n c y was r e i n f o r c e d i n t h e c r e a t i o n o f t h e d a t a s e t by
t h e i n s e r t i o n o f system n o i s e i n t o t h e p r o c e s s such t h a t t h e system noise variance i s proportional t o t h e variance i n concentration attributable t o t i d a l fluctuations. embayments.
The r e v e r s e p r o c e s s o c c u r s a t
A t t h e measurement l o c a t i o n , i n f o r m a t i o n i s a v a i l a b l e f r o m
an observed t i m e s e r i e s t o i m p r o v e t h e e s t i m a t e . The d y n a m i c - s t o c h a s t i c model e s t i m a t e s t h e -p r o b a b i l i t y d i s t r i b u t i o n o f the concentration f i e l d .
Thus, i f t h e p r o b a b i l i t y d i s t r i b u t i o n i s
Gaussian, n o t o n l y t h e c o n c e n t r a t i o n b u t i t s v a r i a n c e must be estimated.
It can be seen f r o m F i g u r e 5 t h a t when d a t a a r e a v a i l a b l e
from 9 sensors d i s t r i b u t e d t h r o u g h o u t t h e e s t u a r y , t h e MSE and v a r i a n c e e s t i m a t e d by t h e d y n a m i c - s t o c h a s t i c model a r e i n much c l o s e r agreement t h r o u g h o u t t h e e s t u a r y t h a n i n t h e one-sensor case o f F i g u r e 4.
Thus,
t h e e s t i m a t e d v a r i a n c e more c l o s e l y a p p r o x i m a t e s t h e MSE o v e r d i s t a n c e as t h e number o f s e n s o r s i s i n c r e a s e d .
-t
"1
F i g u r e 6.
MERN S Q U R R E ERROR
Mean square e r r o r and model -computed v a r i a n c e a v e r a g e d o v e r t h e e s t u a r y as a f u n c t i o n o f t h e number o f mea s urement sensors.
259
The overall effect of spatial sampling density on the uncertainty o f the s t a t e estimates i s summarized in Figure 6. I n t h i s plot the MSE and estimated variance have been averaged over all the grid points i n space as well as over 4 tidal cycles in time. While the model estimated variance i s r e l a t i v e l y invariant with the number of measurement sensors, the MSE decreases approximately as l / m . The MSE approaches the estimated variance asymptotically, b u t for the MSE and estimated variance t o be approximately equal, there must be more t h a n 9 measurement locations (in > 9 ) . I f there i s prior knowledge of the system and measurement noise c h a r a c t e r i s t i c s , a simulation such as that carried o u t in t h i s study can provide a useful tool for experimental design.
For example,
Figures 3 and 4 suggest that placing sensors a t g r i d points 9 , 20 and 37 would reduce the uncertainty of concentration estimates considerably since these grid points are situated in regions of maximum variance. Furthermore, from a plot such as Figure 6, the number of sensors required t o achieve a given level of accuracy can be obtained. 5. CONCLUSIONS A model has been developed t h a t combines a numerical solution t o the one-dimensional advect ion-di spersion equation w i t h a stochastic f i l t e r . The major portion of the variation o f concentration over time and space in open channels can be described by the deterministic numerical model. By constructing a stochastic f i l t e r a r o u n d the numerical model i t i s possible t o compensate for errors incurred d u r i n g the modelling process. A numerical model alone cannot be constrained t o track the t r u e system t h r o u g h the use of observed d a t a . However, time s e r i e s models such as autoregressive moving average s e r i e s (Box and Jenkins, 1976) or simple Kalman f i l t e r s (e.g., Chiu and Isu, 1 9 7 7 ) , i n t o which observations are directly incorporated, are black box approaches t h a t are unrelated t o physics. The dynamic-stochastic model proposed in t h i s paper retains the best features of the conventional determini s t i c and stochastic approaches.
260
The model equations are posed in s t a t e space form. The usual approach of applying a Kalman f i l t e r algorithm t o obtain estimates of the concentration vector and covariance m a t r i x could lead t o f i l t e r i n s t a b i l i t y due t o covariance matrices becoming nonpositive d e f i n i t e . To circumvent t h i s problem, the covariance matrix i s factored into unit upper triangular and diagonal matrices. The time and measurement updates are performed on these factor matrices. This formulation ensures t h a t the covariance matrices will remain positive d e f i n i t e and t h a t the f i l t e r i n g algorithm will remain stable. The dynamic-stochastic model h a s been tested using simul ated s a l i n i t y observations from an Arctic estuary. I t was found t h a t the model estimates of concentration closely track the t r u e s t a t e in the vicinity of observations. The accuracy of the estimates deteriorates with increasing distance from measurement locations, b u t the accuracy i s s t i l l superior t o t h a t of values produced by applying a determinist i c numerical model t o the same data s e t . As the number of measurement locations increases, the mean square error of the dynamic-stochastic model estimates approaches the computed f i l t e r variance. The mean square error decreases approximately as the reciprocal of the number of sensors. If the system and measurement noise s t a t i s t i c s are known, a simulation of the stochastic advection-dispersion process together with the dynamic-stochastic model can be used t o select the number and locations of measurement sensors t o be deployed in f i e l d programs. REFERENCES
Bankoff, S.G. and Hanzevak, E.L., 1975. The adaptive-filtering transport model for prediction and control of pollutant concentration i n an urban airshed. Atmos. E n v i r o n . 9:793-808. Bierman, G.J., 1977. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 241 pp. Box, G.E.P. and Jenkins, G.M., 1976. Time Series Analysis: Forecasting and Control Hol den-Day , San Franci sco, 575 pp. Budgell, W.P., 1976. Tidal P r o p a g a t i o n i n Chesterfield I n l e t , N.W.T. Manuscript Report Series No. 3, Ocean and Aquatic Sciences, Central Region, Environment Canada, Burl i n g t o n , 99 pp.
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261 Budge11 , W.P. , 1981., A S t o c h a s t i c - D e t e r m i n i s t i c Model f o r E s t i m a t i n g Tides i n Branched E s t u a r i e s . Manuscript Report S e r i e s No. 10, Ocean Science and Surveys , F i s h e r i es and Oceans Canada , B u r l in g t o n , 189 PPChiu, C.L. ( E d i t o r ) , 1978. A p p l i c a t i o n s o f Kalman F i l t e r Theory t o Hydrology, H y d r a u l i c s and Water Resources. U n i v e r s i t y o f P i t t s b u r g , P i t t s b u r g , 783 pp. Chiu, C.L.. and I s u , E.O., 1977. A p p l i c a t i o n o f Kalman f i l t e r i n model1 i n g d a i l y stream temperature. I n : Proceedings o f t h e Seventeenth Congress of t h e I n t e r n a t i o n a l A s s o c i a t i o n f o r H y d r a u l i c Research. I.A.H.R. , Baden-Baden, Vol. 3, pp. 463-470. Crank, J. and Nicholson, P., 1947. A p r a c t i c a l method f o r numerical i n t e g r a t i o n o f s o l u t i o n s of p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f heat c o n d u c t i o n type. Proc. Cambridge P h i l o s . SOC. 43: 50-67. DeGuida, R.N., Connor, J.J. and Pearce, R.R., 1977. A p p l i c a t i o n o f e s t i m a t i o n t h e o r y t o design o f sampling programs f o r v e r i f i c a t i o n o f coastal dispersion predictions. I n : Gray, W.G. , Pinder, G.F. and Brebbia, C.A. ( E d i t o r s ) , F i n i t e Elements i n Water Resources. Pentech, London, pp. 4.303-4.334. Desal u, A.A. , Gould, L.A. and Schweppe, F.C. , 1974. Dynamic e s t i m a t i o n o f a i r pollution. I E E E Trans. Autom. Contr. 19:904-910. 1973. L o n g i t u d i n a l d i s p e r i s o n and m i x i n g i n openF i s c h e r , H.B., channel flow. Ann. Rev. F l u i d Mech. 5:59-79. 1976. M i x i n g and d i s p e r s i o n i n e s t u a r i e s . Ann. Rev. F i s c h e r , H.B., F l u i d Mech. 8:107-133. Fronza, G., S p i r i t o , A. and T o n i e l l i , A. 1979. R e a l - t i m e f o r e c a s t o f a i r p o l l u t i o n episodes i n t h e Venetian region. P a r t 2: t h e Kalman p r e d i c t o r . Appl. Math. Model. 3:409-415. U n i v e r s i t y o f Toronto Press, Godin, G., 1972. The A n a l y s i s o f Tides. Toronto, 264 pp. iiann, R.W. and Young, P.J., 1972. Mathematical models o f water q u a l i t y parameters f o r r i v e r s and e s t u a r i e s . Report TR-45, Texas Water Resources I n s t i t u t e , Texas A & M U n i v e r s i t y , C o l l e g e S t a t i o n , Texas. Harleman, D.R.F. , 1971. One-dimensional models. I n : Ward, G.H. and Epsey, W.H. ( E d i t o r s ) , E s t u a r i n e Model1 i n g : An Assessment. Tracor Inc., A u s t i n , pp. 34-89. Hinwood, J.B. and W a l l i s , I.G., 1975. C l a s s i f i c a t i o n o f models o f 101:1315-1331. t i d a l waters. J. Hydraul. Div. Am. SOC. Civ. Engrs. Jazwinski , A.H. , 1970. S t o c h a s t i c Processes and F i l t e r i n g Theory. Academic Press, New York, 376 pp. Kalman, R.E., 1960. A new approach t o l i n e a r f i l t e r and p r e d i c t i o n problems. J. bas. Engng. 82: 35-45. 1961. New r e s u l t s i n l i n e a r f i l t e r i n g and Kalman, R.E. and Bucy, R.S., p r e d i c t i o n t h e o r y . J. bas. Engng. 83:95-108. Koda, M. and S e i n f e l d , J.H., 1978. E s t i m a t i o n o f urban a i r p o l l u t i o n . Automatica. 14: 583-595.
262
Lam, D.C.L., 1977. Comparison of finite-element and finite-difference methods for nearshore advection-diffusion Pinder, G.F. and Brebbia, transport models. I n : Gray, W.G., C.A. (Editors) , Finite Elements in Water Resources. Pentech, London, pp. 1.115-1.129. Roache, P.J. , 1972. Computational F1 uid Dynamics. Hermosa Pub1 i shers , A1 buquerque , 446 pp. Roff, J.C. , P e t t , R.J., Rogers, G.F. and Budge11 , W.P., 1980. A study of p l a n k t o n ecology in Chesterfield I n l e t , Northwest T e r r i t o r i e s : an Arctic estuary. I n : Kennedy, V.S. (Editor) , Estuarine Perspectives. Academic Press, New York, pp. 185- 197. Stone, H.L. and Brian, P.L.T., 1963. Numerical solution of convective transport problems. Amer. Inst. Chem. Engrg. J . 9 :681-688. Thatcher, M.L. and Harleman, D.R.F. , 1972. A mathematical model for the prediction of unsteady s a l i n i t y intrusion in estuaries. Ral ph M. Parsons Laboratory for Water Resources a n d Hydrodynamics, Report No. 144. Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts, 232 pp. LIST OF SYMBOLS
A A B C
Cross-sectional area NxN coefficient matrix NxN coefficient matrix Concentrat ion Vector of length N specifying concentrations a t grid points i n the discretized channel NxN diagonal matrix in covariance matrix factorization Longitudinal dispersion coefficient Nx2 matrix specifying the nature and location of boundary conditions Measurement control vector of length N rnxN observation control matrix specifying the model grid points a t which concentrations have been observed Kalman gain vector of length N Kalman gain vector of length N Nm Kalman gain matrix Length of the open channel
263
m
n N P -
Q Q R -
t U -
U W -
Number of measurement locations (sensors) in the channel Time step index Number of grid points in the discretized channel NxN covariance matrix Volume flow rate NxN system noise covariance matrix Measurement noi se covariance matri x time vector of length 2 containing specified boundary conditions NxN unit upper triangular matrix in covariance matrix factorization Vector of length N specifying system noise input a t each of the model grid points Distance Vector o f length m containing observations from m sensors Time increment Vector o f length m containing measurement noise corresponding to z NxN s t a t e t r a n s i t i o n matrix Nx2 i n p u t control matrix Filtered estimate, e.g. expected value of the variable a t time n A t conditioned on information up t o n A t One step ahead prediction, e.g. expected value a t time ( n + l ) A t conditioned on information up t o n A t
264
THE MEAN AND VARIANCE OF WATER CURRENTS INDUCED BY IRREGULAR SURFACE WAVES
B. DE JONG and A.W.
HEEMINK
Twente U n i v e r s i t y of Technology, Enschede, and Data P r o c e s s i n g D i v i s i o n of R i j k s w a t e r s t a a t , The N e t h e r l a n d s ABSTRACT I r r e g u l a r s u r f a c e waves g e n e r a t e a n e t mean v e l o c i t y o f t h e f l u i d and t h e m a t e r i a l i n i t w h i l e due t o t h e random f l u c t u a t i o n s o f t h i s v e l o c i t y a b o u t i t s mean
v a l u e t h e r e w i l l be a d i s p e r s i o n
o f t h e m a t e r i a l . E x p r e s s i o n s a r e d e r i v e d f o r t h e mean v a l u e and t h e v a r i a n c e of t h e s e c u r r e n t v e l o c i t i e s f o r a one- and twod i m e n s i o n a l i r r e g u l a r wave f i e l d . Numerical r e s u l t s a r e g i v e n f o r a one-dimensional wave f i e l d . I t a p p e a r s t h a t t h e f a m i l i a r d i r e c t i o n a l wave s p e c t r u m f o r a two-dimensional
wave f i e l d i s
insufficient t o derive useful1 r e s u l t s .
INTRODUCTION
T i d e s and winds a r e found t o be major s o u r c e s o f g e n e r a t i o n
of r e s i d u a l c u r r e n t s a s o b s e r v e d i n s e a s and e s t u a r i e s ( A l f r i n k &
V r e u g d e n h i l , 1 9 8 1 ) . These c u r r e n t s g e n e r a t e n o t o n l y a n e t
k c a n s p o r t o f t h e c e n t e r o f g r a v i t y o f s u b s t a n c e s suspended o r d i s s o l v e d i n it b u t a l s o e f f e c t a d i s p e r s i o n of t h i s m a t e r i a l r e l a t i v e t o t h i s c e n t e r of g r a v i t y . I n t h e p r e s e n t p a p e r w e s t u d y o n l y t h e c u r r e n t due t o i r r e g u l a r wind waves which we assume t o have a v e l o c i t y e q u a l t o t h e Lagrangian v e l o c i t y g e n e r a t e d by t h e s e waves. E x p r e s s i o n s f o r t h e Lagrangian v e l o c i t y g e n e r a t e d by a harmonic s u r f a c e wave a r e well-known and can be d e r i v e d i n t h e way a s i n d i c a t e d f o r example by P h i l l i p s 1 9 7 7 . On t h e b a s i s o f t h e s e e x p r e s s i o n s w e d e r i v e t h e Lagrangian v e l o c i t y f o r i r r e g u l a r waves by c o n c e i v i n g t h e
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
265 i r r e g u l a r waves a s a s i m u l t a n e o u s a m p l i t u d e , wave number and frequency v a r i e d harmonic wave w i t h a l o c a l l y and i n s t a n t a n e o u s l y d e f i n e d f r e q u e n c y , p h a s e , wave number and a m p l i t u d e which a r e a l l random v a r i a b l e s .
I n t h i s c o n c e p t t h e s t a t i s t i c a l pro-
p e r t i e s o f t h e i r r e g u l a r wave f i e l d a r e d e f i n e d by t h e j o i n t d e n s i t y of t h e s e q u a n t i t i e s . By assuming t h e wave e l e v a t i o n s normally d i s t r i b u t e d t h i s j o i n t d e n s i t y can be d e r i v e d a s i n d i c a t e d f o r example by R i c e , 1954 a n d Cramer
&
L e a d b e t t e r , 1967.
However, it w i l l be s e e n t h a t f o r a c a l c u l _ a t i o n o f t h e s p e c t r a l moments which a p p e a r a s p a r a m e t e r s i n t h i s d e n s i t y we need an e x p r e s s i o n f o r t h e j o i n t wave number-frequency spectrum w h i l e f o r s e a waves i n g e n e r a l o n l y t h e f r e q u e n c y spectrum i s a v a i l a b l e . By u s i n g t h e d i s p e r s i o n r e l a t i o n which i s o n l y v a l i d f o r c o n s t a n t atmospheric p r e s s u r e an approximate e x p r e s s i o n i s d e r i v e d f o r t h e frequency-wave number spectrum. I n t h e f i r s t p a r t of t h e p a p e r e x p r e s s i o n s a r e d e r i v e d f o r t h e mean and v a r i a n c e o f t h e r e s i d u a l c u r r e n t f o r a one-dimensional wave f i e l d . I n t h e second p a r t e x p r e s s i o n s a r e d e r i v e d f o r t h e two-dimensional c a s e . I t
i s shown i n t h e f i n a l p a r t o f t h e p a p e r t h a t t h e s e e x p r e s s i o n s lead
t o a c c e p t a b l e r e s u l t s f o r t h e one-dimensional wave f i e l d .
However, i t a p p e a r s t h a t t h i s a p p r o x i m a t i o n i s i n s u f f i c i e n t f o r two-dimensional
i r r e g u l a r waves
THE MEAN AND VARIANCE O F THE RESIDUAL CURRENT GENERATED BY ONE-
DIMENSIONAL IRREGULAR SURFACE WAVES W e assume t h e x- and y - a x i s o f a C a r t e s i a n c o o r d i n a t e system
i n t h e s t i l l w a t e r s u r f a c e and t h e z - a x i s p o s i t i v e i n upward d i r e c t i o n . A two-dimensional
normally d i s t r i b u t e d random f i e l d
c ( x , t ) of s u r f a c e e l e v a t i o n s due t o one-dimensional
irregular
s u r f a c e waves p r o p a g a t i n g i n t h e x - d i r e c t i o n can b e r e p r e s e n t e d by
m
m
with k = O < k l < k 0
2
....< . - - I
w
-1
< w
0
= o < w
< w2
0 , s(w) = 0 where V i s t h e wind speed a t a h e i g h t of 1 9 . 5 m.
w
b + s
J = b
It i s a p p a r e n t t h a t i n o r d e r t o s o l v e equ.
t h e model equ.
(3) as f o l l o w s :
( 6 ) must b e e s t i m a t e d .
..... b+s
( 8 ) , t h e o r d e r ( r , s , b ) of
The o r d e r r i s h e r e e s t i m a t e d by
u s i n g a g e n e r a l c h a r a c t e r i s t i c of t h e inflow-outflow p r o c e s s d i s c u s s e d below and t h e adequacy of t h i s estimate i s f u r t h e r confirmed by d a t a analysis.
The o r d e r s s and b , on t h e o t h e r hand, which a r e s p e c i f i c
c h a r a c t e r i s t i c s of t h e h y d r o l o g i c a l system under c o n s i d e r a t i o n a r e e s t i mated by t r i a l and e r r o r d u r i n g t h e c a l i b r a t i o n of t h e model. The o r d e r r i s assumed t o be e q u a l t o t h e o r d e r o f t h e d i f f e r e n t i a l e q u a t i o n which can b e used t o r e p r e s e n t t h e i n f l o w - o u t f l o w r e l a t i o n s h i p i n t h e r e a c h between ( i ) and ( j ) .
When t h e s y s t e m i s u n d i s t u r b e d , t h e
inflow-outflow t r a n s f e r f u n c t i o n i s s i m p l e and l i n e a r , g i v e n by:
(9)
qj,t = gqi,t
w i t h g t h e s t e a d y s t a t e g a i n of t h e system.
For unsteady f l o w s , t h e
302 ) where b t h e t i m e l a g , i s u s u a l l y d i f f e r e n t from j ,t T h i s d i f f e r e n c e (gqi,t-b) obeys t h e d i f f e r e n t i a l e q u a t i o n : ‘j , t
d i f f e r e n c e (gqi,t-b-q zero.
.
dq s d=t L T(
- ‘j,t ) ,
g‘ i , t - b
(10)
where T i s a t i m e c o n s t a n t of the s y s t e m (Box and J e n k i n s ( 1 9 7 0 ) ) . D e f i n i n g dq
j ,t
/dt
=
Dq
j,t
e q u . (10) may b e r e w r i t t e n a s e q u . ( l l ) , which
is a f i r s t
(l+TD)qj,t
= gqi, t-b’
o r d e r d i f f e r e n t i a l e q u a t i o n r e p r e s e n t i n g t h e inflow-outflow r e l a t i o n s h i p
in a reach.
T h e r e f o r e , t h e o r d e r r i s assumed e q u a l t o one.
So f a r , t h e o t h e r i n p u t t o t h e s y s t e m , namely t h e r a i n f a l l o v e r t h e
b a s i n , which p r o d u c e s a l a t e r a l f l o w c o n t r i b u t i o n t o t h e o u t f l o w , has n o t been c o n s i d e r e d i n t h e t r a n s f e r f u n c t i o n model.
In r e l a t i v e l y s m a l l
b a s i n s , s u c h as t h e one u n d e r c o n s i d e r a t i o p , t h e r a i n f a l l o v e r t h e b a s i n
i s r e l a t e d t o i n f l o w s b e c a u s e of t h e u n i f o r m i t y o f s t o r m o c c u r r e n c e s . I n s u c h cases, t h e i n f l o w - o u t f l o w t r a n s f e r f u n c t i o n model t a k e s i n t o a c c o u n t a p a r t of t h e l a t e r a l f l o w c o n t r i b u t i o n , which depends on t h e s i g n i f i c a n c e of t h e r a i n f a l l - i n f l o w r e l a t i o n s h i p .
I n any c a s e , f t i s
d i f f i c u l t t o estimate t h e l a t e r a l f l o w c o n t r i b u t i o n a c c u r a t e l y , m a i n l y b e c a u s e i t i s s t r o n g l y a f f e c t e d by l o c a l c h a r a c t e r of s t o r m s .
The s i g -
n i f i c a n c e of l a t e r a l f l o w estimates c a n be checked by u s i n g t h e r e s i d u a l s S
t
of t h e model i n equ. ( 6 ) .
I f t h e l a t e r a l f l o w s are a predominant
component of o u t f l o w s , t h e n t h e s e r e s i d u a l s would b e s i g n i f i c a n t l y cross-correlated
t o t h e r a i n f a l l over t h e basin.
Otherwise, t h e t r a n s -
f e r f u n c t i o n would a c c o u n t f o r t h e i n f l o w - o u t f l o w r e l a t i o n s h i p s and hence t h e c r o s s - c o r r e l a t i o n s small.
between t h e i n f l o w s and r e s i d u a l s would b e
I f t h e r e s i d u a l s are s t r o n g l y c r o s s - c o r r e l a t e d t o t h e r a i n f a l l
t h e n t h e l a t e r a l f l o w s must b e modeled s e p a r a t e l y and u s e d w i t h t h e t r a n s f e r f u n c t i o n model.
B e s i d e s , t h e t r a n s f e r f u n c t i o n g a i n of t h e
s y s t e m g i v e n by equ. ( 6 a ) , which may b e used as a n i n d i c a t i o n of t h e mean l a t e r a l f l o w c o n t r i b u t i o n from t h e b a s i n (Whitehead e t a l . (1979)), must h e a l s o c o r r e c t e d .
303 The lateral inflow model. The residual series St of the model in equ. ( 6 ) is a combination of is the
lateral inflows and noise and is given by equ. (12), where t'
=
qjyt
<j
7t
(12)
'jyt
the observed outflow. Thus the lateral inflows LSt estimated and q j,t must be filtered from the noise in St. This can be accomplished by a Kalman filter with time varying parameters (Schwartz and Shaw ( 1 9 7 5 ) ) . The model used for latersl inflow estimation is given in equ. (13), for which the mean daily basin rainfall Pt is the input and at and bt LSt
=
atLSt-1
+
btPt'
I
(13)
lat < 1,
parameters of the filter. Based on the following two conditions, Schwartz and Shaw (1975) give the expressions for the estimation of the parameters of the filter:
(1) Input series Pt is a stationary stochastic process. (2) Sample series St can be modeled as a first order autoregressive process driven by a zero-mean white noise w t' St = aSt-l
+ Wt
(134
Thus the noise variance u2 is related to the variance u2 of the sample w
series through the formula: u 2
u
=
u2 (1-a2>, where s
autocorrelation coefficient of the sample series.
a
is the first order
The relation between
sample series St and the input series P
is obscured by an additive t zero-mean observation white noise nt, as shown in equ. (14). Pt
= S
(14)
t +nt
The noise variance u 2 is related to the variances u2 and 0: of series n P Pt and Sty by the expression: u 2 = u 2 - u;, " P Under these assumptions, the expressions for the parameters a and bt t
in equ. (13) are as follows.
304 where
i s a l r e a d y d e f i n e d and A i s t h e r a t i o of t h e v a r i a n c e of t h e
cl
white-noise w
A
= 02/02=
u
n
t o t h e v a r i a n c e of t h e o b s e r v a t i o n n o i s e u2. n
t
a2(1-a2)/02 s P
The estimated b
t
- u2
S
v a l u e s from e q u . (15) h a s v e r y l i m i t e d v a r i a b i l i t y ,
s i n c e i t r a p i d l y converges t o a s t e a d y v a l u e ,
E q u a t i o n (15) i s v a l i d
o n l y when equ. (14) i s r e a l i z e d , i . e . , when n o t h i n g else b e s i d e s n o i s e
n
i n t e r f e r e s between t h e i n p u t series P and t h e s i g n a l St. T h i s s i t u t t a t i o n can b e o n l y r e a l i z e d when t h e b a s i n i s s a t u r a t e d , when a l l com-
p o n e n t s such as i n f i l t r a t i o n , i n t e r c e p t i o n , o v e r l a n d f l o w , e t c . r e a c h t h e i r maximum v a l u e s . P a o t h e r o b s e r v a t i o n i s t h a t as b i s smaller t h a n t i t s s t e a d y v a l u e , equ. (15) i s a n i n c r e a s i n g f u n c t i o n on t h e ( b b ) t ' t-1 plane. S i n c e b can b e p h y s i c a l l y i n t e r p r e t e d a s t h e d a i l y r u n o f f t c o e f f i c i e n t of t h e b a s i n ( r u n o f f - r a i n f a l l r a t i o ) , i t i s s t r o n g l y r e l a t e d t o rainfall.
E x p e r i e n c e h a s shown t h a t t h e r u n o f f c o e f f i c i e n t of t h e
b a s i n i n c r e a s e s (up t o a s t e a d y l i m i t i n g v a l u e ) , when r a i n f a l l i n c r e a s e s o r when i t r e m a i n s c o n s t a n t ,
T h e r e f o r e , equ.
(15) i s more
u s e f u l f o r p e r i o d s of heavy ( c a p a b l e t o s a t u r a t e t h e b a s i n ) and i n c r e a s ing or constant r a i n f a l l .
For t h e o t h e r p e r i o d s of t h e y e a r some modi-
f i c a t i o n s o f equ. (15) a r e n e c e s s a r y . R e p l a c i n g LStTl
i n e q u . (13) by i t s e x p r e s s i o n from t h e same e q u a t i o n
a t t h e p r e v i o u s s t e p and a g a i n d o i n g t h e same f o r L S t - 2 , e t c , g e t t h e f o l l o w i n g e x p r e s s i o n f o r t h e Kalman f i l t e r equ. r
LSt = a t t t - l
k-1
... at-k LS t-k-1
1
(atat-l
L
t h e backwards s h i f t o p e r a t o r . and can b e o m i t t e d .
one c a n
(18) where B i s -
... a t -n Bn+')+l
btPt,
(18)
J
The f i r s t t e r m of equ. (18) i s v e r y small
Therefore, t h e f i l t e r f o r t h e lateral flow contri-
b u t i o n becomes equ. (19) where k i s t h e memory of t h e r a i n f a l l - l a t e r a l inflow process.
1
r
LSt =
1
k- 1
1
n=O
L
(atat-l
... a t -n Bn+')+l
btPt, -J
305 The i n p u t - o u t p u t model f o r d a i l y flow s y n t h e s i s . Adding t h e s i g n a l LS
t
t o t h e r i g h t hand s i d e of equ. ( 6 ) , improved
estimates are o b t a i n e d f o r t h e d a i l y o u t f l o w s
4j,t .
The new r e s i d u a l s
c a l l e d now second stage r e s i d u a l s of j,t’ The e v e n t u a l t h e model, must b e u n c o r r e l a t e d t o t h e r a i n f a l l series P,.
R
from t h e h i s t o r i c a l v a l u e s q
t
a u t o c o r r e l a t i o n s t r u c t u r e of t h i s r e s i d u a l series i s due t o t h e p r e s e n c e of a n o i s e component R
which can b e removed by t h e f o l l o w i n g ARMA
t’
(p,q) model, equ. (20) (Box and J e n k i n s (1970), Kashyap and Rao (1976))
where $ ( B ) , B(B) and
E
t
a r e polynomials of B of o r d e r p and q c o r r e s p o n d i n g l y ,
i s a zero-mean w h i t e n o i s e series r e p r e s e n t i n g t h e f i n a l t h i r d
s t a g e r e s i d u a l s of t h e model.
Thus, f i n a l l y , t h e i n p u t - o u t p u t model f o r
d a i l y f l o w s s y n t h e s i s t a k e s t h e form:
DATA USED I N THE STUDY The model i s t e s t e d by u s i n g t h e d a t a of a r e a c h of Aoos river i n Northern Greece, between t h e i n l e t s t a t i o n ( i ) a t Vovoussa and t h e o u t The a r e a of t h e d r a i n a g e b a s i n i s 4 6 1 km2
let s t a t i o n ( j ) a t Konitsa.
and t h e r e a c h i s 34 k m l o n g .
D a i l y f l o w d a t a are a v a i l a b l e a t t h e
Vovoussa and K o n i t s a s t a t i o n s from 1965 and 1971 r e s p e c t i v e l y .
Daily
r a i n f a l l measured a t P a d e s , D i s t r a t o n and I l i o c h o r i s t a t i o n s i n t h e b a s i n are a l s o a v a i l a b l e .
However, o n l y t h e s t a t i o n a t Pades i s con-
t i n u a l l y working s i n c e 1965, and t h e d a t a from t h e o t h e r two s t a t i o n s a r e discontinuous.
Furthermore, t h e Pades s t a t i o n can b e c o n s i d e r e d as
a r e p r e s e n t a t i v e s t a t i o n , b e c a u s e i t i s l o c a t e d a t t h e c e n t e r of t h e b a s i n and a t an a l t i t u d e c l o s e t o t h e mean b a s i n a l t i t u d e .
Consequently,
t h e d a i l y r a i n f a l l a t Pades s t a t i o n , a f t e r m u l t i p l i c a t i o n by a p o i n t - t o s u r f a c e r a i n f a l l c o e f f i c i e n t of 0 . 9 1 i s assumed t o g i v e t h e mean d a i l y basin r a i n f a l l .
R a i n f a l l v a l u e s were c o n v e r t e d t o m3/sec and u s e d .
y e a r s of d a t a (1971-77)
of d a i l y f l o w s and r a i n f a l l were used i n t h e
Six
3 06 study.
The f i r s t f o u r y e a r s (1971-75)
of d a t a were used f o r t h e c a l i -
b r a t i o n of t h e model and t h e l a s t two y e a r s (1975-77)
of d a t a were used
for verification.
CALIBRATION OF THE MODEL T r a n s f e r f u n c t i o n model c a l i b r a t i o n . The a u t o and c r o s s - c o r r e l a t i o n c o e f f i c i e n t s of t h e i n f l o w , outflow and t h e i r d i f f e r e n c e d series i n d i c a t e d a weakly n o n s t a t i o n a r y behavior. Daily series are u s u a l l y s e a s o n a l w i t h y e a r l y and within-the-year city.
cycli-
I n t h i s p a r t i c u l a r c a s e , t h e a u t o c o r r e l a t i o n s t r u c t u r e of t h e
d a i l y flows i n d i c a t e d only weakly p e r i o d i c behavior.
For t h i s reason
only a nonseasonal f i r s t s r d e r d i f f e r e n c i n g o p e r a t i o n was i n i t i a l l y tried.
The e f f i c i e n c y of t h i s d i f f e r e n c i n g a l o n e t o s i g n i f i c a n t l y re-
duce t h e p e r i o d i c component of t h e s e r i e s i s checked by t h e a u t o and cross-correlograms of t h e d i f f e r e n c e d s e r i e s .
The correlograms of t h e
f i r s t d i f f e r e n c e d series e x h i b i t e d nonperiodic s t a t i o n a r y behavior, and f l u c t u a t e d i n s i d e t h e 97.5% confidence l i m i t s ( J e n k i n s and Watts (1969)) a f t e r t h e f i r s t few l a g s .
B e s i d e s , t h e f i r s t and second o r d e r autocor-
r e l a t i o n c o e f f i c i e n t s of t h e i n f l o w and outflow d i f f e r e n c e d series i n d i c a t e d t h e nonperiodic behavior of t h e a u t o c o r r e l a t i o n c o e f f i c i e n t s . Therefore, t h e nonseasonal f i r s t o r d e r d i f f e r e n c i n g was considered adequate t o render t h e flow s e r i e s s t a t i o n a r y . The a u t o and cross-covariance f u n c t i o n s of t h e d i f f e r e n c e d s e r i e s a r e used t o e s t i m a t e t h e weights of t h e impulse response f u n c t i o n of t h e system equ. ( 7 ) . analysis.
The maximum l a g m i n equ. ( 7 ) w a s found by s e n s i t i v i t y
It w a s found t h a t t h e maximum v a l u e of m was about 10 and
t h e impulse response f u n c t i o n o r d i n a t e s U11, U12, c a l l y zero.
..., e t c ,
were p r a c t i -
The impulse response v a l u e s i n d i c a t e d t h a t t h e f i r s t two
v a l u e s of t h e impulse response f u n c t i o n a r e by f a r more important t h a n the rest. The cross-correlogram of t h e d i f f e r e n c e d inflow-outflow p r o c e s s e s which had t h e h i g h e s t v a l u e a t z e r o l a g , suggested t h a t t h e d e l a y parameter b i s zero.
The memory of t h e inflow-outflow p r o c e s s was s i g n i f i -
c a n t upto t h e t h i r d l a g .
I n o t h e r words, t h e parameter s w a s e q u a l t o
307 three.
It h a s been a l r e a d y shown t h a t t h e o r d e r r of t h e inflow-outflow
t r a n s f e r p r o c e s s can be assumed t o b e e q u a l t o one.
T h i s assumption w a s
a l s o v e r i f i e d by u s i n g t h e e s t i m a t e d impulse response f u n c t i o n , which
U.-GIUj-l=O, w i t h 61=0.36, 3 Besides, examining t h e d i f f e r e n c e equation: Uj-6Uuj-1-62Uj-2=0
s a t i s f i e s t h e d i f f e r e n c e equation:
for j>3. for j>3,
f o r a second o r d e r model, i t w a s found t h a t , f o r j = 4 , and j = 5 , t h i s equation g i v e s a v e r y s m a l l v a l u e of 0.006 f o r 6 2 , which i s i n s i g n i f i cant compared t o 6
1' These r e s u l t s w e r e used t o f i x t h e v a l u e s of r , s and b .
By u s i n g
r = 1, s = 3 , b = 0 , equ. (8) w e r e solved t o - e s t i m a t e t h e parameters Values of t h e s e e s t i m a t e s w i t h t h e i r s t a n d a r d e r r o r s i n j' = 0.36 ( 0 . 0 3 ) , w o = 1.81 (O.O7Y, to1 = 0.54 ( 0 . 0 4 1 , parentheses are: G i and w
w 2 = 0.12
(0.02),
w3 = 0.06
The t r a n s f e r f u n c t i o n g a i n of t h e
(0.01).
(1-0.36)- 1 (1.81-0.54-0.12-0.06)
system i s t h u s g =
= 1.70.
These i n i -
t i a l estimates may be r e f i n e d by u s i n g o p t i m i z a t i o n t e c h n i q u e s (Box and Jenkins (1970)).
But i f t h e t h i r d s t a g e r e s i d u a l s e r i e s of t h e model
a r e a w h i t e n o i s e s e r i e s , t h e s e e s t i m a t e s would be very c l o s e t o t h e optimal estimates.
Estimation of t h e parameters of t h e l a t e r a l i n f l o w model. Applying t h e t r a n s f e r f u n c t i o n model t o t h e d a t a of t h e c a l i b r a t i o n period, one can o b t a i n t h e f i r s t s t a g e r e s i d u a l series S The cross-correlogram of r e s i d u a l s S ed s t r o n g dependence of S
t
t
t
of equ. (12).
and b a s i n r a i n f a l l s e r i e s e x h i b i t -
on r a i n f a l l , which i s p a r t l y due t o t h e pre-
sence of t h e l a t e r a l flow c o n t r i b u t i o n i n t h e r e s i d u a l s .
The c r o s s -
c o r r e l a t i o n between S and i n f l o w s on t h e o t h e r hand, i n d i c a t e d v e r y t weak dependence of t h e S series on t h e i n f l o w s . The correlogram between t f i r s t s t a g e r e s i d u a l s and r a i n f a l l had t h e maximum v a l u e a t l a g z e r o r a p i d l y decayed a f t e r t h e t h i r d l a g , which implied t h a t t h e memory parameter k of t h e model of equ. (19) w a s e q u a l t o t h r e e . The parameters a
t
and bt of t h e model f o r l a t e r a l i n f l o w a r e estimated
from e q u a t i o n s ( 1 5 ) and (16), a f t e r v e r i f y i n g t h a t t h e two p r e v i o u s l y d e s c r i b e d (Schwartz and Shaw (1975)) c o n d i t i o n s a r e met.
The f i r s t
c o n d i t i o n , about t h e s t a t i o n a r i t y and t h e s t o c h a s t i c i t y of t h e i n p u t s e r i e s i s m e t , because t h e series of d a i l y r a i n f a l l a t Pades can be con-
308 s i d e r e d t o be a s t a t i o n a r y s e r i e s .
T h i s was v e r i f i e d by examining t h e
correlogram of t h e r a i n f a l l series which f l u c t u a t e d i n s i d e t h e 97.5% confidence l i m i t s a f t e r t h e f i r s t few l a g s .
The second c o n d i t i o n i s
a l s o m e t , as t h e correlogram of t h e f i r s t s t a g e r e s i d u a l series S
t an e x p o n e n t i a l form and resembles t h e correlogram of a f i r s t o r d e r
had
Therefore, e q u a t i o n s (15) and (6) can b e used t o e s t i -
Markov p r o c e s s .
m a t e t h e parameters a
t
The parameters a and A a r e e s t i m a t e d
and b t .
t o be e q u a l t o 0.36 and 0.10 r e s p e c t i v e l y . Equation (15) can be w r i t t e n a s equ. (22) a f t e r d e f i n i n g L=1/1+A+c12 bt-l'
and is v a l i d f o r time p e r i o d s of i n c r e a s i n g o r c o n s t a n t r a i n f a l l ,
b t = l - L
(22)
g r e a t e r than o r e q u a l t o a v a l u e which s a t u r a t e s t h e b a s i n , a s explained
earlier.
For t h e rest of t h e p e r i o d s , equ. (15) and more s p e c i f i c a l l y
i t s s l o p e on t h e (bt,btel)
p l a n e must be modified,
The m o d i f i c a t i o n s
a r e made i n accordance t o t h e changes which have been observed i n t h e r e l a t i o n s h i p between t h e h i s t o r i c a l d a i l y r a i n f a l l and t h e runoff coeff i c i e n t (which i s t h e p h y s i c a l analog of b t ) of t h e Aoos r i v e r b a s i n . S e v e r a l such m o d i f i c a t i o n s w e r e used i n t h e p r e s e n t s t u d y and a r e d i s cussed helow. The s l o p e of equ. ( 1 5 ) , u s i n g t h e t r a n s f o r m a t i o n L , i s given by equ. (23). dbt/dbt-l
= a2L2
(1) For r a i n f a l l t' v a l u e s less than a c r i t i c a l v a l u e , below which n o d i r e c t s u r f a c e runoff T h e following m o d i f i c a t i o n s a r e made i n e s t i m a t i n g b
i s produced, b
T h i s c r i t i c a l v a l u e w a s found t o t-1' be approximately e q u a l t o 0.5 mm/day f o r t h e p r e s e n t d a t a . (2) Above t
is s e t equal t o b
t h i s c r i t i c a l v a l u e and f o r i n c r e a s i n g o r c o n s t a n t r a i n f a l l up t o a v a l u e which s a t u r a t e s t h e b a s i n , t h e i n c r e a s i n g r a t e of t h e runoff coeff i c i e n t of t h e b a s i n and t h e r e f o r e , b+ i s estimated t o be e q u a l t o t h e s l o p e given by equ. (23)
t o become a(a
2
1) times s m a l l e r ; namely
i t becomes m i l d e r and e q u a l t o :
(dbt/db,-l)
= a3L2
(24)
309 I n t e g r a t i n g e q u . ( 2 3 ) , one g e t s t h e e x p r e s s i o n f o r bt i n e q u . ( 2 5 ) , bt
=
(25)
a(l-L)
The upper zone i n m o d e r a t e l y c o v e r e d b a s i n s , w i t h s t e e p s l o p e s and w i t h reasonably uniform r a i n f a l l throughout t h e y e a r , i s approximately s a t u r a t e d f o r d a i l y r a i n f a l l v a l u e s e q u a l t o : 0.06(4+Ft/8), y e a r l y b a s i n r a i n f a l l i n i n c h (25.4 mm), e q u a l h e r e t o 900 m.
with
Ft
t h e mean
(Crawford and L i n s l e y (1966))
Therefore, t h e r a i n f a l l value f o r t h e s a t u r a t i o n
(3) F o r d e c r e a s -
of t h e b a s i n i s a p p r o x i m a t e l y e q u a l t o 1 2 . 0 m / d a y , i n g r a i n f a l l , t h e r u n o f f c o e f f i c i e n t and t h e r e f o r e b
t
is estimated t o
d e c r e a s e a l s o and t h e s l o p e of e q u . (23) i s c h a n g i n g d i r e c t i o n ; t h e exp r e s s i o n found t o f i t t h e r e c e s s i o n limb o f b (dbt/dbt-l)
=
w e l l i s t h e following:
t
1-m(a2L2)’
(26)
> 12.0 m / d a y , it i s estimated t h a t b i s slowly decreasing, t t a c c o r d i n g t o a r a t e g i v e n by e q u . (26) w i t h m = 1, ii = 1, ( m i l d s l o p e ) ,
For P
and t h e e x p r e s s i o n f o r b
b
= b
t
For P
t-1
t
t
becomes a p p r o x i m a t e l y :
(1-u2L2)
(27)
< 12.0 m / d a y ,
i t i s e s t i m a t e d t h a t b,
is rapidly decreasing,
a c c o r d i n g t o a r a t e g i v e n by e q u , ( 2 6 ) w i t h m = 1, 1~ = 1 / 2 , ( s t e e p s l o p e ) , and t h e e x p r e s s i o n f o r b
t
becomes a p p r o x i m a t e l y :
S t a r t i n g w i t h a n a r b i t r a r y v a l u e f o r bl and u s i n g t h e p a r a m e t e r s a and A t h e t i m e v a r y i n g p a r a m e t e r b and h e n c e a t are e s t i m a t e d . The e f f i c i e n c y t of t h e c a l i b r a t e d f i l t e r o f e q u . (19) t o e x t r a c t t h e l a t e r a l i n f l o w s LS
t
from t h e series S
t
i s t e s t e d by i n v e s t i g a t i n g t h e c r o s s - c o r r e l o g r a m
of t h e second s t a g e r e s i d u a l series R series.
The c r o s s - c o r r e l o g r a m o f R
t
t
o f t h e model w i t h t h e r a i n f a l l
and Pt i n d i c a t e d t h a t t h e s e two
series a r e u n c o r r e l a t e d .
The mean l a t e r a l i n f l o w s i g n a l v a l u e f o r t h e c a l i b r a t i o n p e r i o d w a s found t o b e e q u a l t o 3.70 m 3 / s e c , inflow value.
which a c c o u n t s f o r 40% of t h e mean
I n o t h e r words, t h e i n f l o w - o u t f l o w t r a n s f e r f u n c t i o n g a i n
g found e q u a l t o 1 . 7 0 , must b e c o r r e c t e d t o : g ’
=
1.70
+
0.40 = 2.10.
The i n t e r m e d i a t e b a s i n c o n t r i b u t e s 110% of t h e mean f l o w a t K o n i t s a s t a t i o n and n o t 70%, a s t h e g a i n g i s i n d i c a t i n g .
310 The n o i s e model. The a u t o c o r r e l a t i o n s t r u c t u r e of t h e r e s i d u a l series Rt9 resembled t h e This
a u t o c o r r e l a t i o n s t r u c t u r e of a second o r d e r a u t o r e g r e s s i v e process. means t h a t t h e a u t o c o r r e l a t e d s i g n a l R t ,
i n h e r e n t i n t h e r e s i d u a l s , can
be modeled as i n equ. (29).
The e s t i m a t e d parameters
are: +l = + 2 = 0.25 and
I $ ~ ,
4, and
a
E'
t h e s t a n d a r d d e v i a t i o n of n o i s e ,
= 7.5 m3/sec.
0
Diagnostic checking of t h e model, The w h i t e n o i s e series (21),
E
t
of equ, (29) may be e s t i m a t e d by u s i n g equ.
Without b and B(B), equ. (21) g i v e s t h e f o l l o w i n g e x p r e s s i o n f o r
the residuals
2t :
Applying (30) t o t h e d a t a of t h e c a l i b r a t i o n p e r i o d , series
gt
i s ob-
t a i n e d and then used f o r d i a g n o s t i c checking of t h e model and of t h e e f f i c i e n c y of i t s parameters e s t i m a t e s .
I f t h e model i s c o r r e c t and i t s
parameters have been e f f i c i e n t l y e s t i m a t e d , t h i s r e s i d u a l series must be a zero-mean white n o i s e series.
In addition it has t o s a t i s f y t h e
Darbin-Watson s t a t i s t i c d, given by:
(31) which w a s found (Kendall (1973)) t o b e e q u a l t o z e r o f o r a u t o c o r r e l a t e d sequences and c l o s e t o two f o r random sequences w i t h N v a l u e s . Indeed,
Et
was found t o b e a zero-mean w h i t e n o i s e s e r i e s , w i t h v a r i -
ance e q u a l t o 7.5 m 3 / s e c and w i t h s t a t i s t i c d e q u a l t o 1.96.
The empi-
r i c a l p r o b a b i l i t y d e n s i t y f u n c t i o n of t h e series had a h i g h peak and w a s approximately symmetrical and bounded.
One can assume t h a t i t s high
k u r t o s i s i n comparison w i t h a normal d i s t r i b u t i o n may have r e s u l t e d from over o r under-removal
of harmonics i n p e r i o d i c components of t h e d a i l y
flow series (Yevjevich (1976)).
Based only on t h e skewness t e s t f o r
normality d i s c u s s e d by H i p e l e t a 1 (1977), t h e skewness c o e f f i c i e n t w a s found t o be n o t s i g n i f i c a n t l y d i f f e r e n t from z e r o a t 97.5% confidence
311 T h e r e f o r e , series
level.
E~
i s assumed t o be normal, which can b e gen-
e r a t e d by t h e f o l l o w i n g mechanism, where t i are s t a n d a r d normal v a r i a t e s from N ( 0 , l ) .
The f i n a l model t a k e s t h e form:
(1-0.36B)-1(l,81-0.54B-0.12B
=
2 -0.06B 3 ) q i , t
-I-
‘j,t
S y n t h e s i s of d a i l y f l o w s i n t h e c a l i b r a t i o n p e r i o d . The e f f i c i e n c y of t h e model of ( 3 3 ) i n s y n t h e s i z i n g d a i l y f l o w s i n t h e c a l i b r a t i o n p e r i o d was checked by t h e f o l l o w i n g c r i t e r i a : (1) The mean a c c u r a c y i n s i m u l a t i o n , which i s g i v e n by:
where q
j ,t
and
4j , t a r e
and N t h e sample s i z e .
r e s p e c t i v e l y t h e h i s t o r i c a l and s i m u l a t e d flows T h i s v a l u e w a s found t o b e e q u a l t o 14%. The
a p p l i c a t i o n of t h e model i n ( 3 3 ) w i t h o u t t h e l a t e r a l i n f l o w s and t h e n o i s e model, i n o t h e r words o n l y t h e t r a n s f e r f u n c t i o n model equ, ( 6 ) , had a n M.A.
v a l u e of 27%.
A d d i t i o n of t h e o t h e r models t h u s s i g n i f i c a n t -
l y improves t h e a c c u r a c y o f t h e t r a n s f e r f u n c t i o n model.
(2)
The a b i l i t y of t h e model t o p r e s e r v e some h i s t o r i c a l s t a t i s t i c a l characteristics.
The s t a t i s t i c a l c h a r a c t e r i s t i c s of t h e h i s t o r i -
c a l and e s t i m a t e d d a t a were a l s o e v a l u a t e d f o r d i f f e r e n t y e a r s .
The
n u l l h y p o t h e s i s (Benjamin and C o r n e l l ( 1 9 7 0 ) ) , t h a t each e s t i m a t e d chara c t e r i s t i c i s n o t s i g n i f i c a n t l y d i f f e r e n t from t h e h i s t o r i c a l
one can
be a c c e p t e d a t t h e 90% c o n f i d e n c e l e v e l f o r t h e mean, skewness, k u r t o s i s and f i r s t o r d e r a u t o c o r r e l a t i o n and inflow-outflow
cross-correlation
c o e f f i c i e n t s and a t t h e 95% c o n f i d e n c e l e v e l f o r t h e v a r i a n c e .
MODEL VERIFICATION AND APPLICATIONS
S y n t h e s i s of d a i l y d a t a which were n o t used f o r c a l i b r a t i o n , The model (33) w a s used t o s y n t h e s i z e d a i l y f l o w s which w e r e n o t used to calibrate it.
The s y n t h e t i c o u t f l o w s e s t i m a t e d by u s i n g t h e model The e f f i c i e n c y of t h e model i s
w e r e compared t o t h e observed o u t f l o w s .
312 a g a i n checked by t h e two p r e v i o u s l y d e s c r i b e d c r i t e r i a . and t h e h i s t o r i c a l series were i n good a g r e e m e n t ,
The s i m u l a t e d
The mean a c c u r a c y o f
t h e model w a s good and found t o b e e q u a l t o 17%. The a p p l i c a t i o n of o n l y t h e t r a n s f e r f u n c t i o n e q u . ( 6 ) gave a n a c c u r a c y o f 30%.
Thus
once a g a i n t h e a d d i t i o n of t h e two o t h e r models t o t h e t r a n s f e r f u n c t i o n model s i g n f i c a n t l y improves t h e a c c u r a c y o f d a i l y f l o w e s t i m a t i o n . F i n a l l y , t h e model p r e s e r v e s t h e h i s t o r i c a l s t a t i s t i c a l c h a r a c t e r i s t i c s of t h e series a t t h e 95% c o n f i d e n c e l e v e l f o r t h e v a r i a n c e and a t t h e 90% level f o r t h e rest of t h e c h a r a c t e r i s t i c s . V e r i f i c a t i o n of t h e model f o r d a i l y f l o o d r o u t i n g . S i x s e p a r a t e f l o o d h y d r o g r a p h s , r e c o r d e d a t t h e i n l e t of t h e b a s i n
d u r i n g t h e y e a r 1976-77, were r o u t e d i n o r d e r t o estimate t h e c o r r e sponding o u t f l o w h y d r o g r a p h s a t t h e o u t l e t s t a t i o n .
H i s t o r i c a l outflow
d a t a w e r e n o t used i n t h e c o m p u t a t i o n s and t h e s y n t h e t i c h y d r o g r a p h s
w e r e compared w i t h t h e o b s e r v e d h y d r o g r a p h s .
The mean a c c u r a c y i n
r o u t i n g i s a g a i n measured by ( 3 4 ) , where N i s t h e d u r a t i o n of each hydrograph i n d a y s .
Another measure of a c c u r a c y w a s a l s o u s e d i n t h i s
p h a s e of t h e s t u d y .
It i s t h e "peak accuracy",
d e f i n e d as:
(35)
where qp
i':
and a r e t h e h i s t o r i c a l and s y n t h e t i c peak f l o w s r e s p e c J ,t The mean and peak a c c u r a c i e s e s t i m a t e d from ( 3 4 ) and ( 3 5 ) are
j,t
tively.
g i v e n f o r e a c h of t h e s i x h y d r o g r a p h s .
It w a s a g a i n a p p a r e n t t h a t t h e r e
i s a s i g n i f i c a n t improvement i n t h e a c c u r a c y of t h e model w i t h t h e
a d d i t i o n of t h e l a t e r a l i n f l o w and n o i s e models.
CONCLUSIONS The f o l l o w i n g c o n c l u s i o n s are a r r i v e d a t from t h i s s t u d y ,
(1) The i n p u t - o u t p u t s t o c h a s t i c model developed i n t h i s s t u d y can b e e f f i c i e n t l y a p p l i e d f o r d a i l y f l o w s and d a i l y f l o o d hydrograph s y n e t h e s i s . The model p r e s e r v e s t h e h i s t o r i c a l y e a r l y mean, skewness, k u r t o s i s and
313 f i r s t o r d e r a u t o c o r r e l a t i o n and inflow-outflow c r o s s - 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 t h e 90% confidence l e v e l and t h e v a r i a n c e a t t h e 95% level,
(2)
The a d d i t i o n of t h e l a t e r a l inflow and t h e n o i s e models t o t h e
inflow-outflow t r a n s f e r f u n c t i o n model s i g n i f i c a n t l y improves i t s accuracy i n d a i l y flow e s t i m a t i o n .
The a d d i t i o n of t h e l a t e r a l inflow
model i s necessary whenever t h e l a t e r a l inflow component i s i d e n t i f i e d i n t h e r e s i d u a l s of t h e inflow-outflow t r a n s f e r f u n c t i o n model, (3)
The model connected t o an automatic network w i t h a s m a l l computer
can be o p e r a t i o n a l l y used o n - l i n e f o r r e a l time d a i l y flows e s t i m a t i o n , which i s e s p e c i a l l y u s e f u l a t s i t e s w i t h r e s e r v o i r s i n o p e r a t i o n ,
(4)
The model, because of i t s l i m i t e d d a t a need, i s e s p e c i a l l y u s e f u l
f o r e s t i m a t i n g runoff from watersheds w i t h h i g h l y v a r i a b l e p h y s i c a l c h a r a c t e r i s t i c s (roughness, r a t i n g c u r v e s , e t c . )
and where l i m i t e d d a t a
are available,
REFERENCES Benjamin, T.R. and A.C. C o r n e l l , 1970. P r o b a b i l i t y , S t a t i s t i c s and Decis i o n f o r C i v i l Engineers, McGraw-Hill Co., New York, 684 pp. Box, G.P. and G.M. J e n k i n s , 1970. Time S e r i e s Analysis-Forecasting and Control, Holden-Day Co., San F r a n c i s c o , 553 pp, Crawford, N.H. and R.K. L i n s l e y , 1966. D i g i t a l Simulation i n Hydrology: S t a n f o r d Watershed Model I V , Tech. Rept. No. 39, Stanford U n i v e r s i t y , C a l i f o r n i a , 210 pp. Eagleson, S . P . , PP *
1970. Dynamic Hydrology, McGraw-Hill
Co., New York, 462
H i p e l , K.W., A . I . McLeod and W.C, Lennox, 1977. Advances i n Box-Jenkins Modeling, 1-Model C o n s t r u c t i o n , Water Resour, Res., 1 3 ( 3 ) , 567-576, and D.G. Watts, 1969. S p e c t r a l Analysis and I t s AppliJenkins, G.M. c a t i o n s , Holden-Day Co., San Francisco, 525 pp. Kashyap, R . L . and A.R. Rao, 1976, "Dynamic S t o c h a s t i c Models from Empirical Data", Academic P r e s s , New York, New York, Kendall, M . G . ,
1973. Time S e r i e s , G r i f f i n , London, 330 pp.
314 Nemec, J., 1972. Engineering Hydrology, McGraw-Hill Co:,
England, 316 pp.
Schwartz, M, and L. Shaw, 1975. Signal Processing, Discrete Spectral Analysis, Detection and Estimation, McGraw-Hill Co., New York, 396 pp. Whitehead, P., G . Hornberger and R. Black, 1979, Effects of Parameter Uncertainty in a Flow Routing Model, Hydrol. Sc. Bull., 2414, 4 4 5 - 4 6 3 , Yevjevich, V . , 1976. Structure of Natural Hydrologic Time Processes, In: H.W. Shen (Editor), Stochastic Approaches to Water Resources, Vol, I:2. 1-2.59.
315
ANALYSIS OF FLOOD SERIES BY STOCHASTIC MODELS P. VERSACE, M. FIORENTINO AND F. ROSS1
Dip. Difesa del Suolo, Universita della Calabria, and 1st. Idraulica e Costruzioni Idrauliche, Universita di Napoli , Italy
ABSTRACT Flood analysis for regions, l i k e Southern I t a l y , where the annual flood s e r i e s exhibits o u t l i e r s (and, then, high skewness), associated with disastrous storms, requires building s u i t a b l e stochastic models. I n such cases, the usual simple model (Model A ) , which assumes the largest annual flood t o be the maximum of a Poissonian nunber of indepen dent random variables w t h common exponential distribution function, proves t o be inadequate Better models can be b u i l t b y replacing the hypotheses on which Model A i s based with others, phenomenologically closer t o r e a l i t y , name y, t h a t the number of exceedances i n a year i s s t i l l a non-homogeneous Poisson process, b u t the exceedance values are n o t i d e n t i c a l l y distributed random variables. Of the two models considered, i . e . , a time-dependent distribution f o r the exceedancetxagni tude (Yodel B ) and a mixed exponential distribution (Model C ) , the l a t t e r i s found t o give a b e t t e r s t a t i s t i c a l f i t . There i s also b e t t e r phenomenological support f o r Model C i n t h a t disastrous storms occur more rarely b u t with much larger i n t e n s i t i e s t h a n others , a n d they are accordingly better modelled as belonging t o d i f f e r e n t populations.'
INTRODUCTION The analysis of floods has been the object of investigations by many authors. Aniong the approaches followed, two d i s t i n c t ones, respectively empirical a n d t h e o r e t i c a l , may be i d e n t i f i e d . The former consists in guessing which theoretical d i s t r i b u t i o n best f i t s the observed frequency distribution of the l a r g e s t annual flood peak. Following t h i s approach, f o r example, the l o g Pearson Type-3 distribution has been recommended in the USA (U.S.W.R.C., 1 9 7 7 ) . While t h i s p a r t i c u l a r choice has met w i t h much adverse c r i t i c i s r (Landwehr e t a l . , 1 9 7 8 ) , more generally the empirical approach i s objected t o , i n principle, on several grounds. Thus, i t makes no use 1
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
316 o f t h e p a r t i a l f l o o d s e r i e s , w h i c h r e t a i n s more i n f o r m a t i o n t h a n i s t h e case w i t h t h e a n n u a l f l o o d s e r i e s ( T o d o r o v i c , l 9 7 8 ) . G o o d n e s s - o f - f i t t e s t s , used t o ccmpare t h e p e r f o r m a n c e o f d i f f e r e n t d i s t r i b u t i o n s , y i e l d l a r g e l y i n c o n c l u s i v e r e s u l t s even w i t h t h e l o n g e r r e c o r d s (N.E.R.C., 1 9 7 5 ) . F u r t h e r m o r e , t h e approach t a k e s n o a c c o u n t o f p h y s i c a l a s p e c t s o f t h e phenomena i n v e s t i g a t e d . F i n a l l y , w i t h t h e d i s t r i b u t i o n s most commonly used, one i s u n a b l e t o a c c o u n t f o r t h e h i g h o b s e r v e d v a r i a n c e o f t h e skewness o r f o r t h e p r e s e n c e o f o u t l i e r s , as a r e sometimes t h e c a s e i n t h e d a t a o b s e r v e d ( R o s s i and Versace, 1 9 8 1 ) . By c o n t r a s t , t h e t h e o r e t i c a l a p p r o a c h endeavours t o c o n s t r u c t a model, based on p h e n o m e n o l o g i c a l c o n s i d e r a t i o n s . The d a t a a r e t h e n used m e r e l y t o v e r i f y t h e model and, p o s s i b l y , t o s u g g e s t w h i c h i f any m o d i f i c a t i o n s a r e needed. I n r e c e n t y e a r s t h i s approach has undergone much development and i t w o u l d seem t o o f f e r t h e b e s t b a s i s f o r t h e a n a l y s i s and p r e d i c t i o n o f f l o o d s .
2
MATHEMATICAL MODELS L e t us c o i i s i d e r t h e s t o c h a s t i c p r o c e s s d e s c r i b e d b y t h e s t r e a m - f l o w h y d r o g r a p h c Q ( t ) ; t > O } and l e t us s e l e c t a base l e v e l q,,. The sequence o f t h e h y d r o g r a p h peaks above q o ( r e f e r r e d t o as t h e p r o c e s s o f exceedances) i s a marked p o i n t p r o c e s s ( S n y d e r , 1975) c h a r a c t e r i z e d by : , .where . ~i i s t h e i n s t a n t o f t i n e when - a sequence T , , T ~ , .. . , ~ i., t h e i - t h exceedance o c c u r s ; - a sequence Z,, Z2,. .., Z i , ..., where Z i = g ( ~ i )- q o i s t h e m a g n i t u d e o f t h e exceedance a t t i m e ~ i . B o t h cjccurrence t i m e s and exceedance v a l u e s a r e random v a r i a b l e s . The p r o c e s s i s f u r t h e r c h a r a c t e r i z e d b y t h e random v a r i a b l e K t , t h e number o f exceedances w i t h i n a f i x e d i n t e r v a l [ 0 , t ] , w h i c h can assume, f o r e v e r y t 2 0 , t h Q i n t e g e r v a l u e s k = 0, 1, 2, :
...
K t = max T i ;
so TKt;
T,'t>;
(1)
t201 i s a countino process.
L e t X i d e n o t e t h e m a g n i t u d e o f t h e l a r g e s t exceedance w i t h i n [0, tl, i.e.,
s o X ' i s t h e maximun among a random number o f randon; v a r i a b l e s . t A c c o r d i n g l y t h e d i s t r i b u t i o n o f X i w i l l depend on b o t h t h e c o u n t i n g
p r o c e s s ! K t ; t S 0 1 and t h e d i s t r i b u t i o n o f C Z i l . The p r o c e s s T K t ; t ? O > i s u s u a l l y assumed t o be a non-homogeneous
317 P o i s s o n c o u n t i n g p r o c e s s ( Z e l e n h a s i c , 1970; T o d o r o v i c and Z e l e n h a s i c , 1970; Dauty, 1972; N o r t h , 1980; R o s s i and Versace, 1981) w i t h k (At) exP(-ht) P$k) = P[Kt = k ] = , k=O,1,2, (3) k!
...
where
(4)
At = E [Ktl
i s t h e p a r a m e t e r f u n c t i o n o f t h e P o i s s o n p r o c e s s . The d e r i v a t i v e h ( t ) o f A t i s t h e i n t e n s i t y f u n c t i o n o f t h e process, i.e., At
t
,fo A(U)dU.
=
(5)
F o r a h i g h enough base l e v e l q,, t h e v a r i a b l e s Z i may be assumed t o be m u t u a l l y i n d e p e n d e n t . Many a u t h o r s ( Z e l e n h a s i c , 1970; T o d o r o v i c and Z e l e n h a s i c , 1970; D a u t y , 1972) i n t r o d u c e t h e f u r t h e r a s s u m p t i o n t h a t t h e Z i ' s a r e i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s , t h e i r common d i s t r i b u t i o n b e i n g o f t h e e x p o n e n t i a l t y p e : (0)
where
E[Z]
1/@.
=
(7)
i n t h i s c a s e t h e d i s t r i b u t i o n o f X k , t h e l a r g e s t exceedance w i t h i n
lo,
tl
Y
is (8)
I f t h e i n t e r v a l LO, t] i s a y e a r and we assume A t and B = C Xi,t f o l l o w s from ( 8 ) t h a t
=
exp[a(~-a~)]
where X d e n o t e s the l a r g e s t a n n u a l f l o o d . E q u a t i o n ( 9 ) i s t h e w e l l - k n o w n Gumbel's d i s t r i b u t i o n (model A ) w i t h p a r a m e t e r s cx and E . In niany cases t h e r e i s a good aqreement between Gunibel's d i s t r i b u t i o n and o b s e r v e d a n n u a l f l o o d s e r i e s , i n d i c a t i n g t h a t t h e a s s u m p t i o n s i n t r o d u c e d i n t h e d e r i v a t i o n above a r e b a s i c a l l y c o r r e c t . T h e r e a r e cases, however, when, u s i n g Gumbel's d i s t r i b u t i o n , t h e o b s e r v e d and f i t t e d d i s t r i b u t i o n s of t h e l a r g e s t annual f l o o d s e x h i b i t a p p r e c i a b l e d i s c r e p a n c y , and t h e need f o r more r e f i n e d models a r i s e s . One may p r o c e e d i n t h i s d i r e c t i o n , b y r e m o v i n g t h e s t r o n g e s t o f t h e
318
above hypotheses, namely, t h a t the Zi are dentical ly distributed random variables. As i t has been remarked by many authors (Todorovi c a n d Roussel e , 1 9 7 1 ; Rousselle, 9 7 2 ; North, 1930), the d s t r i b u t i o n o f Z i i s ac ual l y dependent on -ri. This time dependence may be a1 lowed f o r b i retaining a n exponential d i s t r i b u t i o n a n d then assupin? i t s parameter B t o be time dependent, i . e . ,
On t h i s assumption, the d i s t r i b u t i o n of the l a r g e s t exceedance X i within t] will be given by the expression
E),
This distribution will be referred t o as Model B . Another model deserving consideration i s obtained by assuming t h a t Zi a r i s e s as the mixture of two components, b o t h exponentially distributed. I t s d i s t r i b u t i o n i s accordingly written:
Z , and Z, being the component random variables a n d p the proportion o f Z , in the mixture.
The underlying! assumption of t h i s model allows f o r the existence of two d i s t i n c t types o f p r e c i p i t a t i o n , as i s the case in some regions l i k e Southern I t a l y (Penta e t a1 . , 1980). If the numbers of exceedances of the two components in a year, K, a n d K, follow Poisson processes of parar;;eters A , and A 2 respectively, we have
(13) where X ' i s the l a r g e s t exceedance in a year, and A1 A1
+
= P A2
As i s readily shown, the d i s t r i b u t i o n of the l a r g e s t annual flood may be wri tten :
i . e . , as the p r o d u c t of two Gumbel's d i s t r i b u t i o n s o f parameters
al,
319 E~ and a 2 , E~ r e s p e c t i v e l y , i.e., the largest-annual-flood d i s t r i b u t i o n s o f t h e i n d i v i d u a l components. T h i s t h i r d model s h a l l be r e f e r r e d t o as Model C .
3
APPLICATIONS The above t h r e e models were a p p l i e d t o a n a l y s i n g s e v e r a l s e r i e s of l a r g e s t annual f l o o d peaks i n S o u t h e r n I t a l y . As a t y p i c a l example, an a c c o u n t i s h e r e g i v e n o f such a n a l y s i s f o r t h e d a i l y f l o w s a t t h e Amato R i v e r , a t M a r i n o s t a t i o n ( C a l a b r i a ) , f o r w h i c h a 3 6 - y e a r r e c o r d i s a v a i l a b l e . Compared w i t h Gumbel's d i s t r i b u t i o n , t h e annual f l o o d s e r i e s e x h i b i t s an o u t l i e r , t h e l a r g e s t and n e x t l a r g e s t o b s e r v e d v a l u e s b e i n g x ( ~ )= 185 m 3 s e c - l and x ( ~ - ,= ~:l m 3 s e c - I r e s p e c t i v e l y . A s a r e s u l t , t h e o b s e r v e d skewness c o e f f i i e n ( i , = 2 . 8 0 ) i s much t o o h i g h f o r a Gumbel d i s t r i b u t i o n w i t h n = 36, f o r w h i c h t h e e x p e c t e d v a l u e and s t a n d a r d d e v i a t i o n o f t h e sample skewness c o e f f i c i e n t a r e E I T 1 ] = 0.88 and o[T1] = 0.54 r e s p e c t i v e l y ( F l a t a l a s e t a l . , 1 9 7 5 ) . To i n v e s t i g a t e t h e v a l i d i t y o f t h e h y p o t h e s e s on w h i c h Model A i s based, l e t us c o n s i d e r t h e p a r t i a l d u r a t i o n s e r i e s . The number o f i n d e p e n d e n t exceedances o c c u r e d was t a k e n t o e q u a l 74. Observed and t h e o r e t i c a l d i s t r i b u t i o n f u n c t i o n s o f t h e number of exceedances i n a y e a r a r e shown i n F i g . 1 = k = 2 . 0 6 ) . The good agreement between
(n
0.1 0.0
0
1
2
3
4
5
6
F i g . 1. Amato R i v e r a t M a r i n o . Observed ( s o l i d l i n e ) and t h e o r e t i c a l P o i s s o n ( b r o k e n l i n e ) d i s t r i b u t i o n f u n c t i o n s o f t h e number o f exceedances i n a y e a r (i = k = 2 . 0 6 ) . t h e d i s t r i b u t i o n s lends support t o t h e hypothesis t h a t t h e process T K t , t L 01 i s a P o i s s o n c o u n t i n g p r o c e s s . The c o n c l u s i o n i s a l s o w a r r a n t e d b y t h e v a l u e o f t h e t e s t s t a t i s t i c R, e q u a l l i n g t h e r s t i o o f t h e o b s e r v e d v a r i a n c e t o t h e o b s e r v e d mean ( R = 1 . 3 0 a g a i n s t t h e c r i t i c a l v a l u e a t t h e 5% l e v e l , R = 1 . 4 2 ) .
320 Observed and t h e o r e t i c a l ( e x p o n e n t i a l ) d i s t r i b u t i o n f u n c t i o n s f o r t h e m a g n i t u d e o f t h e exceedances a r e shown i n F i g . 2. The e s t i m a t e s
F i g . 2. Amato R i v e r a t M a r i n o . Observed ( b l a c k p o i n t s ) d i s t r i b u t i o n f u n c t i o n o f exceedance v a l u e s . E x p o n e n t i a l ( C u r v e E ) and m i xed e x p o n e n t i a l ( C u r v e ME) t h e o r e t i c a l d i s t r i b u t i o n f u n c t i o n s . D i s t r i b u t i o n f u n c t i o n s f o r t h e i n d i v i d u a l components ( C u r v e s 1 and 2 ) o f t h e ME model. f o r t h e p a r a m e t e r s q o and B i n ( 6 ) were o b t a i n e d by t h e b e s t l i n e a r u n b i a s e d e s t i m a t o r s (Sarhan, 1 9 5 4 ) . The t h e o r e t i c a l d i s t r i b u t i o n (curve E ) i s a poor f i t t o t h e observed data, p a r t i c u l a r l y a t t h e 1 a r g e s t Val ues whi ch a r e s i g n i f i c a n t l y u n d e r e s t i m a t e d . I n F i g . 3 t h e annual f l o o d s e r i e s a l s o i n d i c a t e s a p o o r f i t by Model A . F u r t h e r m o r e , were Model A a p p l i c a b l e , t h e o b s e r v e d l a r g e s t = 185 m3secm1 w o u l d c o r r e s p o n d t o a c u m u l a t i v e exceedance p r o b a bxi i 1n ly c l o s e t o u n i t y b o t h f o r t h e maximum a n n u a l f l o o d d i s t r i b u t i o n F x ( x ) and t h e maximum-in-36-years f l o o d d i s t r i b u t i o n Fn(X) * L e t us now pass t o c o n s i d e r a l t e r n a t i v e models, s t a r t i n g f r o m Model B w h i c h pays t r i b u t e t o t h e u n d e r l y i n g t i m e dependence o f t h e exceedance m a g n i t u d e . I n F i g . 4a a r e shown , superimposed, t h e exceedances o f t h e 36-year record. There a r e i n d i c a t i o n s s u p p o r t i n g t h e assumption of a p i e c e w i s e c o n s t a n t B ( t ) i n ( 1 1 ) ( s e e , e . g . , T o d o r o v i c a n R o u s s e l l e ,
321
0.011
'
qo
I/
50
/
I
1
100
150
x ( m3 s-1)
Fig. 3. Amato River a t Marino. Observed d i s t r i b u t i o n f u n c t i o n of annual flood s e r i e s (black p o i n t s ) . T h e o r e t i c a l d i s t r i b u t i o n f u n c t i o n s of maximum annual flow ( s o l i d c u r v e s ) and of t h e maximum-in-36-year flow (broken c u r v e s ) f o r Models A , B , C . D i s t r i b u t i o n f u n c t i o n s of the maximum annual flow f o r t h e components of Models C (curves 1 and
2) 1 9 7 1 ) and t h e costancy i n t e r v a l s may here be i d e n t i f i e d with one-month periods. Monthly values of the exceedance mean magnitude a n d mean number i n a y e a r may be read i n Figs. 4a and 4b r e s p e c t i v e l y . With such a piecewise c o n s t a n t B ( t ) in ( 1 1 ) the d i s t r i b u t i o n i s e a s i l y eval ua t e d . Consider f i n a l l y Model C . The parameters p , a, and fi2 i n ( 1 2 ) were estimated by t h e maximum-likelihood method (Hasselblad 1969). The d i s t r i b u t i o n f u n c t i o n of t h e individual components and r e s u l t i n g mixed-exponential ( M E ) d i s t r i b u t i o n f u n c t i o n thus obtained a r e shown i n Fig. 2 . I t i s seen t h a t t h e ME d i s t r i b u t i o n f i t s t h e observed d a t a much b e t t e r than i s t h e case of t h e p l a i n exponential d i s t r i b u t i o n .
322
180 Z(t
P"'i (rn3 s-1 8C
60
40
20 0
06,
7
F i g . 4. Amato R i v e r a t Y a r i n o . ( a ) Observed v a l u e s o f exceedances and t h e i r m o n t h l y means; ( b ) o b s e r v e d m o n t h l y means o f number o f exceedances i n a y e a r . As t h e e x p e c t e d v a l u e o f t h e number o f exceedances A = A 1 t A, i s known, a l l p a r a m e t e r s i n ( 1 5 ) may be d e t e r m i n e d . F i g u r e 3 a l s o shows t h e d i s t r i b u t i o n f u n c t i o n s of b o t h t h e maximum annual f l o w and t h e maximum-in-36-year f l o w when Model B o r Model C holds. O f a l l models c o n s i d e r e d , t h e l a t t e r (Model C ) shows t h e b e s t f i t t i n g o f t h e o b s e r v e d d a t a and, i n p a r t i c u l a r , i t w o u l d seem t o account f o r t h e l a r g e s t observed values.
4
CONCLUSIONS
The a n a l y s i s o f f l o o d d a t a f o r t h e Amato R i v e r and o t h e r r i v e r s o f Southern I t a l y suggests t h e f o l l o w i n g c o n c l u s i o n s : 1 ) The f l o o d peaks e x c e e d i n g a g i v e n base l e v e l may be t r e a t e d as
323
a marked p o i n t p r o c e s s . 2 ) The number o f exceedances K t w i t h i n a f i x e d i n t e r v a l o f t i m e [ O , t] i s a non-homogeneous P o i s s o n c o u n t i n g p r o c e s s . 3 ) I n some cases t h e exceedance v a l u e s Z i above a base l e v e l q o do n o t l e n d t h e m s e l v e s t o be m o d e l l e d as i n d e p e n d e n t random v a r i a b l e s w i t h common e x p o n e n t i a l d i s t r i b u t i o n . More r e f i n e d models, i . e . , a t i m e - d e p e n d e n t d i s t r i b u t i o n f o r t h e Zi (Model B) o r a m i x e d e x p o n e n t i a l d i s t r i b u t i o n (Model C) p r o v e t o be more c o r r e c t . 4 ) I n many a r e a s o f S o u t h e r n I t a l y t h e a n n u a l f l o o d s e r i e s e x h i b i t s t a t i s t i c a l o u t l i e r s (and, a c c o r d i n g l y , h i g h v a l u e s o f skewness), a s s o c i a t e d w i t h d i s a s t r o u s s t o r m s . Yodel C, w h i c h a c c o u n t s f o r them b y m o d e l l i n g t h e f l o o d p o p u l a t i o n as t h e m i x t u r e o f two d i s t i n c t populations, i s i n keeping w i t h t h e f a c t t h a t d i s a s t r o u s storms occur more r a r e l y b u t w i t h l a r g e r i n t e n s i t y t h a n o t h e r s . By c o n t r a s t , t h e r e i s l i t t l e p h e n o m e n o l o g i c a l e v i d e n c e i n s u p p o r t o f Model B, f o r d i s a s t r o u s s t o r m s may o c c u r a t any t i m e d u r i n g t h e y e a r . The s u p e r i o r i t y o f Model C i s c o n f i r m e d b y t h e b e t t e r f i t i t p r o v i d e s t o t h e d a t a , i n s p i t e o f t h e f a c t t h a t i t has f e w e r p a r a m e t e r s t h a n i s t h e c a s e o f Model B. ACKNOWLEDGEMENTS T h i s work was s u p p o r t e d b y CNR " P r o g e t t o F i n a l i z z a t o C o n s e r v a z i one d e l S u o l o " s o t t o p r o g e t t o Dinami ca F1 u v i a l e - Pubbl n. 154. REFERENCES Dauty, J., 1972. M6thodes des p r o c e s s u s s t o c h a s t i q u e s p o u r l a d e t e r m i n a t i o n de l o i s de p r o b a b i l i t 6 des c r u e s . A t t i d e l Convegno I n t e r n a z i o n a l e P i e n e : l o r 0 p r e v i s i o n e e d i f e s a d e l s u o l o . Ronia, 11 pp. H a s s e l b l a d , V . , 1969. E s t i m a t i o n o f F i n i t e M i x t u r e s o f D i s t r i b u t i o n s f r o m t h e E x p o n e n t i a l F a m i l y . J . Amer. S t a t i s t . Assoc., 64: 1459-71 Landweher, J . , M a t a l a s , i1.C. and H a l l i s , J.R., 1978. Some Comparisons o f F l o o d S t a t i s t i c s i n Real and Log Space. Water Re o u r . Res., 14: 902-920. M a t a l a s , N . C . , S l a c k , J.R. and W a l l i s , J.R., 1975. Reg o n a l Skew i n Search o f a P a r e n t . Water Resour. Res., 11: 815-826 N a t u r a l E n v i r o n m e n t Research Counci 1, 1975. F l o o d S t u d es R e p o r t . NERC Pub1 i c a t i ons. London. N o r t h , M., 1980. Time-Dependent S t o c h a s t i c Model o f F l o o d s . P r o c e e d i n g s Am. SOC. C i v . Eng., 106: 649-665. Penta, A., R o s s i , F., S i l v a g n i , G., V e l t r i , M. and Versace, P . , 1980. Un m o d e l l o s t o c a s t i c o p e r l ' a n a l i s i d e l l e massime p i o g g e g i o r n a l i e r e i n p r e s e n z a d i g r a n d i n u b i f r a g i . A t t i d e l X V I I Convegno d i I d r a u l i c a e C o s t r u z i o n i I d r a u l i c h e . Palermo, 17 pp.
.
324
Ross”, F. and Versace, P . , 1981. Criteri e metodi per 1 ’ a n a l i s i s t a t i s t i c a d e l l e piene. P u b b l i c a z i o n e Programma F i n a l z z a t o Conservaz i o n e Suolo. Roma, 61 p p . R o u s s e l l e , J . , 1972. On Some Problems o f Flood Analys s . P h . D . Thesis, Colorado S t a t e U n i v e r s i t y . F o r t C o l l i n s , 226 pp. Sarhan, A . E . , 1954. E s t i m a t i o n of t h e mean and s t a n d a r d d e v i a t i o n by o r d e r s t a t i s t i c s . A n n . Math. S t a t i s t . , 25: 317-328. Snyder, D . L . , 1975. Random P o i n t P r o c e s s e s . John Wiley and Sons. New York, 485 pp. Todorovic, P., 1976. S t o c h a s t i c Models o f Floods. Water Resour. Res., 14: 345-356. Todorovic, P . and R o u s s e l l e , J . , 1971. Some Problems o f Flood A n a l y s i s . Water Resour. Res., 7: 1144-1150. Todorovic, P . and Z e l e n h a s i c , E . , 1970. A S t o c h a s t i c Model f o r Flood A n a l y s i s . Water Resour. Res., 6: 1641-1648. U.S. Water Resources C o u n c i l , 1977. G u i d e l i n e s f o r Determining Flood Flow Frequency. Hydrologic Comtxittee, B u l l . 17 A . Washington. Z e l e n h a s i c , E . , 1970. T h e o r e t i c a l P r o b a b i l i t y D i s t r i b u t i o n f o r Flood Peaks. Hydrology Paper 42. Colorado S t a t e U n i v e r s i t y , F o r t C o l l i n s .
325
A MODEL FOR SIMULATING DRY AND WET PERIODS OF ANNUAL FLOW S E R I E S
M . BAYAZIT Department o f H y d r a u l i c s and Water Power,Technical I s t a n b u l , Turkey
University,
ABSTRACT
A two-stage model has been developed w i t h t h e purpose o f s i m u l a t i n g p e r i o d s o f f l o w s o f v a r i o u s magnitudes.
Observed
annual f l o w s o f a r i v e r a r e arranged i n t o n subsets i n v i e w o f t h e i r p o s i t i o n s w i t h r e s p e c t t o t h e s u i t a b l y chosen t r u n c a t i o n levels.
Elements o f t h e t r a n s i t i o n m a t r i x between t h e s t a t e s
are determined f r o m t h e o b s e r v a t i o n s .
I n t h e f i r s t stage o f t h e
s i m u l a t i o n s t a t e s o f f l o w s a r e generated by a Mark3vian process which preserves t h e t r a n s i t i o n m a t r i x .
I n t h e second stage a c t u a l
values o f f l o w s a r e produced by means o f a f i r s t - o r d e r a u t o r e g r e s s i v e model.
Two-state and t h r e e - s t a t e v e r s i o n s o f t h e
model a r e d e s c r i b e d .
Two-state model s i m u l a t e s d r y p e r i o d s a t a
c e r t a i n t r u n c a t i o n l e v e l whereas t h r e e - s t a t e model p r e s e r v e s t h e p o s i t i v e and n e g a t i v e r u n - l e n g t h s o f observed f l o w s which may have d i f f e r e n t values.
Thus these two models may account f o r extreme
droughts and d i f f e r e n t i a l p e r s i s t e n c e , r e s p e c t i v e l y .
Appl i c a t i o n s
o f t h e model t o t h e s i m u l a t i o n o f annual f l o w s o f a r i v e r e x h i b i t i n g d i f f e r e n t i a l p e r s i s t e n c e a r e presented. INTRODUCTION Hydrologic data are p r e r e q u i s i t e s f o r a l l engineering studies aimed a t d e v e l o p i n g w a t e r r e s o u r c e s .
D e c i s i o n s t o be made i n t h e
p l a n n i n g and o p e r a t i o n o f a w a t e r - r e s o u r c e system depend t o a g r e a t e x t e n t on t h e a v a i l a b l e h y d r o l o g i c i n f o r m a t i o n .
Hydrologic
v a r i a b l e s , b e i n g o f random c h a r a c t e r , can o n l y be expressed i n Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
3 26
terms of t h e i r s t a t i s t i c a l p r o p e r t i e s .
Samples o f s u f f i c i e n t s i z e
a r e r e q u i r e d i n o r d e r t o e s t i m a t e t h e s t a t i s t i c a l parameters o f t h e p o p u l a t i o n w i t h an a c c e p t a b l e p r e c i s i o n .
S t o c h a s t i c dependence
i n c r e a s e s t h e r e q u i r e d s i z e o f sample f o r a g i v e n degree o f accuracy.
Streamflows, which a r e t h e most i m p o r t a n t i n p u t s o f
hydrologic studies, usually e x h i b i t considerable sequential dependence.
On t h e o t h e r hand, s e r i e s o f r e c o r d e d s t r e a m f l o w s a r e
generally too short.
T h i s s i t u a t i o n has l e d t o t h e development o f
s y n t h e t i c h y d r o l o g y which a t t e m p t s a t g e n e r a t i n g s y n t h e t i c s e r i e s o f f l o w s based on a mathematical model o f t h e s t o c h a s t i c process. S y n t h e t i c f l o w s e r i e s a r e m o s t l y used i n r e s e r v o i r o p e r a t i o n s t u d i e s where i t i s expected t h a t t h e i n f o r m a t i o n c o n t a i n e d i n t h e o b s e r v a t i o n s w i l l be used more e f f i c i e n t l y and t h e r i s k s c o r r e s p o n d i n g t o v a r i o u s d e c i s i o n s can be e s t i m a t e d , e s p e c i a l l y by s i m u l a t i n g t h e extreme d r y and wet p e r i o d s t h a t m i g h t n o t be c o n t a i n e d i n t h e observed data.
Therefore i t i s e s s e n t i a l t h a t the
generated s e r i e s r e p r e s e n t these p e r i o d s adequately. Serious d i f f i c u l t i e s a r e encountered i n t h e model 1 i n g o f h y d r o l o g i c processes.
The c h o i c e o f t h e model t y p e and t h e
e s t i m a t i o n o f i t s parameters a r e rendered d i f f i c u l t due t o t h e l i m i t e d time-span o f t h e a v a i l a b l e r e c o r d s .
I n order t o minimize
t h e e r r o r s a r i s i n g f r o m t h i s s i t u a t i o n i t has been recommended t o use simple models t h a t have as few parameters as p o s s i b l e .
As no
model can be expected t o r e p r e s e n t a l l aspects o f t h e f l o w process which depends on t h e complex p h y s i c a l c h a r a c t e r i s t i c s o f t h e r i v e r b a s i n , i t should be a t t e m p t e d t o s e l e c t a model which can reproduce t h e p r o p e r t i e s o f t h e f l o w s r e l a t e d t o t h e problem i n hand.
The
model most f r e q u e n t l y used t o generate annual s t r e a m f l o w s i s t h e f i r s t - o r d e r 1 i n e a r a u t o r e g r e s s i v e model : ‘k where Xk and Xk-l a r e t h e flows o f y e a r s k and k - 1 , r e s p e c t i v e l y .
327
The model has three parameters: mean (p), standard deviation ( a ) and lag-one autocorrelation coefficient ( p ) of annual flows. E~ i s the standard normal v a r i a t e . I t was pointed o u t (Askew, e t a l . , 1971; Bayazit, 1974) t h a t dry periods generated by t h i s model were n o t so severe as those recorded i n some r i v e r s . This i s a serious deficiency of the model since reservoir operation i s very s e n s i t i v e t o periods of extreme d r o u g h t . Higher order autoregressive models o r more general ARIMA type of models have t o o many parameters and d o n o t s t i l l guarantee t o preserve the c h a r a c t e r i s t i c s of extreme flows. I n t h i s study a two-stage model i s developed with the purpose of simulating periods of flows o f various magnitudes c o r r e c t l y . I n the f i r s t stage the model generates flow s t a t e s (such as dry, normal , wet). In the next stage actual flows belonging t o these s t a t e s are generated by means o f a modified f i r s t - o r d e r autoregressive process t h a t can preserve mean, standard deviation and lag-one autocorrelation c o e f f i c i e n t of the flows. The advantage of the model i s t h a t i t can preserve the d i s t r i b u t i o n of lengths of dry and wet periods as well. PROPOSED MODEL
Consider a stationary stochastic process consisting of normally distributed variables x k which can be regarded as the flow of year k . An appropriate transformation (such as logarithmic) should be applied f i r s t i f xk are n o t distributed normally. Let the flows be divided into n classes such t h a t the probability of a flow being in class interval i i s q i : (2)
n where, obviously,
2
i =1
qi = 1 .
328
The t r a n s i t i o n m a t r i x o f t h e n - s t a t e Markov p r o c e s s can be = [ a . . I where aij 1J
d e f i n e d as Pij
is the probability o f a flow i n
c l a s s i t o be f o l l o w e d b y a f l o w i n c l a s s j : xi
ai j
1
(3)
Transition p r o b a b i l i t i e s s a t i s f y the following equations:
. . ,n
j
= 1,2,.
i
=1 , 2 , . .n;.
(4)
i=l n
z
aij
=
1
(5)
j=1
a . . v a l u e s can be e s t i m a t e d f r o m t h e r e c o r d e d d a t a b y c o u n t i n g t h e 1J
numbers o f o b s e r v e d t r a n s i t i o n s between t h e s t a t e s . H a v i n g d e c i d e d t h e number o f c l a s s e s n and t h e i r p r o b a b i l i t i e s qi and d e t e r m i n e d t h e e l e m e n t s o f t h e t r a n s i t i o n m a t r i x , s y n t h e t i c f l o w s can be g e n e r a t e d b y t h e f o l l o w i n g t w o - s t a g e scheme.
An i n i t i a l v a l u e xl,i
i s chosen and a sequence o f f l o w s t a t e s o f
d e s i r e d l e n g t h i s g e n e r a t e d b y means o f a random number g e n e r a t o r s i m u l a t i n g t h e n - s t a t e t r a n s i t i o n m a t r i x Pij.
A t t h e end o f t h i s
s t a g e s t a t e s o f s y n t h e t i c f l o w s have been d e t e r m i n e d b u t n o t t h e i r a c t u a l Val ues. Stage 11. Once i t i s d e t e r m i n e d t h a t xk b e l o n g s t o s t a t e j , i t s v a l u e can be computed a s :
3 29
where
'j-1
E
has a t r u n c a t e d normal d i s t r i b u t i o n such t h a t :
k,j
and x . a r e t h e l i m i t s o f t h e c l a s s i n t e r v a l j . J The sequence o f f l o w s generated i n t h i s way p r e s e r v e s t h e
p o p u i a t i o n mean 1-1, s t a n d a r d d e v i a t i o n
5 ,
and t r a n s i t i o n m a t r i x
. I t has a b u i l t - i n a u t o c o r r e l a t i o n c o e f f i c i e n t pi j computed as f o l 1ows : n
p
which can be
n
where P ( i , j ) i s t h e p r o b a b i l i t y of t h e f l o w o f t h e y e a r k-1 t o be i n c l a s s i and t h e f l o w o f t h e n e x t y e a r t o be i n c l a s s j , which i s equal t o : P(i,j)
qi
=
a 1. .J
Expected v a l u e o f t h e p r o d u c t o f x
where
pi
and
respectively. n
p
p
(9)
~ ,i- and ~ x
k ,j
equals:
a r e means o f t h e f l o w s i n c l a s s e s i and j ,
j
S u b s t i t u t i n g these i n t o eq.8: n
r)
v a l u e computed as above w i l l u s u a l l y be l o w e r t h a n t h e observed
a u t o c o r r e l a t i o n c o e f f i c i e n t o f t h e process s i n c e c o r r e l a t i o n s between t h e successive f l o w s a r e n o t c o n s i d e r e d f u l l y i n t h i s scheme.
330
I n o r d e r t o p r e s e r v e t h e observed a u t o c o r r e l a t i o n c o e f f i c i e n t , f o l l o w i n g f i r s t - o r d e r a u t o r e g r e s s i v e model s h o u l d be used:
where x ~ - i~s ,t h~e f l o w o f t h e y e a r k-1 which i s i n c l a s s i, and i s t h e f l o w o f t h e y e a r k which i s i n c l a s s j . T-I~,~ i s a k ,j random v a r i a t e drawn f r o m t h e s t a n d a r d normal d i s t r i b u t i o n w i t h t h e
x
computed by eq. (12) t a k e s indeed a v a l u e k ,j b e l o n g i n g t o c l a s s j . T h i s can be accomplished by means o f a condition that x
random number g e n e r a t o r which produces s t a n d a r d normal v a r i a t e s b u t then r e j e c t s those which do n o t s a t i s f y t h e c o n d i t i o n t h a t x
k ,j
computed by eq.(12) i s i n c l a s s j .
The standard d e v i a t i o n coefficient
pl
0'
and lag-one a u t o c o r r e l a t i o n
o f t h e g e n e r a t i n g scheme g i v e n by e q . ( 1 2 ) can be
expressed i n terms o f o and p o f t h e p o p u l a t i o n o f annual f l o w s . I t can be shown (see Appendix) t h a t u and 0'
and
pl
2
((0')'
-1)
+
2
pl U'
D
=
0
CI
0
where :
D
=
n
w i l l be p r e s e r v e d when
a r e chosen such as t o s a t i s f y t h e f o l l o w i n g e q u a t i o n s :
-
(1-0' )
p
n Y
qi
aij
dij
d . . i n eq.(15) a r e d e f i n e d as f o l l o w s : 1J
331 where Eij
denotes t h e expected v a l u e o f t h e v a r i a b l e i n b r a c k e t s
f o r t h e subset o f f l o w s i n c l a s s i f o l l o w e d by those i n c l a s s j . For a c e r t a i n v a l u e o f n, dij
.
2 , .. ,n)
a r e f u n c t i o n s o f p ' and
qi
( i = 1,
, and can be determined e x p e r i m e n t a l l y as w i 11 be d e s c r i b e d
l a t e r on. O b v i o u s l y t h e number o f s t a t e s n t o be used i n t h e model s h o u l d be small i n o r d e r t o be a b l e t o e s t i m a t e aij observed d a t a w i t h a s u f f i c i e n t accuracy.
values f r o m t h e Below, t w o - s t a t e and
t h r e e - s t a t e v e r s i o n s o f t h e model a r e g o i n g t o be discussed. TWO-STATE MODEL The s i m p l e s t case o f t h e model developed i n t h e p r e v i o u s s e c t i o n i s t h e t w o - s t a t e model where one o f t h e s t a t e s corresponds t o d r y p e r i o d s below a c e r t a i n t r u n c a t i o n l e v e l x1 ( w i t h p r o b a b i l i t y ql) and t h e o t h e r t o wet ( o r normal) p e r i o d s above t h a t l e v e l ( w i t h p r o b a b i l i t y q2 = l - q l ) .
A model o f t h i s k i n d was i n t r o d u c e d by
Jackson (1975 a) w i t h t h e purpose o f p r e s e r v i n g t h e observed Her model , however, d i f f e r s f r o m t h a t
persistence o f droughts. g i v e n by eq.(12)
i n t h a t a c t u a l f l o w values a r e generated by t h e
f o l l o w i n g scheme:
where
ai
and a
j
are standard d e v i a t i o n s o f the flows i n classes
i and j , r e s p e c t i v e l y .
E~
i s t h e s t a n d a r d normal v a r i a t e .
The
t r o u b l e w i t h t h i s model i s t h a t eq.(17) does n o t guarantee t h a t t h e value o f xk generated by t h i s scheme w i l l belong t o s t a t e j indeed as p r e s c r i b e d by t h e t r a n s i t i o n m a t r i x . eq.(12) where
pl
I t should be r e p l a c e d by
and u ' a r e t o be computed f r o m e q s . ( l 3 ) and ( 1 4 ) .
T h i s model w i l l p r e s e r v e p , a and
as w e l l as P
Expected values ij' o f t h e n e g a t i v e and p o s i t i v e r u n - l e n g t h s a t t h e t r u n c a t i o n l e v e l p
x1 a r e r e l a t e d t o t h e t r a n s i t i o n p r o b a b i l i t i e s all
and a22 by t h e
332
f o l l o w i n g e q u a t i o n s ( B a y a z i t and Sen, 1979): all
=
~-U/E(N~)),
=
I-(I/E(N~))
Therefore E(Nn), mean l e n g t h o f d r y p e r i o d s , and E ( N ) , mean l e n g t h P of wet p e r i o d s , w i l l a l s o be p r e s e r v e d b y t h i s g e n e r a t i n g scheme. ( i , j = 1 ,2) values d e f i n e d by e q . ( 1 6 ) have been determined as di,j f u n c t i o n s o f p ' by t h e d a t a g e n e r a t i o n method f o r two cases: q,
=
0.4,
q2
=
0.6
( F i g . 1 ) and q1
=
q2
=
0.5 ( F i g . 2 ) .
THREE-STATE MODEL L e t t h e f l o w s be d i v i d e d i n t o t h r e e c l a s s i n t e r v a l s , such as low f l o w s below x l ( w i t h p r o b a b i l i t y q ) , normal flows between x1 and x2 ( w i t h p r o b a b i l i t y q 2 ) , and h i g h flows above x2 ( w i t h p r o b a b i l i t y q3 = l-ql-q2).
T h i s process can be r e p r e s e n t e d by a t h r e e - s t a t e
model where s t a t e s correspond t o d r y , normal and wet p e r i o d s .
The
t r a n s i t i o n m a t r i x have 9 elements, o n l y 4 of which a r e independent as t h e y have t o s a t i s f y t h e r e l a t i o n s expressed by eq.(4) and ( 5 ) .
all and a33 r e p r e s e n t t h e p r o b a b i l i t i e s o f t r a n s i t i o n s f r o m d r y t o d r y and wet t o wet s t a t e s , r e s p e c t i v e l y , which a r e r e l a t e d t o E(Nn) a t t h e l e v e l x1 and E ( N ) a t t h e l e v e l x2 as f o l l o w s : P all
=
l-(l/E(Nn)),
a33
=
l-(l/E(Np))
(19)
I t can be concluded t h a t t h i s model w i l l p r e s e r v e t h e expected
values o f n e g a t i v e and p o s i t i v e r u n - l e n g t h s a t chosen t r u n c a t i o n l e v e l s , and hence i t can be used t o s i m u l a t e f l o w s e r i e s w i t h d i f f e r e n t i a l persistence.
Jackson (1975 b ) showed t h a t some annual
f l o w r e c o r d s e x h i b i t e d d i f f e r e n t i a l p e r s i s t e n c e , i.e. t h e l o w f l o w s were more p e r s i s t e n t t h a n h i g h f l o w s , and she proposed a b i r t h - d e a t h model t o s i m u l a t e such sequences. I n o r d e r t o generate such flow sequences u s i n g t h e p r e s e n t model, elements o f t h e t r a n s i t i o n m a t r i x Pij
=
[aij]
(i,j =
1,2,3)
are
computed from t h e data, and successive f l o w s t a t e s a r e f i r s t generated
333
Fig.1.
Fig. 2
values of the two-state model with ij q = 0.4 , q = 0.6 1 2
d.
dij
values of the two-state model with
q = q = 0.5 1 2
334
Then a c t u a l f l o w values a r e computed b y eq.(12) where
0'
and
p'
a r e t o be determined f r o m e q s . ( l 3 ) and ( 1 4 ) w i t h a t r i a l - a n d - e r r o r procedure such as t o p r e s e r v e t h e observed s t a n d a r d d e v i a t i o n and lag-one a u t o c o r r e l a t i o n c o e f f i c i e n t o f annual f l o w s . dij
( i ,j
=
1,2,3)
values o f t h e t h r e e - s t a t e model have been
determined by t h e d a t a g e n e r a t i o n method f o r t h e f o l l o w i n g cases: q1 = q3 = 0.3,
q2 = 0.4 ( F i g . 3 ) , and q1 = q3 = 0.4, q2 = 0.2
(Fig.4). APPLICATIONS The proposed model has been a p p l i e d t o annual f l o w s o f S t . M a r y ' s r i v e r i n Canada.
Observed f l o w s o f t h i s r i v e r i n t h e p e r i o d s 1860-
1964 were pub1 ished by Unesco (1971). These f l o w s a r e n o r m a l l y 2 3 d i s t r i b u t e d (x = 3.63 f o r 7 degrees o f freedom) w i t h li = 2103 m / s , 3 o = 326 m / s and p = 0 57. Two-state model The t r u n c a t i o n l e v e f o r t h e t w o - s t a t e model was chosen as 3 x1 = 2021 m / s which corresponds t o a p r o b a b i l i t y o f exceedence o f q2 = 0 . 6 .
Thus f l o w s below 2021 m3/s a r e i n c l a s s 1 ( d r y f l o w s w i t h 3 p r o b a b i l i t y q1 = 0 . 4 ) , and f l o w s above 2021 m / s a r e i n c l a s s 2 (normal f l o w s w i t h q2
=
0.6).
Elements o f t h e t r a n s i t i o n m a t r i x
were determined by c o u n t i n g t h e number o f t r a n s i t i o n s between t h e s t a t e s i n t h e recorded s e r i e s w i t h t h e f o l l o w i n g r e s u l t s : all
=
p'
0.63, a12 and
CT'
=
0.37,
a21
=
0.24,
a22
=
0.76
values were determined f r o m e q s . ( l 3 ) and ( 1 4 ) w i t h t h e
3 i d o f F i g . 1.
I n t h i s case above e q u a t i o n s a r e s a t i s f i e d
3pproxirnately when p ' = p and u ' = u .
5000 y e a r s l o n g s y n t h e t i c
f l o w t r a c e generated u s i n g e q . ( 1 2 ) has t h e f o l l o w i n g s t a t i s t i c s :
335
d. values of the three-state model with 11 q = q = 0.3, q = 0.4 1 3 2
Fig.3. 1.2
0.6
0.4
-__
0
0.2
0.1
Fig.4.
0.3
d..
3
0.5
0.6
0.7
2
9
0.8
values of the three-state model with q = 0.2
17 q = q = 0.4,
1
0.4
P'
336
2102,
=
p
0.62,
al 1
290,
=
a
0.38,
al 2
=
p
0.52 0.23,
a21
0.77
a22
E(N ) = 4.34 P For comparison, s t a t i s t i c s of t h e s y n t h e t i c f l o w s e r i e s o f equal
E(Nn)
=
2.74,
l e n g t h generated by t h e simple f i r s t - o r d e r a u t o r e g r e s s i v e model a r e g i v e n below: 1-1
=
2106,
o
=
319,
p
=
0.69
all
=
0.68,
a12
=
0.32,
aZ1
=
0.21,
E(Nn)
=
3.37,
E(Np)
aZ2
=
0.79
5.55
=
Mean n e g a t i v e and p o s i t i v e r u n - l e n g t h s of t h e observed f l o w s e r i e s a r e 2.73 and 4.28,
r e s p e c t i v e l y , which agree f a v o r a b l y w i t h
those generated by t h e p r e s e n t model. T h r e e - s t a t e model
As mentioned b e f o r e , t h e t h r e e - s t a t e model can be used t o generate s t r e a m f l o w t r a c e s w i t h d i f f e r e n t i a l p e r s i s t e n c e .
In
a p p l y i n g t h i s model t o t h e annual f l o w s o f S t . M a r y ' s r i v e r , t h e t r u n c a t i o n l e v e l s were chosen such t h a t q1
=
q3
=
0.3,
i.e. the
l o w e r 30% o f t h e annual f l o w s belonged t o d r y y e a r s and t h e upper 30% t o wet y e a r s .
Flows e x h i b i t e d s t r o n g d i f f e r e n t i a l p e r s i s t e n c e
as was evidenced by t h e f a c t t h a t t h e observed mean n e g a t i v e r u n l e n g t h a t t h i s l e v e l was 2.73 y e a r s whereas t h e observed p o s i t i v e r u n - l e n g t h was 1.87 y e a r s , a1 though t h e p r o b a b i l i t i e s were equal T r a n s i t i o n p r o b a b i l i t i e s f r o m d r y - t o - d r y and (ql = q3 = 0.3). wet-to-wet s t a t e s can be computed by e q . ( 1 9 ) a s : all
=
1-(1D.73)
=
0.64,
a33
=
1-(1/1.87)
=
0.47
Elements of t h e t h r e e - s t a t e t r a n s i t i o n m a t r i x were computed f r o m t h e observed d a t a w i t h t h e f o l l o w i n g r e s u l t s :
337
'ij
1
=
0.64 0.13 0.18
::E
0.27 0.54 0.35
0.47
It should be remarked t h a t all
j
and a33 values computed i n t h i s way
are t h e same as those o b t a i n e d through t h e use o f e q . ( 1 9 ) . U'
and
p
'
o f t h e model a r e computed from eqs. ( 1 3 ) and ( 1 4 )
u s i n g as F i g . 3 u '
=
365 and
p'
= 0.67.
S y n t h e t i c f l o w s o f 5000
years generated by e q . ( 1 2 ) w i t h t h e above values o f
5'
and
p'
have
the f o l l o w i n g s t a t i s t i c s :
u
2108, u
=
E(Nn) Pij
=
2.95,
=
0.65 0.13 0.16
=
327,
p =
E(N ) = P 0.26 0.53 0.36
0.49 1.85
::;: 0.48
F i r s t o r d e r a u t o r e g r e s s i v e model w i t h
p =
0.57 generated a s e r i e s
o f equal l e n g t h w i t h t h e s t a t i s t i c s : 2106, u = 319,
=
E(Nn) P.. 1J
2.66,
=
=
1
0.60 0.26 0.04
p =
E(N ) = P 0.35 0.52 0.32
0.69 2.90
0:;
0.64
i
It i s seen c l e a r l y t h a t t h e simple f i r s t - o r d e r a u t o r e g r e s s i v e
model cannot s i m u l a t e t h e d i f f e r e n t i a l p e r s i s t e n c e i n t h i s case whereas t h e t h r e e - s t a t e model can. CONCLUSIONS I t has been shown t h a t a two-stage Markov model can be employed
s u c c e s s f u l l y t o generate s y n t h e t i c t r a c e s o f annual f l o w s which preserve t h e mean, v a r i a n c e , lag-one a u t o c o r r e l a t i o n c o e f f i c i e n t
338
of the process a s well a s the t r a n s i t i o n matrix between the s t a t e s of flows. The two-state version of the model generates sequences with the desired mean length of droughts. The three-state version can be used in modeling d i f f e r e n t i a l persistence. APPENDIX
Expressions f o r the variance and lag-one autocorrelation coefficier of the variable x k generated by the scheme of eq.(12) can be derived as follows: Squaring b o t h sides of eq. ( 1 2 ) :
Variance.
Expected values of the terms in eq.(A.l) can be computed as follows:
=
u
n
n
2
2
aij
qi
dij
=
OD
(A.3)
j=1
were defined by eqs. ( 1 5 ) a n d ( 1 6 ) .
where D and d i
Substituting these i n t o eq. ( A . l ) : u2
=
p12a2
+
2p'a'
aD
+ (~'~(1-p'~)
D i v i d i n g by u 2 and rearranging: (1-p'Z)
((01)2-1) U
+
2p'
5'D 5
=
0
339
Autocorrelation coefficient.
Multiplying both sides
o f eq. ( 1 2 )
by ' k - l , i :
'k-l,i X
k-1,i
Xk , j
=
'k-l,i
lJ
+
'k-l,i
'i,j
('k-l,i
-lJ
1 +
d(1-& (A.7)
E x p e c t e d v a l u e s of t h e t e r m s i n t h i s e q u a t i o n can be computed as f o l l o w s :
(A.lO)
S u b s t i t u t i n g t h e s e i n t o eq. ( A . 7 ) :
( A . 11)
(A.12)
(A.13)
ACKNOWLEDGMENT The a u t h o r i s g r e a t f u l t o Mrs. Beyhan Oguz f o r h e r a s s i s t a n c e i n computer programming f o r t h i s s t u d y .
340
REFERENCES and H a l l , W.A. (1971): " A Comparative Askew, A.J., Yeh, W.W.-G, Study o f C r i t i c a l Drought S i m u l a t i o n " , Water Resources Research , Vol.7, No. 1 , p . p. 52-62. B a y a z i t , M. (1974): " S t a t i s t i c a l A n a l y s i s o f Dry P e r i o d s i n T u r k i s h Rivers", B u l l e t i n o f the Technical U n i v e r s i t y o f Istanbul , V01.27, No.2, pp.24-35. B a y a z i t , M., and Sen, Z. ( 1 9 7 9 ) : "Dry P e r i o d S t a t i s t i c s o f M o n t h l y Flow Models", Modeling H y d r o l o g i c Processes, ed. by H.J. Morel-Seytoux e t . a l . , Water Resources P u b l i c a t i o n s , L i t t l e t o n , Col orado. Jackson, B.B. (1975 a) : "Markov M i x t u r e Models f o r Drought Lengths", Water Resources Research, Vol.11, No. 1, pp.64-74. Jackson, B.B. (1975 b ) : " B i r t h - D e a t h Models f o r D i f f e r e n t i a l P e r s i s t e n c e " , Water Resources Research, V o l . l l , No.1, pp.75-95. Unesco (1971): Discharge o f S e l e c t e d R i v e r s o f t h e World, Vol.11, Paris.
341
A COMBINED SNOWMELT AND RAINFALL RUNOFF KAZUMASA MIZUMURA
Kanazawa I n s t i t u t e o f Technology, Ishikawa, Japan
1.
INTRODUCTION The area faced to the sea of Japan are known as the region with
heavy snowfall in Japan. The main cause is the monsoon blowing from the high pressure developed over the Siberia to the low pressure over the Pacific Ocean. The monsoon becomes contained much moisture during passing over the sea of Japan and makes heavy snowfall when it rises along the high mountaneous zone in the Honshu Island. The snowmelt becomes very important water resources such as electricity, rice growing, and drinking water. The snowmelt runoff is storaged in reservoirs and the accurate prediction of that is necessary for water level controls in reservoirs. During snowmelt period the rainfall and snowmelt runoffs are the dominant source of streamflow. The study area used €or the rainfall-runoff process and the snow accumulation-snowmelt runoff process is the Sai river watershed located in the southeast of Kanazawa city in Japan. The watershed area is 56.1
!an2
and the
meteolorogical data are observed at the measuring station. The elevation of this station is almost 300 m above the sea and many mountains from 1000 m to 1500 m exist within the watershed.
2.
TANKS MODEL SYSTEM
In this study four tanks model (Sugawara, 1978) was used €or rainfall-runoff analysis. The reason is dependent on the watershed area. An additional tank for snowfall is located on the upper position
*
of the four tanks as illustrated in Fig.1. The snowfall is storaged in
the upper tank and it melts when air temperature becomes higher than 0 c. The snowmelt runoff or/\and the rainfall runoff are poured into
the second tank. The most part of the snowmelt runoff and rainfall
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
0
342 runoff is discharged by the side outlets and the remains infiltrate
I
!L!
into the third tank. Therefore, there is no interaction between rainfall
-1
and snowfall, that is, the rain does not melt the snow. This process is also reported by Sugawara (1978). Let us define the rainfall and the c12\
cx3
Y
2
-h Y
snowmelt in the tank r and y o . The n n snowmelt does not occur when air temperature T is less than 0 c or the snowfall accumiilation ho in the first n tank is zero. Accordingly, the snowmelt can be expressed by the following equations.
Fig. 1. Tanks model if h,,”,= 0 or Tn< 0 ,
0
if :h if :h
n in which :h and
Si
=
=;?;
< mT and T > 0 , n n 2 mTn and Tn> 0.
(XSi-yy) , m and X
=
the parameters to be identified,
snowfall at i-step. A s the snowfall data in this watershed
the data at the measuring station are used, but in general the average snowfall in this watershed can be several times of that at the measuring station. So, AS. is considered to be the average snowfall in this watershed. To simplify the model, the snowmelt can be assumed to be proportional to the air temperature T and it is expressed by mT n n Eq.(l) can be written as
-
.
343 in which the sign min {A, B} means the selection of smaller one, If xn
=
:y
+
rn, the runoff from the second tank can be obtained as:
a: tXA - hi) n
=
B1
(X; - hi)
x,:
- xn -
+
if X1 I hi, n 1 if hi < X1 I h2, n if h: < XA.
1
(3)
(4)
- y , : - z ln + x n+l
(5)
in which z 1 = discharge from the tank bottom, X1 = storage in the tank, n n n = time step, a : , a : , and = discharge coefficients, hi and hi = the elevations of outlets from the tank bottom, and the superscript 1 shows the second tank. The third tank is also formulated as
y;
=
i"
a2 (X'
- h2)
if X2 I h2, n if X2 > h2. n
in which z2 = discharge from the tank bottom, X2 = storage in the n n tank, n = time sfep, a2 and B 3 = discharge coefficients, and h2 = the elevation of a outlet from the tank bottom. The calculations in the forth and fifth tanks are same as that in the third tank..The used data for this procedure are rainfall, snowfall, snowfall accumulation, runoff, and air temperature at 9 a.m. at the measuring station. The rainfall, snowfall, and runoff are daily averaged from 9 a.m. to 9 p.m. The temperature is also much influenced by the sea of Japan and the minimum of that in a year appears in February. And it remarkably increases during the snowmelt period from March to May. The higher temperatures than 20 c found in April are caused by the foehn phenomenon. Fig.2 shows rainfall and snowfall at the measuring station. This watershed belongs to the heavy snowfall district in Japan and the precipitation in January and in February is principally due to snowfall. The snowfall at the measuring station starts in the first part
3 44
G .r(
..........
Snowfa11
- Rainfall u
I
d
Nov. Dec. Jan. Feb. Mar. Apr. May June 1980
2
Fig. 2. Observed precipitation of December and ends in the last part of March. But in mountains of this watershed it starts in the first part of November.
3.
MAXIMUM A POSTERIORI ESTIPWTION
The maximum a posteriori estimation i s equivalent to an appropriate least-squares curve fit, using as weighting matrices the inverses of the plant- and measurement-noise covariances. We use for the optimum estimate the value of 8 which maximizes p sage and observation models are given by
in which s(n)
=
(01 Z). The discrete mes-
N-dimensional state vector, Q i ( s ( n ) , n)
vector-valued function, y(n) =
812
=
=
N-dimensional
M-dimensional plant-noise vector, z(n)
R-dimensional observation vector, & ( X ( n ) , n)
valued function, and y ( n )
=
=
R-dimensional vector-
R-dimensional observation noise vector.
For the discrete-estimation model, y ( n ) and v(n)
are assumed t o be
independent zero-mean Gaussian white sequences such that
345
in which 6 (n-j) is the Kronecker delta function, and V,(n) and Vv(n) k are non-negative definite MxM and RxR covariance matrices, respective-
ly. The estimate is derived from maximizing the conditional probability function. The one-stage prediction is given by
The priori error-variance algorithm is
The filter algorithm becomes
-
ft(n+l)
= ft(n+lln)
K(n+l)
=
-
+K(n+l){z(n+l) -
VX(n+l){ -
- _h(g(n+lln), _
h'l'(ft(n+lln), as(n+lIn)-
-
n+l)}
n+l)}
346 The flow chart of the maximtim a posteriori estimation is given in the reference (Sage et al., 1971). To apply the maximun a posteriori estimation to the tanks model described in the previous section, let us define the state vector as follows:
The vector function 9 can be calculated from the relation in the tanks
h(X(n) ,
model and
4.
n) becomes as:
NUMERICAL EXAMPLE The initial vector s(0) and the initial covariance matrix V-(0) must
X
be given beforehand to apply the algorithm of the maximum a posteriori estimation. In the consideration of the watershed area the result of Sugawara (1978) was used for the initial values of the parameters of the tanks model. Further we assumed m
=
3 and X
=
2 as the initial
values of the parameters related with snowmelt. So the initial state vector was
X(0)
-
=
(0, 0, 0 , 0 , 0, 0, 0 , 0 , 0 . 2 , 0 . 2 , 0 . 0 4 , 0.01, 0.001, 0 . 2 , 0 . 0 4 ,
0.01, 10, 2 0 , 10, 10, 3 , 2 )
T
(19)
The determination of the initial matrix V-(0) was made by the followX ii ing way. For simplicity the orders of y and X for i = 1 to 4 were n n assumed as: Order of X;
=
1
Order of y 1 n Order of X2 n Order of y2
=
1
=
Order of @’Xi
=
Order o f a’X:
=
Order of a2X;
n
Order of X 3 n
347 Order of y 3 = n Order of X 4 = n Order of y4 = n And the order
Order of a3XA Order of P 3 X 3 n Order of a4X4 n of the other parameters were assumed to be the initial
values. For example, the (1, 10) component of the covariance matrix Vit(0) was 0.2. The sign of each component of V - ( 0 )
X
was determined as
follows. By considering that the increase of yT corresponds to the n increase of a:, the correlation was positive and so is the sign. If there is no correlation between them such as a: and a:, the component is zero. So the (1, 10) component of V (0) was 0.2. Next, we assumed
it
that Vw(n)
=
0 and V (n) V
=
1, because the variation range of runoff
data was approximately from 1 to 100. And the results of the calculations with V (n) V
=
1 and V (n) V
=
10 were better in the following four
predictions (V (n) = 0.1, 1, 10, 100). The numerical calculation V
started from the first of September.
5.
PARAMETER IDENTIFICATION AND RUNOFF PREDICTION The parameters to construct the tanks model are a:, a:, a2,a 3 , B1,
B2, B 3 , hi, h:, h2, h 3 , A, and m. In the tanks model introduced by Sugawara (1978) these parameters are assumed to be constant. In this study these parameters are identified step by step by the algorithm of the maximum a posteriori estimation. The parameters such as the coefficients of the third, forth, and fifth tanks were almost constant. For small discharge these parameters do not show remarkable changes, but for large discharge a : and a: increase and hi, hi, and B' decrease. And X and m are correlated with snowfall and snowmelt, respectively, also increased. In Fig. 3 the observed snowfall accumulation at the measuring station was compared with the estimated average snowfall accumulation in this watershed. The former is much smaller than the latter, because of the elevation difference. There is still much snowfall in mountains even if there is no snowfall at the measuring station. In the estimation snowfall exists in the last of May and it explains the condition of remaining snow very well. Fig. 4 represents the observed and predicted runoff. The prediction was made by using
348
E
1500
t
Nov. Dec. J a n . .
Mar.
May
1980
F i g . 3. Measured and c a l c u l a t e d snow a c c u m u l a t i o n s
- Measured
7, rd
- - - - -.- C a l c u l a t e d
100
c m
1
1'
Nov. Dec. J a n . F e b . Mar. Apr. May J u n e 1980
F i g . 4 . Measured and e s t i m a t e d snowmelt h y d r o g r a p h
349 the data in the previous day. The prediction from November to February exceeded the observed runoff around peaks and it was below the observed one around receding runoff. The prediction of runoff becomes in good agreement from March. It can be explained that the precipitation in mountaneous area becomes snow already in November and it melts gradually, but it still rains at the measuring station in the same season because of the elevation difference and it discharges immediately.
6.
SUMMARY AND CONCLUSIONS
Through :hi
i
.,tudy the follawing r k u l ts were obtained.
The parameters in the tanks model were identified by the algorithm of the maximum a posteriori estimation on each time step. By using the method described herein it become possible to predict snowfall in mountains during winter. It becomes possible that the prediction of mean dai1.y runoff combined rainfall and snowfall runoff in the previous day by meteorological factors. Since the measuring station is located 300 m above the sea, it snows in mountains and it rains at the measuring station in the same time in November or December. The incorrect description of precipitation gives the error in runoff. Therefore, in future study an additional parameter will be considered during this period. The usage of air temperature at 9 a.m. results in error, for example, in the case of which the air temperature at 9 a.m. is less than 0 c and the maximum temperature in a day more than 0 c the runoff is zero in calculations but it has some value in data. In the application of the tanks model to predict runoff the model in this study is suggested by Sugawara (1978) and the parameters are constant. But since the parameters are estimated step by step, the problem on over parameterization occurs. This will be discussed in the future study.
3 50 REFERENCES Sage, A.P.
and Melsa, J . L . ,
1971. Estimation theory with applications
t o communications and c o n t r o l . M c G r a w - H i l l
Book Company, N e w York,
pp. 4 4 1 - 4 5 7 . Sugawara, M . ,
1 9 7 8 . Runoff a n a l y s e s . K y o r i t s u Shuppan Book Company,
Tokyo ( i n J a p a n e s e ) .
351 ANALYSIS OF CURRENT METER DATA FOR PREDICTING POLLUTANT DISPERSION
PHILIP J.W.
ROBERTS
School o f C i v i l E n g i n e e r i n g , G e o r g i a I n s t i t u t e o f T e c h n o l o g y
INTRODUCTION Although c u r r e n t meter data a r e f r e o u e n t l y c o l l e c t e d d u r i n g t h e d e s i g n o f m a j o r ocean o u t f a l l s , t h e d a t a a r e r a r e l y s u b j e c t t o e x t e n s i v e a n a l y s e s t o a i d i n t h e d e s i g n o f t h e s e systems.
An
exception t o t h i s occurred d u r i n g oceanographic i n v e s t i g a t i o n s f o r o u t f a l l s p r o p o s e d f o r t h e C i t y o f San F r a n c i s c o , C a l i f o r n i a .
In
t h i s c a s e c u r r e n t m e t e r d a t a c o l l e c t e d were s u b j e c t e d t o f a i r l y e x t e n s i v e a n a l y s e s w h i c h a i d e d c o n s i d e r a b l y i n t h e d e s i g n and p r e d i c t i o n o f performance o f t h e o u t f a l l s .
The p u r p o s e o f t h i s
paper i s t o p r e s e n t t h e r e s u l t s o f some o f t h e s e ana1;tses.
STUDY
SITE The s t u d y s i t e and p r o p o s e d o u t f a l l d e s i g n a r e shown i n
F i g u r e 1 ( R o b e r t s , 1980).
The p r o p o s e d d i s c h a r g e s i t e l i e s i n
t h e P a c i f i c Ocean o f f San F r a n c i s c o , j u s t S o u t h o f t h e G o l d e n Gate B r i d g e .
Uhen c o m p l e t e d , t h e o u t f a l l w i l l d i s c h a r g e b o t h
d o m e s t i c and i n d u s t r i a l sewage, and d u r i n g w e t w e a t h e r , a m i x t u r e o f sewage and s t c r n w a t e r r u n o f f .
N o t e t h a t t h e d e s i g n has been
m o d i f i e d f r o m t h a t d i s c u s s e d p r e v i o u s l y ( R o b e r t s , 1980) i n t h a t t h e o l d w e t w e a t h e r o u t f a l l s have been e l i m i n a t e d , and now a l l f l o w s w i l l d i s c h a r g e t h r o u g h t h e one l o n g o u t f a l l . The l o c a t i o n o f t h e moored c u r r e n t m e t e r s a r e a l s o shown i n F i g u r e 1.
S i t e 7 c o n t a i n s two m e t e r s , 7A n e a r e r t h e s u r f a c e ,
and 78 n e a r e r t h e b o t t o m .
O f t h e s e s i t e s , numbers 7, 8, and 12
were o c c u p i e d c o n t i n u o u s l y f o r one y e a r , and t h e r e s t were o c c u n i e d Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
352
\
SAN F R A N C I S C O
.I
T"
0 6
-
SCALE: f l 0 5000 10000
-
CURRENT
METER
STATION
DIFFUSER
SECTION
OUTFALL
PIPE
F i g u r e 1. Stud)( s i t e . component d i r e c t i o n s .
V e c t o r s on S t a t i o n 7 a r e t h e a r i n c i p a l
i n t e r m i t t e n t l y f o r p e r i o d s o f one t o two months d u r i n g t h e y e a r . The t o t a l v e r i o d o f i n v e s t i g a t i o n was February 1977 t o January 1978. The c u r r e n t meters were Endeco t y p e 105 s e t t o r e c o r d speed and d i r e c t i o n averaged e v e r y h a l f - h o u r ; was n o m i n a l l y one month.
t h e d u r a t i o n o f each d a t a s e t
The analyses d i s c u s s e d below a r e f o r t h e
p e r i o d September 2 t o September 30, 1977, when seven meters were operating. CURRENT NETER ANALYSES The speed and d i r e c t i o n were f i r s t c o n v e r t e d t o o r t h o g o n a l speed components, one i n a N o r t h e r l y d i r e c t i o n and one i n an Easterly direction.
As these d i r e c t i o n s do n o t have any i n h e r e n t
3 53
p h y s i c a l s i g n i f i c a n c e , t h e d i r e c t i o n s o f t h e p r i n c i p a l axes were computed.
These a r e d e f i n e d as t h e d i r e c t i o n s o f t h e e i g e n v e c t o r s
o f t h e m a t r i x formed by t h e c o v a r i a n c e s between t h e two speed component t i m e s e r i e s .
These axes a l s o maximize and m i n i m i z e ,
r e s p e c t i v e l y , t h e v a r i a n c e o f t h e c u r r e n t s p r o j e c t e d o n t o them. I t was found t h a t t h e d i r e c t i o n s o f t h e f i r s t p r i n c i p a l components
a l l p o i n t towards t h e Golden Gate, w i t h t h e i r v a r i a n c e d e c r e a s i n g w i t h d i s t a n c e f r o m t h e Gate.
The f i r s t p r i n c i p a l component i s
s t r o n g l y t i d a l , t h e second l e s s so, and t h e f i r s t p r i n c i p a l compon e n t c o n t a i n s much more energy t h a n t h e second.
These preliminar;.
f i n d i n g s a r e d i s c u s s e d i n Roberts, 1980. Oceanic motions occur over a v e r y wide range o f t i m e s c a l e s , each o f which has d i f f e r e n t e f f e c t s on t h e f a t e o f t h e d i s c h a r g e d wastewater.
To i l l u s t r a t e t h i s , t i m e s e r i e s p l o t s o f t h e f i r s t
and second p r i n c i p a l components a t S t a t i o n 7A a r e shown i n F i g u r e 2, and a power s p e c t r a l e s t i m a t e o f t h e f i r s t p r i n c i p a l component i n F i g u r e 3.
(The s p e c t r a was computed by an FFT
a l g o r i t h m a f t e r a p p l y i n g an a p p r o x i m a t i o n t o t h e Parzen window t o 1024 p o i n t s o f t h e raw data.
No a v e r a g i n g o f t h e r e s u l t i n g c o e f f i -
c i e n t s was employed, a l t h o u g h t h e spectrum was smoothed b y one pass o f a recursive f i l t e r . )
The spectrum shows t h e energy t o be
s t r o n g l y peaked a t t h e d i u r n a l and s e m i d i u r n a l t i d a l f r e q u e n c i e s , w i t h most energy a t t h e s e m i d i u r n a l frequency.
Relatively l i t t l e
energy i s c o n t a i n e d i n t h e h i g h e r f r e q u e n c i e s .
The l o w frequency
shows i n c r e a s i n g energy w i t h d e c r e a s i n g frequency. I n order t o b e t t e r discuss the e f f e c t s o f the d i f f e r e n t t i m e s c a l e s on t h e w a s t e f i e l d , t h e c u r r e n t s were d i v i d e d i n t o t h r e e frequency bands by t h e a p p l i c a t i o n o f d i g i t a l f i l t e r s . frequency bands a r e shown i n F i g u r e 3.
The
To do t h i s , t h e raw
N o r t h e r l y and E a s t e r l y speed components were b o t h s u b j e c t e d i n t t r n to:
F i r s t , a lon-pass f i l t e r w i t h a c u t - o f f frequency o f
0.84 cpd; Second, a band pass f i l t e r w i t h h a l f - p o w e r c u t o f f s a t 2.40 cpd and 0.84 cpd, and t h i r d , a high-pass f i l t e r w i t h a h a l f power c u t o f f a t 2.40 cpd.
The f i l t e r s used were o f t h e l i n e a r
3 54
70
60
50
40
-
30
3
20
\ VI
0
w
Ln
I0
0
-10
-l I
I
-30-
-4
o;
:oo
I
I 150
I0
250
300
400
350
450
500
550
I
600
650
i i u i (tirs)
F i g u r e 2. P r i n c p a l components o f c u r r e n t s a t S t a t i o n 7A The second p r i n c p a l compc:ient ( t o p ) i s d i s p l a c e d 50 cm/s above t h e f i r s t b o t t o m ) . phase f i n i t e impulse response t y p e u s i n g a K a i s e r window.
After
f i l t e r i n g , each frequency band p a i r o f t i m e s e r i e s were s u b j e c t e d t o a p r i n c i p a l component a n a l y s i s .
For S t a t i o n 7A, t h e r e s u l t i n g
d i r e c t i o n s o f t h e p r i n c i p a l axes o f t h e l o w , t i d a l ,
and h i g h
frequency bands a r e shown i n F i g u r e 1, and t h e t i m e s e r i e s o f t h e
low frequency c u r r e n t s i n F i g u r e 4.
An a l t e r n a t i v e a n a l y s i s would
be t o compute t h e r o t a r y s p e c t r a o f t h e c u r r e n t s .
Filtering i s
used here t o p r e s e r v e t h e t i m e s e r i e s , p a r t i c u l a r l y o f t h e l o w frequency c o n t e n t . s e p a r a t e l y below.
The c h a r a c t e r i s t i c s of each band a r e d i s c u s s e d
I JOO
355
FREOUENCY ( H r - ’ )
F i g u r e 2. Povler s y e c t r a l e s t i m a t e o f f i r s t p r i n c i p a l component o f F i g u r e 2. Also h a l f - p o w e r c u t o f f s o f f i l t e r f r e q u e n c i e s used t o s e p a r a t e c u r r e n t s i n t o frequency bands.
The f i r s t few minutes f o l l o w i n g r e l e a s e o f t h e sewage c o n s i s t s o f t h e i n i t i a l d i l u t i o n phase.
As t h i s i n i t i a l d i l u t i o n i s s t r o n g l y
i n c r e a s e d by ambient c u r r e n t s , i t would be expected t h a t t h e most e f f e c t on t h e sewage f o r s h o r t times would r e s u l t f r o m t h e most energetic currents. band.
These c u r r e n t s a r e c o n t a i n e d i n t h e t i d a l
Because d i f f u s e r s p l a c e d p e r p e n d i c u l a r t o a c u r r e n t r e s u l t
i n t h e h i g h e s t i n i t i a l d i l u t i o n (Roberts, 1979), t h e d i f f u s e r was
placed perpendicular t o the f i r s t p r i n c i p a l c u r r e n t d i r e c t i o n (see F i g u r e 1).
The d i r e c t i o n o f t h e t i d a l p r i n c i p a l components
shown i s an average o v e r t h e d i u r n a l and s e m i d i u r n a l components. A l t h o u g h t h i s would n o t g e n e r a l l y be d e s i r a b l e , i n t h i s case t h e
3 56
80 IU
61, Liu
40 30 \
E
0 c o m p l e t e l y by t h e i Walsh f u n c t i o n s , i t i s n e c e s s a r y t h a t t h e number o f d a t a p o i n t s i n t h e time s e r i e s be e q u a l t o t h e minimum s e q u e n c y o r d e r , N, H e n c e i n g e n e r a l , Walsh r e p r e s e n t a t i o n i s
.
,
393 N
where C, a r e t h e c o e f f i c i e n t s t o b e c h o s e n s u c h t h a t t h e J
mean-square
a p p r o x i m a t i o n e r r o r i s minimum; t h a t i s N
.3
i go
a s a r e s u l t o f which one can o b t a i n I N C = X .WAL(,j,i) (6) j N j=O j T h e f i r s t W a l s h c o e f f i c i e n t , Cg, i s , i n f a c t , e q u a l t o t h e mean v a l u e o f t h e time s e r i e s c o n c e r n e d , s i n c e WAL ( 0 , i ) = 1 f o r a l l i values. $en(1981) ha8 r e p r e s e n t e d m o n t h l y f l o w time s e r i e s p e r i o d i c p a r t b y a c o m p l e t e s e t o f Walsh f u n c t i o n s w i t h a maximum s e q u e n c y number N = 16; Hence, t h e p e r i o d i c s t o c h a s t i c p r o c e s s t u r n s o u t t o h a v e t h e f o l l o w i n g mathematical form,
-1
N
+ &,
C.*WAL(j,k)
J
-
(7)
where E i s tke s t o c h a s t i c c o m p o n e n t a n d k = i 15[i/Ig; here t h e i n t e g e r p a r t o f t h e argument. The r e l e v a n t Walah c o e f f i c i e n t c a n b e c a l c u l a t e d a c c o r d i n g t o E q . ( 6 ) . T h e a p p l i c a t i o n o f Walsh d e c o m p o s i t i o n t o C o l s l b i a r i v e r ( U S A ) m o n t h l y f l o w s y i e l d s t h e c o e f f i c i e n t s shown i n F i g u r e 2. T h e s e p e r a t i o n o f t h e p e r i o d i c c o m p o n e n t a c c o r d i n g t o Eq.(7) g i v e s t h e s t o c h a s t i c p a r t . Fitting o f t h e f i r s t o r d e r Markov p r o c e s s t o t h i s p a r t r e s u l t s i n a satisfactory solution. I n time series a n a l y s i e , t h e Walsh f u n c t i o n s a r e c a p a b l e o f d e p i c t i n g t h e p e r i o d i c component w i t h minimum e f f o r t o f c o m p u t a t i o n a n d g r e a t accuracy.
p]is
20
J
10 -
0
9
-
6
I
1-1
8
I
I
I
I
i
16
F i g u r e 2. Walah c o e f f i c i e n t s f o r C o l u m b i a R i v e r .
394
POROUS MEDIA D E S C R I P T I O N
I n o r d e r t o r e p r e s e n t s t a t i s t i c a l l y t h e p o r o u s medium it m u s t be r e p l a c e d b y a c o n v e n i e n t mathematical a b s t r a c t i o n . T o t h i s e n d , t h e p o r o u s medium w i l l be r e p r e s e n t e d by a c h a r a c t e r i s t i c f u n c t i o n , f i ( a > , w h i c h is d e f i n e d a8 a random s e q u e n c e o f + I t s a n d - I t s . Herein, i d e n o t e s t h e i - t h r e a l i z a t i o n o u t o f an ensemble and s t h e a r c l e n g t h o f a n y p o i n t on t h e l i n e f r o m a n a r b i t r a r i l y chosen o r i g i n . I n such a representation, t h e occur e n c e s o f + ? I s and -1'8 imply g r a i n and v o i d s p a c e s , respectively. Such a f u n c t i o n i s r e f e r r e d t o a s t h e Sam p l e characteristic function. Fara a n d S c h e i d e g g a r ( 1 9 6 1 ) made a n a t t e m p t t o char a c t e r i z e a g i v e n p o r o u s medium f r o m a p h o t o m i c r o g r a p h i c a l l y read o f f data. I n t h e i r s t u d y , t h e s p e c t r a l anal y s i s i n terms o f h a r m o n i c f u n c t i o n s a n d o f o t h e r o r t h o gonal functions together with a s p e c t r a l a n a l y s i s o f a s p e c i a l l y c o n s t r u c t e d f u n c t i o n o f t h e p o r o u s medium h a v e been c o n e i d e r a d as p o s s i b l e d e s c r i p t o r s o f t h e medium g e o m e t r y . However, t h e s e m e t h o d s c a n n o t a c c o u n t f o r d i s c o n t i n u i t i e s i n t h e p o r o u s media s a m p l e f u n c t i o n d u e In order t o alleviate t h i s situat o G i b b s phenomenon, t i o n t h e u s e o f Walsh f u n c t i o n 8 a r e c a p a b l e t o d i g e s t e f f e c t i v e l y t h e e x i s t i n g d i s c o n t i n u i t i e s i n t h e sample function. To i l l u s t r a t e t h e Walsh f u n c t i o n a p p l i c a t i o n , t h e
p o r o u s medium i s a s s u m e d t o h a v e a s a m p l e c h a r a c t e r i s t i c f u n c t i o n g i v e n i n F i g u r e 3a. T h e W a l s h a s w e l l a s t h e F o u r i e r a n a l y s i s a r e t h e n a p p l i e d t o t h i s s a m p l e charact e r i s t i c f u n c t i o n which y i e l d s mathematically o b t a i n e d F i g u r e 3b s h o w s c o u n t e r p a r t s a8 i n F i g u r e 3 b , c a n d d. t h e t r a n s f o r m a t i o n a n d r e c o n s t r u c t i o n o f t h e s a m p l e char a c t e r i s t i c f u n c t i o n by Walreh f u n c t i o n s o f t h e s e q u e n c y
395
o r d e r 32,
(d)
(b)
F i g u r e 3a, An i l l u s t r a t i v e s a m p l e c h a r a c t e r i s t i c b e Walsh t r a n s f o r m w i t h 3 2 terms. c. F o u r i e r t r a n e f o r m w i t h 18 terms, d. F o u r i e r t r a n s f o r m w i t h 44 terms.
function,
The f i r a t t e n Walsh c o e f f i c i e n t s a r c p r e s e n t e d i n
F i g u r e 4, On t h e o t h e r h a n d , F i g u r e s 3c a n d d show t h e F o u r i e r approximation t o t h e sample characteristic funct i o n w i t h 24 a n d 3 2 h a r m o n i c s , r e s p e c t i v e l y , C o m p a r i s o n o f F i g u r e s 3a, b , c a n d d y i e l d how e f f e c t i v e a r e t h e W a l s h f u n c t i o n s i n t h e p o r o u s madium d e s c r i p t i o n , Virtuall y , a l l o f t h e CharaCt@ri8tiC f u n c t i o n is r e p r e s e n t e d w i t h i t s d i s c o n t i n u i t i e s t o t a l l y b y t h e W a l s h s e r i e s , However, F o u r i e r approach g i v e s a g e n e r a l p a t t e r n similar t o t h e o r i g i n a l characteristic f u n c t i o n b u t lacks i n d e p i c t i n g t h e d . i s c o n t i n u i t i e 8 i,e, c o r n e r s ,
F i g u r e 4. W a l s h c o e f f i c i e n t s o f s a m p l e f u n c t i o n . REAL-TIME
PREDICTION
Real-time p r e d i c t i o n o f a n y p e r i o d i c d a t a r s q u i r e s c o n s t r u c t i o n o f a r e c u r s i v e m o d e l ( $ e n , 1 9 8 0 ) . I n order t o produce euch a r e c u r a i v e model o f e p e r i o d i c a t o c h a r t l c p r o c e m i t h e d i f f e r e n c e Xi XiOl is performed by coneideri n g Eqe(7) l e a d i n g t o ,
-
15
xi
= x
i-l
+
k=l
Cke[WAL(kei)
- WAL(k,i-I)]
+
ei
(8)
3 96
..
where i=2,3, ,16n ; n b e i n g t h e number o f y e a r s a n d e i s a r a n d o m v a r i a b l e w i t h zero mean. By d e f i n i n g t h e i Walah d i f f e r e n c e a s , WAD(k,i)
= WAL(k,i)
- WAL(k,i-l)
Eqo(8) can t h e n be r e w r i t t e n s u c c i n c t l y
--
(9) Ck.WAD(k,i) + ei k =I H e r e i n , t h e c o e f f i c i e n t s , C k , a r e unknown a n d n e e d t o b e e a t i s e t e d from t h e a v a i l a b l e m o n t h l y d a t a . H o w e v e r , i t i s a s s u m e d t h a t t h e c o e f f i c i e n t s a r e time i n d e p e n d e n t i o e o t h e y d o n o t c h a n g e w i t h time. Hence, Eq.(lO) c a n b e w r i t t e n i n m a t r i x n o t a t i o n 8s : e h) w \d15 2 .i 0 0 0 0 1 0 0 0 xi
+
88,
15
.
.
.
0
0
0
0
1
0
.
.
(10)
0 a
-i
w h e r e w l s a r e s h o r t v e r s i o n s o f W A D ( k , i ) ' s . On hand, i n a n i m p l i c i t m a t r i x n o t a t i o n Eq.(lO) becomes, yi=
0i,i-I.
Y i-1
+ w
i
(11)
w h e r e Y i i s a (17x1) v e c t o r o f s t a t e v a r i a b l e s , T h e t r a n i-l f r o m s t a t e ( i - I ) t o a t a t e i h a s a e i t i o n matrix d i m e n s i o n o f (17x13) a n d W i i s ( 1 7 x 1 ) v e c t o r o f i n d e p e n d e n t errors i n c l u d i n g 16 e l e m e n t s w h i c h a r e a l l e q u a l t o
Q,
zero. Halman f i l t e r c a n b e a p p l i e d t o t h e s y s t e m e q u a t i o n given i n Eq.(ll) p r o v i d e d t h a t a s u i t a b l e measurement e q u a t i o n i s s u p p l i e d , ( K a l m a n , 1 9 6 0 ) . A t t h e time i n s t a n t i t h e r e i s o n l y one measured state v a r i a b l e t h a t is t h e monthly f l o w v a l u e , Hence, t h e measurement e q u a t i o n which r e n d e r s t h e s t a t e v a r i a b l e s i n t o m e a s u r e m e n t s c a n be as,
397
= H
i where H zi
.Y
i
i
+ V
(12)
i
i s t h e measurement dynamics v e c t o r w i t h i t s first
e l e m e n t e q u a l t o 1 o t h e r s b e i n g a l l zero. A s t h e a c c u r a c y o f t h e m e a s u r e m e n t i n c r e a s e s t h e error c o n t r i b u t i o n , Vi, diminishes. H e r e i n , t h e m e a s u r e m e n t s a r e a s s u m e d t o be p e r f e c t w h i c h i s t h e case when V i = 0. W i t h E q s . ( l l ) a n d ( 1 2 ) a t h a n d , t h e Halman f i l t e r a p p l i c a t i o n i s s t r a i g h t f o r w a r d ( s e e G e l b , l 9 7 4 ) . The s t a t e e s t i m a t e , Y i / i - l , and e x t r a p o l a t i o n s are, error covariance, P i/i-l'
respectively.
T h e Halman g a i n m a t r i x ,
T = pi/i-lo H Ti H i.Pi/i-l.Hi
+
-1 Ri
Ki,
is (15)
F i n a l l y , t h e state e s t i m a t i o n and e r r o r covariance updates a f t e r t h e measurement8 t u r n s o u t t o be,
-
yi/i 'i/i-I respectiv e l y
.
+
Hi
Zi
H i .Y i / i - 1
(16)
A p p l i c a t i o n o f t h e m e t h o d is p r e s e n t e d f o r t h e Seyh a n river i n t h e s o u t h e r n T u r k e y . The s u c c e s s i v e execu= t i o n o f Eqs.(13)-(17) on a d i g i t a l c o m p u t e r r e q u i r e i n i a n d 9. T h e d i a g o n a l e l e m e n t 8 o f t h e t i a l v a l u e s Yo/o, P 0 /o c o v a r i a n c e m a t r i x are a l l t a k e n a s 1000's a n d o f f d i a g o n a l e l e m e n t s are e q u a l t o 100. A l l o f t h e i n i t i a l s t a t e v e c t o r elemente are adopted as zeros. T h e system n o i s e v a r i a n c e is s e l e c t e d a s Q=IOOO. However, t h e measurement n o i s e i s t a k e n a s zero i.e. t h e m e a s u r e m e n t s a r e a s s u m e d p e r f e c t . With these i n i t i a l v a l u e s t h e f i l t e r i n g e q u a t i o n s (Eqe. 13-17) a r e e x e c u t e d o n e b y o n e a n d f i n a l l y t h e Walah c o e f f i c i e n t s a r e o b t a i n e d a s i n T a b l e 1.
398
TABLE 1,
Seyhan r i v e r Walsh c o e f f i c i e n t e , Coefficient Sequency 1 -3.49 0-56 2 -0 60 3 94-03 4 -0 60 5 -3.60 6 7 94-12 8
98-27
9 10 11
-4, 12
12
13 14 15 16
-2.22 05-66 99-13 95-66 -5.47 -3.90 4.85
F i g u r e 5 r e p r e s e n t s t r u e and filtered monthly r u n o f f v a l u e s o f t h e Seyhan r i v e r P l o u s f o r t h e first f i v e
F i g u r e 5, S e y h a n r i v e r t r u e a n d p r e d i c t e d m o n t h l y f l o u a ,
P e r i o d i c i t y i n t h e o b s e r v e d s e q u e n c e i s p r e s e r v e d slmliG:::. l a r l y i n t h e p r e d i c t e d v a l u e s , T h e t r a c e o f t h e error c o v a r i a n c e u p d a t e s c h a n g e is shown i n F i g u r e 6 w h i c h e x i b i t s a s t e a d y decrease a n d i t t h e n s t a b i l i z e s ,
F i g u r e 6, Trace o f e r r o r c o v a r i a n c e u p d a t e m a t r i x .
399
DIFFERENTIAL EQUATION SOLUTION A n o t h e r v e r y p o t e n t i a l a p p l i c a t i o n area of t h e Walsh f u n c t i o n s i n t h e d i f f e r e n t i a l e q u a t i o n s o l u t i o n a8 a n a l t e r n a t i v e t o t h e c l a s s i c a l f i n i t e e l e m e n t t e c h nique, A p p l i c a t i o n s t o t h i s end have a l r e a d y been undert a k e n i n eyerterns e n g i n h a r i n g by Chen a n d H s i a o ( 1 9 7 5 ) ; Para8keVOpOUlO8 a n d a o u n a s ( 1 9 7 8 ) a n d S h i h a n d Han(1978). S i n c e , t h e b a s i c Walsh f u n c t i o n s a r e p i e c e w i s e & o n s t a n t a t e i t h e r + I or -1 t h e i r i n t e g r a t i o n y i e l d 8 s i m p l e pieceu i s e l i n e a r f u n c t i o n s u h i c h a r e i n f a c t t r i a n g l e s . Tho i n t e g r a t i o n s a r e shown i n F i g u r e 7 for a e q u e n c y o r d e r o f 3 2 The Walsh i n t e g r a t i o n s c a n be e x p r e s s e d i n terms o f b a s i c Walsh f u n c t i o n a . A f t e r p e r f o r m i n g t h e n e c e s s a r y a n a l y t i c a l e v a l u a t i o n s o n e c a n o b t a i n for N d 3 t h e fallowing equations :
.
/ W A L ( O , t ) d t = ( 1/2)WAL(O, t>-(1/4)WAL( 1, t)-( 1 / 8 ) W A L ( 2 , t ) /WAL(l,t)dt
(WAL(2,t)dt (WAL(3,t)dt
(WAL(4,t)dt
-(1/16)WAL(4,t) = (1/4)WAL(O, t>-( 1 / 8 ) W A L ( 3 , t)-(1/16)WAL(5,t) = ( 1 / 8 ) W A L ( O , t ) - ( l / l 6 > W AL ( 6,t ) = (1/8)WAL(l,t>-(I/16)WA L (7 ,t) = (1/?6)WAL(O, t )
(18)
= (1/16)WAL(l,t) /WAL(G,t)dt = (1/16)WAL(Z,t) / WAL( 7, t )d t = ( 1/16) WAL ( 3 , t 1 o r i n matrix notation succinctly, /WAL(5,t)dt
fW8dt = T ( 8 ~ 8 1 ~ ~ 8
(19)
u h e r e We is (8x11 v e c t o r o f t h e b a s i c Walah f u n c t i o n s is t h e t r a n s i t i o n matrix. The g e n e r a l f o r m and T ( 8 x 8 )
400
m a t r i x is g i v e n by Chen a n d H s i a o ( 1 9 7 5 ) a s ,
of this
...................
I
:
f :
%/a
ON/4 - ( 1/2N) = I n L ...........................
(1/N)eI~/4
.................................. ( 1/2N e
L
'(1/N)eIN/4
...................
.........................................
T ( p1 xN1"
1
............
.
(Z/N).I~/~
. . i .
INl2
ON/Z
a r e t h e i d e n t i t y a n d z e r o matrices, and 0 N/2 respectively. I t is e v i d e n t f r o m t h e a b o v e c a l c u l a t i o n s t h a t i f a n y mathematical expression i s w r i t t e n i n terms o f Walsh f u n c t i o n s t h e n t h r o u g h t h e a f o r e m e n t i o n e d t r a n s i t i o n m a t r i x i t s i n t e g r a t i o n c a n b e a c h i e v e d by t h e Walah f u n c t i o n s . Hence, t h e i n t e g r a t i o n p r o c e d u r e become8 t h e p r o b l e m o f f i n d i n g t h e r e l a v a n t Wal8h c o e f f i c i e n t a . Let U B now c o n s i d e r a s i m p l e i l l u s t r a t i o n as i n t h e f o l l o w i n g example, where I
b! / 4
d x / d t = ZX
(21)
u i t h t h e b o u n d a r y c o n d i t i o n s ( x = l a t t-0). t i v e i s e x p a n d e d i n t o Walah 8 e t w i t h N=2 2
If t h e d e r i v a -
, then
d x / d t = CoWAL(O,t)+C 1WAL(l,t)+CzWAL(2,t)+C3WAL(3,t) t a k i n g t h e i n t e g r a t i o n leads t o ,
x =
Co WAL(O,t)dt+C1
or b r i e f l y
x = C
4
OT
WAL(l,t)dt+CZ WAL(Z,t)+C3 WAL(3,t)dt
,
(4x4)
+
ow
4
+
xo
1
T
where C4=[Co c1 c 2 C 3] i s t h e c o e f f i c i e n t s v e c t o r . E x p l i c i t l y , t h e 8 U b S t i t U t i O n o f t h e n e c e s e a r y Waleh f u n c t i o n i n t o Eq.(21) y i e l d s , C4eW8
= -4C 4' T ( 4 x 4 ) ' w 4 +"4
0
0
0
w4J
The o n l y unknoldn is t h e c o e f P i c i e n t s v e c t o r w h i c h c a n be
401
f o u n d ae,
Exact a n d a p p r o x i m a t e Walsh s o l u t i o n s are shown i n F i g u r e 8 . I t i s o b v i o u a t h a t i n c r e e a e i n t h e Waleh a e q u e n c y M i l l r e e u l t i n more r e f i n e d a p p r o x i m a t i o n s . T h e o r e t i c a l l y , i n f i n i t y o f a e q u e n o y number c o r r e s p o n d s w i t h t h e e x a c t solution. Unsteady one-dimensional flow i n an unconfined a q u i f e r u i t h o u t r e c h a r g e i s g i v e n by t h e f o l l o w i n g p a r t i a l d i f f e r e n t i a l equation,
where a ( x , t ) , S a n d T are t h e drawdown, s t o r a t i v i t y a n d t r a n s m i a a i v i t y o f t h e a q u i f e r , r e s p e c t i v e l y ; x a n d t are t h e s p a c e a n d time v a r i a b l e s . T h e r e is n o g e n e r a l s o l u t i o f o r t h i s d i f f e r e n t i a l e q u a t i o n a n d o n l y f o r s p e c i f i c bound a r y c o n d i t i o n s i t e s o l u t i o n is p o s s i b l e . Herain, t h e Walah a o l u t i o n i s p r e s e n t e d a s a b r i e f summary. Let u a i n t e g r a t e Eq.(22) twice w i t h r e s p e c t t o x a n d o n c e w i t h r e s p e c t t o t w i t h b o u n d a r y c o n d i t i o n s e(x,O>=O e n d a@,t) s n i p . Aftar sohe a l g e b r a i c m a n i p u l a t i o n s i t h a d 8
tot
7f
-
At
/s(x,t)dt + a e(x,t)dxdx = 0 (23) 0 0 0 where a S/T a n d a s s u m e d t o b e k n o m . Two v a r i a b l e
-
f u n c t i o n , s ( x , t ) c a n b e e x p a n d e d i n t o a d o u b l e Walsh series a s ,
m m
where (P,(t) a n d q ' ( x ) a r e t h e Walsh f u n c t i o n s w i t h 9 I r e s p e c t t o v a r i a b l e s t a n d x, r e s p e c t i v e l y ; 'id t h e d o u b l e WaOsh c o e f f i c i e n t s w h i c h a r e g i v e n b y ,
402
1/G 0
WAL(3,B)dt
I /e 0
WAL(4,8)dt
1/8
WAL ( 5 , 8 ) d t
1/ a
WAL(6,B)dt
0
WAL ( 7,8)d t Figure.
7. Idalsh f u n c t i o n s i n t e g r a l s f o r N 4 .
_ _ _ _ Exact \
'\, 2/3 \
0.5
\
I
Walsh s o l u t i o n w i t h N
\ \
0
solution
I
\
\
'. --- 2/9
1
I
----
2/27
I
2/8 1 I
)
= 4
403 1
1
However, a n a p p r o x i m a t i o n o f s ( x , t )
5r k l
H-I
a(x,t)
&osiJ.
w i l l t h e n be,
(P,(t). cp y x )
j=
o r i n matrix notation
where s u p e r s c r i p t T d e n o t e s t h e m a t r i x t r a n s p o s i t i o n a n d 'MN
i s (MxN) m a t r i x o f drawdown v a r i a b l e ,
a
I n t e g r a t i o n o f Eq.(25) times w i t h r e s p e c t t o time time8 w i t h r e s p e c t t o t h e s p a c e h a s b e e n d e r i v e d and b y P a r a s k e v a p o u l o s a n d Bounas a s , x x t t (26) . ,( ( x 1 SMN.@ ( t 1d t dt times time o r s u c c i n c t l y t h i s i n t e g r a t i o n is g i v e n i n i t s i m p l i c i t m a t r i x n o t a t i o n as,
B
&/
-..i 6
.
.... a
p
U n d e r t h e l i g h t o f these e x p l a i n a t i o n s Eq.(23) w r i t t e n i n Walsh e x p a n s i o n form a s ,
c a n be
Hence, t h r o u g h t h e m a t r i x a l g e b r a t h e unknown, S M N , c a n be solved t h r o u g h a s i m i l a r p r o c e d u r e p r e s e n t e d b y P a r a s k e v a p o u P a s a n d Bounas.
404 CONCLUSIONS The Walsh f u n c t i o n e x p a n s i o n s have been a p p l i e d t o v a r i o u s p r o b l e m s encountered w i t h i n t h e c o n t e x t o f hydroscience.
They decompose
s u c c e s s f u l l y a g i v e n t i m e s e r i e s i n t o l i n e a r and s i m p l e components. T h e i r m a t h e m a t i c a l m a n i p u l a t i o n s a r e based on s i m p l e a d d i t i o n a n d / o r subtraction.
A c o m p l e t e s e t o f Walsh f u n c t i o n s i s t h e most c o n v e n i e n t
t r a n s f o r m a t i o n f o r r e p r e s e n t i n g t h e p o r o u s media sample c h a r a c t e r i s t i c function.
The p e r i o d i c - s t o c h a s t i c d a t a can be e f f e c t i v e l y a s s e s s e d b y
an a d a p t i v e Walsh p r o c e d u r e .
T h i s p r o c e d u r e does n o t g i v e o n l y t h e
p a r a m e t e r e s t i m a t i o n s a t e a c h t i m e i n s t a n t b u t a1 s o s i m u l t a n e o u s l y decompose s e r i e s i n t o p e r i o d i c and s t o c h a s t i c p a r t s .
The Walsh
f u n c t i o n s can be e f f e c t i v e y a p p l i e d i n t h e s o l u t i o n o f o r d n a r y o r p a r t i a l d i f f e r e n t i a l e q u a t ons. REFERENCES Brown, R.D. 1977. A r e c u r s i v e a l g o r i t h m f o r s e q u e n c y - o r d e r e d f a s t Walsh t r a n s f o r m s . I E E E T r a n s . on Comp. , C - 2 6 ( 8 ) : 8 7 9 - 8 8 2 . Chen, C.F., and H s i a o , C . H . , 1975. Walsh s e r i e s a n a l y s i s i n o p t i m a l c o n t r o l . V o l . 21, No. 6 : 881-897. Fara, H.D., and S c h e i d e g g e r , A.E., 1961. S t a t i s t i c a l g e o m e t r y o f p o r o u s media. J o u r . o f Geophys. Res. , Vol .66, No. 10: 3279-3284. Gelb, A., 1974. A p p l i e d o p t i m a l e s t i m a t i o n . M . I . T . P r e s s Cambridge, Mass.: 379 pp. Kalman, R . E . , 1960. A new a p p r o a c h t o l i n e a r f i l t e r i n g and p r e d i c t i o n t h e o r y . T r a n s . ASME, S e r . D., J o u r . B a s i c Engrg., V o l . 83: 35-45. P a r a s k e v o p o u l o s , P.N. , and Bounas, A.C. , 1978. D i s t r i b u t e d p a r a m e t e r system i d e n t i f i c a t i o n v i a Walsh f u n c t i o n s . I n t . J o u r . Systems S c i . , V o l . 9, No. 1 : 75-83. 1932. A r e m a r k a b l e s e r i e s o f o r t h o g o n a l f u n c t i o n s . Paley, R.E.A.C., P r o c . London Math. SOC., V o l . 34: 241-279. S h i h , Y . , and Han, J . , 1978. Double Walsh s e r i e s S o l u t i o n o f f i r s t o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . I n t . J o u r . Systems S c i . , Vol . 9, NO. 5 : 569-578. Sen, Z . , 1980. A d a p t i v e F o u r i e r a n a l y s i s o f p e r i o d i c - s t o c h a s t i c h y d r o l o g i c sequences. J o u r . Hydro1 . , Vol . 46: 239-249. Sen, Z . , 1981. Walsh a n a l y s i s o f m o n t h l y f l o w volumes. I n t . Symp. on R a i n f a l l - R u n o f f Modeling, M i s s i s s i p p i . Walsh, J . L . , 1923. A c l o s e d s e t of o r t h o g o n a l f u n c t i o n s . Amer. J o u r . Math. , Vol . 45: 5-24.
405
MODELLING THE ERROR I N FLOOD DISCHARGE MEASUREMENTS
KENNETH W. POTTER AND JOHN F. WALKER Department o f C i v i l and Environmental Engineering, U n i v e r s i t y of Wisconsin-Madison
ABSTRACT
The measurement of peak discharge is typically composed of three distinct processes.
Low magnitude floods are determined with an
established rating curve.
Intermediate magnitude floods are
inferred by extrapolating the established rating curve.
High mag-
nitude floods are usually determined through a field survey.
The
measurement error characteristics for each process are different, a phenomenon termed discontinuous measurement error (dme).
Monte Carlo
experiments with an error model that approximates the peak measurement process reveal a bias in the measured coefficients of variation, skewness, and kurtosis.
This bias is significant and has
important implications with regard to-flood frequency analysis.
INTRODUCTION
As the statistical methods available to the hydrologist become more sophisticated, i t is essential that closer attention be paid to the operational procedures by which hydrologic data are collected.
Failure to do so can lead to serious misinterpretation of
statistical results. Hurst phenomenon.
A notable example of such a failure is the
Clearly in many instances high Hurst coeffici-
ents are merely artifacts o f consistent measurement error. such as results from the relocation of a precipitation gage.
We believe
that similar artifacts may result from the way in whicli flood discharge records are constructed. For low magnitude floods, peak stages are recorded at the gage and the corresponding discharges are estimated from a rating curve established by current meter measurements.
For high magnitude
floods, peak stages are usually inferred from high-water marks and Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
406 the discharges are estimated by rating curve extension or by an indirect means, such as the slope-area method.
Clearly the vari-
ance of the discharge estimates is much higher for high magnitude floods.
We have termed this phenomenon "discontinuous measurement
error" (drne), and have shown that it causes the coefficients of variation, skewness, and kurtosis of the measured flood distribution to be much higher than the corresponding coefficients of the parent flood distribution (Potter and Walker, 1981). There are two limitations to our initial study of dme.
First,
we documented biases in population coefficients of variation, skewness, and kurtosis, rather than in small-sample coefficients.
The
latter, of course, are also subject to bias due to small-sample boundedness (Wallis et al., 1974).
Second, our model of dme was
based on the assumption that errors in estimates of high-magnitude floods are homescedastic and independent.
Such a model might be
appropriate if the slope-area method were independently applied to all high magnitude floods.
This is not, of course, the case.
For
many high magnitude floods the discharges are estimated by simple rating-curve extension.
This results in errors which are both cor-
related and heteroscedastic.
Furthermore, if the extended rating
curve is adjusted to be consistent with available slope-area estimates, the error variance also decreases with time.
The problem
may be further complicated by temporal changes in the measurement procedures and in the hydraulic conditions at the gage.
In this
paper we develop a model which more realistically mimics this complex measurement process.
We then use this model in a limited
exploration of the small-sample effects of dme.
NEW MODEL OF DME
Our new model is based on a three-tiered measurement process, as depicted in Figure 1.
For stages below a certain value (S ) , the
1
rating curve established by current meter measurements applies. Typically S1 would be bankful stage.
Between S
curve is linearly extended in log-space.
1
and S2, the rating
(In Figure 1 t h e sloping
407
Ind i r e c t
iiie as u r c m e n
ts
Rating curve extensions
E s t a b l i s h e d ratin::
F i g . 1.
A three-tiered
curve
e r r o r model
dashed l i n e r e p r e s e n t s t h e e x t e n d e d r a t i n g c u r v e ; t h e s l o p i n g s o l i d l i n e r e p r e s e n t s t h e a c t u a l , b u t unknown r a t i n g c u r v e . ) s t a g e s beyond S
2
i n d i r e c t measurement i s assumed.
For
Our model of
t h i s t h r e e - t i e r e d p r o c e s s i s b a s e d on t h e f o l l o w i n g a s s u m p t i o n s :
1.
S t a g e measurements a r e made w i t h o u t e r r o r .
2.
The l o w e r r a t i n g c u r v e i s l i n e a r i n l o g - s p a c e ,
i s known
w i t h o u t e r r o r , and i s u n c h a n g i n g i n t i m e .
3.
The t r u e ( b u t unknown) r a t i n g c u r v e e x t e n s i o n i s l i n e a r i n l o g - s p a c e and i s u n c h a n g i n g i n t i m e .
4.
T h e i n i t i a l e s t i m a t e of t h e i n v e r s e o f t h e s l o p e of t h e
r a t i n g c u r v e e x t e n s i o n i s a random v a r i a b l e w i t h a t h r e e parameter lognormal d i s t r i b u t i o n .
408 5.
The e s t i m a t e d r a t i n g c u r v e e x t e n s i o n i s c o n t i n u o u s l y u p d a t e d t o comply w i t h i n d i r e c t d i s c h a r g e m e a s u r e m e n t s .
6.
The e r r o r s i n i n d i r e c t d i s c h a r g e m e a s u r e m e n t s a r e i n d e p e n d e n t and h o m e s c e d a s t i c .
and a b o v e S t h e new model i s t h e same a s t h e p r e v i 1 2’ ous one, except t h a t t h e v a r i a n c e of t h e low-discharge e r r o r i s Below S
assumed t o b e z e r o ( s e e P o t t e r and W a l k e r , 1 9 8 1 , f o r d e t a i l s of t h e simple model).
T h i s is done t o g i v e t h e extended r a t i n g c u r v e
a f i x e d p o i n t from which t o b e g i n .
Because r a t i n g - c u r v e e r r o r s
a r e g e n e r a l l y much s m a l l e r t h a n t h e e r r o r s a s s o c i a t e d w i t h r a t i n g curve e x t e n s i o n s o r i n d i r e c t measurements, it i s b e l i e v e d t h a t t h i s simplification is reasonable.
In t h e i n t e r m e d i a t e r e g i o n b e t w e e n S is l i n e a r l y extended i n log-space.
where Q
m
and S the r a t i n g curve 1 2’ T h i s c a n b e r e p r e s e n t e d by
i s t h e estimated d i s c h a r g e with s t a g e S, ql
charge associated with S extended r a t i n g curve.
is the dis-
a n d l / &i s t h e assumed s l o p e o f t h e 1’ E q u a t i o n (1) c a n b e r e w r i t t e n as
where l/m i s t h e t r u e ( b u t unknown) s l o p e o f t h e e x t e n d e d r a t i n g curve.
L e t t i n g X = i / m , w e a s s u m e t h a t X i s a random v a r i a b l e
having a +parameter e q u a l t o m,/m, Thus l o g (X
E[X]
=
-
lognormal d i s t r i b u t i o n w i t h a s h i f t parameter
where l / m ,
is t h e s l o p e of t h e lower r a t i n g curve.
i s normally d i s t r i b u t e d . 2 1 and V [ X ] = u m,/m)
X
W e a l s o assume t h a t
.
The a s s u m p t i o n o f u n i t mean i n s u r e s t h a t estimates o f d i s c h a r g e b a s e d on r a t i n g - c u r v e e x t e n s i o n s a r e u n b i a s e d . assumed e r r o r s t r u c t u r e i n s u r e s t h a t
i, t h e
Furthermore, t h e
i n v e r s e of t h e e s t i -
mated s l o p e o f t h e e x t e n d e d r a t i n g c u r v e , i s a l w a y s b e t w e e n m, infinity.
and
The l o w e r bound i s c o n s i s t e n t w i t h t h e u s u a l f l a t t e n i n g
of t h e r a t i n g c u r v e beyond b a n k f u l s t a g e .
409
In actual field situations, rating curve extensions are adjusted as additional information is obtained by indirect-discharge measurements.
This facet of the measurement process is incorporated
in our model by adjusting made.
whenever an indirect measurement is
The adjustment is made by simple linear regression in log-
space on all available indirect measurements.
Thus the error in
;
decreases (on average) with each additional indirect measurement. The new model of dme can be summarized by
and t are random variables.
Note that
The variable t represents
the multiplicative error associated with indirect measurement, with E[t]
=
1 and V[t]
the estimate of
=
o
2
.
As indirect measurements are made,
is modified.
EFFECTS OF DME ON SMALL SAMPLES Monte Carlo experiments were conducted to explore the effects of dme on small-sample statistics. Our new model of dme involves seven parameters: S1
q1 m
- stage of first error discontinuity; - discharge corresponding to S
*
1’ - inverse of slope of lower rating curve;
m>,; - inverse of slope of true extended rating curve;
o
X
- variance of ;/m,
where
1/6 is the estimated slope of the
extended rating curve; S2 G
2
- stage of second error discontinuity;
- variance of multiplicative error associated with indirect measurements.
In order to reduce the parameter space, S was fixed at the stage
2 at which the variance of a rating curve extension just equalled
the variance of an indirect measurement.
This is a reasonable
410 c o n s t r a i n t , s i n c e u s e o f t h e r a t i n g c u r v e e x t e n s i o n beyond t h i s p o i n t would, on a v e r a g e , y i e l d l a r g e r e r r o r s t h a n u s e o f i n d i r e c t measurements.
s2
=
s1
Based on t h i s c o n s t r a i n t ,
+
expC[loge(l
1/2
a2) ]
/moxl
(4)
I n a d d i t i o n t o t h e model p a r a m e t e r s , which h a v e b e e n r e d u c e d t o s i x , i t i s n e c e s s a r y t o s p e c i f y t h e mean and c o e f f i c i e n t of v a r i a t i o n of t h e p a r e n t f l o o d d i s t r i b u t i o n ,
and o a f p a , and t h e l e n g t h 'a A s i n t h e c a s e of o u r i n i t i a l model of dme, t h e
of t h e sample, n .
p a r e n t f l o o d d i s t r i b u t i o n i s assumed t o b e l o g n o r m a l . Monte C a r l o r u n s w i t h numerous p a r a m e t e r c o m b i n a t i o n s i n d i c a t e d t h a t t h e s m a l l s a m p l e c o e f f i c i e n t s of v a r i a t i o n , s k e w n e s s , and k u r t o s i s of t h e measured f l o o d d i s c h a r g e s a r e u n a f f e c t e d by t h e c h o i c e of S l y m , m k , and p is ql,
o 2,
5,
X
.
a o a f p a , and n .
Therefore t h e r e l e v a n t parameter set
F i g u r e 2 i l l u s t r a t e s s e l e c t e d r e s u l t s o f t h e Monte C a r l o e x p e r i ments.
These r e s u l t s a r e e x p r e s s e d i n t e r m s o f t h e r e l a t i v e b i a s
i n t h e small-sample
c o e f f i c i e n t s of v a r i a t i o n , s k e w n e s s , and k u r -
t o s i s , where r e l a t i v e b i a s i s d e f i n e d as t h e r a t i o o f t h e a v e r a g e small-sample estimate of t h e p o p u l a t i o n c o e f f i c i e n t t o t h e population value.
Average s m a l l - s a m p l e
realizations.
estimates a r e b a s e d o n 1000
A l s o shown a r e t h e r e l a t i v e b i a s e s which r e s u l t
when t h e r e i s s a m p l i n g w i t h o u t e r r o r , d u e t o t h e boundedness of t h e sample c o e f f i c i e n t s . charge threshold of
6 / m (ax)
For t h e case i l l u s t r a t e d , t h e l o w e r d i s -
( q ) i s t h e 2-year
1
event; the standard deviation
i s 1 . 0 ; t h e s t a n d a r d d e v i a t i o n o f t h e i n d i r e c t measure-
ment e r r o r ( a ) i s 0 . 2 ; a n d t h e p a r e n t f l o o d p o p u l a t i o n c o e f f i c i e n t of v a r i a t i o n ( 5 / p ) i s 0 . 4 .
a
b i a s curves
-
a
I n e a c h p l o t , t h e r e a r e two r e l a t i v e
o n e f o r t h e dme e r r o r model and o n e f o r t h e case of
no measurement e r r o r .
I n t h e l a t t e r case, w i t h i n c r e a s i n g n t h e
r e l a t i v e b i a s c u r v e c o n v e r g e s from below t o t h e z e r o - b i a s r e p r e s e n t e d by t h e d a s h e d l i n e .
case,
T h i s , of c o u r s e , r e € l e c t s t h e
e a s i n g of t h e e f f e c t s o f s m a l l - s a m p l e boundedness w i t h i n c r e a s i n g sample s i z e .
411
0
5
10
15
20
25
30
35
40
45
50
55
6q.2
Fig. 2. Coefficients of variation, skewness, and kurtosis, from t o p to bottom, for oa/pa = 0 . 4 . The square and triangle s y m b o l s represent no error and the dme error model, respectively.
412 Figure 2 illustrates that for parent flood distributions with
/ v = 0 . 4 ) , the small-sample a a For all three coefficients the
average coefficients of variation ( u effects of dme are very striking.
relative biases increase with sample size.
In two cases the
effects of small-sample boundedness are offset for low sample sizes.
Thus for n greater than 30, dme leads to expected coeffi-
cients of variation and skewness greater than the population values.
In all cases the relative bias for the dme error model is
considerably higher than the zero-error case. By varying the model parameters, it was discovered that the small-sample biases due to dme diminish as the parent population coefficient of variation increases.
Thus for smaller coefficients
of variation, the biases are even more dramatic than the biases shown in Figure 2.
Furthermore, the coefficient of skewness
proved to be the most sensitive to the effects of dme.
Clearly
fitting techniques relying on the coefficient of skewness are highly suspect!
CONCLUSIONS AND RECOMMENDATIONS (1) It is clear from our results that dme has an important effect on the coefficients of variation, skewness, and kurtosis of mea-
sured flood discharges, particularly when the coefficient of variation of the parent flood distribution is low.
This effect is
to offset downward bias due to small-sample boundedness.
Further-
more the bias due to dme increases with sample size, unlike the bias due to most other sources.
(2)
Immediate attention should be focused on estimating the vari-
ance of errors in rating-curve extensions and indirect measurements, in order to determine the magnitude of the problem caused by dme.
(3)
If, as expected, dme proves to be an important problem in
flood-frequency estimation, ways must be developed to deal with it.
One obvious way is to abandon the coefficient of skewness.
413 (4)
The new model o f dme p r e s e n t e d i n t h i s p a p e r seems t o be a t
l e a s t c o n c e p t u a l l y a d e q u a t e f o r t h e case o f r i v e r s w i t h s t a b l e s e c t i o n s a t t h e i r gages.
It i s c l e a r l y n o t adequate f o r t h e
u n s t a b l e case.
ACKNOWLEDGEMENTS
Funding f o r t h i s r e s e a r c h w a s p r o v i d e d by t h e G r a d u a t e School and t h e E n g i n e e r i n g Experiment S t a t i o n of t h e U n i v e r s i t y of Wisconsin.
We would a l s o l i k e t o t h a n k Ruth Wyss f o r h e r u s u a l
splendid j o b of typing.
REFERENCES P o t t e r , K.W. and Walker, J o h n F . , 1 9 8 1 . A model of d i s c o n t i n u o u s measurement e r r o r a n d i t s e f f e c t s on t h e p r o b a b i l i t y d i s t r i b u t i o n of f l o o d d i s c h a r g e m e a s u r e m e n t s , Water R e s o u r . R e s . , 1 7 ( 5 ) , 1505-1509. Wallis, J . R . , Matalas, N . C . and S l a c k , J . R . , 1 9 7 4 . J u s t a moment!, Water R e s o u r . R e s . , 10(2), 211-219.
414 INFORMATION THEORETICAL CHARACTERISTICS OF SOtlE STATISTICAL MODELS I N THE HYDROSCIENCES W.F.
CASELTON
The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, B.C.,
1
Canada
INTRODUCTION W i t h i n t h e hydrosciences, t h e r e i s a growing need t o model
complex environmental phenomena which a r e b o t h s p a t i a l and random i n nature.
S t a t i s t i c a l models capable o f accommodating t h e l a r g e
number o f v a r i a b l e s i n v o l v e d are, i n themselves, h i g h l y complex. A broad u n d e r s t a n d i n g of t h e c h a r a c t e r i s t i c s o f such models, t h e
i n f l u e n c e o f t h e i r u n d e r l y i n g assumptions, t h e imp1 i c a t i o n s o f n o t meeting t h e s e assumptions i n p r a c t i c e , and t h e i r p o t e n t i a1 performance, a r e a l l d e s i r a b l e t o t h e p r a c t i t i o n e r b u t d i f f i c u l t t o achieve.
Theoretical investigations i n t o the characteristics
o f any p a r t i c u l a r m u l t i v a r i a t e model a r e o f t e n i n v o l v e d and t h e r e s u l t s d i f f ic u l t t o general ize. The a u t h o r o r i g i n a l l y encountered q u e s t i o n s c o n c e r n i n g t h e performance c h a r a c t e r i s t i c s o f mu1 t i v a r i a t e s p a t i a l models i n c o n n e c t i o n w i t h t h e design of m o n i t o r i n g networks.
There i s a
d i s t i n c t r i s k of p r o d u c i n g a network design which i s more a r e f l e c t i o n o f t h e t y p e o f s p a t i a l model adopted i n t h e design method r a t h e r than an o p t i m a l c o l l e c t o r o f i n f o r m a t i o n f o r t h e r e g i o n served.
Caselton and Husain C19801 have shown t h a t t h e
need t o adopt any form o f s p a t i a l model can be avoided when t h e network performance o b j e c t i v e i s s p e c i f i e d i n i n f o r m a t i o n t h e o r e t i c terms and i n f o r m a t i o n t r a n s m i s s i o n i s maximized. I n f o r m a t i o n t h e o r e t i c approaches t o e s t a b l i s h i n g e s t i m a t i o n performance bounds have been presented by Weidemann and S t e a r C19691 f o r t h e case o f parameter e s t i m a t i o n and by Tomita e t a l . C19761 i n c o n n e c t i o n w i t h t h e Kalman f i l t e r .
The l a t t e r showing
t h a t t h e o p t i m a l f i l t e r i s a l s o t h e optimal i n f o r m a t i o n Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) - Printed in The Netherlands
8 1982 Elsevier Scientific Publishing Company, Amsterdam
415
transmitter.
Both o f these papers a l s o d e s c r i b e aspects o f t h e
general case o f e s t i m a t i o n i n i n f o r m a t i o n terms and these a r e summarized here.
T h i s summary shows t h e r e l a t i o n s h i p between t h e
e n t r o p i e s o f t h e p r i n c i p a l v a r i a b l e s i n v o l v e d i n e s t i m a t i o n and t h e i n f o r m a t i o n transmissions between these v a r i a b l e s .
Some
s p e c i f i c types o f models which a r i s e i n t h e hydrosciences are then considered and i n c l u d e :
a simple form o f s p a t i a l e s t i m a t i o n ;
model s i n v o l v i ng s e r i a1 dependency; and models invol v i ng Gaussian errors. 2
A MEASURE OF INFORMATION
The i d e a o f q u a n t i t a t i v e l y measuring i n f o r m a t i o n has considerable appeal i n many s c i e n t i f i c and e n g i n e e r i n g s i t u a t i o n s . Many concepts o f i n f o r m a t i o n measures have been proposed b u t t h e t h r e e most o f t e n encountered a r e F i s h e r ' s , Shannon's, and t h e one which a r i s e s i n s t a t i s t i c a l d e c i s i o n a n a l y s i s .
A useful
comparative review o f these t h r e e measures i n a hydrosciences c o n t e x t has been p r o v i d e d by Dyhr-Nielsen C19721.
Only Shannon's
measure w i l l be discussed here. Shannon's i n f o r m a t i o n i s t i e d d i r e c t l y t o t h e concept o f message u n c e r t a i n t y p r i o r t o r e c e i v i n g a t r a n s m i t t e d s i g n a l and a f t e r receipt o f t h i s signal.
The q u a n t i t a t i v e measure o f
u n c e r t a i n t y used i s t h e f u n c t i o n H(X)
- 1 P(Xi)lOg
=
P(xi)
i
where
x
i
i s , f o r example, a d i s c r e t e message o r outcome c
random source
X.
The q u a n t i t y
H(X)
i s r e f e r r e d t o as t h
entropy o f
X.
source
t h e p r o b a b i l i t i e s concerning the message a r e am
Y,
Upon r e c e i p t o f a s i g n a l
y
jy
from t h e s i g n
p r o b a b i l it i e s and u n c e r t a i n t y i s now r e f 1 e c t e d by t h e condi entropy
416 The amount o f i n f o r m a t i o n t r a n s m i t t e d about
X
by
is
Y
and t h e r e d u c t i o n i n u n c e r t a i n t y a t t r i b u t a b l e t o t h e
I(X;Y)
signal. I(X,Y)
T h i s i s given by =
- H(XIY)
H(X)
(2.3)
o r equivalently I(X;Y)
=
H(X) + H(Y)
-
H(X,Y)
(2.4)
when t h e j o i n t entropy i s d e f i n e d as
(2.5) These d e f i n i t i o n s extend n a t u r a l l y t o t h e case where
and
X
Y
.,X m
a r e random v e c t o r s so t h a t , i f t h e message v e c t o r i s X1 ,X2,.. and t h e s i g n a l v e c t o r i s Y1,Y2, Y (where n i s n o t n
...
n e c e s s a r i l y equal t o m), then t h e i n f o r m a t i o n t r a n s m i t t e d i s
(2.6) 3 3.1
INFORMATION AND ESTIMATION Estimation Error The process o f e s t i m a t i o n w i l l be described i n a simple c o n t e x t
o f an m
dimensional s t a t e v e c t o r
o f t h e q u a n t i t y t o be estimated. vector
Z
X
An
representing the t r u e value n
dimensional measurement
w i l l r e p r e s e n t measurements which a r e i n some way A
related t o X
X
and upon which an
w i l l be based.
i s represented by
m
dimensional e s t i m a t e X
o*f
The e s t i m a t o r used t o produce t h e e s t i m a t e F(Z).
No r e s t r i c t i o n s as t o t h e form o f
X
F(
w i l l be imposed o t h e r than being unbiased. I n i n f o r m a t i o n terms
X
i s t h e message which i s b e i n g
t r a n s m i t t e d f i r s t by t h e i n t e r m e d i a t e s i g n a l
Z
which i s f u r t h e r
A
processed t o produce an o u t p u t s i g n a l estimation e r r o r vector
E
X.
i s d e f i n e d by
The
m
dimensional
417 A
E
=
x - x
(3.1)
=
X
-
(3.2)
or E
F(Z)
Because o f t h e dependency o f the estimate on t h e measurement i t follows t h a t p(X,Z)
=
p(X-F(Z) ,Z)
P(E,Z)
=
(3.3)
so that, from t h e d e f i n i t i o n o f entropy H(X,Z)
=
H(E,Z)
=
H(X,X)
A
H( E,X)
A
An i n f o r m a t i o n a l q u a n t i t y o f p a r t i c u l a r i n t e r e s t i s
I(X,^x),
t h e i n f o r m a t i o n t r a n s m i t t e d by the estimate about the t r u e state. From Equation 2.5 I(X;i)
=
H(X) + H(X^)
Adding and s u b t r a c t i n g
-
H(X,i)
(3.7) o f equation 3.7 and
from the R.H.S.
H(E)
s u b s t i t u t i n g e q u i v a l e n t j o i n t entropies from Equation 3.6 y i e l d s A
I(X;X)
=
H(X)
=
H(X)
-
A
A
-
H(E) + H(X) + H( E)
H( E,X)
A
H(E) + I(E;X)
(3.8)
Equation 3.8 expresses the i n f o r m a t i o n t r a n s m i t t e d by e s t i m a t i o n i n terms o f the estimate and i t s e r r o r only.
An a l t e r n a t i v e
expression i n v o l v i n g t h e measurement Z i s obtained by adding and subtracting I(X;i)
from the RHS o f Equation 3.8 which y i e l d s
H(Z)
-
=
H(X) + H(Z)
=
I(X;Z) t H(X,Z)
=
I(X;Z)
H(E)
-
H(Z) + I ( & ; ? ) A
~(x;i)
-
[I(E;z)
-
I(E;Z)
-
-
~(E;ji)
H( E;Z) + I ( E;X)
1
(3.9)
F i n a l l y combining equation 3.6 and 3.9 y i e l d s an expression s i m i l a r t o Equation 3.8. A
I( €;XI
From 3.9 I(X;Z)
=
I ( X ; i ) + I(E;Z)
-
These e x p r e s s i o n s w i l l be used i n t h e f o l l o w i n g s e c t i o n t o e s t a b l i s h bounds f o r i n f o r m a t i o n t r a n s m i s s i o n and e r r o r e n t r o p y . 3.2
PERFORMANCE BOUNDS
3.2.1
I n f o r m a t i o n Transmission
A t each s t e p i n a f e e d f o r w a r d e s t i m a t i o n process i n f o r m a t i o n
can o n l y be p r e s e r v e d o r l o s t , i t cannot be increased.
In
general I(X;Y) z I(X;G(Y)) where
1 i s any f u n c t i o n o f Y ( G a l l a g e r [19681).
G(
Thus t h e
in f o r m a t i o n t r a n s m i t t e d by t h e e s t i m a t e cannot exceed t h e i n f o r m a t i o n t r a n s m i t t e d by t h e measurement, i.e. A
I(X;Z)
2
(3.1 1 )
I(X;X)
Hence t h e upper bound o f i n f o r m a t i o n t r a n s m i s s i o n by t h e e s t i m a t e i s e s t a b l i s h e d by t h e i n f o r m a t i o n , o r l a c k o f information,
i n t h e measurement.
I(X;Z)
I(X;i)
and
An o v e r a l l upper bound t o b o t h
i s t h e e n t r o p y o f t h e message
i s o n l y achieved when a l l u n c e r t a i n t y c o n c e r n i n g
H(X) X
and t h i s
i s resolved
A
by
o r by
Z
3.2.2
X.
E r r o r Entropy
I n f o r m a t i o n t r a n s m i s s i o n i s always non-negative so t h a t i n Equations 3.8 and 3.10 h
I(E;Z) a 0
and
I(E;X) a 0
and e q u a t i o n s 3.8 and 3.10 can be w r i t t e n as i n e q u a l i t i e s A
H(E)
b
H(X)
-
I(X;X)
(3.12)
419 Because I(X;Z)
2
I(X;X)
then e q u a t i o n 3.12 w i l l g e n e r a l l y y i e l d a h i g h e r v a l u e f o r t h e lower bound o f e r r o r entropy.
Equation 3.13 has t h e advantage,
however, o f b e i n g a b l e t o p r e d i c t t h e l o w e r bound o f e s t i m a t i o n e r r o r e n t r o p y w i t h o u t t h e need t o d e f i n e t h e o p t i m a l e s t i m a t o r o r even r e s t r i c t t h e form o f t h e e s t i m a t o r .
T h i s e q u a t i o n was
proposed by Weidemann and S t e a r [19691 f o r purposes o f The l o w e r bounds
performance p r e d i c t i o n i n parameter e s t i m a t i o n .
o f e r r o r e n t r o p y a r e b o t h achieved when t h e e r r o r i s independent o f b o t h measurement and estimate.
Under t h i s (commonly assumed) A
condition, the information transmission
I(X;X)
i s maximized and
equations 3.8 and 3.10 become I(X;Z)
I(X;i)
=
=
H(X)
-
H(E)
(3.14)
This i n d i c a t e s t h a t t h e e s t i m a t o r preserves a l l i n f o r m a t i o n contained i n t h e measurement.
I n e f f e c t , t h e e r r o r a r i s e s from an
addi t i ve n o i se source i n t h e measurement process. 4
SPATIAL ESTIMATION An elementary case o f s p a t i a l e s t i m a t i o n i n v o l v e s t h e
e s t i m a t i o n o f events a t
q
d i s c r e t e l o c a t i o n s i n a r e g i o n on t h e
b a s i s o f measurements made a t j u s t a few o f these l o c a t i o n s , say n (where n K o r Ft > K and Ft+l < K , t where Ft i s t h e n o r m a l i z e d f l o w a t t i m e t . S i n c e F* i s n o r m a l i z e d , t t h e l e v e l s K are t h e number of s t a n d a r d d e v i a t i o n s from t h e mean, tween t i m e s t and t
*
1if F
and t h e r e g i o n of i n t e r e s t i s t y p i c a l l y -3 2 K 2 3; f o r j K ( 2 3 few crossings occur.
A s i n g l e estimated crossing d i s t r i b u t i o n i s
428
e s t i m a t e d f o r e a c h flow s e q u e n c e , s y n t h e t i c o r h i s t o r i c ; € o r synt h e t i c flows t h e average over t h e e s t i m a t e d c r o s s i n g d i s t r i b u t i o n s i s computed.
The s i g n i f i c a n c e of t h e c r o s s i n g d i s t r i b u t i o n i s t h a t i t repr e s e n t s t h e combined e f f e c t of t h e m a r g i n a l d i s t r i b u t i o n ( v a r i a b i l i t y a t a g i v e n t i m e ) and p e r s i s t e n c e .
For i n s t a n c e , changes i n skewness
a l t e r t h e c r o s s i n g d i s t r i b u t i o n by changing t h e number of low l e v e l r e l a t i v e t o high l e v e l crossings, while increasing t h e persistence d e c r e a s e s t h e number of c r o s s i n g s a t a l l l e v e l s .
Persistence effects
a r e p e r h a p s of g r e a t e r i n t e r e s t b e c a u s e t h e y a r e r e l a t e d t o d r o u g h t frequency.
IMPLEMENTATION A computer program w a s d e v e l o p e d t o p r o v i d e g r a p h i c a l d i s p l a y s
of t h e s t a t i s t i c a l and performance measures d i s c u s s e d f o r s i n g l e and multiple site analysis.
For s i n g l e s i t e s , c o e f f i c i e n t s of v a r i a t i o n
and skew c o e f f i c i e n t s a r e e s t i m a t e d on a s e a s o n a l and a n n u a l b a s i s . S e a s o n a l l a g one c o r r e l a t i o n c o e f f i c i e n t s are computed f o r a d j a c e n t s e a s o n s , and a n n u a l l a g one c o r r e l a t i o n c o e f f i c i e n t s and H u r s t c o e f f i c i e n t s are estimated.
For t h e l a t t e r , H u r s t ' s K e s t i m a t o r i s u s e d ,
p r i m a r i l y b e c a u s e i t r e q u i r e s much l e s s computer t i m e t h a n t h e GH e s t i m a t o r e v a l u a t e d by Wallis and Matalas (1970).
Sequent peak
s t o r a g e ( F i e r i n g , 1 9 6 7 ) and c r i t i c a l e x t r a c t i o n r a t e a r e computed f o r t h e e n t i r e ( s e a s o n a l ) h i s t o r i c and s y n t h e t i c r e c o r d s , where a n n u a l demand i s a f i x e d q u a n t i t y a p p o r t i o n e d e q u a l l y t o e a c h s e a s o n .
Annual
demand l e v e l f o r t h e s e q u e n t peak s t o r a g e d e t e r m i n a t i o n , and s t o r a g e capacity f o r t h e c r i t i c a l e x t r a c t i o n determination are s p e c i f i e d a priori.
E x p e r i e n c e h a s shown t h a t t h e most u s e f u l r e s u l t s a r e
produced when s e q u e n t peak demand i s r e l a t i v e l y h i g h , and c r i t i c a l e x t r a c t i o n s t o r a g e i s low, so t h a t t h e h y p o t h e t i c a l s y s t e m s are relatively highly stressed.
F i n a l l y , c r o s s i n g d i s t r i b u t i o n s are
computed f o r t h e i n d i v i d u a l s i t e s a s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n , For m u l t i p l e s i t e v a l i d a t i o n , many of t h e same measures used f o r s i n g l e s i t e s are a p p l i e d t o a h y p o t h e t i c a l a g g r e g a t e r e c o r d formed by
429 adding t h e p r e d i c t e d f l o w s a t s e l e c t e d s i t e p a i r s .
Two o p t i o n s are
p r o v i d e d ; t h e f i r s t t a k e s t h e s i m p l e a v e r a g e of t h e f l o w s a t t h e two s i t e s , w h i l e t h e second computes an a g g r e g a t e f l o w e q u a l t o one h a l f of t h e sum of t h e f l o w a t t h e f i r s t s i t e and t h e w e i g h t e d f l o w a t t h e second s i t e , where t h e w e i g h t i s t h e r a t i o of t h e a n n u a l mean a t t h e f i r s t s i t e t o t h e a n n u a l mean a t t h e second s i t e .
The l a t t e r
o p t i o n h a s t h e a d v a n t a g e t h a t d i f f e r e n c e s i n mean f l o w do n o t a l l o w e i t h e r s t a t i o n t o d o m i n a t e , a s s u r i n g t h a t m u l t i p l e s i t e e f f e c t s are r e p r e s e n t e d i n t h e a g g r e g a t e flow. The s t a t i s t i c a l i n d i c a t o r s computed f o r s t a t i o n p a i r s a r e t h e c o e f f i c i e n t s of v a r i a t i o n and skew c o e f f i c i e n t s f o r s e a s o n a l and a n n u a l a g g r e g a t e f l o w , l a g one c o r r e l a t i o n c o e f f i c i e n t s and e s t i m a t e d H u r s t c o e f f i c i e n t of a g g r e g a t e a n n u a l f l o w s , and s e a s o n a l and a n n u a l lag zero cross correlations.
Sequent peak s t o r a g e , c r i t i c a l e x t r a c -
t i o n r a t e , and c r o s s i n g d i s t r i b u t i o n s are a l s o e s t i m a t e d f o r t h e aggregate flows.
A t t h e i n d i v i d u a l s i t e s , c o e f f i c i e n t s of v a r i a t i o n
and skew c o e f f i c i e n t s are estimates f o r s e a s o n a l and a n n u a l f l o w s , i n a d d i t i o n t o s e a s o n a l l a g one c o r r e l a t i o n s and a n n u a l H u r s t c o e f f i c i e n t s and l a g one c o r r e l a t i o n c o e f f i c i e n t s .
Sequent peak s t o r a g e ,
c r i t i c a l e x t r a c t i o n r a t e , and c r o s s i n g d i s t r i b u t i o n s are a l s o e s t i mated a t t h e i n d i v i d u a l s i t e s . The r e s u l t s of t h e a n a l y s e s are p l o t t e d as e m p i r i c a l c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n s on a normal p r o b a b i l i t y s c a l e .
The normal
p r o b a b i l i t y s c a l e h a s no p a r t i c u l a r s i g n i f i c a n c e o t h e r t h a n i t s wide f a m i l i a r i t y and i t s a b i l i t y ( i n common w i t h o t h e r p r o b a b i l i t y s c a l e s ) t o t r a n s f o r m e x t r e m e e v e n t s f o r ease of g r a p h i c a l comparison.
No
i m p l i c a t i o n i s made t h a t t h e d i s t r i b u t i o n s e s t i m a t e d a r e , o r s h o u l d b e , normal.
The g r e a t e s t a d v a n t a g e i n p r e s e n t i n g f u l l d i s t r i b u t i o n s
of v a l i d a t i o n m e a s u r e s , r a t h e r t h a n summary measures ( s u c h a s mean and v a r i a n c e ) i s t h a t t h e a n a l y s t i s made i m m e d i a t e l y aware of how f a r t h e h i s t o r i c e s t i m a t e s l i e from t h e predominance of s y n t h e t i c estimates.
For i n s t a n c e , i n e x t r e m e c a s e s i t may b e t h a t a l l of t h e
s y n t h e t i c e s t i m a t e s l i e above o r below t h e h i s t o r i c , which would b e a rare o c c u r r e n c e i f t h e s y n t h e t i c model were i n f a c t r e p r e s e n t a t i v e
430 of t h e h i s t o r i c p r o c e s s .
Comparison of means and v a r i a n c e s of syn-
t h e t i c and h i s t o r i c measures d o e s n o t p r o v i d e n e a r l y a s c l e a r an i n d i c a t i o n of model p e r f o r m a n c e , as s h o u l d become c l e a r from t h e r e s u l t s presented i n the following section.
APPLICATION
The v a l i d a t i o n p r o c e d u r e d i s c u s s e d above w a s a p p l i e d t o t h r e e t w o - s i t e models of t h e Cedar R i v e r , and t h e North Fork of t h e Snoq u a l m i e R i v e r , Washington.
These models w e r e e s t i m a t e d from 48
y e a r s of c o i n c i d e n t r e c o r d a t t h e two s t a t i o n s .
The r a w f l o w r e c o r d
c o n s i s t e d of r e c o r d e d monthly volumes and estimates p r o v i d e d by t h e U.S.
Army Corps of E n g i n e e r s d u r i n g p e r i o d s when o b s e r v a t i o n s w e r e
n o t made.
The monthly f l o w volumes were a g g r e g a t e d t o t h r e e s e a s o n s
p e r (water) year:
October-February,
The r u n o f f r e s p o n s e i n b o t h b a s i n s
March-June,
and July-September.
i s dominated by snow a c c u m u l a t i o n
d u r i n g t h e w i n t e r months, and m e l t d u r i n g t h e s p r i n g and e a r l y summer. T h e r e f o r e , t h e s e a s o n s were chosen t o r e f l e c t p e r i o d s of dominant snow a c c u m u l a t i o n , m e l t , and snow-free c o n d i t i o n s .
The t h r e e models
c o n s i d e r e d f o r g e n e r a t i o n of a n n u a l flow volumes w e r e A) m u l t i v a r i a t e l a g one Markov ( C l a r k e , 1 9 7 3 ) ; B) m u l t i - s i t e ARMA (1,l) ( L e d o l t e r , 1 9 7 8 ) , a maximum l i k e l i h o o d a p p r o a c h , and C) m u l t i p l e s i t e ARMA (1,l) w i t h m o d i f i e d maximum l i k e l i h o o d e s t i m a t i o n ( L e t t e n m a i e r , 1 9 8 0 ) . Each a n n u a l model was used t o g e n e r a t e 100 s e q u e n c e s of l e n g t h 48 y e a r s i n n o r m a l i z e d ( z e r o mean, u n i t v a r i a n c e , normal m a r g i n a l d i s t r i b u t i o n ) form.
These n o r m a l i z e d s e q u e n c e s were t h e n d i s a g g r e g a t e d
u s i n g L a n e ' s (1979) m u l t i p l e s i t e d i s a g g r e g a t i o n model i n t h e normal domain, and s u b s e q u e n t l y t r a n s f o r m e d t o s y n t h e t i c f l o w s u s i n g t h r e e p a r a m e t e r l o g normal t r a n s f o r m a t i o n s .
Th er ef o r e, each s y n t h e t i c
model c o n s i s t s of an a n n u a l model c o u p l e d w i t h a s e a s o n a l d i s a g g r e g a t i o n model, where t h e l a t t e r i s t h e same f o r a l l t h r e e a n n u a l models. AlthouLh t h e p u r p o s e of t h i s p a p e r i s t o d i s c u s s a v a l i d a t i o n t e c h n i q u e , and n o t t o assess models per s e , a b r i e f comment r e g a r d i n g t h e a n n u a l models s h o u l d b e made.
The l a g one Markov model (Model A )
431 is a s h o r t t e r m p e r s i s t e n c e model which h a s a r e l a t i v e l y s m a l l low frequency component, h e n c e l o n g p e r i o d s of f l o w above o r below t h e mean a r e n o t r e p r e s e n t e d .
For t h i s r e a s o n , c o n s i d e r a b l e work i n
s t o c h a s t i c h y d r o l o g y h a s b e e n d i r e c t e d t o w a r d s development of models
similar t o t h e second two, which may b e d e s c r i b e d as l o n g t e r m p e r s i s t e n t (Models B and C ) .
These models a r e c a p a b l e of g e n e r a t i n g
very l o n g p e r i o d s of d e f i c i t ( d r o u g h t ) and e x c e s s f l o w s .
One of t h e
d i f f i c u l t i e s w i t h m u l t i p l e s i t e g e n e r a t i o n of f l o w s w i t h long t e r m p e r s i s t e n c e i s a s s u r i n g t h a t t h e flows a t t h e i n d i v i d u a l s i t e s are
s i m i l a r w i t h t h o s e t h a t would have b e e n g e n e r a t e d had t h e s i t e s b e e n r e p r e s e n t e d b y a u n i v a r i a t e model.
T o accomplish t h i s , Lettenmaier
(1980) proposed a m o d i f i e d maximum l i k e l i h o o d e s t i m a t i o n t e c h n i q u e , i n c o r p o r a t e d i n model C , which p e n a l i z e s p a r a m e t e r estimates t h a t g i v e r i s e t o a u t o c o r r e l a t i o n s t r u c t u r e s much d i f f e r e n t from t h o s e r e p r e s e n t e d by s i n g l e s i t e e s t i m a t e s of t h e H u r s t c o e f f i c i e n t .
RESULTS The computer program developed t o p e r f o r m t h e model v a l i d a t i o n can p r o v i d e o u t p u t i n t h e form of l i n e p r i n t e r p l o t s on a computer t e r m i n a l o r h a r d copy, o r as c o n t i n u o u s p l o t s on a g r a p h i c s t e r m i n a l or ink p l o t t e r .
Given t h e l a r g e number of p l o t s i n v o l v e d i n m u l t i p l e
s i t e a p p l i c a t i o n s , w e have found t h e f i r s t o p t i o n s p r e f e r a b l e .
Due
t o s p a c e l i m i t a t i o n s h e r e , o n l y a s m a l l s u b s e t of t h e p l o t s g e n e r a t e d f o r e a c h model can b e shown.
I n p r a c t i c e , g i v e n t h e g r e a t number of
p l o t s g e n e r a t e d i n v a l i d a t i o n of a m u l t i s i t e , m u l t i s e a s o n model, w e f i n d i t i s much q u i c k e r t o r e v i e w a group of l i n e p r i n t e r
p l o t s as
h a r d copy, r a t h e r t h a n i n d i v i d u a l p l o t s on a computer t e r m i n a l s c r e e n F i g u r e s la-d
show t h e e m p i r i c a l d i s t r i b u t i o n s of t h e c o e f f i c i e n t
of v a r i a t i o n f o r t h e a g g r e g a t e f l o w s g e n e r a t e d from Model A f o r seas o n s 1-3 and f o r a n n u a l f l o w volumes.
I n t h e s e f i g u r e s , as i n t h o s e
t h a t f o l l o w , t h e ( s i n g l e ) h i s t o r i c estimate i s p l o t t e d as a dashed l i n e between c u m u l a t i v e p r o b a b i l i t y l e v e l s 10 and 90 p e r c e n t .
If
t h e s y n t h e t i c s e q u e n c e s were i n d e p e n d e n t of t h e h i s t o r i c r e c o r d , t h e e m p i r i c a l c u m u l a t i v e d i s t r i b u t i o n from t h e s y n t h e t i c f l o w s would b e
432
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433 e x p e c t e d t o c r o s s t h e d a s h e d l i n e w i t h a b o u t 80 p e r c e n t c o n f i d e n c e . S i n c e h i s t o r i c and s y n t h e t i c f l o w s e q u e n c e s a r e n o t i n d e p e n d e n t , t h e t r u e c o n f i d e n c e r e g i o n r e p r e s e n t e d by t h i s l i n e s h o u l d b e c o n s i d e r a b l y h i g h e r t h a n 80 p e r c e n t . S i n c e t h e s e a s o n a l c o e f f i c i e n t s of v a r i a t i o n are l a r g e l y d e t e r mined by t h e d i s a g g r e g a t i o n model, and n o t t h e a n n u a l g e n e r a t o r ,
s i m i l a r r e s u l t s t o t h o s e shown i n F i g u r e s l a - c were i n d i c a t e d f o r A s shown i n F i g u r e l c , t h e s e a s o n 3 c o e f f i c i e n t o f
models B and C.
v a r i a t i o n w a s s l i g h t l y underestimated.
T h i s may b e a r e s u l t of t h e
method used i n t h e d i s a g g r e g a t i o n model t o c o n s e r v e mass, which somet i m e s r e s u l t s i n s l i g h t i n c o n s i s t e n c i e s i n t h e f i r s t a n d / o r l a s t sea-
s o n of t h e y e a r .
F i g u r e I d i n d i c a t e s t h a t a n n u a l c o e f f i c i e n t s of
v a r i a t i o n from t h e s y n t h e t i c s e q u e n c e s w e r e s i m i l a r t o t h e h i s t o r i c v a l u e (median a p p r o x i m a t e l y e q u a l t o h i s t o r i c a n n u a l c o e f f i c i e n t of variation).
However, r e s u l t s f o r models B and C ( n o t shown) i n d i c a t e d
t h a t t h e s y n t h e t i c c o e f f i c i e n t s of v a r i a t i o n were s l i g h t l y l o w e r t h a n the historic.
T h i s may b e t h e r e s u l t of b i a s i n g of t h e e s t i m a t o r
by t h e h i g h e r a u t o c o r r e l a t i o n s p r e s e n t i n mqdels B and C . F i g u r e s 2a-d show r e s u l t s from Model A a g g r e g a t e f l o w s f o r s e a s o n s 1-3 and a n n u a l t o t a l s .
A s w i t h t h e c o e f f i c i e n t of v a r i a t i o n ,
t h e r e s u l t s are q u i t e s i m i l a r f o r a l l models.
The e m p i r i c a l d i s t r i -
b u t i o n s f o r s y n t h e t i c f l o w s g e n e r a l l y r e f l e c t t h e downward b i a s i n g of t h e e s t i m a t o r of t h e skew c o e f f i c i e n t ( W a l l i s , e t a l . , 1 9 7 4 ) . Although t h i s b i a s c o u l d b e c o r r e c t e d a t t h e p a r a m e t e r e s t i m a t i o n s t a g e , c o r r e c t i o n of moments f o r b i a s may b e c o u n t e r p r o d u c t i v e , as shown by S t e d i n g e r (1980); t h e r e f o r e n o b i a s c o r r e c t i o n w a s a t t e m p t e d . A s f o r t h e c o e f f i c i e n t of v a r i a t i o n ,
p a r e n t l y anomolous.
t h e r e s u l t s f o r s e a s o n 3 a r e ap-
We b e l i e v e t h a t t h i s i s a l s o r e l a t e d t o t h e
m a s s c o n s e r v a t i o n a d j u s t m e n t made i n t h e d i s a g g r e g a t i o n model.
F i g u r e s 3a-d
show t h e e m p i r i c a l d i s t r i b u t i o n of K - e s t i m a t o r s
of t h e H u r s t s t a t i s t i c , as w e l l as t h e h i s t o r i c estimates f o r a n n u a l f l o w volumes a t b o t h s i t e s , f o r a l l t h r e e models. ( F i g u r e s 3a and 3b) y i e l d s estimates t h a t
The Markov model
g e n e r a l l y a r e more com-
p a t i b l e ( l o w e r ) t h a n t h e ARMA models, a l t h o u g h t h e m o d i f i e d ARMA
1.50
.................................................................................
1.50
SKEW C O E F F I C I E N T I S U Y l E D F L W S l CEDAR 1 I N F SNW SEASON 1 SYWTHETIC FLOW
. ccccccc
1 .a).
........
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E S T I M A T E FROM H l S T O R t C RECORD
1.10..
.......
ESTIMATE FROM H I S T O R I C RECORD
1.20..
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1.05:.
SKEW C O E F F I C I E W T I S W E D FLOW81 CEDAR 1 1 1 F S N W SEASON 1 SYNTHETIC FLOW
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cc ccc cc ccc PP cccc ................................. c ........................................ ............................ .......................................... o....oi..o~.i.a..si..z...s...~o..zo.ao.~oso.~o.~o.zo..~o..~...z..~..s..z.~.os..o~. 0 . . . . o i . . o s . i . I . . s i . .I.. .5... l o . . l o . s o . ~ o s o . ~ o . a o . ~ o 1 0.. .. o . . .z..I . .s. .z.t . 0 5 .
15'
--
C W U L A T I V E PROBABILITY I P I
.IS..
EXCEEOANCE P R O B A O I L I T Y 1 1 - P I
Season 1
2a. 1.50
CUMULATIVE P R O B A B I L I T Y I P I
2b.
.................................................
..............................
C. SKEW C O E F F I C I E N T lSULYED F L W S l CEDAR 111 F S N W SEASON S S V N T H E T I C [ L O W
1.50
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EXCEEDAWCE P R O B A B I L I T Y 1 1 - P I
Season 2 ..................................................................
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1.05..
1.05..
.DO..
.90:.
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0
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C C C
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.lo..
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cc C cc ccc c c . . ...I.I....1.. . . . .. .6.. .. .. .t .o.. ..io.ao.40~0.~0.~0.10.. . . . . . c c . . ................................ o .. .. .....o.i. .. .. .o.s...~... z. .. ...~.i..... z. .. .....s..... .. ~. .o.....~.o. .. .a .o...~.o.s. o. .. .~.o...~. o. .. .~.o.....~. o. .. ...s....... z. .. ...1 . . ~ . . 1 . ~ . o ~ .0...o..~.....o. i. .. ..o.s...1..z. 10. .s.. .I.. I. . a , . 2. 1 . 0 5 .
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CUMULATIVE P R O 8 A B l L I T Y I P I
2c.
Season 3
F i g u r e 2.
EXCEEDANCE P R O B A B I L I T Y 1 1 - P I
..........................
CUYULATIVE P R O B A B I L I T Y I P I
EXCEEDAWCE P 1 1 0 B A B l L I T Y I t - P I
21. Annual E m p i r i c a l c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n s of skew c o e f f i c i e n t f o r a g g r e g a t e d f l o w s , Model A , S e a s o n s 1-3 and Annual.
,ii :
1.00
.......................................................
.95..
1.00
HURST C O E F F I C I E N T I K METHOD1 CEDAR INFLOW 1 ANNUAL SYNTHETIC FLOWS
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I
9s
E S T I M A T E FROM H I S T O R I C RECORD
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HURST C O E F F I C I E N T 111. METHOD1 N F SNOQUALYIE ANNUAL SYNTHETIC FLOWS
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c c c
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cc . . . . . . . . . . . . . . . . . . . . . . . . ccc ............................................... . . . . . .....0.5. .. .1 .. ..~...s .i .. ..z. .. .....s.. . .cc~ o. .....~. o. ...a.o. .. .~.o.~.o. .. .~.o... a. .o...~. o. .....~.o. .. ...s... .. ...z... .. .~.....s . . ~ . ~ . o s . ~~0....01..05.1.1..51..1...5...10..20.J0.4050.k0.30.20..10..~...2..1..S..2.1.05..01. .o~.
so ....01
CWULATIVE PROBABILITY I P I
3b.
Cedar E5ver, Xodel A
3a. 1.00
.................................................................................
.95..
1.00
MURST COEFFlClEW? IK METHOD1 CEDAR INFLOW 1 AWYUAL SYNTHETIC FLOWS
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........
....................................................
ESTIMATE F l l W H I S T O R I C RECORD
HURST C O E F F I C I E N T I K METHOD1 Y F SWWUALMIE ANNUAL SYNTHETIC FLOWS
........
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..............
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. . . . . . . . . . . . c . . c cc ...................................................... . ..oi..os.i.~..5i..~...s...io..~o.ao.~oso.~o.ao.~o..~o..5...~..1..s..z.1.o1..o~. CWULATIVE PROBABILITY I P I
ESTIUATE FROM H I S T O R I C RECORD
.15:.
cccc
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,101.
50..
11-PI
North 3ork Snoqualnbe Xiver, Kodel A
.so..
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EXCEEDANCE P R O B A B I L I T Y 1 1 - P I
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EXCEEDANCE P R O B A B I L I T Y
11-PI
3c. Cedar River, '::ode1 B 3d. North Pork Snocualnie Xiver, :Podel B Figur'e 3. Empirical cumulative distribution functions of Hurst coefficient (K estimator), Cedar River and North Fork Snoqualmie River, Models A and B.
436 model ( n o t shown) a p p e a r e d p r e f e r a b l e t o model B ( f i g u r e s 3c and 3d) i n t h i s respect.
S i n c e t h e primary m o t i v a t i o n behind long t e r m per-
s i s t e n c e models i s t o p r e s e r v e t h e s o - c a l l e d H u r s t e f f e c t , r e p r e s e n t e d by t h e H u r s t c o e f f i c i e n t , i t i s n o t s u r p r i s i n g t h a t Models B and C w i l l g e n e r a l l y have h i g h e r H u r s t c o e f f i c i e n t s , s i n c e t h e e s t i mator K i s b i a s e d upwards f o r s m a l l
( 5 0 . 7 ) v a l u e s of H.
The r e s u l t s
do emphasize, however, t h a t b i a s i n t h e e s t i m a t o r may i t s e l f b e a s u f f i c i e n t e x p l a n a t i o n of t h e H u r s t e f f e c t , s i n c e Model A , i n expect a t i o n , h a s an H of 0 . 5 ,
c o n s i d e r a b l y less t h a n t h e h i s t o r i c e s t i m a t e
a t e i t h e r s i t e , even a l t h o u g h t h e median s y n t h e t i c K estimate i s approximately equal t o t h e h i s t o r i c value. F i g u r e s 4a-d
show e m p i r i c a l d i s t r i b u t i o n s of s e a s o n a l l a g one
c o r r e l a t i o n s f o r Model A , s e a s o n s 2 and 3 a t b o t h s i t e s .
In these
f i g u r e s t h e c o r r e l a t i o n f o r t h e season i n d i c a t e d r e p r e s e n t s t h e l a g one c o r r e l a t i o n w i t h t h e s u b s e q u e n t s e a s o n .
S i n c e t h e s e a s o n a l cor-
r e l a t i o n s a r e d e t e r m i n e d by t h e d i s a g g r e g a t i o n model, r e s u l t s are e s s e n t i a l l y i d e n t i c a l f o r a l l t h r e e a n n u a l models.
Season 1 c o r r e l a -
t i o n s ( n o t shown) were n e a r z e r o f o r b o t h t h e h i s t o r i c estimate and t h e s y n t h e t i c median a t b o t h s i t e s .
Although t h e low c o r r e l a t i o n
between s e a s o n 1 ( w i n t e r ) and s e a s o n 2 ( s p r i n g ) may s e e m c o u n t e r i n t u i t i v e , i t i s t h e r e s u l t of a m i x t u r e of e f f e c t s .
During w i n t e r
s e a s o n s w i t h normal o r below normal t e m p e r a t u r e s , low r u n o f f o c c u r s
as much of t h e p r e c i p i t a t i o n i s s t o r e d i n t h e snowpack, and s u b s e quently c o n t r i b u t e s t o s p r i n g runoff suggesting a negative correlation.
I f p r e c i p i t a t i o n i s below normal a n d / o r t e m p e r a t u r e s a r e
above n o r m a l , on t h e o t h e r hand, a p o s i t i v e c o r r e l a t i o n w i t h s p r i n g runoff is indicated.
The n e t e f f e c t i s t o make t h e c o r r e l a t i o n be-
tween t h e s e two s e a s o n s a p p r o x i m a t e l y z e r o .
Season 2 and s e a s o n 3
c o r r e l a t i o n s are p o s i t i v e , i n d i c a t i v e of f l o w p e r s i s t e n c e which i s u s u a l l y observed i n rain-af f e c t e d watersheds.
The o n l y p o s s i b l e
model inadequacy i n d i c a t e d by F i g u r e 4 i s t h a t s e a s o n 3 c o r r e l a t i o n s f o r t h e second s i t e a r e s l i g h t l y o v e r e s t i m a t e d .
G e n e r a l l y , however,
t h e s y n t h e t i c and h i s t o r i c f l o w s a p p e a r t o b e c o m p a t i b l e w i t h r e s p e c t o seasonal correlations.
....................................................
1.15
LAO ONE CORRELATION C O E F F I C I E N T INFLOW 1 SEASON 1 SYNTHETIC FLMS
SEASONAL CEDAR
i.ia..
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........
E S T I M A T E TROY H I S T O R I C RECORD
1.00..
1.15
................................................................................. SEASONAL LAO ONE C O l R E L A T I O N C O E F F I C I E N T
1. I a
:
,
I N F L O W1
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SEASON
a
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F R M I H l S T O R I C RECORD
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4a.
Cedar R i v e r , Season 2
1.25
........................................................................
1.1s..
1.15
SEASONAL LAO OWE CORRELATION C O E F F I C I E N T N F SWOPUALMIE SEASON 1 SYNTHETIC FLOWS
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0....01..05.1.1..51..1...5...10..10.~0.4050.40.~0.10..10..5...1..1..5..1.1.05..01. CUMULATIVE P R O B A B I L I T Y ( P I
4c.
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SEASONAL LAO ONE CORRELATION C O E F F I C I E N T N F SNOPUALMIE SEASON 3 SYNTHETIC FLOWS
. .
i 1a.
E S T I Y A T E F R W H I S T O R I C RECORD
n
.................................................................................
EXCEEDANCE P R O B A B l L I T Y 1 1 - P I
..........................................................................
o....o~..o~.~.~..5i..i...5...io..io.~o.~o~o.~o.~o.io..io..5...i..i..5..~.i.o~..oi. CUMULATIVE P R O B A B I L I T Y 1P1
EXCEEDANCE PROBABIL1TV 1 1 - P I
North Fork Snoqualnie River, Season 2 4d. North Fork S n o q u a l n i e E i v e r , Season 3 F i g u r e 4 . E m p i r i c a l cumulative d i s t r i b u t i o n f u n c t i o n s of s e a s o n a l l a g one c o r r e l a t i o n c o e f f i c i e n t € o r Model A , Seasons 2 and 3, Cedar River and North Fork Snoqualmie River.
rp
w
4
438 F i g u r e s 5a-c show a n n u a l l a g z e r o c r o s s c o r r e l a t i o n s f o r a l l t h r e e models.
The r e s u l t s r e p r e s e n t one of t h e most s i g n i f i c a n t
d i f f e r e n c e s between models.
The median Model A s y n t h e t i c l a g z e r o
c o r r e l a t i o n s are q u i t e c l o s e t o t h e h i s t o r i c e s t i m a t e , w h i l e t h e Model B and Model C c o r r e l a t i o n s a r e much l e s s t h a n t h e h i s t o r i c . T h i s i s d i r e c t l y a t t r i b u t a b l e t o t h e p a r a m e t e r e s t i m a t i o n methods used by t h e t h r e e models:
t h e annual cross c o r r e l a t i o n matrix i s
an e x p l i c i t p a r a m e t e r s e t i n t h e Markov model, w h i l e i t i s n o t i n t h e ARMA models which u s e maximum l i k e l i h o o d e s t i m a t o r s .
Sensitivity
a n a l y s i s s u g g e s t s t h a t t h e m u l t i s i t e ARMA models a c h i e v e l o n g t e r m p e r s i s t e n c e a t t h e i n d i v i d u a l s i t e s by r e d u c i n g c r o s s c o r r e l a t i o n s , t h e r e f o r e a t r a d e o f f i s i n d i c a t e d i n t h e s e models between p r e s e r v a t i o n of s i n g l e s i t e and c r o s s s i t e p r o p e r t i e s . F i g u r e s 5d-f
show a n n u a l l a g one c o r r e l a t i o n c o e f f i c i e n t s f o r
t h e aggregate flows.
These f i g u r e s do n o t p r o v i d e a complete p i c t u r e
of t h e d i f f e r e n c e s i n c o r r e l a t i o n s t r u c t u r e between models, f o r i n s t a n c e a l t h o u g h l a g one s y n t h e t i c c o r r e l a t i o n s f o r Models A and C a r e s i m i l a r , t h e a u t o c o r r e l a t i o n f u n c t i o n f o r Model A ( F i g u r e 5d)
decays much more r a p i d l y t h a n f o r Model C .
These f i g u r e s do i n d i -
c a t e , however, t h a t i n t e r m s of h i g h f r e q u e n c y e f f e c t s models A and C a p p e a r t o b e p r e f e r a b l e t o model B , which g e n e r a l l y o v e r e s t i m a t e s
low l a g c o r r e l a t i o n s . F i g u r e s 6a-f
show a g g r e g a t e f l o w s e q u e n t peak s t o r a g e and c r i t i c a l
e x t r a c t i o n f o r t h e t h r e e models.
A s d i s c u s s e d e a r l i e r , s e l e c t i o n of
t h e demand l e v e l f o r t h e s e q u e n t peak a l g o r i t h m , and s t o r a g e c a p a c i t y f o r t h e c r i t i c a l e x t r a c t i o n computation d e t e r m i n e s e n s i t i v i t y of t h e s e i n d i c a t o r s t o p o s s i b l e model i n a d e q u a c i e s .
It w a s determined
t h a t a demand l e v e l of 0.90 t i m e s mean a n n u a l f l o w w a s a p p r o p r i a t e f o r t h e s e q u e n t peak c o m p u t a t i o n s , and a r e s e r v o i r s t o r a g e of 0.25 t i m e s t h e mean a n n u a l i n f l o w f o r c r i t i c a l e x t r a c t i o n d e t e r m i n a t i o n s .
The i n d i c a t e d model d i f f e r e n c e s are s i m i l a r t o t h o s e s u g g e s t e d by t h e H u r s t c o e f f i c i e n t ; s t o r a g e r e q u i r e m e n t s f o r Model A a r e s l i g h t l y u n d e r e s t i m a t e d , w h i l e Models B and C a p p e a r t o b e more c o m p a t i b l e with the h i s t o r i c record i n requiring larger storage.
A l s o of
439
..........................
c c
FECCIICCCOICCC
c
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c c c
111
5e.
Annual Lag One Corre3.ation, Model B.
Figure 5.
5f.
Apnual L E F One ~ Ccrrelation, Model C.
Annual lag zero cross correlation coefficients, Model A-C (Figures 5a-5c) and annual lag correlation coefficients, aggregated f l o w s , Models A-C (Figures 5d-5f).
440
I 00
I 10
6e.
Critical Extraction, Model B.
Figure 6.
6E.
Critical Extraction, Model C.
Empirical cumulative distribution functions of sequent peak storage, aggregated flows, Models A-C (Figures 6a6c) and critical extraction rate (Figures 6d-6f).
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442 s i g n i f i c a n c e i s t h e s t e e p e r s l o p e of t h e Model B and C d i s t r i b u t i o n s ; t h i s i s c o n s i s t e n t w i t h t h e r e s u l t s of e a r l i e r woric (Burges and L e t t e n m a i e r , 1977) showing t h a t r e q u i r e d s t o r a g e a t h i g h r e l i a b i l i t y (e.g.,
e x c e e d a n c e p r o b a b i l i t y 2 % ) can b e much h i g h e r f o r l o n g t e r m
p e r s i s t e n c e models.
C r i t i c a l e x t r a c t i o n d i s t r i b u t i o n s show r e l a t i v e l y
l i t t l e d i f f e r e n c e between m o d e l s , most l i k e l y b e c a u s e t h e s m a l l s t o r -
age s i z e used emphasizes w i t h i n - y e a r
f l o w p r o p e r t i e s , which a r e
s i m i l a r f o r a l l t h r e e models. F i n a l l y , F i g u r e s 7a-c show c r o s s i n g d i s t r i b u t i o n s f o r a g g r e g a t e f l o w s u s i n g a l l t h r e e models.
Crossing d i s t r i b u t i o n s f o r each of
t h e models a r e q u i t e s i m i l a r , s u g g e s t i n g t h a t t h e form of t h e m a r g i n a l d i s t r i b u t i o n s , d e t e r m i n e d by t h e d i s a g g r e g a t i o n model, d o m i n a t e s . The e s t i m a t e d d i s t r i b u t i o n f o r t h e h i s t o r i c d a t a h a s f e w e r mid l e v e l crossings than t h e average s y n t h e t i c d i s t r i b u t i o n , e s p e c i a l l y i n t h e vicinity F
:t
=:
0.5.
h i s t o r i c estimates.
T h i s may b e s i m p l y a r e s u l t of v a r i a b i l i t y i n t h e The smooth form of t h e a v e r a g e s y n t h e t i c d i s t r i -
b u t i o n , as opposed t o t h e j a g g e d h i s t o r i c e s t i m a t e s , s u p p o r t s t h e view t h a t sample v a r i a b i l i t y may b e t h e most i m p o r t a n t c o n t r i b u t o r t o d i f f e r e n c e s i n t h e two d i s t r i b u t i o n s .
SUMMARY AND CONCLUSIONS
T h e u s e of g r a p h i c a l t e c h n i q u e s f o r v a l i d a t i o n of m u l t i v a r i a t e s y n t h e t i c s t r e a m f l o w models i s a d v o c a t e d . d a t i o n measures a r e s u g g e s t e d :
Two g e n e r a l t y p e s of v a l i -
s t a t i s t i c a l and p e r f o r m a n c e - b a s e d .
Although p r e s e r v a t i o n of low o r d e r moments, p a r t i c u l a r l y t h e mean, w i l l o f t e n b e a n e c e s s a r y c o n d i t i o n f o r model a c c e p t a n c e , b i a s i n g of h i g h e r o r d e r moment e s t i m a t o r s c o m p l i c a t e s t h e i r u s e f o r v a l i d a t i o n purposes.
A l t h o u g h moment e s t i m a t o r s may b e c o r r e c t e d f o r b i a s ,
t h i s d o e s n o t n e c e s s a r i l y r e s u l t i n improvement o f a s t o c h a s t i c model from a p e r f o r m a n c e s t a n d p o i n t .
T h e r e f o r e , performance-based
model v a l i d a t i o n m e a s u r e s , p a r t i c u l a r l y s e q u e n t p e a k s t o r a g e , may b e more s i g n i f i c a n t f o r o p e r a t i o n a l v a l i d a t i o n . A p p l i c a t i o n of t h e t e c h n i q u e s s u g g e s t e d t o t h r e e t w o - s i t e ,
three
s e a s o n models of t h e Cedar and N o r t h F o r k Snoqualmie R i v e r , Washington
443 i n d i c a t e d p o s s i b l e i n a d e q u a c i e s i n t h e s e a s o n a l d i s t r i b u t i o n of f l o w s , a s w e l l as d i f f e r e n c e s r e l a t e d t o l o n g t e r m p e r s i s t e n c e structure.
The g r a p h i c a l r e s u l t s a l s o p o i n t e d o u t a t r a d e o f f i n
t h e m u l t i v a r i a t e l o n g t e r m p e r s i s t e n c e models between c r o s s - s i t e c o r r e l a t i o n s and a u t o c o r r e l a t i o n s a t t h e i n d i v i d u a l s i t e s . d i s t r i b u t i o n s of moments and a u t o - and c r o s s - c o r r e l a t i o n s
Empirical at the
s e a s o n a l l e v e l were u s e f u l i n v a l i d a t i n g t h e m u l t i - s i t e d i s a g g r e g a t i o n model, w h i l e t h e s e q u e n t p e a k a l g o r i t h m w a s most u s e f u l f o r o v e r y e a r validation.
The l a t t e r i n d i c a t o r i s , however, s e n s i t i v e t o t h e demand
p a t t e r n imposed.
C r i t i c a l e x t r a c t i o n r a t e and c r o s s i n g d i s t r i b u t i o n s
were l e s s u s e f u l model v a l i d a t i o n m e a s u r e s .
REFERENCES Akaike, H . , "A New Look a t S t a t i s t i c a l Model I d e n t i f i c a t i o n " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l , Vol. AC-19, No. 6 , Dec. 1 9 7 4 , pp. 716-723. Burges, S . J . and D.P. L e t t e n m a i e r , "A Comparison of Annual S t r e a m f l o w Models", J o u r n a l o'f t h e H y d r a u l i c s D i v i s i o n , ASCE, Vol. 1 0 3 , No. H Y 9 , 1 9 7 7 , pp. 991-1006. Burges, S . J . and D.P. L e t t e n m a i e r , " R e l i a b i l i t y Measures f o r Water Supply R e s e r v o i r s and t h e S i g n i f i c a n c e of Long-Term P e r s i s t e n c e " , P a p e r p r e s e n t e d a t I n t e r n a t i o n a l Symposium on Real T i m e O p e r a t i o n of Hydrosystems, U n i v e r s i t y of W a t e r l o o , J u n e 1981. C l a r k e , R . T . , M a t h e m a t i c a l Models i n H y d r o l o g y , I r r i g a t i o n and D r a i n a g e P a p e r No. 1 9 , Food and A g r i c u l t u r e O r g a n i z a t i o n , U n i t e d N a t i o n s , Rome, 1973. F i e r i n g , M.B., S t r e a m f l o w S y n t h e s i s , H a r v a r d U n i v e r s i t y P r e s s , Camb r i d g e , M a s s a c h u s e t t s , 1 9 6 7 , p . 11. J o n e s , D . A . , P.E. O ' C o n n e l l and E. T o d i n i , "A Model V a l i d a t i o n Framework f o r S y n t h e t i c Hydrology", P a p e r p r e s e n t e d a t C o n f e r e n c e on Water R e s o u r c e s P l a n n i n g i n E g y p t , C a i r o , J u n e 1979. K l e m e s , V . , R. S r i k a n t h a n and T.A. McMahon, "Long Memory Flow Models i n R e s e r v o i r A n a l y s i s : What Is T h e i r P r a c t i c a l V a l u e ? ' ' Water R e s o u r c e s R e s e a r c h , Vol. 1 7 , No. 3 , pp. 737-751, J u n e 1981. Lane, W . , "Applied S t o c h a s t i c T e c h n i q u e s User Manual", U.S. Bureau of R e c l a m a t i o n , Denver 1979. L e d o l t e r , J . , "The A n a l y s i s of M u l t i v a r i a t e T i m e S e r i e s A p p l i e d t o Problems i n Hydrology", J o u r n a l of H y d r o l o g y , Vol. 36, pp. 327352, 1978. L e t t e n m a i e r , D.P., "Parameter E s t i m a t i o n f o r M u l t i v a r i a t e S t r e a m f l o w S y n t h e s i s " , P r o c e e d i n g s , J o i n t A u t o m a t i c C o n t r o l C o n f e r e n c e , San F r a n c i s c o , August 1980. ~~
444 Palmer, R.N. and D.P. Lettenmaier, "Indexing Multiple Site Synthetic Streamflow Sequences Using Reliability Measures", Paper presented at ASCE Specialty Conference, Technical State of the Art Exchange, San Francisco, August 1981. Stedinger, J.R., "Parameter Estimation, Streamflow Model Validation, and the Effects of Parameter Error and Model Choice on Derived Distributions", Paper presented at American Geophysical Union Fall Meeting, San Francisco, December 1979. Stedinger, J.R., "Fitting Log Normal Distributions to Hydrologic Data'', Water Resources- Research, Vol. 16, No. 3 , pp. 481-490, June 1980. Wallis, J.R. and N.C. Matalas, "Small Sample Properties of H and K Estimators of the Hurst Coefficient h", Water Resources Research, Vol. 6, No. 6, December 1970, pp. 1583-1594. Wallis, J . R . , N.C. Matalas, and J.R. Slack, "Just a Moment", Water Resources Research, V o l . 10, No. 2, April 1974, pp. 211-219.
445
OBSERVATION AND SIMULATION OF THE SOOKE HARBOUR SYSTEM D.P.
KRAUEL, F. MILINAZZO, M. PRESS, AND W.W.
Royal Roads M i l i t a r y Col ege, V i c t o r i a , B.C.
WOLFE (Canada)
ABSTRACT The f i n d i n g s o f a c o n t n u i n g p h y s i c a l oceanographic s t u d y o f t h e Sooke I n l e t System on t h
West Coast o f Vancouver I s l a n d a r e d e s c r i b e d
The system c o n s i s t s o f a s h a l l o w harbour, f r e e l y connected t o ' a n i n l a n d b a s i n w i t h s e a s o n a l l y v a r y i n g f r e s h w a t e r i n f l o w a t t h e mouth o f the basin.
A summary o f s a l i n i t y , temperature, w a t e r c u r r e n t , and
t i d a l e l e v a t i o n d a t a i s presented.
A two-dimensional, b a r o t r o p i c
t i d a l model i s used t o p r e d i c t c i r c u l a t i o n w i t h i n t h e b a s i n .
A com-
p a r i s o n between t h e s e c a l c u l a t i o n s and t h e observed c u r r e n t s i s made.
IIjTRODUCT I O N The Sooke Harbour-Basin system i s a small i n l e t about 30 km west of V i c t o r i a , B.C.
on t h e S t r a i t o f Juan de Fuca ( F i g 1 ) .
The B a s i n
i s about 4 km l o n g and 3 km wide w i t h a depth v a r y i n g f r o m a 37
m
deep h o l e near t h e mouth o f t h e Basin t o t i d a l mud f l a t s a t t h e mouth.
The average d e p t h i s 17 m.
The B a s i n i s connected t o Juan
de Fuca S t r a i t v i a Sooke Harbour, a broad (=1 km), s h a l l o w s i l l about 3 km l o n g , h a v i n g a mean depth o f about 3.5 m.
The Sooke
R i v e r p r o v i d e s a source o f f r e s h water a t B i l l i n g s S p i t , t h e boundary between t h e B a s i n and t h e Harbour.
There a r e no s i g n i f i c a n t
f r e s h water i n f l o w s d i r e c t l y i n t o the Basin i t s e l f .
The r i v e r f l o w
shows a s t r o n g w i n t e r maximum, e s t i m a t e d a t 50 rn3s-l i n January and becoming n e g l i g i b l e d u r i n g August ( E l l i o t t , 1969).
The t i d e s
a r e mixed, m a i n l y s e m i - d i u r n a l w i t h an average range o f about 2 m
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
446
Fig. 1 .
Sooke Harbour and Basin;
tidal staff
x
c u r r e n t meter.
i n t h e S t r a i t . The hydrography permits a r e l a t i v e l y f r e e exchange of water between the system and t h e S t r a i t and thus the t i d e s a r e a t t e n u a t e d and delayed very l i t t l e . The system i s e x p e r i e n c i n g p r e s s u r e s from a broad spectrum of u s e r s . There a r e e x t e n s i v e moorage and r e p a i r f a c i l i t i e s f o r the West Coast f i s h i n g i n d u s t r y and a planned marine i n d u s t r i a l park. Extensive log booming a r e a s a r e a t the n o r t h end of the Basin and a t t h e harbour s i d e o f Wiffen S p i t which c o n t r i b u t e l a r g e q u a n t i t i e s o f wood fragments and bark t o t h e w a t e r . The Basin provides a l a r g e h a r v e s t of shrimp, o y s t e r s , and clams. F i n a l l y , the Sooke r e g i o n i s a n important r e c r e a t i o n c e n t r e because of i t s proximity t o V i c t o r i a There a r e s e v e r a l marinas c a t e r i n g t o t h e r e c r e a t i o n a l f i s h i n g i n d u s t r y . Extensive beaches lend t o beach combing and clam-digging. The e f f e c t s o f t h i s r a p i d l y developing t o u r i s t i n d u s t r y and the r e s u l t a n t d i s t u r b a n c e t o t h e environment a r e of g r e a t conc'irn t o
447 t h e l o c a l governments and l o n g t e r m r e s i d e n t s . Because o f t h e s e e n v i r o n m e n t a l and economic c o n c e r n s , and because t h e system i s o f academic i n t e r e s t as a n o n - t y p i c a l e s t u a r i n e system,
a n u m e r i c a l model o f t h e system i s b e i n g d e v e l o p e d t o p r e d i c t t h e c i r c u l a t i o n , f l u s h i n g , d i s p e r s i o n o f p o l l u t a n t s , and t h e e f f e c t s t h a t d r e d g i n g o r c o n s t r u c t i o n o f b r e a k w a t e r s and h a r b o u r s m i g h t have on t h e s e p a r a m e t e r s and on w a t e r q u a l i t y .
To c a l i b r a t e and
v e r i f y t h e m o d e l , we have s t a r t e d a s e r i e s o f s p a t i a l and t e m p o r a l measurements o f w a t e r c i r c u l a t i o n , t e m p e r a t u r e , and s a l i n i t y . P r o f i l e s o f s a l i n i t y and t e m p e r a t u r e as a f u n c t i o n o f d e p t h a l o n g t h e a x i s o f t h e i n l e t f r o m t h e S t r a i t t o t h e e a s t e r n end o f t h e B a s i n have been t a k e n i r r e g u l a r i l y f o r a p e r i o d o f a b o u t a y e a r . D u r i n g J a n u a r y , when t h e f r e s h w a t e r r u n - o f f f r o m t h e Sooke R i v e r i s s i g n i f i c a n t , t h e t e m p e r a t u r e i s q u i t e u n i f o r m a t 7.5',
with
s l i g h t l y c o o l e r w a t e r a t t h e s u r f a c e due t o t h e c o l d f r e s h w a t e r . The exchange o f h e a t e n e r g y between t h e w a t e r and t h e atmosphere has been d i s c u s s e d b y E l 1 i o t t ( 1 9 6 9 ) .
The c o r r e s p o n d i n g s a l i n i t y
d i s t r i b u t i o n has a marked s t r a t i f i c a t i o n due t o t h e l a r g e f r e s h w a t e r i n f l o w , w i t h s a l i n i t i e s v a r y i n g f r o m 16% a t t h e s u r f a c e t o
297; a t t h e b o t t o m . D u r i n g t h e summer months t h e s e d i s t r i b u t i o n s a r e s i g n i f i c a n t l y altered.
The f r e s h w a t e r i n f l o w i s now n e g l i g i b l e and t h e r e i s a Thus d u r i n g A u g u s t
l a r g e a b s o r p t i o n o f s o l a r energy a t t h e surface.
t h e e a s t end o f t h e B a s i n , w h i c h i s s h a l l o w and f a r removed f r o m t h e l a r g e c u r r e n t s e x p e r i e n c e d a t t h e B a s i n ' s mouth, can be a b o u t 5
0
4' warmer t h a n t h e d e e p e s t p a r t of t h e B a s i n ( 1 7 v e r s u s 13 ) and more t h a n 7' wariiier t h a n t h e S t r a i t .
The s a l i n i t y i n t h e B a s i n
d u r i n g t h i s p e r i o d i s v e r y u n i f o r m a t a p p r o x i m a t e l y 31.5'
a
and
o n l y m a r g i n a l l y below t h a t o f t h e S t r a i t . The c i r c u l a t i o n s t r u c t u r e i n t h e B a s i n i s q u i t e complex, c o n s i s t i n g o f several c o u n t e r - r o t a t i n g gyres which m i g r a t e i n t h e Basin.
A
s e r i e s o f s u b s u r f a c e d r o g u e s were employed t o o b s e r v e t h e c i r c u l a t i o n .
A
c l o c k w i s e g y r e was n o t i c e d i n t h e c e n t r a l p a r t o f t h e B a s i n and
a counter-clockwise gyre i n the north-west region.
Along t h e
448 s o u t h - w e s t s h o r e t h e w a t e r c o n t i n u e s f l o w i n g n o r t h and w e s t even w h i l e t h e f l o o d i n g t i d e i s moving e a s t .
This c i r c u l a t i o n pattern
i s much more complex t h a n t h a t o b s e r v e d b y E l l i o t t ( 1 9 6 9 ) .
Perhaps
t h e s i n g l e c l o c k w i s e g y r e he o b s e r v e d c o r r e s p o n d s t o t h a t n o t e d i n t h e c e n t r a l p a r t o f t h e Basin. -
,he t i d e s are being monitored a t several p o s i t i o n s i n t h e I n l e t
(Fig 1).
The government w h a r f , l o c a t e d on t h e n o r t h - w e s t s h o r e o f
t h e harbour i s a r e f e r e n c e t i d e - r e p o r t i n g s i t e ; thus l o n g term records are available.
I n a d d i t i o n , t i d a l d a t a a r e being recorded
a t two l o c a t i o n s i n s i d e t h e B a s i n : one j u s t s o u t h o f t h e mouth and a second a t t h e n o r t h s h o r e .
These r e c o r d s i n d i c a t e t h a t t h e t i d a l
e x t r e m a i n s i d e t h e B a s i n a r e d e l a y e d b y t h e o r d e r o f an h o u r f r o m t h o s e a t Sooke H a r b o u r . S e v e r a l Aanderaa r e c o r d i n g c u r r e n t m e t e r s have been p l a c e d i n t h e B a s i n t o o b t a i n l o n g t e r m r e c o r d s o f t h e c u r r e n t , s a l i n i t y and temperature.
Two of t h e s e m e t e r s have been moored i n t h e d e e p e s t
p a r t o f t h e B a s i n n e a r t h e mouth where t h e c u r r e n t s s h o u l d be s t r o n g e s t (one a t 10m d e p t h , t h e o t h e r a t 20m), and a t h i r d m e t e r was p l a c e d east o f B i l l i n g s Spit, i n the north-west corner o f the Basin (Fig I ) . These m e t e r s d i g i t a l l y r e c o r d c u r r e n t speed and d i r e c t i o n , t e m p e r a x r e , c o n d u c t i v i t y , and p r e s s u r e and can be l e f t u n a t t e n d e d f o r a 2 month p e r i o d , a t t h e end of w h i c h t h e y a r e r e c o v e r e d t o have t h e i r m a g n e t i c t a p e s and b a t t e r y packs r e p l a c e d . The model b e i n g d e v e l o p e d i s based on t h e L e e n d e r t s e ( 1 9 6 7 ) model as m o d i f i e d b y W i l l i s
(1977).
It i s 2-dimensional,
based on t h e
v e r t i c a l l y i n t e g r a t e d e q u a t i o n s o f n o t i o n and c o n t i n u i t y . c a n n o t model t w o - l a y e r f l o w .
Thus i t
However, d u r i n g t h e summer, when
f r e s h w a t e r i n f l o w i s n e g l i g i b l e and t h e B a s i n i s w e l l - m i x e d , 2 - d i m e n s i o n a l model s h o u l d be adequate.
a
P r o v i s i o n s a r e made f o r
the e f f e c t s o f t h e e a r t h ' s r o t a t i o n , bottom f r i c t i o n , t i d a l f o r c i n g , and w i n d s t r e s s on t h e s u r f a c e .
The s h o r e l i n e i s a p p r o x i m a t e d b y
a s q u a r e g r i d w i t h a s p a c i n g o f a b o u t 210m and t h e dynamics a r e c a l c u l a t e d w i t h a 60 second t i m e s t e p .
C a l i b r a t i o n o f t h e model
i s a c h i e v e d b y a d j u s t i n g t h e b o t t o m f r i c t i o n v i a a Chezy c o e f f i c i e n t .
449
The r e s p o n s e o f t h e model t o s i m u l a t e d t i d a l d a t a and a c t u a l t i d a l d a t a has been d e t e r m i n e d .
The c a l c u l a t e d f l o w p a t t e r n s show some
A c o m p a r i s o n of t h e h a r m o n i c
o f t h e complex f e a t u r e s e x p e c t e d .
c o m p o s i t i o n o f t h e w a t e r speeds w i t h t h a t o f t h e d r i v i n g f o r c e has been made t o d e t e r m i n e t h e b e h a v i o u r o f t h e m o d e l .
THE MODEL -
i h e model i s based on t h e l o n g wave a p p r o x i m a t i o n o f t h e v e r t i c a l l y
i n t e g r a t e d , s h a l l o w w a t e r e q u a t i o n s o f m o t i o n and c o n t i n u i t y :
du
= - -I vp-gk+F P dt div (pi) = 0
where
i is
the v e l o c i t y vector,
p
i s the density, g i s the acceler-
a t i o n due t o g r a v i t y , p i s p r e s s u r e , and
F
i s the resultant external
f o r c e composed o f t i d e , w i n d , b o u n d a r y f r i c t i o n , and C o r i o l i s . The b o u n d i n g s u r f a c e o f t h e f l u i d i s g i v e n b y
~ ( x , y , z , t ) = 0. The d e p t h b e l o w t h e r e f e r e n c e p l a i n i s h ( x , y ) w h i l e q ( x , y , t gives t h e e l e v a t i o n o f w a t e r above t h e p l a i n . s u r f a c e is t i m e dependent.
Note t h a t o n l y t h e f r e e
The n a t u r a l b o u n d a r y c o n d i t i o n
s
so t h a t , a t t h e f r e e s u r f a c e , t h e c o n d i t i o n becomes t h e k i n e m a t i c
lit
condition
+
Unx
+
vlly = w
where u, v , and w a r e t h e v e l o c i t y components i n t h e x , y, and z directions, respectively.
Since pressure
is assumed t o be h y d r o -
s t a t i c and a l i n e a r f u n c t i o n o f d e p t h , t h e model w i l l be v a l i d o n l y for unstratified fluids. -
he model a l s o a l l o w s f o r v i s c o s i t y t e r m s f r o m w i n d s h e a r - s t r e s s
and t h e e f f e c t s o f b o t t o m roughness
The l a t t e r f o r c e i s a p p r o x i m a t e d
i n t h e e q u a t i o n s b y t h e Chezy c o e f f c i e n t , C . can be p r e s e n t e d as
aatu t -
u -a ut ax
v a - f v + g aY
The d e r ved e q u a t i o n s
450
-av+ at
u -av+ ax
v -av+ aY
fu
+
g
aq
--$
ay
g v(u2+v2)' C 2 (h+n)
where f i s t h e C o r i o l i s p a r a m e t e r and F(')
and
- ,(y)
are the horizontal
components o f w i n d s t r e s s and b a r o m e t r i c p r e s s u r e . The p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e a p p r o x i m a t e d b y a f i n i t e d i f f e r e n c e scheme o v e r a s p a c e - s t a g g e r e d g r i d . are c a l c u l a t e d a t t h e f u l l g r i d steps ( i , j ) ;
The w a t e r e l e v a t i o n s
the u i s calculated
a t t h e h a l f h o r i z o n t a l and f u l l v e r t i c a l s t e p ( i + $ , j ) ; a n d
the v i s
c a l c u l a t e d a t t h e f u l l h o r i z o n t a l and h a l f v e r t i c a l s t e p ( i , j + & ) . The scheme i s s e m i - i m p l i c i t , m u l t i - o p e r a t i o n a l , time-step.
On t h e f i r s t h a l f t i m e - s t e p ,
and v i m p l i c i t l y .
using a double
n and u a r e s o l v e d e x p l i c i t l y
Then, on t h e f u l l t i m e - s t e p ,
the calculation o f
t h e v e l o c i t y components i s done i n r e v e r s e o r d e r .
The r e s u l t a n t
l i n e a r system i s s o l v e d u s i n g a d e c o m p o s i t i o n o f t h e s p a r s e , trid i a g o n a l f i n i t e - d i f f e r e n c e m a t r i x i n t o u p p e r and l o w e r t r i a n g u l a r f a c t o r s , and t h e n p e r f o r m i n g a f o r w a r d - b a c k w a r d s u b s t i t u t i o n t o solve t h e equations. The c l o s e d b o u n d a r y a t t h e s h o r e l i n e s i s assumed t o be a v e r t i c a l w a l l where t h e normal v e l o c i t y component i s z e r o and t h e d e p t h i s finite.
A t t h e f o r c i n g b o u n d a r y t h e w a t e r l e v e l s a r e g i v e n as t i m e
varying water elevations.
Near t h e s e b o u n d a r i e s , b u t w i t h i n t h e
c o m p u t a t i o n f i e l d , t h e n o r m a l b o u n d a r y c o n d i t i o n c a n n o t be a p p l i e d and some t e r m s a r e u n d e f i n e d i n t h e d i f f e r e n t i a l e q u a t i o n .
This
p r o b l e m i s overcome b y u s i n g a l i n e a r a p p r o x i m a t i o n . A l t h o u g h t h e s t a b i l i t y a n a l y s i s i s made d i f f i c u l t b y t h e C o r i o l i s and b o t t o m - s t r e s s t e r m s , t h e model has been shown t o be s t a b l e u n d e r a number o f r e a s o n a b l e c o n d i t i o n s ( L e e n d e r t s e , 1967, W i 11 i s , 1977, K r a u e l and B i r c h , 1 9 7 9 ) . The NRC F o r t r a n programme ( W i l l i s , 1977) was m o d i f i e d t o s i m p l i f y I / O and t o i l l u s t r a t e t h e l o g i c f l o w , b u t t h e c o m p u t a t i o n s were n o t
altered.
The model was r u n on an I B M 3780 c o m p u t e r .
The g r i d f o r
Sooke B a s i n was d e v e l o p e d u s i n g d i g i t i z e d b a t h y m e t i c d a t a and
451 an automated i n t e r p o l a t i o n f o r d e p t h s a t t h e d e s i r e d p o i n t s . The g r i d b o u n d a r i e s were s e l e c t e d t o r e f l e c t t h e r e g i o n ' s g e o m e t r y . The s h o r e l i n e a t t h e mouth was matched t o a h i g h w a t e r l e v e l t o r e f l e c t t h e f l o o d i n g d u r i n g h i g h t i d e when i t was a n t i c i p a t e d t h a t most v e l o c i t y f e a t u r e s w o u l d be c r e a t e d .
T h i s semi-automated
process e n a b l e s t h e e a s y c o n v e r s i o n t o d i f f e r e n t g r i d s .
A t present
a 23 x 18 m a s t e r g r i d w i t h a g r i d s i z e o f 200m i s b e i n g used.
This
g i v e s 202 e l e v a t i o n p o i n t s w i t h i n t h e c o m p u t a t i o n f i e l d , b u t i s t o o coarse t o r e c o g n i z e t h e i s l a n d s w i t h i n t h e Basin.
A time-step o f
1 m i n u t e was chosen as a r e a s o n a b l e compromise between n u m e r i c a l s t a b i l i t y and c o m p u t a t i o n a l speed. -
[ h e r e i s some f r e e d o m i n t h e s e l e c t i o n o f a p p r o p r i a t e Chezy
coefficients.
Although t h e bottom o f t h e Basin v a r i e s from s i l t -
l a d e n p l a i n t o a v e r y s t e e p , r o c k y h o l e , t h e l o n g p e r i o d waves may n o t see any s m a l l s c a l e v a r i a t i o n s and a c o n s t a n t c o e f f i c i e n t t h r o u g h o u t t h e B a s i n may be v a l i d .
With l i t t l e j u s t i f i c a t i o n other
t h a n success b y o t h e r u s e r s , a Chezy c o e f f i c i e n t o f 50 m i s - l chosen.
was
The s e n s i t i v i t y o f t h e model t o t h e c o e f f i c i e n t s was
measured b y v a r y i n g t h e v a l u e used b y 20%.
A simple sinusoidal
i m p u l s e was used as t h e f o r c i n g t i d e a t t h e mouth t o t h e B a s i n . A l t h o u g h n o t c o m p l e t e l y a n a l y s e d , t h e d a t a show l i t t l e e f f e c t due t o such v a r i a t i o n . A t s e l e c t e d t i m e i n t e r v a l s t h e model o u t p u t s a r e c o r d o f w a t e r e l e v a t i o n s and v e l o c i t i e s a v e r a g e d o n t o t h e e l e v a t i o n
(Q)
grid.
V e l o c i t i e s a r e p l o t t e d as a v e c t o r f i e l d . The v e c t o r m a g n i t u d e d a t a a t s e l e c t e d p o i n t s has been t r a n s f o r m e d t o a f r e q u e n c y s p e c t r u m p l o t u s i n g a waveform a n a l y s i s package d e v e l o p e d f o r t h e H e w l e t t - P a c k a r d 9325 c a l c u l a t o r ( K r a u e l e t a l , 1982:
S INUSOI DAL D R I V IMG FORCE
As a f i r s t a p p r o x i m a t i o n o f t h e t r u e f o r c i n g f u n c t i o n a t Sooke, t h e model was d r i v e n w i t h t h e f u n c t i o n f ( t ) = 1.2
-r
0.5584 s i n ( 0 . 0 0 0 0 7 2 9 t )
+ 0.4415 s i n ( 0 . 0 0 0 1 4 0 5 t )
T h i s f u n c t i o n c o r r e s p o n d s t o t h e two m a i n components o f t h e t i d e as
452
identified by studies a t the I n s t i t u t e of Ocean Sciences, P a t Bay, B.C. The system was allowed t o r u n f o r the equivalent o f one t i d a l cycle t o i n i t i a t e the forcing i n t o the f i e l d . T h i s appears t o be adequate as v e l o c i t i e s appear t o d i s s i p a t e regularly with slack t i d e s . The calculated currents display many of the large scale counter-rotating gyres observed i n the Basin. TIDAL DRIVING FORCE Tidal records from the gauge a t Sooke Harbour f o r the period from June 23 t o July 5 , 1981 were digitized a t 5 minute i n t e r v a l s a n d used as a forcing function f o r the model. The v e l o c i t i e s predicted by the model were plotted (Fig 2 ) . These graphically i l l u s t r a t e the persistance of the counter-rotating gyres well into the ebb t i d e even during great t i d a l ranges. Also there i s evidence t h a t the outflow i s mainly from the southern portion of the Basin.
/-
Fig. 2.
-
Current v e l o c i t i e s predicted by model f o r mid-ebb t i d e . The maximum speeds are in the order of 2 ms-'.
453
A
FFT of t h e c u r r e n t speeds a t g r i d p o i n t s n e a r t h e l o c a t i o n of
t h e two c u r r e n t m e t e r s i n t h e deep h o l e show t h e t w o m a i n f r e q u e n c i e s which appear i n t h e t i d a l d r i v i n g f o r c e and t h e s p e c t r a o f t h e r e c o r d e d c u r r e n t m e t e r speeds ( F i g 3 ) .
T h e r e i s such s i m i l a r i t y i n
t h e s p e c t r a a t p o i n t s a c r o s s t h e mouth t h a t i t a p p e a r s t h e t i d a l f o r c i n g o c c u r s i n a w i d e band w i t h i n t h e B a s i n as p o r t r a y e d i n t h e vector p l o t s .
1
a
16
32
48
r
~~
80
64
c o EE
0
4. 17e-05
B. 33e-05
----96
CIENTS--
FI, r-r--
1. 25e-04
112
128
1. 67e-04
FREQUENCY CHzl
F i g . 3.
Frequency s p e c t r a f o r ( a ) m o d e l - p r e d i c t e d c u r r e n t speeds a t a g r i d p o i n t n e a r t h e mouth; ( b ) c u r r e n t speeds a t c u r r e n t m e t e r n e a r t h e rtiouth; and ( c ) t h e d r i v i n g t i d e r e c o r d e d a t Sooke H a r b o u r .
CONC L U S I ON The model a p p e a r s t o be a v a l i d p r e d i c t o r o f c u r r e n t s w i t h i n Sooke B a s i n d u r i n g p e r i o d s o f l o w f r e s h w a t e r i n p u t and n o n - s t r a t i f i c a t i o n . The system i s c h a r a c t e r i z e d b y n o n - s t a t i o n a r y c o u n t e r - r o t a t i n g g y r e s w h i c h a r e b o r n on t h e f l o o d t i d e and p e r s i s t i n t o t h e ebb t i d e . S p e c t r a l a n a l y s i s a p p e a r s t o be an e f f e c t i v e method o f v e r i ' y i n g
the
454 model and f o r p r e d i c t i n g h a r m o n i c components i n t h e c i r c u l a t i o n .
The
model a l s o p r e d i c t s a r e a s where c r i t i c a l c u r r e n t b e h a v i o u r s h o u l d be m o n i t o r e d .
To f u r t h e r c a l ib r a t e t h e model
,
d a t a sampl i n g w i l l
have t o c o n t i n u e , and w i n d f o r c i n g and f r e s h w a t e r i n f l o w d a t a w i l l be i n c l u d e d . REFERENCES K r a u e l , D.P.,
and B i r c h , J.R.,
1979.
Wind and F r e s h Water I n f l o w
E f f e c t s on t h e C i r c u l a t i o n o f t h e M i r a m i c h i E s t u a r y , N.B.-A N u m e r i c a l Model.
Coastal Marine Science Laboratory, Manuscript
R e p o r t 79-2, Royal Roads M i l i t a r y C o l l e g e , FMO V i c t o r i a , B.C. K r a u e l , D.P.,
M i l i n z a a o , F.,
P r e s s , M.,
Model o f Sooke H a r b o u r and B a s i n .
and W o l f e , W.W.,
1982.
A
C o a s t a l M a r i n e S c i e n c e Lab-
o r a t o r y N o t e , i n p r e s s , Royal Roads M i l i t a r y C o l l e g e , FMO V i c t o r i a , B.C. L e e n d e r t s e , J.J.
, 1967.
A s p e c t s o f a c o m p u t a t i o n a l model f o r l o n g -
p e r i o d water-wave p r o p a g a t i o n . Santa Monica, Ca., W i l l i s , D.H.,
1977.
RM-5294-PRY t h e Rand C o r p o r a t i o n ,
165 pp. M i r a m i c h i Channel s t u d y h y d r a u l i c i n v e s t i g a t i o n .
H y d r a u l i c s L a b o r a t o r y T e c h n i c a l R e p o r t LTR-HY-56,
Vol.
I and 11,
D i v i s i o n o f M e c h a n i c a l E n g i n e e r i n g , N a t i o n a l Research C o u n c i l o f Canada , Ottawa.
455
RAINFALL-FLOW RELATIONSHIP IN SOME ITALIAN RIVERS
BY MULTIPLE STOCHASTIC MODELS
;:'
ELPIDIO CARONI R e s e a r c h e r , C . N . R . / I . R.P. I . , T o r i n o FRANCESCO MANNOCCHI R e s e a r c h e r , U n i v e r s i t y of P e r u g i a , I n s t . A g r i c u l t u r a l H y d r a u l i c s
LUC I 0 UBERT IN I Director of C . N .R . / I . R . P I . , P e r u g i a P r o f e s s o r , U n i v e r s i t y of P e r u g i a , I n s t . A g r i c u l t u r a l H y d r a u l i c s
.
ABSTRACT The h o u r l y r a i n f a l l - f l o w relationship was studied by m u l t i p l e t r a n s f e r p l u s n o i s e methodologies. The f o r m u l a ( 1 ) s h o w s t h e g e n e r a l form of t h e models. T h i s f o r m u l a , w h e r e o n e o r two i n p u t s c a n b e e l i m i n a t e d , w a s u t i l i z e d t o b u i l d o p e r a t i v e models f o r flow s i m u l a t i o n a n d flow r e a l time f o r e c a s t . I n t h i s work we p r e s e n t t h e models a n d t h e r e s u l t s f o r some e v e n t s of two I t a l i a n b a s i n s ( S i e v e , 831 k m 2 ; Toce, 1535 km2) a n d of two e x p e r i m e n t a l b a s i n s ( M a r c h i a z z a , 5 k m 2 ; Fosso d e g l i I m p i c c a t i , 7.6 k m * ) . We p r e s e n t , b e s i d e s , p r o p e r t i e s , limits a n d possible f u t u r e developments of m u l t i p l e t r a n s f e r p l u s n o i s e methodologies i n t h i s f i e l d .
1. INTRODUCTION
D u r i n g t h e l a s t few y e a r s , a n e f f o r t w a s m a d e i n o r d e r t o a s s e s t h e p o s s i b i l i t i e s of u s i n g s t o c h a s t i c models i n flood s i m u l a t i o n a n d r e a l time f o r e c a s t . T h e s e models a r e of p a r t i c u l a r interest for t h e i r c a p a b i l i t y to estimate the rainfall-flow r e l a t i o n s h i p s from a n o b s e r v e d s a m p l e b y m e a n s of s t a t i s t i c a l
*
C.N.R.
Special P r o j e c t f o r S o i l Conservation, Sub-project F l u v i a l Dynamics, paper
No. 155.
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
0
456 e s t i m a t o r s . T h i s f a c t p e r m i t s t h e r e d u c t i o n of t h e i n f l u e n c e of p e r s o n a l j u d g e m e n t s a b o u t t h e p h y s i c a l phenomenon o v e r t h e model c h a r a c t e r i s t i c s . I n t h i s c o n t e x t a c r i t e r i o n f o r p a r a m e t e r p a r s i m o n y must b e considered i n o r d e r t o o b t a i n a p r a c t i c a l o p e r a t i v e tool. The u s e of m u l t i p l e l i n e a r t r a n s f e r f u n c t i o n models (Anselmo e t a l i i , 1981) a l l o w s u s t o t a k e i n t o a c c o u n t i n some w a y t h e n o n l i n e a r i t y of t h e phenomenon. P a r t i c u l a r r e s e a r c h e s were c a r r i e d o n i n t h i s s e n s e ; i n f a c t t h e s e r i e s , h e r e c o n s i d e r e d , a r e composed of h o u r l y ( o r more f r e q u e n t ) d a t a , which u s u a l l y show a more e v i d e n t n o n l i n e a r b e h a v i o u r t h a n other hydrologic samples with longer gaging intervals ( d a i l y , weekly a n d so o n ) . I n o u r p a p e r we d i s c u s s t h e a p t i t u d e of s e v e r a l i n p u t v a r i a b l e s i n a m u l t i p l e t r a n s f e r f u n c t i o n p l u s n o i s e model t o e x p l a i n the rainfall-runoff phenomenon, p a r t i c u l a r l y within t h e peak hydrograph.
2 . THE MODEL Following t h e n o t a t i o n u s e d b y Box a n d J e n k i n s ( 1 9 7 0 ) a n d w i d e l y a d o p t e d f o r s t o c h a s t i c modeling ( H i p e l e t a l i i , 19771, t h e g e n e r a l form of t h e model i s :
+ v (B) Z + v ( B ) St-b (T) 2 t-b 3 1 2 3 where v . ( B ) (for i=1,2,3) i s Yt
=
v l ( B ) Xt-b
L
+ N
t ’
2 2 -1 B ) (1 - 6 B - 6 B ) 2 1 2 a n d B i s t h e b a c k w a r d s h i f t o p e r a t o r s u c h t h a t BX, = X t - 1 . T h e o ‘ s a n d 6 ‘ s a r e called respectively input a n d output p a r a m e t e r s ; Yt i s t h e d i s c h a r g e a t time t ; X t - b l i s t h e t o t a l r a i n inflow d u r i n g t h e time i n t e r v a l between t - b r l a n d t-b, ( w h e r e is a delay parameter) ; Zt-b2 is the cumulated r a i n f a l l effect; ~ [i slt h& e i n t e r v e n t i o n v a r i a b l e ; N, i s t h e n o i s e t e r m . vi(B)
=
(ao -
wlB
--w
...
...
9
A r e c e n t work b y Anselmo a n d U b e r t i n i s i m p l e l i n e a r model w i t h o n l y one i n p u t :
(1979) l e d t o a ( 2)
With t h i s model t h e n o n l i n e a r e f f e c t s i n t h e r a i n f a l l - f l o w r e l a t i o n s h i p s a r e n o t a c c o u n t e d f o r . I n f a c t we f o u n d t h a t : - t h e h i g h e s t v a l u e s of t h e r e s i d u a l s c l a s h w i t h t h e p e a k s , w h e r e t h e e f f e c t s of n o n l i n e a r i t y a r e more r e l e v a n t ;
457 - i n r e a l time f o r e c a s t , underestimates of f u t u r e d i s c h a r g e s occur whenever t h e time o r i g i n for f o r e c a s t s l i e s n e a r t h e beginning of the h y d r o g r a p h r i s i n g l i m b . These underestimates diminish a s long a s t h e s t a r t i n g point for forecasts a p p r o a c h e s t h e flood p e a k . I n o r d e r to improve t h e r e s u l t s of t h i s model, two o t h e r i n p u t v a r i a b l e s were e x p l o r e d . We sought a second i n p u t v a r i a b l e Zt , a function of r a i n f a l l X t , which w a s named "cumulated r a i n f a l l effect"; a f t e r w a r d s we i n t r o d u c e d a n i n t e r vention v a r i a b l e a c t i n g a t time t=T when r a i n f a l l , X T , o r "cumulated r a i n f a l l " , Z T , exceeds a g i v e n s a f e t y t h r e s h o l d E , according to t h e formula:
1 if X
(T)
T
> E or Z > E
T
L
0
otherwise
( 3)
.
2 . 1 Cumulated r a i n f a l l effect ~~~
~
~~
Discharge from a r i v e r may b e considered dependent on s e v e r a l f a c t o r s b e s i d e s r a i n f a l l . Among t h e s e , a n important r o l e i s assumed by the g l o b a l amount of r a i n f a l l f a l l e n i n t h e b a s i n d u r i n g a previous time period. T h i s amount may b e q u a n t i f i e d by means of a weighted sum of antecedent p r e c i p i t a t i o n s i n t h e following way: m
Z
t
= C w X . k t-k k=O Since a d e c r e a s i n g importance of p r e v i o u s r a i n f a l l s on Zt may b e assumed going b a c k i n time, a s u i t a b l e system of weights may b e g i v e n by w
=
.
m
k
c w = 1 k k=O expressed i n terms of
(1-c) c k , with Oiponents h a d been hidden i n t h e harmonic a n a l y s i s . The r e s u l t s o f t h e s e
468
TABLE 1.
Mean v a l u e , r a n g e o f v a l u e s , s t a n d a r d d e v i a t i o n and c o e f f i c i e n t o f v a r i a t i o n f o r t h e r e c o r d mean and t h e a m p l i t u d e a n d phase a n g l e o f t h e f i r s t f i v e h a r m o n i c s determined by examination o f t h e 24-year w a t e r temperature r e c o r d on a c a l e n d a r y e a r b a s i s . Water T e m p e r a t u r e (OC) Mean Range M~ in Max _ _
Standard Deviation
Coefficient o f Variation
Year Average
15.57
16.84
14.66
0.647
0.042
1 s t Harmonic Ampl it u d e Phase A n g l e
11 .61 4.91
12.98 4.12
10.01 4.36
0.702 0.057
0.060 0.014
2nd Harmonic Ampl it u d e Phase A n g l e
0.794 3.90
1.82 6.25
0.14 0.04
0.423 1.37
0.53 0.35
3 r d Harmonic Ampl it u d e Phase A n g l e
0.53 2.68
1.27 6.19
0.09 0.11
0.315 2.01 7
0.59 0.75
4 t h Harmonic Ampl it u d e Phase A n g l e
0.486 3.19
0.96 6.22
0.08 0.03
0.234 1.875
0.48 0.59
5 t h Harmonic Ampl it u d e Phase A n g l e
0.44 3.14
0.99 6.09
0.06 0.37
0.221 1.746
0.50 0.56
e f f o r t s , summarized i n T a b l e 3, s u g g e s t t h a t s u n s p o t s and o t h e r s o l a r phenomena m i g h t be a f f e c t i n g t h e w a t e r t e m p e r a t u r e r e c o r d s l i g h t l y , b u t t h e data a r e h a r d l y c o n c l u s i v e (Hsieh, 1979).
There i s much
s t r o n g e r e v i d e n c e t o document t h e l a c k o f any s i g n i f i c a n t peak i n t h e v a r i a n c e s p e c t r u m c o r r e s p o n d i n g t o t h e l u n a r c y c l e ( 2 9 . 5 day p e r i o d ) o r the semi-lunar cycle.
T h i s i s somewhat s u r p r i s i n g s i n c e o t h e r
s t u d i e s have shown t h a t s t r a t i f i c a t i o n , and t h e l a c k o f i t , i n t h e York R i v e r e s t u a r y i s t i e d t o t h e s p r i n g t i d e t o neap t i d e l u n a r c y c l e (Haas _ et _ a l , 1981).
I t had been e x p e c t e d t h a t t h i s s t r a t i f i c a t i o n -
m i x i n g c y c l e w o u l d a f f e c t t h e w a t e r t e m p e r a t u r e s as w e l l . I n s h o r t , t h e dominant p e r i o d i c f e a t u r e o f t h e t i m e s e r i e s i s t h e seasonal v a r i a t i o n i n w a t e r temperatures.
The r e s i d u a l s i g n a l a p p a r -
e n t l y i s s t o c h a s t i c i n n a t u r e s i n c e n e i t h e r harmonic a n a l y s i s n o r s p e c t r u m a n a l y s i s was a b l e t o show any c o n s i s t e n t b e h a v i o r f o r o t h e r
469
TABLE 2 .
Variance ( i n degrees c e n t i g r a d e s q u a r e d ) and p e r c e n t o f v a r i a n c e a t t r i b u t e d t o t h e f i r s t harmonic, second t o f i f t h harmonics, and s i x t h and h i g h e r o r d e r harmonics determined by a n a l y s i s of t h e 24-year w a t e r temperature record on a calendar year basis.
Variance (oc') Total Variance Variance c o n t r i b u t e d by: 1s t Harmonic 2nd-5th Harmonics 6 t h and Higher Harmonics
Average 69.77
Maximum aa. 64
Minimum 51.70
67.72 0.86 1.19
84.32 2.33 1.99
50.10 0.21 0.63
97.06 1.22 1 .72
98.58 3.40 2.68
95.13 0.30 0.87
% o f Total Variance ( % )
1s t Harmon i c 2nd-5th Harmonics 6 t h and Higher Harmonics
components o r t o r e l a t e t h e s e t o physical processes p r e v i o u s l y believed t o be i m p o r t a n t . DETERMINISTIC - STOCHASTIC MODEL The p r e l i m i n a r y a n a l y s i s showed t h a t t h e seasonal v a r i a t i o n o f water temperatures could be approximated by a s i n u s o i d a l f u n c t i o n , b u t t h a t t h i s simple d e t e r m i n i s t i c model d i d not account f o r a p o r t i o n
On t h e o t h e r hand a purely s t o c h a s t i c approach does not seem a p p r o p r i a t e f o r a time s e r i e s which c o n s i s t e n t l y shows a s t r o n g annual c y c l e . rlcMichae1 and Hunter (1972) found t h a t a combination o f d e t e r m i n i s t i c and s t o c h a s t i c models "provided s u c c i n c t and useful f o r e c a s t f u n c t i o n s f o r t h e temperature and flow changes i n t h e Ohio River". Their approach, which was based on t h e work o f Box and Jenkins ( 1 9 7 6 ) , was used t o f u r t h e r s t u d y t h e York (about 5%) o f t h e v a r i a n c e i n t h e r e c o r d .
River water t e m p e r a t u r e r e c o r d . The d e t e r m i n i s t i c p o r t i o n o f t h e model was taken t o be t h e record mean (15.6OC) and t h e annual harmonic having an amplitude of 11 .6OC and a phase l a g o f 240'. ARIMA (AutoRegressive, I n t e g r a t e d Moving Average) models were t e s t e d t o s e e i f t h e y f i t t h e r e s i d u a l s e r i e s which was o b t a i n e d by s u b t r a c t i n g t h e d e t e r m i n i s t i c model values from t h e a c t u a l time s e r i e s r e c o r d .
The a u t o c o r r e l a t i o n f u n c t i o n ( A C F ) f o r
t h e r e s i d u a l s e r i e s e x h i b i t e d an exponential decay f o r t h e f i r s t 83
470 TABLE 3 .
The p o r t i o n o f t h e t o t a l v a r i a n c e c o n t r i b u t e d b y t h e c y c l i c a l , s e a s o n a l and i r r e g u l a r components o f t h e 24-year w a t e r temperature r e c o r d ( a f t e r Hsieh , 1979). Component C y c l ic
Seasonal Irregular
*
*
Intensity (% o f Total Variance)
Period ~ 2 Year 4
0.44
26 R o n t h s
0.21
1 3 - 1 4 Months
0.30
6
Fonths
0.24
a2
Fonths
0.06
1 2 Months
95.84
-
2.81
O n l y t h o s e h a r m o n i c s w h i c h were s i g n i f i c a n t a t 95% p r o b a b i 1 it y 1 i m i t s have been in c l uded.
values f o l l o w e d by a s i n e w a v e - l i k e o s c i l l a t i o n a t l o w v a l u e s .
The ACF
f o r t h e s e r i e s m o d i f i e d b y t h e f i r s t d i f f e r e n c e o p e r a t o r had no v a l u e s g r e a t e r t h a n 0.05 f u n c t i o n u n i t s a f t e r t h e f i r s t f o u r l a g s .
The
second d i f f e r e n c e s e r i e s had an i n i t i a l v a l u e o f a b o u t 0.5 w i t h a l l s u b s e q u e n t v a l u e s l e s s t h a n 0.04.
The c h a r a c t e r i s t i c s o f t h e f o u r
models i d e n t i f i e d as b e i n g a p p r o p r i a t e f o r t h e g i v e n ACF p a t t e r n s (Sox and J e n k i n s , 1 9 7 6 ) a r e summarized i n T a b l e 4 . A l l f o u r models were f o u n d t o s i m u l a t e t h e d a t a r e a s o n a b l y w e l l and t o i n c o r p o r a t e e s s e n t i a l l y t h e same p o r t i o n o f t h e v a r i a n c e .
All f o u r
r e s i d u a l s e r i e s , w h i c h r e s u l t e d when b o t h t h e d e t e r m i n i s t i c and s t o c h a s t i c model v a l u e s were s u b t r a c t e d f r o m t h e r e c o r d , a p p r o x i m a t e d w h i t e noise.
A b o u t o n e - t e n t h o f t h e ACF v a l u e s were o u t s i d e t h e 95%
confidence l i m i t s .
Q v a l u e s f o r t h e f o u r models a l s o were somewhat
g r e a t e r t h a n t h e 90% l i m i t s ( H s i e h , 1 9 7 9 ) .
Considering the p r i n c i p l e
o f maximum s i m p l i c i t y , o r Occam's R a z o r , t h e b e s t c h o i c e i s t h e (1,0,0) model.
T h i s f i n d i n g i s s i m i l a r t o t h a t o f Mehta e t a1 ( 1 9 7 5 ) who
found t h a t t h e " c o m b i n a t i o n o f f i r s t - h a r m o n i c
e l i m i n a t i o n and f i r s t -
o r d e r a u t o r e g r e s s i v e model p r o d u c e s a 99% r e d u c t i o n i n t h e v a r i a n c e " o f t e m p e r a t u r e d a t a f o r t h e P a s s a i c R i v e r . The value f o r t h i s s t u d y ( 0 . 9 1 Y ) was h i g h e r t h a n t h e y found a p p r o p r i a t e f o r t h e P a s s a i c
471 TASLE 4.
A R I MA Type
C h a r a c t e r i s t i c s o f t h e ARIMA models used t o s i m u l a t e the stochastic p o r t i o n o f the time series record. ( A f t e r H s i e h , 1979) Parameter Values
(1 ,O,O)
=
Percent o f T o t a l Variance I n c l u d e d i n Model
Model
0.919
(1
- $1 B
-. )Zt = a t
99.42
2 (2,0,0) $ 1 = 0 . 9 1 0 $ 2 = 0.00919
(l,O,l)
(0,2,1)
(1 -
$1 = 0 . 9 2
el
=
-0.008
=
0.99
v2
R i v e r (0.818 t o 0 . 8 6 1 ) , (0.915)
99.42
$2B ) Zt=a t
(1 - $ l B -
-
B) Zt = ( 1
Zt = (1 -
el)
+ e l B)
at
99.42
at
99.33
b u t was c l o s e t o t h a t f o r t h e Ohio R i v e r
(McMichael and H u n t e r , 1 9 7 2 ) .
The (1,0,0)
model i m p l i e s t h a t on any g i v e n day t h e d e v i a t i o n f r o m
t h e normal a n n u a l c y c l e i s a w e i g h t e d f u n c t i o n o f a l l p r e v i o u s d e v i a t i o n s f r o m t h a t c y c l e p l u s any new " s h o c k " r e c e i v e d t h a t day.
Further-
more, t h e w e i g h t i n g f u n c t i o n s f o r t h i s p a r t i c u l a r case d e c r e a s e as t h e t i m e i n t e r v a l i n c r e a s e s w i t h a f a c t o r o f 0.919. A1 t h o u g h t h e combined d e t e r m i n i s t i c - s t o c h a s t i c model can p r o d u c e v e r y r e a l i s t i c s y n t h e t i c r e c o r d s (McMichael and H u n t e r , 1972) l e s s we1 1 s u i t e d f o r p r e d i c t i n g a c t u a l t e m p e r a t u r e s .
,
it i s
Since f u t u r e
shocks o r random f a c t o r s can be s i m u l a t e d b u t n o t f o r e c a s t , t h e model p r e d i c t s d e c r e a s i n g d e v i a t i o n s from t h e s e a s o n a l c y c l e as t h e length o f the prediction increases.
F o r t h e York R i v e r t h e p r e d i c t i o n s
approach t h e h a r m o n i c c u r v e a f t e r a b o u t 1 5 days l e a d t i m e . t h a t t i m e frame, p r o j e c t i o n s a r e r a t h e r good.
Within
F i f t e e n day f o r e c a s t s
made u s i n g a c t u a l t e m p e r a t u r e r e c o r d s and s t a r t i n g w i t h a r b i t r a r i l y s e l e c t e d s t a r t i n g d a t e s i n w i n t e r , s p r i n g and summer a r e shown i n F i g u r e 2.
The a c t u a l o b s e r v a t i o n s u s u a l l y f e l l w i t h i n t h e 50% proba-
b i l i t y limits.
472
lo[ 8 A
-
Winter
J
0
321 C
20 Days
-
22 25
40
I
-
1
I
I
L
0
20
40
60
Days
Summer
4
A c t u a l Values
*rcu~x
Forecast
bbbna
50% P r o b a b i 1 it y Lim t s
.*-*
0
20
Spring
40
90% P r o b a b i l i t y Lim t s
60
Days
FIGURE 2 .
F i f t e e n day f o r e c a s t s f o r a r b i t r a r i l y s e l e c t e d days i n a ) w i n t e r , b ) s p r i n g , and c ) summer.
473 CONCLUSIONS B o x - J e n k i n s models a r e u s e f u
t o o l s f o r t h e a n a l y s i s and i n t e r p r e t a -
t i o n o f w a t e r t e m p e r a t u r e t i m e s e r i e s f o r e s t u a r i e s , and f o r c r e a t i n g s y n t h e t i c temperature s e r i e s .
F o r t h e York R i v e r e s t u a r y t h e e a r t h ' s
r o t a t i o n a b o u t t h e sun r e s u l t s i n a s t r o n g s e a s o n a l v a r i a t i o n w h i c h can be a p p r o x i m a t e d b y a s i n g l e s i n e wave. c o n s i s t e n t o v e r t h e 24 y e a r s o f r e c o r d .
T h i s b e h a v i o r was shown t o be Therefore the best prediction
o f w a t e r t e m p e r a t u r e s f a r i n t o t h e f u t u r e ( s a y months o r y e a r s ) i s t h a t o b t a i n e d u s i n g t h e r e c o r d mean and t h e annual c y c l e .
A number o f f a c t o r s i n t r o d u c e a s t o c h a s t i c a s p e c t t o t h e r e c o r d . F o r t h e York R i v e r e s t u a r y f o u r ARIMA models p r o v i d e d a good f i t t o the data.
The s i m p l e s t o f t h e s e , t h e f i r s t o r d e r a u t o r e g r e s s i v e p r o -
cess, was s e l e c t e d as t h e b e s t .
T h i s model i s c a p a b l e o f p r o j e c t i n g
f u t u r e w a t e r t e m p e r a t u r e s u p t o a b o u t 1 5 days i n advance.
For l o n g e r
p e r i o d s t h e d e v i a t i o n d e c r e a s e s and t h e p r e d i c t i o n approaches t h e a nn ua 1 harmo n
L.
RE FE REN CES
Box, George E P . and G w i l y m M. J e n k i n s , Time S e r i e s A n a l y s i s : f o r e c a s t i n g and c o n t r o l , R e v i s e d E d i t i o n , 575 pp., H o l den-Day , San F r a n c i s c o , 976 Haas, Leonard W., F r e d e r i c k J . H o l d e n and C h r i s t o p h e r S . Welch, " S h o r t Term Changes i n t h e V e r t i c a l S a l i n i t y D i s t r i b u t i o n o f t h e York R i v e r E s t u a r y A s s o c i a t e d w i t h N e a p - S p r i n g T i d a l C y c l e " . I n : E s t u a r i e s and N u t r i e n t s , B. J. N e i l s o n and L . E . C r o n i n , E d i t o r s , Humana P r e s s , C l i f t o n , New J e r s e y , 1981. H s i e h , B e r n a r d B., V a r i a t i o n and P r e d i c t i o n o f Water T e m p e r a t u r e i n t h e York R i v e r E s t u a r y a t G l o u c e s t e r P o i n t , V i r g i n i a , M.A. T h e s i s , C o l l e g e o f W i l l i a m and Mary, W i l l i a m s b u r g , VA, 1979. Kothandaraman, Veerasamy, A n a l y s i s o f Water T e m p e r a t u r e V a r i a t i o n s i n L a r g e R i v e r , J o u r . San. E n g ' g . D i v i s i o n Amer. SOC. C i v i l Eng., 97 ( S A l ) , 19-31, 1971. Mehta, B . M., R . C . A h l e r t and S . L. Yu, S t o c h a s t i c V a r i a t i o n o f Water Q u a l i t y o f t h e P a s s a i c R i v e r , Water Resour. Res., 11 ( 2 ) , 300-308, 1975. McMichael, F r a n c i s C l a y and J . S t u a r t H u n t e r , S t o c h a s t i c M o d e l i n g o f T e m p e r a t u r e a n d Flow i n R i v e r s , Water Resour. Res., 8 ( 1 1 , 87-98, 1972. Thomann, R o b e r t V., T i m e - S e r i e s A n a l y s e s o f Water Q u a l i t y Data, J o u r . San. Eng. D i v . Amer. S O C . C i v i l Eng., 93 ( S A l ) , 1 - 2 3 , 1967.
474
STOCHASTIC ARIMA MODELS FOR MONTHLY STREAMFLOWS SRINIVAS G. RAO AND EDWIN W. QUILLAN School o f Civil Engineering, Georgia Institute o f Technology, Atlanta, Georgia 30332
ABSTRACT The n o n - s t a t i o n a r y and seasonal n a t u r e of t h e monthly flows a r e modeled using ARI:;A c l a s s of models. A s t e p - b y - s t e p procedure i s designed t o o b t a i n v a l i d models through proper model i d e n t i f i c a t i o n , parameter e s t i m a t i o n , performance e v a l u a t i o n , model ?arsimon:f and v a l i d a t i o n of r e s i d u a l s . Several s t a t i s t i c a l i n d i c e s of n e r f o r mance, and s t a t i s t i c a l t e s t s a r e used t o p r o p e r l y screen t h e c a n d i d a t e models f o r o b t a i n i n g t h e b e s t models.
The imyortance of
o b t a i n i n g parsimonious models and v a l i d a t i o n of r e s i d u a l s f o r whiteness i s emphasized. The procedure i s demonstrated f o r t h e monthly flows of t h e Chattahoochee River i n Georgia i n o b t a i n i n ? a v a l i d model capable of a c c e p t a b l e p r e d i c t i o n r e s u l t s . IiiTRODUCT I O i l The world today f a c e s an expanding p o y l a t i o n and an accompanying i n c r e a s i n g demand f o r v a t e r r e s o u r c e s .
E f f e c t i v e design
and o p e r a t i o n of water r e s o u r c e s systems f o r e f f i c i e n t use of a v a i l a b l e water r e q u i r e s t h e a b i l i t y t o f o r e c a s t streamflows.
Of
t h e s e v e r a l methods a v a i l a b l e f o r streamflow f o r e c a s t i n g , s t o c h a s t l c models seem a t t r a c t i v e and a p n r o n r i a t e i n d e a l i n q v i t h t h e u n c e r t a i n t y i n measurement and randomness i n t h e p r o c e s s . Construction of v a l i d s t o c h a s t i c models f c r f o r e c a s t i n g monthly streamflo1.l i s t h e s u b j e c t of t h i s paper. llonthly flows e x h i b i t i n ? nonstationarit!,
and s e a s o n a l i t y can
be modeled using A?IiiA models d e s c r i b e d by Box and Jenkins ( 1 ? 7 C ) . Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterb: o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Editors)
475
Such models have been used i n the p a s t , t o a l i m i t e d e x t e n t t o model monthly and annual streamflows (McKerchar and D e l l e u r , 1972; tiipel e t a l . , 1977). These models a b r e v i a t e d ARIMA (P, D,
a), c o n s i s t s
(0,
d , o)x
of a seasonal ARUA ( P , Q ) model f i t t e d t o t h e
D - t h seasonal d i f f e r e n c e of d a t a coupled with an ARMA (p, 0 ) model f i t t e d t o the d - t h d i f f e r e n c e of the r e s i d u a l s of t h e former model. Following t h e t h r e e - s t e p procedure of i d e n t i f i c a t i o n , e s t i m a t i o n and checking a s given by Box and J e n k i n s , we w i l l i l l u s t r a t e the r e s u l t s of b u i l d i n g a v a l i d , a c c u r a t e and parsimonious ARIPAA model f o r t h e monthly streamflows of t h e Chattahoochee River near A t l a n t a , Georgia. DATA USED I!i THE STUDY The d a t a used i n t h i s study were t h e monthly mean flows of t h e Chattahoochee River a t kJest Point near A t l a n t a , Georgia f o r water y e a r s 1898-1978. Buford Dam, 149 miles upstream from t h e gage a t ",'est P o i n t , began r e g u l a t i o n i n January 195€. To determine i f t h i s dam had a f f e c t e d t h e homogeneity of the d a t a , t h e d a t a were ana yzed i n t h r e e p e r i o d s , 1898-1978, 1398-1955, and 1957-1978. Table 1 includes several b a s i c s t a t i s t i c s of t h e observed d a t a f o r t h e t h r e e p e r i o d s . The mean v a l u e has not changed s i g n i f i c a n t l y w h l e t h e standard d e v i a t i o n of t h e flows a f t e r t h e beginning of requ a TABLE 1 : Pcri od of record
1898-1978 1898-1955 1957-1978
STATISTICS OF OBSER\!ED
I4umber of data
972 696 264
MONTHLY FLO!IS
Nean cfs
Standard deviation
Skewness
Kurtosis
5665 5644 5841
3763 3986
1.79 1.85
3126
1.39
5.02 5.18 2.07
t i o n by t h e Dan i s decreased by 22 F e r c e n t .
However p l o t s of
continuous means and standard d e v i a t i o n s show s l i g h t d r i f t around t h e i r long-term v a l u e s . A comparison of histograms f o r t h e two periods ( b e f o r e and a f t e r t h e c o n t i n u a t i o n of t h e dam) i n d i c a t e an i n c r e a s e i n t h e frequency of occurrence i n t h e mid-range and a
476
decrease i n t h e frequency o f occurrence i n t h e h i g h values o f flows.
The c o r r e l o g r a m s and power s p e c t r a o f t h e d a t a f o r t h e
t h r e e p e r i o d s a r e a p p r o x i m a t e l y s i m i l a r and do n o t i n d i c a t e any changes i n t h e s e f l o w c h a r a c t e r i s t i c s due t o t h e dam.
A strong
p e r i o d i c i t y o f 12 months i s e v i d e n t i n t h e s e p l o t s f o r a l l t h e three periods.
These and o t h e r d e t a i l s can be f o u n d i n Q u i l l i a n
(1930). Thus some o f these d a t a analyses i n d i c a t e t h a t t h e B u f o r d Dam has changed t h e f l o w c h a r a c t e r i s t i c s , p a r t i c u l a r l y t h e s t a n d a r d deviation o f flows.
LJhether t h e s e changes a r e s i g n i f i c a n t t o
cause nonhomogeneity i n t h e d a t a need t o be pursued f u r t h e r and i s n o t d i s c u s s e d here.
Because o f these observed changes due t o
c o n s t r u c t i o n o f t h e dam, d a t a p r i o r t o 1956 which i s homogeneous i s used f u r t h e r i n t h i s paper. MODEL DESCRIPTION The g e n e r a l f o r m o f a seasonal A R I M A model t a k e s t h e f o r m
where Z t i s t h e v a l u e o f t i m e s e r i e s , t
=
1, 2,
... N;
at i . i . d
normal v a r i a t e w i t h z e r o mean and v a r i a n c e oa2; B i s t h e backward s h i f t o p e r a t o r , s i s t h e s e a s o n a l i t y , V d and :V a r e r e s p e c t i v e l y t h e r e g u l a r and seasonal d i f f e r e n c i n g o p e r a t o r s , d and D a r e r e s p e c t i v e l y t h e degrees o f r e g u l a r and seasonal d i f f e r e n c i n g ,
I$ ( B ) , 0 ( B ) , 8 ( B ) and 0 ( B ) a r e r e s p e c t i v e l y t h e r e g u l a r AR polynomial o f degree p, seasonal AR polynomial o f degree P, r e g u l a r
MA polynomial o f degree o, and seasonal F.lA ?olynomial o f degree Q. ZESULTS OF MODEL BIJILDIAG A I D D I A G N O S T I C C H E C K I N G
A s t e p - b y - s t e p procedure f o r model b u i l d i n g and d i a g n o s t i c c h e c k i n g was a p p l i e d f o r s e l e c t i n g a seasonal ARIPlA model t o t h e m o n t h l y f l o w o f t h e Chattahoochee R i v e r .
The p e r i o d o f water year;
1313-1953 g i v e s a homogeneous d a t a f o r e s t i m a t i n g and f o r e c a s t i n g and excluded t h e e f f e c t of Buford Dam which was b u i l t i n 1C56. The s t e p - b y - s t e p procedure was a p p l i e d as f o l l o w s :
477
Step 1 SELECTION OF s,d and D The ACF and PACF o f t h e m o n t h l y mean f l o w o f t h e Chattahoochee R i v e r , 1313-1952 were c a l c u l a t e d up t o 48 l a g s ( F i g u r e 1). Since t h e ACF has a r e c u r r i n g s i n u s o i d a l F a t t e r n w i t h a p e r i o d o f 12 months, i t was conclLIded t h a t seasonal ARILIA models w i t h s e a s o n a l i t y , s, equal t o 12 would be adequate.
Since t h e ACF o f t h e observed t i m e
s e r i e s d i d n o t d i e o u t q u i c k l y by e x p o n e n t i a l decay, damped s i n c e wave, o r d e c i s i v e c u t - o f f i t was expected t h a t a n o n - s t a t i o n a r y model was necessary. 1,2 and D = 0, 1,2.
T h e r e f o r e d i f f e r e n c i n g was a p p l i e d w i t h d=O, The A C F ' s and PACF's o f t h e d i f f e r e n c e d s e r i e s
showed improvement i n t h a t t h e y came c l o s e r t o t h e use o f pure AR,
MA o r ARMA processes.
The combination d=O, D = l seemed b e s t o f a l l
because i t s ACF and PACF, which a r e shorm i n F i g u r e 1 d i e o u t somewhat more t h o r o u g h l y t h a n those o f t h e o t h e r combinations. Step 2
SELECTIOIj OF p , q , P and Q
Although one c o m b i n a t i o n o f degrees o f d i f f e r e n c i n g was suggested as b e s t values f o r p y a , P and
Q were chosen f o r each o f
t h e f o u r combinations by examinin? t h e p l o t s o f ACF and PACF. These combinations i n c l u d e d d=O, D=O,
d=O D = l , d = l D = l .
I n addi-
t i o n , s e v e r a l o t h e r processes such as A R ( l ) , ARFIA(2,2),
ARIFlA(l,O,
1 ) x (1,0,1)12y Step 3
and A R I K A ( l , l , l )
SELECTIOil OF 1:lITIAL
x ( l y l y l ) 1 2 were chosen. PARAKETER ESTIMATES
It was found t h a t t h e method used f o r f i n a l e s t i m a t i o n o f parameters was i n s e n s i t i v e t o t h e v a l u e o f t h e i n i t ' a l
paraneter
e s t i m a t e s (Rao and Rao, 1974) , t h e r e f o r e i n i t i a l e s t i m a t e s were chrjsen a r b i t a r i l y and had an a b s o l u t e v a l u e l e s s t h a n one.
For t h e i n i t i a l
e s t i m a t e of t h e mean, which was r e a u i r e d f o r models w i t h o u t differencing,
t h e v a l u e of t h e samole mean of t h e m o n t h l y f l o w f o r
t h e p e r i o d 1913-1952 (5574 c f s ) was used. Step 4
FILIAL ESTIt'iA IC)N OF PARAiiETER VALUES
F o r each model f i n a l e s t i m a t e s o f F a r a n e t e r v a l u e s and t h e i r s t a n d a r d e r r o r s were o b t a ned u s i n ? t h e I l a r o u a r d t n o n l i n e a r l e a s t squares procedure.
The f n a l e s t i m a t e s o f some o f t h e v a r i o u s models
a r e shown i n T a b l e 2.
478
d=O;D=l, $ = I 2
+loAC F
.
+a5
ACF
ta5-
I
x
2
0.
0
a -0.5-
05.
LAG k
LAG k 1.0
12
-1.0-
~
a -05LAG k
Figure I :
LAG k
ACF and PACF of observed and d i f f e r e n c e d s e r i e s
Step 5 STUDY OF RESIDUAL AllD FORECAST ERROR PROPERTIES The r a t i o s of mean, mean scluare and v a r i a n c e of r e s i d u a l s t o t h e corresponding q u a n t i t i e s of Z ( t ) (R1, R2, and R3 r e s p e c t i v e l y ) were c a l c u l a t e d f o r r e s i d u a l s and f o r f o r e c a s t e r r o r s . The oarameters were estimated using d a t a f o r 1913-1952 and the one-step-ahead f o r e c a s t s (Box and J e n k i n s , 1976) were obtained f o r t h e twelve months of water y e a r 1953. The R r a t i o s f o r some of t h e models considered a r e given i n Table 2.
W-I
w
I-
s. L
H
IW-)
w
w w I-
!&!
3 4
a
-J
5 H
LL
N W
1 M
4: I-
. L+J
o m w Z L E
7
U
W m N
. .
-
o w w o
7
. .
m w 0 m 0
h U
0
W N
m N
m
7
7
W
N
c
m
co 0 7
7
0
m
0
m h m
m 0 0
0
,
0 N
h
W
.r
N h
m N m 0 0
, \D
U
m m N
I
N N 0
0, 0
m
U N
N
7
N U
m 0
N N
m
0
m 0
N
I
I
U r m .
I
I
7
m N m
m
N
2
I
Ln
01 0
.r
-
N
ln
-
7 7
0
ln
0
7
m
7
0
m
0
h L n
7
"9
0 h 0
N
m
N
m
W m
W
-
0
7
m
N
m
h
m
0 0
m m W
0 m
I
m
m m u3
m
0 h
m
d
W
-
I
0
N W
0
N m
-
0
. .
U h U
N m N
7
0
I
d ln N r-
W d I
-
W .lrn
m
N
h
w
m
m m II
c
w h 0
h 0
L W
m
2 m
479
T k b L E 3.
S T W D A R D ERROF! TEST AR Parameters
Model No. 6
10
16
20
A
A
$1
$2
$1
Model Name
(2SE)
0 0
ARMA
1.576 (.loo)
- .841 (.083)
S
S
(2,2)
ARIMA (l,O,l)
ARIMA (2,0,0,)
ARIMA ( l , l , l )
.633 (.210) S
x (1,0,1)12
x (0.1.1)12
x (1,1,1)12
19
Code:
MA Parameters
A
S = The parameter i s s i g n i f i c a n t ; NS
=
.118 (.094) S
.233 (.108) S
-
A
A
%
0
(2SE)(2SE) 1.254 (.137) S
.970 (.024) S
.259 ( .095) S
A
1
0
-. 598 (.127) S
.363 ( .254)
.890 (.092) 5 .g02
(.040) 5
- .022
.906 (.044)
( .096) NS
-
.922 ( .038)
S
5
0.794 (0.056) S
0.905 (0.028) 5
The parameter i s n o t s i g n f i c a n t
481
Stey G Ir.!ITIAL SCREENING OF CANDIDATE ilODELS The c a n d i d a t e models s e l e c t e d f o r f u r t h e r stud:! were t h o s e wi.:: the s m a l l e s t R v a l u e s . Of t h e t h r e e R v a l u e s , R z and R3 a r e bettei n d i c a t o r s of accuracy of models and a c c o r d i n g l y t h e t h e e models with t h e best ( s m a l l e s t ) R2 and R 3 values f o r t h e r e s i d u a l s were :,lo. 10 ARIIIA (1,0,0) x ( 1 , 0 , 0 ) 1 2 NO. 16 ARIriA (2,0,0) x ( 0 , 1 , 1 ) 1 ~ The three models w i t h the b e s t R 2 ' and R3' v a l u e s f o r the f o r e c a s t e r r o r s were No. 6 ARIMA ( 2 , 2 ) N O . 19 ARINA (0,1,1) x ( 0 , 1 , 1 ) 1 2 Yo. 20 ARIPIA ( 1 , l ,1) x ( 1 , 1 , 1 ) 1 2 Consideration of both r e s i d u a l and f o r e c a s t e r r o r ? r o ? e r t i e s l e d t o models, 6 , 1 0 , 1 6 , 1 9 , and 20 f o r f u r t h e r s t u d y . S t e p 7 SELECTION OF PARSIPIONOUS MODELS Parsimony i s t h e degree t o which a model uses t h e s m a l l e s t nurober of parameters t o g i v e t h e b e s t p o s s i b l e r e s u l t s . To d e t e r n i n e i f a parameter i s extraneous ( a ) Standard E r r o r T e s t ( b ) Variance Ratio T e s t and ( c ) Akaike Information c r i t e r i a a r e used. a . Standard E r r o r Test The s t a n d a r d e r r o r of each parameter i s corn?ared w i t h t h e napn-it u d e of t h a t narameter. I f t h e magnitude o f a r a r a n e t e r i s l e s s ttlan twice i t s s t a n d a r d e r r o r then i t i s d e c l a r e d n o t s i g n i f i c a n t and i s d e l e t e d t o o b t a i n a more parasinonous model. Table 3 summarizes t h e r e s u l t s of Standard Error Test.
For Model
l o . 20, A?Irl,l ( 1 , 1 , 1 ) x ( 1 , 1 , 1 ) 1 2 , t h e seasonal A R parameter was not
s i c l n i f i c a n t t h e r e f o r e i t was d e l e t e d r e s u l - t i n g i n t h e Model [lo. 23a ?121NA ( l , l , l ) x ( 0 , 1 , 1 ) 1 2 w i t h parameters A $1 = .241, A e l = . C 2 7 , and $1 = .SC2. All o t h e r parameters t e s t e d a s s i g n i f i c a n t . b. Variance R a t i o T e s t The Variance Z a t i o T e s t i s a method of comparing t y o models, c:c having fe!.cer n a r a n e t e r s than the o t h e r ( B a r t l e t t , 1 9 6 6 ) . A nodel having n parameters i s considered adecuate i n comFarison w i t h a nodel having n+m p a r a x e t e r s (n.- 1 ) i f t h e s t a t i s t i c S ~ L X : ~ , '-a vo!:ere :'I
482
was computed as given in Table 4. The results of the Variance Ratio Test for all of the comparisons are summarized in Table 4. In all cases the model with the qreater number of parameters gave a significantly better fit. This left the following models for further consideration: No. 6 ARrlA (2,2) No. 10 ARIriA (l,O,l) x (1,0,1)12 No. 16 ARIPiA (2,0,1) x (0,1yl)12 No. 20,ARIPIA (l,l,l) x (0,1,1)12 c. Akaike Information Criteria Akaike information criteria (AIC) can be computed using the estimated residual variance and the model which gives minimum AIC is selected as the parsimonious model. The AIC was computed using AIC = N Ln(a$) + 2 ( p + q t P + Q ) The results of AIC test were similar to the above tests. TABLE 4.
VARIANCE RATIO TEST No. o f Parameters*
Models Beinq Comoared
No. 6 b
ARMA ( 2 , l )
vs No. 6
ARMA ( 2 , 2 )
Deg. o f Freedom
Statistic
X2
S:
4
R
1
26.773
3.841
5
No. 10b ARIMA ( 1 , 0 , 0 ) x ( 1 , 0 , 1 ) 1 2
A
4
R
1
vs
Decision
5.972
3.841
No. 10
ARIMA ( l , O , l ) x (1,0,1)12
5
A
No. 16
ARIMA (2,0,0)
x (0,1,1)12
3
A
No. 19 vs
ARIMA ( 0 , 1 , 1 ) x ( 0 , 1 , 1 ) , 2
2
No. 20a
ARIMA ( l , l , l ) x ( 0 , 1 , 1 ) 1 2
S1
=
N
R
13.84
3
(Vat-,, - Var,+,,,) 2 Var, , accept i f SlzX m , l - a .' A
=
3.841
A
accept, R = r e j e c t , a = 5%
483
Step 8
CHECKING FOR IIHITENESS OF RESIDUALS
For t h e model t o be v a l i d , t h e r e s i d u a l should be u n c o r r e l a t e d , i.e.,
w h i t e and s h o u l d n o t c o n t a i n any p e r i o d i c i t i e s .
If residuals
a r e n o t w h i t e and/or c o n t a i n p e r i o d i c i t i e s t h i s i n d i c a t e s t h a t t h e t i m e s e r i e s has n o t been a d e q u a t e l y modeled and t h a t t h e r e s i d u a l s s t i l ; c o n t a i n process i n f o r m a t i o n .
The Portmanteau Lack o f F i t Test, t h e
A u t o c o r r e l a t i o n Check and t h e Cumulative Periodgram Check a r e used t o t e s t f o r r e s i d u a l whiteness.
D e t a i l s o f these t e s t s can be found Zn
Box and J e n k i n s (1976). Table 5 summarizes t h e r e s u l t s o f Portmanteau T e s t .
Exceot f o r
t h e case o f Model No. 6, t h e r e s i d u a l s o f a l l t h e models t e s t e d as For Model No. 6, ARMA ( 2 , 2 ) , t h e Portmanteau T e s t i n d i c a t e s t h e
white.
r e s i d u a s a r e n o t w h i t e and t h u s s t i l l c o n t a i n unused i n f o r m a t i o n . TABLE 5.
Model No.
Model Name ARVA
6
PORTMANTEAU LACK OF F I T TEST
(2,Z)
Degrees of Freedom
Statistic
43
67.746
59.304
NW
X2
S2
Decision
10
A R I H A ( l , O , l ) x (1,0,1)12
43
32.708
59.304
W
16
ARIMA (2,0,0) x (0,1,1),2
45
36.622
61.656
W
20a
ARIMA ( l , l , l )
45
43.112
61.656
W
W
=
s2
x (O,l,l)lz
2 The residuals are white, NW = non white. White if S2zXf, l-a = (YO.
of parameterslx
L c rk2; L k=l
= max
NO. of Laqs, rk = ACF at lag K
F i g u r e 2 shows p l o t s o f t h e r e s i d u a l s ACF's w i t h t2SE bounds vhere
SE has been d e f i n e d as
4-
For Rodel Yo. 10 and Model No. 16 a l l
t h e r e s i d u a l ACF values f e l l on o r w i t h i n t h e SE bounds i n d i c a t i n g t h a t t h e r e s i d u a l s f o r t h e s e two models a r e w h i t e .
F o r Plodel Yo. 6 t h e
r e s i d u a l ACF v a l u e a t l a g 12 g r e a t l y exceeded t h e bound and f o r Plodel
Jo. 20a t h e r e s i d u a l ACF v a l u e a t l a g 24 l i e o u t s i d e t h e bound t h e r e f c r e t h e r e s i d u a l s of these two models were i n d i c a t e d t c he n n t w h i t e .
484
The normalized cumulative periodograms of the residuals of Model No. 10 and 16 are shown in Fig. 2. Also shown in this figure are the Kolmogorov-Smirnov probability limit lines at 5% significance level. Since all the estimated cumulative periodogrom fall within these limits, it is concluded that the residuals from these models can be considered white and do not contain any significant periodicities. Based on the results of the above tests the following models \:ere accepted as valid models. ilodel No. 10 ARIHA ( l , O , l ) x ( 1 , 0 , 1 ) 1 2 Nodel No. 16 ARIHA (2,0,0) x ( 0 , 1 , 1 ) 1 2 MODEL NO. 16
f
+O.'r
z 0 -
to.1 0.6
24
z
0
LAG k
-0 2
to2
-2SE
FREQUENCY
-
+ai .
0
G
d LT
0 -
a
0
0
p
0.1 0.2 0 3 0.4 0.5
-01.
3
4
00
485 Step 9
FINAL RODEL SELECTION
A f i n a l model s e l e c t i o n was made between t h e r e m a i n i n g two models. ilodel No. 10 had a s m a l l e r r e s i d u a l v a r i a n c e b u t more parameters t h a n iiodel iio. 16.
The Variance R a t i o T e s t as seen e a r l i e r i n d i c a t e d t h a t
Model No. 10 d i d indeed g i v e a s i g n i f i c a n t l y b e t t e r f i t a t t h e 5% l e v e l . Therefore, Model No. 10 was d e c l a r e d t h e b e s t model and Model No. 16, as an a l t e r n a t i v e b e s t model.
The performance o f Model No. 10 and Flodel No. 16 i n f o r e c a s t i n q monthly f l o w s up t o 12 months f o r t h e w a t e r y e a r 1953 was found t o be very good ( Q u i l l i a n , 1980).
The 95% c o n f i d e n c e i n t e r v a l s o f t h e
f o r e c a s t s i n c l u d e d t h e observed values i n a l l cases.
A l s o t h e seasonal Thus
r i s e and f a l l o f t h e m o n t h l y f l o w was e f f e c t i v e l y forecasted.
b o t h models show promise as p r e d i c t i v e t o o l s f o r t h e m o n t h l y mean f l o w o f t h e Chattahoochee R i v e r . V.
SUNNARY AND CONCLUSIONS
A n i n e - s t e p procedure f o r model i d e n t i f i c a t i o n , parameter e s t l m a t i o n , performance e v a l u a t i o n , model parsimony and v a l i d a t i o n o f r e s i d u a l s which emphasis on t h e l a t t e r t h r e e i t e m was implemented f o r o b t a i n i n g v a l i d s t o c h a s l i s models f o r t h e m o n t h l y f l o w s o f t h e C h a t t a hoochee R i v e r a t Nest P o i n t , Georgia. and (2,0,0)x (0,1,1)12
The A R I M A ( 1 , O , l )
x (1 , O , l ) ~ ~
model a r e o b t a i n e d as v a l i d models.
It i s
i n t e r e s t i n g t o n o t e t h a t ElcKerchar and D e l l e u r (1972) f i t t e d an ARIMA
(2,@,0) x (0,1,1)12
model t o t h e m o n t h l y mean flows o f t h e B l u e R i v e r
i n I n d i a n a and H i p e l , e t a1 . (1977) f i t an AR ( 3 ) model t o t h e annual mean f l o w s o f t h e S t . Lawrence R i v e r . REFERENCES
B a r t l e t t , N.S., S t o c h a s t i c Processes. Press, 1966.
New York:
Cambridge U n i v e r s i t y
Box, G.E.P. and Jenkins, G.M., "Time S e r i e s A n a l y s i s " F o r e c a s t i n g and C o n t r o l San F r a n c i s c o , Holden-Day, I n c . , 1976. "Advances i n Box-Jenkins H i p e l , K.E., FlcLeod, A . I . , and Lennox, I.I.C., i l o d e l i n ? . L. ilodel C o n s t r u c t i o n : and " 2 . Anpl i c a t i o n s . " \ l a t e r iiesources Research. 13 ( june 1S77), X7-586.
486
ilcKerchar, A. I . , and D e l l e u r , J.W. , " S t o c h a s i t c Analysis of Plonthl:! Flow Data A p p l i c a t i o n t o Lower Ohio T r i b u t a r i e s . Technical Report No. 26. LaFayette, Indiana, Hater Resources Research Center, Purdue U n i v e r s i t y , 1972. G u i l l i a n , E.H. , " S t o c h a s t i c ARIFIA Models f o r Monthly Streamflows 0:' t h e Chatta hoochee R i v e r , Georgia I n s t i t u t e of Techno1 ogy , At1 a n t a , Georgia (unpublished Special P r o j e c t f o r H.S. d e g r e e ) , 1380. Rao, R . A . and Rao, R.G.S., Analyses of t h e E f f e c t of Urbanization oti R a i n f a l l C h a r a c t e r i s t i c s - I . Technical Report No. 50. blest LaFayette, Indiana: Hater Resources Research Center , Purdue U n i v e r s i t y , 1974. 'I
487
THE LINEAR RESERVOIR WITH SEASONAL GAMMA-DISTRIBUTED MARKOVIAN INFLOWS E.H. LLOYD AND D. WARREN (UNIVERSITY OF LANCASTER, U.K.)
ABSTRACT Earlier work has established a method of tabulating the outflow distribution (by numerically inverting its Laplace Transform) from a linear reservoir fed by a discrete-time gamma-distributed inflow process.
The present paper is concerned with an extension of this
line of enquiry to the case where the inflow process is seasonal. Instead, however of seeking to tabulate the outflow distribution, it concentrates on the outflow skewness, and develops a technique for obtaining this by direct calculations.
A detailed study is made of
a special 2-season case, and the resulting skewness values are tabulated in terms of the reservoir constant, the ratio of the two seasonal mean inflows, and the inflow correlation coefficient.
1.
INTRODUCTION This is a drastically abbreviated version (necessitated by the
exigencies of publication) of an examination of. a simplified model of a Klemes catchment (Klemes, 1973), with particular reference t o its behaviour when the inflows are skewed, autocorrelated, and seasonal The simplifications referred to make the model operate in discrete time, with a particular continuous-valued two-season Markovian inflow process, and with outflow proportional t o storage.
This is thus an
extension of the considerable body of existing work on the linear reservoir with independent non-seasonal inflows (Langbein, 1958; Moran, 1967; Brockwell, 1972; Klemes, 1973 and 1974; Klemes and
Boruvka, 1975) and of a study of the case of skewed autocorrelated non-seasonal inflows (Anis, Lloyd and Saleem, 1979).
Reprinted from Time Series Methods in Hydrosciences, by A.H. ElShaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
488 THE LINEAR RESERVOIR IN DISCRETE TIME
2.
If we denote the quantity of water in store at time t by S(t), inflow and outflow quantities in the interval (t,t+l) by X(t),
the
Y(t)
respectively, (t=O,l,. . . ) , with the linear response
the water balance equation Y(t+l)
= bY(t)+cX(t),
S(t+l)-S(t)
t = O,l,. . . ,
= X(t)-Y(t)
becomes
f
Oo)
(3.1)
with Laplace Transform (L.T.). E(e-ex(t))
= (l+ae)-’.
(3.2)
The first three moments of X(t) E{X(t)}
= pa, E{X(t)}’
are
= p(p+1)a2,
E(X(t)j3
=
p(p+l)(p+2)a3
(3.3)
while the variance and the coefficient of skewness are var{X(t)}
= pa2,
= 2p-’.
skew{X(t)}
(3.4)
The transition density is defined in terms of its L.T. as E{e-eX(t) IX(t-l)=x}
E{X(t)X(t-r)}
=
=
pa2(p+pr)
corr{X(t) ,X(t-r) 1 = p E{X2(t)X(t-r)}
H(e)exp{-G(e)x}
,
(3.5)
,
r
= E{X(t)X’(t-r)}
= a3p(p+l) (p+2pr),
’ (3.7)
and E{ X( t-r)X( t-s)X(t-u) } = a3{p3+p2 (pS-r+pU-S)+p(p+2)pU-r},
r<su
4.
I
OUTFLOW SKEWNESS IN THE NON-SEASONAL CASE With the aid of the results summarised in (3.7) it is not difficult
to obtain the outflow skewness for the linear reservoir of 52 subject
490 to the non-seasonal inflow process described in 53.
To avoid the
frequent repetition of constant multipliers we make a change of scale, setting U(j)
j=1,2,. . . ;
= X(j)/a,
Z(k)
...
= Y(k)/ca,
= skew {Y(k)},
It should be noted that skew {Z(k)}
(4.1)
k=1,2,
...
k=1,2,
,
We then have, instead of (2.3),
Z(t+l) with
m
= Ir=,brU(
t-r)
E{U(j)}=p,
and
(4.2)
E{U2(j)}=p(p+l), = p
corr{U(j),U(j-k)]
The mean value of Z(t+l)
A(1)
= E{Z(t+l)}
E{U3(j>}=p(p+l)(p+2),
k
(j,k = 1,2, . . .)
.
is thus
= pCbr = p/(l-b)
.
(4.3)
The second moment is
A(2) = E{Z2(t+l)}
= l:=,Ls=,b
= 1rb2rE{U2 (t-r)}
r+s E{U(t-r)U(t-s)}
.m
+ 2crls br+SE{U(t-r)U(t-s)} (r<s)
m
= p(p+1)
1 b2r
m
+ 2
o
br+s(p2+pps-r) r=O s=r+l
r=O by (3.7).
o
1 1
This reduces to
A(2) = p2/(1-b)2
+ p(l+bp)/(l-b2)
(1-bp)
(4.4)
Equivalently the variance is var{Z(t+l)}
=
u2, say,= p(l+bp)/(l-b2)
The third moment is E{Z3(t+l)}
(1-bp)
.
.
(4.5)
r+s+u E{U( t-r)U (t-s)U (t-u) }
= Ir~s~ub
This may be evaluated by a technique similar to that employed f o r X(2),
that is, by splitting the triple sum into a single sum
involving E{U3(t-r)
3,
a double sum involving E{U2 (t-r)U(t-u)}'
rfu), and a triple sum involving E{U(t-r)U(t-s)U(t-u)} r#s%uzr).
(with
After not inconsiderable reduction we are left with the
result skew {Z(t+l)}
(with
=
UJ-~'~{A(~)-~X(~)A(~)
+ 2A3(l)}
491 where X ( 1 )
and X(2)
(1-b3)X(3)
= p(p+l) (p+2)
+2p/(l-bp)
+ 3p(p+l)b{p/(l-b)
+ pb/(l-b2>
+ 2bp/(l-b2p) 1 + 6pb3{p2/(1-b> (1-bz>
pp/(l-b) (l-b’p)
5.
are as defined in (4.3) and (4.4)) while
+ pp/(l-bp) (1-b‘)
+
+ (p+2)P2/(1-bp> (1-b2P>l.
THE SEASONAL GAMMA-DISTRIBUTED MARKOV CHAIN The process described in 53 generalizes easily to a seasonal
version with k seasons (k=2,3,. . . ) .
For example, the 3-season
version has the three season-to-season transition L.T.‘s
where, for n=0,1,2,. . . , we have H(3n+j,e) = H(j,e) = {l+a.(l-p J
J
and
where a
-1 of X(3n+j)
Then, for n=0,1,2,. . . , the marginal distribution 2’ (i.e. the j-season inflow) is gamma with shape parameter
= ct
a . and shape parameter p , (the shape parameter not being J
seasonalisable in this process), while = p . J corr{X(3n+j) ,X(3n+j-2)} = p . p J j-1 corr{~(3n+j) ,~(3n+j-3)} = p .p p
corr{X(3n+j),X(3n+j-l)}
J
etc. (where P - = ~ p2, p-2
- pl,
j-1 j-2 etc.).
We have to accept the restriction that, necessarily, in a k-season year, the lag-k correlation coefficient is a constant;
thus if for
example the “seasons“ are months, the January-January correlation is the same as the May-May correlation, etc.
6.
OUTFLOW SKEWNESS INDUCED BY A 2-SEASON INFLOW In this exploratory study we take as inflow process the seasonal
chain described in 95, taking however only two seasons and taking p =p =constant ( = p ) .
1
2
The obvious extension of the techniques used
492 i n 54 t o s t u d y t h e o u t f l o w skewness i n d u c e d by n o n - s e a s o n a l
As b e f o r e w e
d i s t r i b u t e d Markovian i n f l o w s may now b e a p p l i e d .
r e p l a c e Y ( t ) by t h e s t a n d a r d i z e d v e r s i o n Z ( t ) = Y ( t ) / c . X ( t ) , X(t-2)) X(t-4), X(t-1),
...
X(t-3))
...
gamma-
We r e g a r d
a s o c c u r r i n g i n s e a s o n " l " , and
a s o c c u r r i n g i n s e a s o n "2"
.
The a p p r o p r i a t e
standardized versions a r e = X(t-2r)/a
U(t-2r)
,r=O,l,... 1
and V(t-2s-1)
= X(t-2s-l)/a2,
s=O,l,
... .
Then t h e U ' s and V ' s a l l h a v e gamma d i s t r i b u t i o n s w i t h u n i t s c a l e p a r a m e t e r and w i t h s h a p e p a r a m e t e r p , and t h e s e q u e n c e
i s a f i r s t - o r d e r homogeneous Markov C h a i n w i t h t r a n s i t i o n L . T . in (3.5).
as
W e t h e n have, a s t h e 1-season o u t f l o w ,
Z(t+l) = allrb
2r
U(t-ar)+ct
1 b 2 s + lV(t-2s-1)
(6.1)
2 s
from which t o e v a l u a t e t h e s k e w n e s s .
U t i l i s i n g r e s u l t s from 5 5 3 , 4
( i . e . homogeneous) i n f l o w s , w e o b t a i n t h e f i r s t
f o r non-seasonal
moment o f t h e 1 - s e a s o n o u t f l o w w i t h n o d i f f i c u l t y a s
X1
= E{Z(t+l) } = p(al+ba2)/(1-b2).
(6.2)
The s e c o n d moment r e q u i r e s a l i t t l e c a l c u l a t i o n ;
it t u r n s out t o
be 2 = E{Z'(t+l))
=
(a1'+b2a2'){p2/(1-b')2
+
p ( l + b 2 p 2 ) / ( 1 - b 4 ) (l+b'p2) } +
+
2{a ct /(1-b4)}{p(p+p)b+2b3p~/(1-bz)+b3pp(l+p2)/(l-b2pz)~. 1 2
(6.3)
The l a b o u r r e q u i r e d t o c a l c u l a t e t h e t h i r d moment, and h e n c e t h e s k e w n e s s , i s of a t o t a l l y d i f f e r e n t o r d e r of m a g n i t u d e , b e i n g i n f a c t s u f f i c i e n t t o discourage r a t h e r e f f e c t i v e l y t h e authors' o r i g i n a l hope o f s t u d y i n g t h e t h r e e - s e a s o n c a s e . cubing ( 6 . 1 ) w e have E { Z 3 ( t + l ) } = a13A+a23b3A+3a tc' 1
B+3a ct *C 2 1 2
I n o u t l i n e , on
493 where A=E{Cb2rU(t-2r)}3,
B=Er{Cb
2r
U(t-2r)}2{Cb2s+1 V ( t - 2 s - 1 )
2s+l C=Er{Eb2’LJ(t-2r) }{Cb Vt (t-2s-1)
11
>*1
(6.4)
The terms A , B and C must b e s e p a r a t e l y e v a l u a t e d .
The r e s u l t s ,
e x p r e s s e d i n terms of t h e a u x i l i a r y f u n c t i o n f ( u , v ) = l / ( l - b 2 u z ) ( l - b 4 u 2 ) (1-b‘).
(6.5)
are
7.
CONCLUSIONS An a b b r e v i a t e d r e p r e s e n t a t i v e s e l e c t i o n of t a b u l a t e d v a l u e s of
t h e o u t f l o w skewness i s p r e s e n t e d i n t h e T a b l e .
Whilst i n p r i n c i p l e
t h e s e v a l u e s a r e f u n c t i o n s of t h e f i v e p a r a m e t e r s
c1l ’ a 2 ’ b ’ P > P !
t h e r e a r e two s i m p l i f y i n g f e a t u r e s which r e d u c e t h e number of parameters t o t h r e e .
One of t h e s e f e a t u r e s r e f e r s t o t h e d e p e n d e n c e on a w i l l b e s e e n from ( 6 . 2 ) t h a t , w r i t i n g
x1
(a1 ,a,>
= a2A1 ( u p 2 , 1 ) .
S i m i l a r l y , from ( 6 . 3 ) and ( 6 . 4 ) )
’
X 2 ( a 1 , a 2 ) = a 2 2 x2 ( 1 a / 2a ’ 1 ) and
XI a s X ( a l , a 2 )
and a
2’ w e have
1
It
494
I t f o l l o w s t h a t t h e o u t f l o w skewness depends n o t on a s e p a r a t e l y b u t on t h e s e a s o n a l i n d e x r a t i o a /a 1 2’ mean i n f l o w i n s e a s o n 1 t o t h a t i n s e a s o n 2 .
and a 1 2 t h e r a t i o of t h e
The s e c o n d s i m p l i f y i n g f e a t u r e r e l a t e s t o t h e dependence of t h e o u t f l o w skewness on t h e p a r a m e t e r p .
I n t h e s p e c i a l c a s e where
t h e i n f l o w s a r e m u t u a l l y i n d e p e n d e n t , s i m p l e c a l c u l a t i o n s show t h a t skew{Y(t+l)} =
2 h(al/a2,b), P
(7.1)
J-
where
S i n c e 2 / J p = skew(X
) , ( 7 . 1 ) shows t h a t t h e r a t i o o f o u t f l o w t+l skewness t o i n f l o w skewness i s i n d e p e n d e n t o f p when t h e i n f l o w s
a r e independent.
I t t u r n s o u t t h a t t h i s i n d e p e n d e n c e of p s t i l l
h o l d s when t h e i n f l o w s a r e c o r r e l a t e d .
T h i s r e s u l t o n l y emerged
when a d e t a i l e d t a b u l a t i o n o f t h e o u t f l o w skewness v a l u e s was studied :
f o r a l l v a l u e s of a /a b and p , t h e r a t i o 1 2’
o u t f l o w skewness i n f l o w skewness
(7.2)
was i n d e p e n d e n t o f p .
Thus by p r e s e n t i n g o u r r e s u l t s i n terms of
t h i s r a t i o i t i s p o s s i b l e t o eliminate t h e parameter p . The T a b l e g i v e s v a l u e s o f t h e r a t i o ( 7 . 2 ) f o r v a r i o u s v a l u e s of t h e r e s e r v o i r c o n s t a n t b ( i n t r o d u c e d i n ( 2 . 1 ) ) , t h e lag-1 season-tos e a s o n c o r r e l a t i o n c o e f f i c i e n t 0 , and t h e s e a s o n a l i n d e x r a t i o a1/a2. I n r e a d i n g t h e T a b l e c a r e i s needed i n e n t e r i n g t h e a p p r o p r i a t e The c o n v e n t i o n a d o p t e d i n C h a p t e r 6 made a l t h e v a l u e of a /a 1 2’ a - p a r a m e t e r o f X t , Xt-2, , and a 2 t h a t o f Xt-4, Now t h e T a b l e g i v e s t h e skewness r a t i o of Xt+l, Xt-l, Xt-3,
.. .
... .
i . e . o f t h e o u t f l o w d u r i n g a s e a s o n d u r i n g which t h e mean Yt+l I f . t h i s is the inflow is proportional ( i n our convention) t o a 2’ d r i e r o f t h e two s e a s o n s w e must have a < a o r a /a > 1. In 1’ 1 2 2
495 o t h e r words t h e t a b u l a t e d o u t f l o w skewness r a t i o i s t o be r e a d o f f a g a i n s t v a l u e s of a /a s u c h t h a t al/a2 > 1 i f o n e i s c o n c e r n e d w i t h 1 2 t h e ” d r i e r ” s e a s o n i . e . t h e o n e which h a s s m a l l e r mean i n f l o w ; w h i l e i f o n e i s c o n c e r n e d w i t h t h e o u t f l o w d u r i n g t h e ”wetter” s e a s o n t h e a p p r o p r i a t e v a l u e o f a /a s a t i s f i e s a /a < 1. (The 1 2 1 2 T a b l e a l s o g i v e s v a l u e s f o r a /a = 1, which i s o f c o u r s e t h e non1 2 seasonal case. ) I t w i l l be s e e n from t h e s e t h a t t h e skewness o f t h e o u t f l o w is
always l e s s t h a n t h a t o f t h e i n f l o w , a n d t h a t t h e r a t i o of o u t f l o w t o i n f l o w skewness i n c r e a s e s w i t h i n c r e a s i n g c o r r e l a t i o n c o e f f i c i e n t p and d e c r e a s e s w i t h d e c r e a s i n g p .
A t t h e e x t r e m e s , t h e case p=O
c o r r e s p o n d s t o i n d e p e n d e n t i n f l o w s , f o r which skew (Y
)/skew(X ) t +1 t+l I n t h e c a s e p=1,
i s g i v e n by t h e e x p r e s s i o n h ( a /a , b ) of ( 7 . 1 ) . 1 2 t h e 2 - s e a s o n v e r s i o n o f ( 5 . 1 ) shows t h a t t h e t r a n s i t i o n L a p l a c e
Transform then reduces t o
(using t h e convention t h a t t h e “ t “ season i s an a
and s o o n .
s e a s o n ) whence
The i n f l o w s e q u e n c e t h u s d e g e n e r a t e s when p = l t o a n
a l t e r n a t i n g sequence of c o n s t a n t s . similarly.
1
Then, f o r e a c h t ,
The o u t f l o w s e q u e n c e b e h a v e s
and X both have zero skewness, Yt+l t+l
t h e r a t i o t e n d i n g t o u n i t y when b z l . O t h e r l i m i t i n g v a l u e s o f i n t e r e s t i n c l u d e t h e c a s e s b=O, b = l ,
I n t h e case b = O ,
a /a = 0 , a2/a1=0. 1 2
(2.1) reduces t o
t h e o u t f l o w i s i d e n t i c a l l y t h e same as t h e p r e c e d i n g i n f l o w .
The
skewness r a t i o i s t h e r e f o r e e q u a l t o u n i t y f o r a l l v a l u e s of a /a 1 2 When b = l , t h e o u t f l o w i s i d e n t i c a l l y z e r o and t h e o u t f l o w skewness When a /a =O o r when a /a =O w e h a v e i n f l o w s 1 2 2 1 i n o n l y o n e of t h e two s e a s o n s i n e a c h y e a r . Then ( t a k i n g
r a t i o tends t o zero.
Xt,
Xt-2,
Xt-4,
...
t o be t h e non-zero inflows) t h e outflow
Jecomes
496 Y
t+l
= (1-b)Cb
when X t ,
X
t-2’
c o r r (X t,Xt-2)
2r
Xt-2r
Xt-4,
...
are identically distributed with
Thus t h e o u t f l o w p r o c e s s i n t h i s c a s e
= p2.
c o r r e s p o n d s t o t h a t f o r n o n - s e a s o n a l i n f l o w s ( a / a =1) w i t h p 1 2 r e p l a c e d by p‘ and b by b 2 .
TABLE showing t h e r a t i o o f o u t f l o w skewness t o i n f l o w s k e w n e s s as a f u n c t i o n of t h e r e s e r v o i r c o n s t a n t b , t h e l a g - 1 s e a s o n - t o s e a s o n i n f l o w c o r r e l a t i o n c o e f f i c i e n t p , and t h e i n f l o w s e a s o n a l i t y i n d e x
a1/a2. d e n o t e s t h e r a t i o o f mean i n f l o w i n s e a s o n 1 t o mean inflow i n season 2. al/cxa = 1 c o r r e s p o n d s t o n o n - s e a s o n a l i t y (al/a2
al/a2
1
corresponds t o outflows occurring i n t h e wetter season
a1/a2 > 1
corresponds t o outflows occurring i n t h e d r i e r season.
b
0.1
P
0.6
0.7
0.8
0.9
0 0.5 0.91
.9860 .9482 .8922 .8225 .7423 .6531 .5544 .4426 .3056 . 9 9 6 7 .9882 . 9 7 5 5 . 9 5 8 7 . 9 3 6 4 . 9 0 5 4 . 8 5 8 9 . 7 8 1 6 . 6 3 1 0 . 9 9 9 9 . 9 9 9 6 . 9 9 9 1 . 9 9 8 4 . 9 9 7 2 . 9 9 5 3 . 9 9 1 3 .9812 . 9 4 2 4
--
-
______._--______-
1
.9962 .9989 0.5 3.9 __ ~ _ .9989 3 .9996 3.5 3.9 0
4
0.5
0 0.5 0.9
--
2
0.4
.8515 .7182 .7076 .7327 .7423 .7184 .6556 .5509 .3933 . 9 7 6 8 . 9 6 2 2 . 9 5 9 2 .9552 . 9 4 2 3 . 9 1 4 8 . a 6 6 3 . 7 8 3 8 . 6 2 7 6 . 9 9 9 3 , 9 9 8 9 . 9 9 8 8 .9984 . 9 9 7 5 . 9 9 5 5 . 9 9 1 4 . 9 8 1 1 . 9 4 2 1 -- _~ . 9 5 0 3 . 8 4 9 6 . 7 5 8 0 . 6 9 5 3 .6521 . 6 0 9 7 . 5 5 1 9 . 4 6 6 4 . 3 3 6 8 .9903 .9749 .9619 .9497 .9334 .5069 .8620 .7838 .6308 . 9 9 9 7 . 9 9 9 2 . 9 9 8 8 .9982 . 9 9 7 3 . 9 9 5 4 . 9 9 1 4 . 9 8 1 2 . 9 4 2 3
0 0.25 0 . 5 0.9
1
0.3
______ . 7 0 7 0 . 8 0 3 1 . 8 7 5 3 .8956 . 8 7 6 4 . 8 2 1 3 . 7 3 0 5 . 6 0 1 9 . 4 2 3 1 . 9 6 2 2 . 9 7 1 4 . 9 7 7 1 .9722 . 9 5 4 6 .9214 . 8 6 7 5 . 7 8 0 6 . 6 2 2 4 . 9 9 9 0 . 9 9 9 1 . 9 9 9 2 . 9 9 8 8 , 9 9 7 7 . 9 9 5 5 . 9 9 1 2 , 9 8 0 7 .9417
0 0.5 0.9
0.5
0.2
.9839 . 9 5 9 4 . 9 1 7 9 .9948 .9858 .9699 .9998 .9993 .9986 _ _ _ _ .9941 .9807 .9524 .9973 .9905 .9764 .9999 .9995 .9987
.8551 .7687 .6583 .5236 .3561 .9456 .9105 .8597 .7795 .6281 .9973 .9952 .9911 .9809 .9422 ~
__.
.go27 .8266 .7214 .5843 .4042 .9523 ,9156 .8616 .7779 .6242 .9974 .9952 .9910 .9807 .9419
497 REFERENCES A n i s , A . A . , L l o y d , E . H . and S a l e e m , S . D . , 1 9 7 9 . The l i n e a r r e s e r v o i r w i t h Markovian i n f l o w s . Water R e s . R e s e a r c h , 1623-1627. B r o c k w e l l , P . J . , 1 9 7 2 . A s t o r a g e model i n which t h e n e t growthJ . Appl. P r o b . , 129-139. r a t e i s a Markov C h a i n . K l e m e s , V . , 1973. Watershed a s s e m i - i n f i n i t e s t o r a g e r e s e r v o i r . J . I r r i g . D r a i n . D i v . A m e r . S O C . C i v i l E n g . , 99, 477-491. K l e m e s , V . and B o r u v k a , L . , O u t p u t from a c a s c a d e of d i s c r e t e 1-13. l i n e a r r e s e r v o i r s with s t o c h a s t i c i n p u t . J. Hydrol., Lampard, D . G . , 1 9 6 8 . A s t o c h a s t i c p r o c e s s whose s u c c e s s i v e i n t e r v a l s between e v e n t s form a f i r s t o r d e r Markov C h a i n . J . Appl. P r o b . , 5 , 648-668. L a n g b e i n , W . B . , 1 9 5 8 . Queuing t h e o r y and w a t e r s t o r a g e . J . H y d r a u l . D i v . h e r . S O C . C i v i l Eng. HY5, 1811/1 - 1 8 1 1 / 2 4 . Lloyd, E . H . , 1963. R e s e r v o i r s w i t h s e r i a l l y c o r r e l a t e d i n f l o w s . T e c h n o m e t r i c s , 3, 83-93. L l o y d , E . H . , and S a l e e m , S . D . , 1 9 7 9 . A n o t e on s e a s o n a l Markov C h a i n s w i t h gamma o r gamma-like d i s t r i b u t i o n s . J . A p p l . P r o b . , 1 6 , 117-128. Moran, P . A . P . , 1 9 6 7 . Dams i n s e r i e s w i t h c o n t i n u o u s r e l e a s e . J . A p p l . P r o b . , 4 , 330-388. P h a t a r f o d , R . M . , 1 9 7 6 . Some a s p e c t s of s t o c h a s t i c r e s e r v o i r t h e o r y . J . H y d r o l . , 30, 199-217.
15,
s,
27,
498
ON
T H E S T O R A G E SIZE-DEPIAND-RELIABILITY R E L A T I O N S H I P RAVINDRA M , PHATARFOD
1.
INTRODUCTION I n v e s t i g a t i o n s by h y d r o l o g i s t s and e n g i n e e r s show t h a t i n a
l a r g e number o f c a s e s monthly streamflows f i t a model o f Markov dependence ( o f v a r i o u s o r a e r s ) w i t h monthly v a r y i n g t r a n s i t i o n probab i l i t i e s ( s e e e . g . Kottegoda, 1 9 7 0 ) .
The o r d e r o f Markov dependence
i n most c a s e s i s one o r two, b u t i n some c a s e s , t h r e e .
I t would
appear then t h a t i f one t a k e s time-periods o r seasons o f about two months, a model o f Markov dependence (of o r d e r one) with s e a s o n a l l y v a r y i n g t r a n s i t i o n p r o b a b i l i t i e s would f i t t h e flows i n most c a s e s . In t h i s paper we c o n s i d e r t h e s t o r a g e p r o c e s s with such an i n f l o w model, and compare t h e procedures t h a t can be used t o determine t h e storage size-demand-reliability
relationship.
L e t us t h e r e f o r e f i r s t c o n s i d e r , b r i e f l y , t h e procedures t h a t
a r e being used ( i n p r a c t i c e ) and can be used, t o determine t h e r e l a t i o n s h i p , and then c o n s i d e r i n d e t a i l s , o n l y t h o s e f o r which a meaningful comparison can be made, when t h e i n p u t p r o c e s s i s of t h e kind d e s c r i b e d above. I t i s g e n e r a l l y accepted ( s e e e . g . McMahon and Mein, 1978) t h a t
t h e s e procedures can be p u t i n t o t h r e e d i s t i n c t groups.
The f i r s t
group ( c a l l e d C r i t i c a l Period Techniques) i n c l u d e s t h o s e procedures which r e l y e n t i r e l y on t h e h i s t o r i c a l d a t a .
These procedures
i n c l u d e R i p p l ' s Mass Curve Method ( t h e e a r l i e s t method known), Sequent Peak Algorithm, Minimum flow method, and o t h e r s ( s e e McMahon and Mein, 1978, f o r d e t a i l s )
The second group ( P r o b a b i l i t y
Methods) i n c l u d e s t h o s e procedures which use t h e c a l c u l u s o f p r o b a b i l i t y t o t h e s t r u c t u r e o f t h e problem, r e s u l t i n g i n a p r o b a b i l i t y d i s t r i b u t i o n of t h e s t o r a g e c o n t e n t .
The t h i r d group ( S y n t h e t i c
Hydrology) c o n s i s t s of t h o s e procedures which a r e based on generated d a t a , t h e r e l a t i o n s h i p being o b t a i n e d by simple s i m u l a t i o n .
The
l a t t e r two groups o f procedures depend on t h e formulation and f i t t i n g Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
499 of a s t a t i s t i c a l model ( a s t o c h a s t i c process) t o t h e given h i s t o r i c a l d a t a , and a r e o f t e n regarded as being s t a t i s t i c a l l y more r i g o r o u s than t h e procedures i n t h e f i r s t group.
The s y n t h e t i c hydrology
procedure i s j u s t one procedure-simulation;
t h e bulk of t h e r e s e a r c h
a s s o c i a t e d with t h i s procedure being done i n t h e a r e a of formulating and f i t t i n g of c o r r e c t models f o r inflows. use a r e
The models i n common
t h e Thomas-Fiering model, Matalas l o g Normal model, Broken
l i n e model, e t c .
On
t h e o t h e r hand, t h e group of p r o b a b i l i t y
methods i n c l u d e s a v a r i e t y of methods w i t h varying mathematical sophistication
-
from numerical t o a n a l y t i c a l , t h e i n p u t models
p o s s i b l e being somewhat more r e s t r i c t i v e than those f o r t h e s y n t h e t i c hydrology group (Savarenskiy, 1940; K r i t s k i y and Menkel, 1940; Moran, 1954; Prabhu, 1958; Lloyd, 1963; Klemes, 1970; Phatarfod, 1981a). W e s h a l l n o t be considering here any procedures belonging t o t h e
f i r s t group.
A comparison of t h e s e procedures and some from t h e
second group with worked o u t examples has been given i n McMahon and Mein (1978).
There, t h e comparison between t h e procedures i s
made n o t only i n terms of t h e i r l i m i t a t i o n s , and underlying assumptions e t c . , b u t a l s o i n t e r m s of t h e f i n a l answer obtained - t h e
s i z e of
t h e r e s e r v o i r r e q u i r e d w i t h a s p e c i f i c d r a f t and r e l i a b i l i t y of supply.
In t h i s paper, we are i n t e r e s t e d i n comparing t h o s e
methods which a r e a p p l i c a b l e when t h e i n p u t process i s of t h e kind described e a r l i e r . The methods compared are: A.
Simulation.
B.
P r o b a b i l i t y Matrix Method ( K r i t s k i y and 14enke1, 1940; Dearlove and H a r r i s , 1965; K l e m e s , 1970).
This i s a seasonal extension of
Lloyd's (1963) procedure ( f o r Markov b u t non-seasonal and involves c o n s t r u c t i o n of m a t r i c e s .
inputs) ,
The procedure i s
e n t i r e l y numerical. C . Bottomless Dam a n a l y t i c a l method Mark 1.
(Phatarfod, 1981a).
W e assume t h e dam t o be bottomless and d e r i v e an a n a l y t i c a l
500 s o l u t i o n f o r t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e d e p l e t i o n of t h e dam. D.
Bottomless D a m a n a l y t i c a l method Mark 2 .
(Phatarfod 1981b).
Here we make t h e f u r t h e r assumption t h a t t h e i n p u t s a r e uncorrelated from year t o y e a r , although they a r e c o r r e l a t e d and ( s e a s o n a l l y Markov) dependent w i t h i n a year.
This assumption makes a d r a s t i c
reduction i n t h e computational complexities involved i n Method C . A common f e a t u r e of t h e methods B , C and D i s t h a t f o r t h e s e
methods we c o n s i d e r t h e i n p u t s , s t o r a g e , r e l e a s e e t c . , as d i s c r e t e q u a n t i t i e s , whereas f o r A , t h e s e a r e continuous. I t i s customary, i n hydrology and engineering d i s c i p l i n e s , t o
t a k e a month as t h e time u n i t o f o p e r a t i o n -most flows a r e , i n f a c t , published a s monthly values.
r e c o r d s o f streamFor reasons
given before,we s h a l l consider a p e r i o d of 2 months a s our u n i t of For s i m p l i c i t y of p r e s e n t a t i o n o n l y , w e s h a l l assume h e r e
time.
a time u n i t o f s i x months, so t h a t we have only two seasons p e r year.
I t i s f a i r l y e a s y t o s e e how t h e procedures can be extended
t o t h e case of s i x seasons. O f course, none of t h e above methods w i l l provide a c o r r e c t
answer
-
t h e answers t h e y a l l provide are approximations.
One
reason f o r comparing them i s t o f i n d o u t t o what e x t e n t t h e answers d i f f e r because o f t h e approximating devices employed i n t h e p r o b a b i l i t y methods.
A comparison between t h e answers o b t a i n e d by A and B and
those by C and D would i n d i c a t e how t h e assumption o f t h e bottomlessness of t h e dam has a f f e c t e d t h e answer, and under what c o n d i t i o n s on t h e d r a f t r a t i o , s a y , t h e answers a r e comparable.
A comparison
between A on one hand and B , C and D on t h e o t h e r would i n d i c a t e t h e effect- o f d i s c r e t i z a t i o n ; t h i s should t e l l us how s m a l l o r l a r g e t h e u n i t of measurement we should choose t o give comparable r e s u l t s . This, i n t u r n , would t e l l us something about t h e amount o f e f f o r t required by using B , C , D.
Comparing answers o b t a i n e d by C and D
would i n d i c a t e t o what e x t e n t t h e answers d i f f e r i f w e n e g l e c t
501 i n t e r - y e a r dependence of flows, and so on. For t h e inflow model assumed, Method B - P r o b a b i l i t y
Matrix method
seems t o be t h e mathematically c o r r e c t procedure, because t h e answer it provides i s f r e e from t h e sampling e r r o r s a s s o c i a t e d w i t h Method
However, i t seems t h a t it i s n o t i n popular u s e , t h e reason
A.
being t h a t it i s an a b s t r a c t method and t h e general (erroneous) impression t h a t it involves very unwieldy m a t r i c e s . The question a r i s e s , t h e r e f o r e , what i s t h e use of Methods C and D , p a r t i c u l a r l y s i n c e t h e y are mathematically more a b s t r a c t than
Method B.
A p a r t i a l answer t o t.%s question i s provided by t h e
following c o n s i d e r a t i o n s .
Whilst it i s c e r t a i n l y t r u e t h a t using
Method A (and p o s s i b l y Method B) would give an engineer a b e t t e r i n s i g h t i n t o t h e r e s e r v o i r performance i n a given s i t u a t i o n , i . e . f o r p a r t i c u l a r values of t h e parameters of t h e inflow model, only an a n a l y t i c a l method would give u s an i n s i g h t i n t o t h e general problem considered here.
W e s h a l l s e e , f o r example, t h a t f o r Method D,
t h e e q u i l i b r i u m d i s t r i b u t i o n of t h e d e p l e t i o n of t h e r e s e r v o i r depends on a few c r i t i c a l v a l u e s .
S p e c i f i c a l l y , i t i s o f t h e form:
Probability {depletion = j ) = C 6 1
1 1
+
C202j
+
C303j
+ ... .
The
0's
depend upon t h e parameter v a l u e s o f t h e annual flows and t h e
C's
depend upon t h e t r a n s i t i o n p r o b a b i l i t i e s of t h e seasonal flows
and t h e seasonal r e l e a s e s . of
8
and
C
In p r a c t i c e t h r e e o r f o u r values each
a r e s u f f i c i e n t f o r reasonable accuracy.
The importance o f t h i s r e s u l t i s as follows.
The inflow model
considered h e r e has 2 4 parameters (6 values of means, standard d e v i a t i o n s , skewness, and s e r i a l c o r r e l a t i o n s ) .
I t i s f a i r l y obvious
t h a t c o n s t r u c t i o n of c h a r t s and t a b l e s (showing t h e r e s e r v o i r s i z e d r a f t - r e l i a b i l i t y r e l a t i o n s h i p f o r v a r i o u s v a l u e s of t h e parameters) by Methods A o r B , i s v i r t u a l l y impossible.
On
t h e o t h e r hand
Method D shows t h a t t h e r e l a t i o n s h i p i s governed by a few values of annual flow parameters and some t r a n s i t i o n p r o b a b i l i t i e s , and n o t on a l l t h e 2 4 seasonal parameters.
I t would seem t h e r e f o r e t h a t
,
-
502 t h i s approach might b e more f r u i t f u l , i f n o t f o r c o n s t r u c t i n g c h a r t s , etc.
,
t h e n a t l e a s t f o r p r o v i d i n g an a l g o r i t h m f o r working o u t dam
sizes.
DESCRIPTION OF THE METHODS
2.
A s mentioned i n t h e I n t r o d u c t i o n , w e s h a l l , f o r s i m p l i c i t y of
p r e s e n t a t i o n , c o n s i d e r f i r s t t h e c a s e o f o n l y two s e a s o n s . Suppose t h e y e a r i s d i v i d e d i n t o two s e a s o n s , Summer ( d r y ) and Winter ( w e t ) , s a y .
L e t t h e i n f l o w s ( c o n t i n u o u s random v a r i a b l e s )
and
respectively.
Wn
= x 'n-1 m a r g i n a l d e n s i t y of
Sn
given
I
e ( x y)
and
density of
given
k (x), a ( y l x)
functions
a ( y l x ) , and t h e s t a t i o n a r y
b e d e n o t e d by
k(x).
Similarly, l e t
denote t h e s t a t i o n a r y marginal o f Wn
h i s t o r i c a l data.
n '
Let t h e t r a n s i t i o n p r o b a b i l i t y d e n s i t y of
b e d e n o t e d by W
y e a r be d e n o t e d by
nth
d u r i n g t h e summer and w i n t e r o f t h e
Sn = y .
R(y)
and t h e c o n d i t i o n a l
S
I n any p r a c t i c a l s i t u a t i o n t h e
e t c . , a r e o b t a i n e d by f i t t i n g them t o t h e
For example, w e may t a k e t h e s e f u n c t i o n s t o b e
as given by t h e Thomas-Fiering
(two-season) model.
L e t t h e r e l e a s e s d u r i n g Summer and Winter be
M1
and
M2 r e s p e c t i v e l y , and l e t t h e c o n t e n t s of t h e r e s e r v o i r a t t h e b e g i n n i n g
o f Summer and Winter o f t h e
and (n) r e s p e c t i v e l y , so t h a t w e have t h e w a t e r b a l a n c e e q u a t i o n ,
C W (n)
(with
nth
= c
+ S - M if ~ ( n ) n 1
replacing
O G C S(n)
= K
if
C
= o
if
C
and a similar r e l a t i o n c o n n e c t i n g
A
C
S
as t h e r e s e r v o i r s i z e ) ,
K
C w(n)
Method
y e a r be d e n o t e d by
M1
s( n )
s(n)
C
+ S
n
- M I G K
+Sn - M 1 > K + S
w(n-1)
n
- M
with
1
G O C
S(n)
with
M2
-
: Simulation
I n t h e s i m u l a t i o n p r o c e d u r e , w e g e n e r a t e a sequence o f v a l u e s o f Sn
and
W
n
a c c o r d i n g t o t h e assumed i n f l o w model, i g n o r i n g t h e
503 i n i t i a l v a l u e s , and then using t h e w a t e r balance equation above, simulate t h e r e s e r v o i r behaviour, with varying values f o r t h e say. The general p r a c t i c e i s t o generate S(0)' about 1000 o r so sequences each sequence as long a s t h e h i s t o r i c a l i n i t i a l content
C
sequence. L e t u s now consider t h e t h r e e p r o b a b i l i t y procedures.
For a l l
of them we approximate t o t h e t r u e s i t u a t i o n by working i n d i s c r e t e W e take a suitable
q u a n t i t i e s r a t h e r than i n continuous ones. u n i t of water
6 , and express a l l t h e q u a n t i t i e s such a s t h e i n f l o w s ,
d r a f t s and t h e r e s e r v o i r s i z e , e t c . i n terms of t h i s u n i t . t h e ranges of
and
Wn
Sn
be
(0,s
i v e l y , i . e . t h e p r o b a b i l i t i e s of
+&
r
and
Wn
+)
and
Wn
and
(0,r
Sn
+
( 0 , 1/2), (1/2, 3 / 2 ) by
'...,
(s
- 1/2,
s
respect-
+)
exceeding
r e s p e c t i v e l y a r e n e g l i g i b l e i f n o t zero.
the i n t e r v a l s variable
+
Let
s
+ 4
Let us denote
+ 1/2)
of t h e
0 , 1, 2,..., s , and s i m i l a r l y f o r t h e v a r i a b l e
Then t h e expressions k ( x ) , a ( y / x ) g ( y ) and e ( x l y ) a r e 'n. r e s p e c t i v e l y , so t h a t we have replaced by k i f a . . 2 . and e . 13' 7 Ii i+? k . = Pr{W=i}= ( y ) dy k ( x ) d x , & = Pr{S=j}= 1 i-1 j j
['+J?, -+
I,
j++ i+$ a , . = Pr{S =jlW =i}= k(x)a(ylx)dxdy/ki 11 n n-1 I j-+J i-4
[
e . . = Pr{W = i l s = j ] = 17 n n
J?,
1
( y ) e ( x y ) dxdy/R
. I
.
With t h i s d i s c r e t i z a t i o n of t h e i n p u t d i s t r i b u t i o n s , t h e Markov dependence of t h e sequence the t r a n s i t i o n matrices
A = I
a . . = Pr{W = i 13 n-1
+
W S W i s now s p e c i f i e d by 1' 1' 2' 2 ' " ' ( a . . ) and g = ( e . . ) where 17 17 and e . = P r { S n = i -+ W = j } . ij n
S
S = j} n
Method B : Probability Matrix Method Let t h e c o n t e n t space 0 5 101
K of
+
Emptiness, 1
1 2 {KI W
Fullness.
into the intervals
[O,K]
be d i s c r e t i z e d i n t o
(0,l)'... i
f
,...
(i - 1,i)
K K
Let t h e d i v i s i o n of t h e space 0 5 (Of$)
,
1
2
(3,3/2),
+
2
states,
(K-lfKlf
(0,s +
3)
...,s ~ ( s - & , s ~ ~ )
504 be as b e f o r e .
(1,2),...
(s,sA)
transition
of
by
W
{Cw(n-l),Wn-l)
0 ' , l',
-+
...,s'.
Consider now t h e
i.e.
{Cs(n) ,S,)
beginning o f Winter, Winter i n p u t }
{Contents a t t h e
{Contents a t t h e beginning o f
-+
Summer, Summer i n p u t 1 , w i t h t h e i n i t i a l s t a t e s {o,S'},
(0,l),
I n a d d i t i o n l e t u s denote t h e i n t e r v a l s
{O,O') 0,l')
{K,O}, ... {K,s}, {K+l,O')
{1,0},... {l,s},...
and w i t h f i n a l s t a t e s as above with
r
replaving
...
s.
...
{K+l,s'}
W e construct
t h e t r a n s i t i o n p r o b a b i l i t y matrix (t.p.m.) of t h e t r a n s i t i o n {Cw(n-l) lwn-l}-+{C,(n) , S n ) , a
(K+2) (s+l)X(K+2) ( r + l ) m a t r i x
W e a l s o construct the t.p.m.
p2
{CS(,)
bl.
of the t r a n s i t i o n
T h i s h a s t h e dimensions (K+2)(r+l)X(K+2)(s+l).
,Sn)+{Cw(n) ,Wn}.
For d e t a i l s o f c o n s t r u c t i o n see P h a t a r f o d and S r i k a n t a n ( 1 9 8 1 ) .
G2g1
The product Markov c h a i n
g i v e s t h e annual t.p.m.
o f t h e homogeneous
E C s ( n )t S n ) .
L e t u s denote t h e e q u i l i b r i u m d i s t r i b u t i o n of t h e p a i r
by
rij
i.e.
T.
ij Summer i n p u t = 1 ) . 71
(Cs,S)
= Pr.{Contents a t t h e beginning o f Summer = i ,
The e q u i l i b r i u m d i s t r i b u t i o n v e c t o r ,
...
= (ro0,Tro1,... Tr0r,7110
...
7rlr
i s o b t a i n e d by powering t h e m a t r i x
T
K+1 ,0'..
IT
K+l,r )
till i t s rows have i d e n t i c a l r values. Summing o v e r groups o f r + l v a l u e s , Vsi = 1 T . . g i v e s j=O 1 3 To o b t a i n t h e equilibri-um d i s t r i b u t i o n o f t h e u s Pr.{CS = i ) .
L2gl
c o n t e n t a t t h e beginning o f w i n t e r we e v a l u a t e
=
?r2,
a vector
S
of
(K+2)(s+l) v a l u e s .
The sum
Vwi
C p.. j=O 1 7
=
gives us
Pr.iCW=i).
Method C : Bottomless Dam Model (Mark 1 ) H e r e w e c o n s i d e r t h e d e p l e t i o n s of t h e dam, assuming i t t o be
bottomless.
Defining
t o the contents Dw(n)
= D
= o
s(n)
D
s ( n ) IDw(n)
as t h e d e p l e t i o n s corresponding
Cs ( n ) ,Cw ( n ) , w e have now
+
MI
-
Sn
if
Ds(n)
+ M 1 - Sn > 0
D
+ M
S(n)
1
- S n Q O
505 and a s i m i l a r r e l a t i o n connecting replacing
with
D W (n-1)
D
S (n)
with
M
2
The mathematical t h e o r y behind t h i s procedure i s
M1.
given i n Phatarfod (1981a).
W e give below t h e s t e p s r e q u i r e d t o
obtain a solution. 1. From t h e m a t r i c e s i E ( 0 ) = ( e . .8 ) , i . e . 13
each element of t h e
-
A
and
A(8)
i
A ( 0 ) = (a, . 0 ) , 13 i s a matrix formed from A by m u l t i p l y i n g
-
E,
ith row by
form t h e m a t r i c e s
Oi,
(0
1.2 the lower BOUND results in a slightly higher religeneral, for D ability, whereas the higher BOUND results in a higher reliability in the vicinity of a draft ratio o f 0.8 or 1.0. As the draft ratio becomes small, of the order o f 0.4, the reliability tends to 100% regardless o f the operating rule. For monthly flows the number of reservoir failures is very small with
519
s m a l l BOUND on t h e r e l e a s e s and i n c r e a s e s as BOUND i n c r e a s e s .
The
monthly r e l i a b i l i t y o f t h e releases i s very s e n s i t i v e t o t h e d r a f t r a t i o b u t i s i n f l u e n c e d i n v a r y i n g ways b y t h e BOUND v a l u e s o f t h e r e 1 ease r u l e . ACKNOWLEDGEMENTS T h i s m a t e r i a l i s based upon work s u p p o r t e d b y t h e N a t i o n a l S c i e n c e F o c n d a t i o n u n d e r G r a n t No. CME 7916819. P r o f e s s o r M.H.
The w r i t e r s w i s h t o t h a n k
Houck f o r h i s a s s i s t a n c e t h r o u g h o u t t h e r e s e a r c h .
REFERENCES Box, G.E.P. and Cox, D.R. A n a l y s i s o f T r a n s f o r m a t i o n s , J . Roy S t a t i s t . SOC. S e r . B . 26, 211-252, 1964. Hashino, M. and D e l l e u r , J.W., I n v e s t i g a t i o n o f t h e H u r s t C o e f f i c i e n t and O p t i m i z a t i o n o f ARMA Models f o r Annual R i v e r F l o w s , Tech. R e p t . CE-HSE-81-1, School o f C i v i l E n g i n e e r i n g , Purdue U n i v e r s i t y , 1981. Kararnouz, CI. and Houck, M . H . , Annual O p e r a t i n g R u l e s G e n e r a t e d b y D e t e r m i n i s t i c O p t i m i z a t i o n f o r a S i n g l e Mu1 t i p u r p o s e R e s e r v o i r , School o f C i v i l E n g i n e e r i n g , Purdue U n i v e r s i t y , Tech. Rept. CE-HSE-8' 11, 1981, a. Karamouz, 11. and Houck, M.H., " G e n e r a t i o n of M o n t h l y and Annual R e s e r v o i r O p e r a t i n g R u l e s " , Tech. Rept. CE-HSE-81-16, School o f C i v i E n g i n e e r i n g , Purdue U n i v e r s i t y , Dec. , 1981, b . K i t a g a w a , G. , On a S e a r c h P r o c e d u r e f o r t h e O p t i m a l AR-MA O r d e r , Ann. I n s t . , S t a t i s t . Math., V o l . 29, P a r t B , pp. 319-332, 1977. McLeod, A . I . and H i p e l , K.W., S i m u l a t i o n P r o c e d u r e s f o r B o x - J e n k i n s Models, Water Resources Research, V o l . 14, No. 5, 1978. M e j i a , J.M. dnd R o u s s e l l e , J. , D i s a g g r e g a t i o n Models i n H y d r o l o g y R e v i s i t e d , J o u r . Water Res. Res. 1 2 ( 2 ) , pp. 185-186, 1976. S a l a s , J.R., D e l l e u r , J.W., Y e v j e v i c h , V., and Lane, W.L., A p p l i e d M o d e l i n g o f H y d r o l o g i c Time S e r i e s , Water Resources P u b l i c a t i o n s , L i t t l e t o n , Col o r a d o , 1980. Y e v j e v i c h , V . , F l u c t u a t i o n s o f Wet and D r y Y e a r s , P a r t I , Research Data Assembly and M a t h e m a t i c a l M o d e l s , H y d r o l o g y Paper No. 1, C o l o r a d o S t a t e U n i v e r s i t y , 1963.
520
AN ANNUAL-blONTHLY STREAPIFLOW FIODEL FOR INCORPORATING PARAPETER UNCERTAINTY INTO RESERVOIR S IFILTLATION J I l R Y STEDINGER AND DANIEL PEI
Department of E n v i r o n m e n t a l E n g i n e e r i n g , C o r n e l l U n i v e r s i t y ,
I t h a c a , N . Y . 14853 ABSTRACT A m o n t h l y s t r e a m f l o w model i s d e v e l o p e d which c a n r e p r o d u c e t h e v a r i a n c e and y e a r - t o - y e a r c o r r e l a t i o n o f an a n n u a l streamf l o w s u r r o g a t e w i t h a modest number o f p a r a m e t e r s . The m o d e l ' s simple s t r u c t u r e f a c i l i t a t e s t h e i n c o r p o r a t i o n of streamflowmodel-parameter u n c e r t a i n t y i n t o s y n t h e t i c streamflow sequences The l a r g e impact o f p a r a m e t e r u n c e r t a i n t y on d e r i v e d r e s e r v o i r c a p a c i t y - r e l i a b i l i t y - d e m a n d r e l a t i o n s h i p s i s i l l u s t r a t e d by comparing t h e r e l a t i o n s h i p s o b t a i n e d i g n o r i n g p a r a m e t e r u n c e r t a i n t y w i t h t h o s e o b t a i n e d when s t r e a m f l o w - m o d e l p a r a m e t e r unc e r t a i n t y i s i n c o r p o r a t e d i n t o s y n t h e t i c flow s e q u e n c e s .
INTRODUCTION S y n t h e t i c s t r e a m f l o w s e q u e n c e s h a v e l o n g been viewed as a means o f i m p r o v i n g o u r a b i l i t y t o e s t i m a t e t h e l i k e l y p e r f o r mance of r e s e r v o i r s y s t e m s and t o f r e e s y s t e m p l a n n i n g s t u d i e s from a t o t a l r e l i a n c e on t h e p a r t i c u l a r flows which o c c u r r e d
et _ a l . , 1962). d u r i n g t h e p e r i o d o f r e c o r d (Flaass _
However, t h e
p a r a m e t e r s o f s , y n t h e t i c s t r e a m f l o w models a r e s u b j e c t t o sampl i n g o r unavoidable e r r o r s o f e s t i m a t i o n (Stedinger, 1980a, 1981; Loucks e_t a_l . , 1 9 8 1 , Appendix 3C) and t h e s e e r r o r s c a n
have a m a j o r impact on e s t i m a t e s o f r e s e r v o i r p e r f o r m a n c e V
(Klemes e t_a l . , 1981; Burges and L e t t e n m a i e r , 1981; Kleme?, 1979; S t e d i n g e r a n d T a y l o r , 1982) a n d t h e moments o f g e n e r a t e d monthly f l o w s [Kleme? and B u l u , 1979; S t e d i n g e r , 1 9 8 0 b ) .
This
p a p e r d e v e l o p s a new s t r e a m f l o w model which can b e u s e d t o g e n e r a t e s y n t h e t i c m o n t h l y s t r e a m f l o w s e q u e n c e s which i n c o r p o r a t e t h e u n a v o i d a b l e u n c e r t a i n t y i n s t r e a m f l o w model p a r a m e t e r s .
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
521 PIODE L STRUCTURE
A reasonable model of monthly and annual flows might reproduce (1) the mean, variance and other parameters of the marginal distribution of flows in each month, (2) the month-tomonth correlation of flows in consecutive months, (3) the variance of the total flow within water years, and (4) the year-toyear correlation of annual flows. Such a model should be reasonable for many applications. Like the Thomas-Fiering model (Thomas and Fiering, 1962), it reproduces the correlation of flows in consecutive months.
By reproducing the year-to-year
correlation of annual flows it should provide a reasonable description of the persistence of those flows; high order ARblA(p,q) models are seldom necessary (Hipel and PlcLeod, 1978; Wallis and O'Connell, 1973; Klemez _ et _ al., 1981; Burges and Lettenmaier, 1981). It is difficult to achieve these objectives in the form
articulated when monthly and annual flows have other than a normal distribution. If monthly flows q eter log normal distribution so that x
Yt
Yt
have a three-param-
= Rn(qyt
-
T ~ )has
a
normal distribution, streamflow models are most conveniently It is then difficult to insure formulated in terms of the x Yt * that the generated annual flows have the desired properties
(Loucks et al., 1981, p. 303). This difficulty can be circumvented if instead of modelling the annual flows, attention is I _
focused on an annual streamflow surrogate. In particular, consider the first-order approximation of the annual flows 12
1 2
are the expected value of the derivatives 2 dq /dxyt. For x = kn[qyt - T ~ ] ,wt = exp(pt + at/2). Yt Yt T o develop a simple model to generate x 's which yields Yt values of 2 with a particular variance, note that Y where the weights w
t
522
Hence, for a model which reproduces the variance of E[(x 2
p )
] for each x
-
Yt it is only necessary to reproduce the co-
Yt ’ t-1 variance between each x and C w (x - u ) to reproduce the s yt s=l s ys variance of Z as well. Y A reasonable model of monthly flows that can reproduce the t
mean, variance and month-to-month correlation of monthly flows as well as the variance and year-to-year correlation of the annual streamflow surrogate is xy,l = a1
+
9 xy-l,12
xy,2 -- a2 and for t
zy-1
+ vy,l
+
y1
+
y2 zy-l + 62 xy,l
(3)
+
(4)
vy,2
3
t-1 xy,t = at
+
6, XyJ-1
+
Yt ZY-l
+
6t
w s=l s
YS
+
v Y,t
(5)
where v
are independent zero-mean normal random variables. Y,t The coefficients of Z can be selected to reproduce the coY-1 variance between x and Zy-l, thus reproducing the covariance Yt of annual flow surrogates: 12 ““zy - v p z - Fiz)l = C wtE“xyt - Fit)(Zy-l - uz)l (6) Y-1 t=l blODE L- PARNIETER UNCERTA INTY
Given the finite and often short length of historical streamflow sequences, parameters of annual and monthly streamflow models can be estimated with only limited precision. The annual-monthly streamflow model provides a convenient structure for generating monthly streamflow sequences which incorporate the uncertainty in the parameters describing the joint distribution of monthly streamflows. Except for Beard‘s proposal
523 ( B e a r d , 19/35), e a r l i e r s t u d i e s h a v e c o n s i d e r e d o n l y t h e unc e r t a i n t y i n s t a t i s t i c s d e s c r i b i n g t h e d i s t r i b u t i o n o f annual
et _ a l . , 1977; IYood, 1978; flows (Vicens c t a l . , 1975; Valdes _ PlcLeod and H i p e l , 1975; S t e d i n g e r and T a y l o r , 1 9 8 2 ) . Methods for i n c o r p o r a t i o n o f model p a r a m e t e r u n c e r t a i n t y i n t o t h e s t r e a m f l o w g e n e r a t i o n p r o c e s s may b e d e v e l o p e d u s i n g Bayesian i n f e r e n c e t h e o r y for t h e normal r e g r e s s i o n model ( Z e l l n e r , 1971, C h a p t e r 3 ) .
Because t h e i n n o v a t i o n terms
i n E q u a t i o n s 3 t h r o u g h 5 a r e d i s t r i b u t e d i n d e p e n d e n t l y of Y,t one a n o t h e r , t h e p a r a m e t e r v e c t o r s = ( a t , Bt, y t , 6 t ) T f o r
v
e a c h month w i l l a l s o b e d i s t r i b u t e d i n d e p e n d e n t l y p r o v i d e d t h e i r p r i o r d i s t r i b u t i o n s a r e independent.
flence, a n a l y s i s o f t h e
‘ s f o r each t is e s s e n t i a l l y Yt e q u i v a l e n t t o a n a l y s i s o f i n d i v i d u a l normal r e g r e s s i o q models
model r e q u i r e d t o g e n e r a t e x
y=xe+y
(71
-
where f o r e a c h t > 3 , t h e i t h row o f X i s (1, x . 2. 1 , t - 1 ’ 1-1’ t-1 Z w s x 1. , s ) a n d V = [v1 , t ’ . . * j V n , t
IT.
s=l
In t h i s i n i t i a l work, a non-information o r J e f f r e y ’ s p r i o r d i s t r i b u t i o n i s u s e d t o i n d i c a t e t h a t l i t t l e i s known a b o u t and
0
a p a r t from t h e i n f o r m a t i o n p r o v i d e d by t h e h i s t o r i c a l flow
r e c o r d (Box a n d T i a o , 1 9 7 3 ) .
This i s a reasonable choice i f
a v a i l a b l e p r i o r i n f o r m a t i o n i s d o m i n a t e d by t h a t p r o v i d e d by t h e streamflow record. S t e d i n g e r and P e i (1981) summarize t h e B a y e s i a n a n a l y s i s o f
2 2 where k i s t h e number o f columns i n 5 , ( n - k ) s t / a t h a s a C h i 2 . s q u a r e d d i s t r i b u t i o n where a t i s t h e unknown v a r i c e o f t h e 2 v. ‘ s . For given at ’ 1, t
524
where
et a r e
t h e unknown model p a r a m e t e r s f o r month t .
S t r e a m f l o w s e q u e n c e s which r e f l e c t b o t h t h e n a t u r a l h y d r o l o g i c v a r i a b i l i t y o f s t r e a m f l o w s a n d what i s known a b o u t t h e m o d e l ' s p a r a m e t e r s were g e n e r a t e d i n two s t e p s .
F j r s t , N com-
p l e t e s e t s o f model p a r a m e t e r s were drawn from t h e i r p o s t e r i o r distribution.
Each c o m p l e t e s e t of model p a r a m e t e r s was u s e d t o
g e n e r a t e one s t r e a m f l o w s e q u e n c e .
These flow sequences r e f l e c t
b o t h what t h e t r u e v a l u e s o f t h e s t r e a m f l o w m o d e l ' s p a r a m e t e r s may be a n d t h e c h a r a c t e r i s t i c s of flow s e q u e n c e s t h a t t h e model would p r o d u c e w i t h t h o s e p a r a m e t e r v a l u e s (PlcLeod and H i p e l , 1978; D a v i s , 1 9 7 7 ) .
SIPILJLATIOP; RESULTS The a n n u a l - m o n t h l y m_odel was u s e d t o d e s c r i b e t h e c h a r a c t e r o f monthly f l o w s i n t h e Upper Delaware R i v e r B a s i n i n ?Jew York
State.
The m o n t h l y flows were m o d e l l e d by a 3 - p a r a m e t e r l o g
normal d i s t r i b u t i o n u s i n g t h e q u a n t i l e l o w e r bound e s t i m a t o r d e v e l o p e d by S t e d i n g e r (1980a). The 5 0 - y e a r h i s t o r i c a l f l o w r e c o r d p r o v i d e d t h e s t a t i s t i c s used t o g e n e r a t e streamflow sequences.
Flows were g e n e r a t e d
assuming t h e h i s t o r i c a l r e c o r d was of l e n g t h m = 25 or 50 y e a r s ; 2 T i n t h e f o r m e r c a s e , t h e v a l u e s o f s t , B and ( -X _X/n) were t h o s e -t o b t a i n e d w i t h t h e e n t i r e 5 0 - y e a r f l o w r e c o r d . fiere m may k viewed as an e f f e c t i v e r e c o r d l e n g t h i f i n f o r m a t i v e p r i o r d i s t r i b u t i o n s were u s e d t o d e r i v e t h e p o s t e r i o r d i s t r i b u t i o n s o f R -t
and o f . L
S t r e a m f l o w s e q u e n c e s were a l s o g e n e r a t e d which r e f l e c t o n l y t h e h y d r o l o g i c v a r i a b i l i t y o f flows t h a t o n e wot:ld e x p e c t i f t h e m o d e l ' s p a r a m e t e r s assumed t h e e s t i m a t c d v a l u e s . are considered.
I n t h e f i r s t , denoted m
ass gned t h e v a l u e
kt
with
=
m
n-1'
each
Two c a s e s
?-t was
525 2
x
Ot = (Y -t
- -t
A -t
1
T
x -t6 ) / ( n - l ) (Lt - -t
(11)
T h i s i s t h e v a l u e of o 2 needed t o r e p r o d u c e t h e o b s e r v e d s a m p l e t I n t h e second c a s e , denoted m = v a r i a n c e of t h e y t ' s . n-k' t h e B was a g a i n a s s i g n e d t h e v a l u e B w h i l e t h e r e s i d u a l v a r i -t -t2 a n c e u e q u a l l e d s2 ( E q u a t i o n 9 ) : Beard (1365) makes u s e o f t t t h i s unbiased e s t i m a t o r of t h e r e s i d u a l variance.
-
The s e q u e n t peak a l g o r i t h m was u s e d t o d e t e r m i n e S the req' r e s e r v o i r c a p a c i t y r e q u i r e d t o r e g u l a t e e a c h of 1000 g e n e r a t e d s y n t h e t i c f l o w s e q u e n c e s s o as t o p r o v i d e an a n n u a l d i v e r s i o n D o f 3 0 % , SO%, 70% a n d 90% of t h e h i s t o r i c a l mean a n n u a l f l o w , A 25-year p l a n n i n g p e r i o d
assuming t h e r e s e r v o i r s t a r t e d f u l l . i s assumed.
To make t h e r e s u l t s d i m e n s i o n l e s s , S
req
i s repor-
t e d a s a f r a c t i o n o f t h e h i s t o r i c a l mean a n n u a l flow. T a b l e 1 r e p o r t s t h e mean and s t a n d a r d d e v i a t i o n o f S ' s distribution. req and As m T h e r e i s l i t t l e d i f f e r e n c e between m = m n-k' n- 1 g o e s from m t o 50 a n d 25, t h e a v e r a g e v a l u e o f S inreq n- 1 creases s t e a d i l y . The s t a n d a r d d e v i a t i o n i n c r e a s e d d r a m a t -
-
i c a l l y ; except f o r D = 0.90, t h e increase f o r m e x c e s s of 75% o f t h e v a l u e f o r m
=
50 i s i n
S t c d i r l g e r and P e i n-1' (1981) show t h a t t h i s i n c r e a s e i s p r i m a r i l y due t o i n c r e a s e s i n =
w
t h e u p p e r q u a n t i l e s o f t h e d i s t r i b u t i o n o f Sreq. TABLE 1 .
Average and S t a n d a r d D e v i a t i o n of S €or L'arious re9 Demand L e v e l s . Standard Deviation
Average
m -
m =
Demand Level (%bliZF) ~
_n-1
_n-k
50 --
25 -
n-1
30% 50% 70% 90%
0.08 0.18 0.34 0.89
0.08 0.18 0.34 0.89
0.09 0.19 0.37 0.92
0.09 0.20 0.39 0.97
0.018 0.032 0.088 0.394
-.____-
co
W
m
__ n-k
50
-25
0.019 0.033 0.091 0.395
0.03 0.08 0.20 0.53
0.04 0.11 0.28 0.67
For s e v e r a l r e s e r v o i r c a p a c i t y - d e m a n d combinnti o n s , t h e sys-
tem's p e r f o r m a n c e was a l s o summarized by t h e e x p e c t e d v a l u e and s t a n d a r d d e v i a t i o n o f two s t a t i s t i c s :
(1) t h e o c c u r r e n c e - b a s e d
526
is t h e frequency o f f a i l u r e y e a r s during a t h e planning period; (2) t h e quantity-based f a i l u r e s t a t i s t i c f a i l u r e frequency F
is the t o t a l s h o r t f a l l o r d e f i c i t during t h e planning period a V d i v i d e d by t h e a n n u a l demand (Klemes e t_a l . , 1 9 8 1 ) .
V
The d i s t r i b u t i o n o f S
allows determination o f t h e probreq abi l i t y o f f a i l u r e - f r e e r e s e r v o i r o p e r a t i o n d u r i n g t h e e n t i r e p l a n n i n g p e r i o d and t h e f r e q u e n c y - m a g n i t u d e r e l a t i o n s h i p f o r t h e worst s h o r t f a l l t h a t o c c u r s .
These q u a n t i t i e s are of major
i m p o r t a n c e i n t h e s t u d y o f r e s e r v o i r s y s t e m s which f a i l i n f r e q u e n t l y , such as municipal water supply r e s e r v o i r s .
Other
s y s t e m s m e e t i n g a g r i c u l t u r e demands may b e d e s i g n e d t o f a i l , on a v e r a g e , one i n t e n y e a r s ( F
= 0.10). In t h i s instance, a a s a h y d r o l o g i c c r i t e r i o n f o r comparing s t o c h a s t i c
use of S req s t r e a m f l o w models may b e i n a p p r o p r i a t e ; u s e o f F
may a provide a b e t t e r assessment o f t h e frequency and c h a r a c t e r o f a
and V
system f a i l u r e s . S t e d i n g e r and f'ei (19911 r e p o r t t h e e x p e c t e d v a l u e s and standard deviations of F
and Va o b t a i n e d w i t h t h e 1000 25-year a s y n t h e t i c s t r e a m f l o w s e q u e n c e s f o r s t o r a g e c a p a c i t i e s S = 0.125, 0 . 2 5 , 0 . 5 0 , 1 . 0 0 and 2 . 0 0 times t h e h i s t o r i c mean a n n u a l f l o w (PNF) and w i t h demand l e v e l s D = 5 0 % , 70% and 90% o f t h e MAF.
T a b l e s 2 and 3 summarize t h e s e v e n c a s e s f o r which I:
(with m = a m ) f e l l between 0.1'0 a n d 20'0, t h e r e g i o n o f g r e a t e s t p r a c n- 1 t i c a l i n t e r e s t . In t h e first t h r e e cases i n Table 2 , F ina c r e a s e d by f a c t o r s o f 3 t o 5 f o r m = 25 o v e r t!ieir v a l u e s f o r In subsequent c a s e s with i n i t i a l l y h i g h e r f a i l u r e n-1' r a t e s , p a r a m e t e r u n c e r t a i n t y had l e s s d r a m a t i c , t h o u g h s t i 11
, = m
s u b s t a n t i a l , an i m p a c t . T a b l e 3 shows t h a t t h e i m p a c t o f p a r a m e t e r u n c e r t a i n t y on t h e moments o f V
a
i s much l a r g e r t h a n i t was on t h e moments o f
1:or S = 0 . 5 0 and D = 0 . 9 0 , t h e a v e r a g e v a l u e o f V between Fa. a the m = m a n d m = 25 c a s e s i n c r e a s e d by 18% o f t h e smaller n- 1 mean; t h e c o r r e s p o n d i n g i n c r e a s e i n I' ' s mean was s l i g h t l y a
527 o v e r 1%. V
a
i s an important index because it is a d i r e c t
measure o f t h e m a g n i t u d e o f water d e f i c i t s a n d t h u s p e r h a p s o f t h e h a r d s h i p t h a t m i g h t b e i n c u r r e d by t h o s e who e x p e c t e d t h a t
-
The l a r g e i n c r e a s e s ( g e n e r a l l y e x c e e d i n g 50% between
water.
m =
a n d m = 25 a n d a v e r a g i n g 520%) i n V Is mean a r e n- 1 a accompanied by e v e n l a r g e r i n c r e a s e s i n V ' s s t a n d a r d d e v i a ation. Average a n d S t a n d a r d d e v i a t i o n o f S i m u l a t e d Annual F a i l u r e F r e q u e n c y F f o r S e l e c t e d Cases. a
TABLE 2.
Standard Deviation
Average
m =
m = m
S -
D -
n- 1 __
0.25 2.00 0.50 1.00 0.25 0.50 0.13
0 .5 0.9 0.7 0.9 0.7 0.9 0.5
0.001 0.002 0.004 0.037 0.112 0.163 0.171
50 -
25 -
0.005 0.005 0.008 0.041 0.133 0.163 0.192
0.007 0.009 0.012 0.047 0.144 0.165 0.204
co
n- 1 __
so -
25 -
0.007 0.015 0.016 0.073 0.076 0.120 0.087
0.024 0.037 0.036 0.083 0.097 0.129 0.105
0.032 0.055 0.051 0.098 0.104 0.132 0.107
Average and S t a n d a r d D e v i a t i o n o f Va f o r S e l e c t e d Cases.
TABLE 3.
Average m = m
Standard Deviation
m =
___ co
S -
-
D
n- 1 __
50 -
-
25
n- 1 __
50 -
25 -
0.25 2.00 0.50 1.00 0.25 0.50 0.13
0.5 0.9 0.7 0.9 0.7 0.9 0.5
0.001 0.005 0.007 0.13 0.19 0.53 0.25
0.015 0.029 0.036 0.17 0.28 0.58 0.33
0.026 0.056 0.057 0.22 0.32 0.63 0.38
0.012 0.050 0.041 0.29 0.19 0.50 0.17
0.16 0.28 0.24 0.47 0.38 0.67 0.35
0.24 0.45 0.36 0.65 0.50 0.82 0.44
CONCLUSIONS A model was d e v e l o p e d which c a n r e p r o d u c e t h e mean, v a r i -
a n c e and month-to-month c o r r e l a t i o n o f m o n t h l y f l o w s and t h e mean, v a r i a n c e a n d y e a r - t o - y e a r c o r r e l a t i o n o f a n a n n u a l streamflow s u r r o g a t e .
The new model h a s a n a u t o r e g r e s s i v e s t r u c t u r e
528
which allows the application of Bayesian inference theory. 'This facilitates development of streamflow generation algorithms which incorporate the uncertainty in estimated streamflow-model-parameters. The new model was used to illustrate the impact of parameter uncertainty on derived reservoir capacity-demand-reliability relationships. With a 25 or 50-year historical flow record, streamflow-model-parameter uncertainty can have an appreciable impact on our best estimate of system reliability and especially on our assessment of possible failure magnitudes. AC KNOIVLF DGPlENTS This work was supported by NSF Grant CblE-8010889 REFE f<E:N CII S Beard, L . R . , 1965, Use of interrelated records to simulate streamflow, Jour. Hydr. Div. (ASCE), 91(HY5), 13-22. Box, G.E.P., and Tiao, G.C., 1973. Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading, Mass. Burges, S . J . , and Lettenmaier, D . P . , 1981. Reliability measures for water supply reservoirs and the significance of long-term persistence, International Symposium on RealTime Operation of Hydrosystems, Waterloo, Ontario, June. Davis, D.R., 1977. Comment on 'Bayesian generation of synthetic streamflows', by G . J . Vicens, I. Rodriguez-Iturbe, and J.C. Schaake, Jr., Water Resour. Res. 13(5), 853-854. flipel, K.W., and McLeod, A.I., 1978. Preservation of the rescaled adjusted range 2. Simulation studies using Box.Jenkins models, Water Resour. Res. 14(3), 509-516. Kleme3, V., 1979. The unreliability of reliability estimates of storage reservoir performance based on short streamflow records, in Re1 iability in Resources Planagement, edited by E.A. blcBean, K . W . flipel, and T . E . Unny, Water Resources Publications, Littleton, Colorado. Kleme?, V . , and Bulu, A., 1979. Limited confidence in confidence limits derived by operational stochastic hydrologic models, <Jour. of Hydrology 42, 9-22. Klemez, V., Srikanthan, R., and McMahon, T.A., 1981. Longmemory flow models in reservoir analysis: What is their practical value?, Water Resour. Res. 1 7 ( 3 ) , 737-751. Loucks, D . P . , Stedinger, J.R., and Haith, D.A., 1981. Water f?esource Systems Planning and Analysis, Prentice-Hall,
529
Englewood Cliffs, New Jersey. Maass, A., et al., 1962. Design of Water Resource Systems, Harvard University Press, cdmbridge, MA. McLeod, A. I., and Hipel, K . W . , 1978. Simulation procedures for Box-Jenkins models, Water Resour. Res. 14(5), 969-975. Stedinger, J.R., 1980a. Fitting log normal distributions to hydrologic data, Water Resour. Res. 16(3), 481-490. Stedinger, J.R., 1980b. Comment on 'Limited confidence in confidence limits derived by operational stochastic hydrologic models', Jour. of Hydrology 43, 377-380. Stedinger, J.R., 1981. Estimating correlations in multivariate streamflow models, Water Resour. Res. 17(1), 200-208. Stedinger, J.R., Taylor, M.R., 1982. Synthetic streamflow generation, Part 11: Parameter uncertainty, to appear in Water Resour. Res. 17( ) , 000-000. Stedinger, J.R., and Pei, D., 1981. An annual-monthly streamflow model for incorporating parameter uncertainty into reservoir simulation, Technical Report, December, Department of Environmental Engineering, Cornell University. Thomas, H.A., and Fiering, E1.B , 1962. Mathematical synthesis of streamflow sequences for the analysis of river basins by simulation, in Design of Water Resource Systems by A . Flaass _ et _ al., Harvard University Press, Cambridge, Mas. Valdes, J.R., Rodriguez-Iturbe, I., and Vicens, G.J., 1977. Bayesian generation of synthetic streamflows 2. The multivariate case, Water Resour. Res. 13(2), 291-295. Valencia, D., and Schaake, J.C., Jr., 1973. Disaggregation process in stochastic hydrology, Water Resour. Res. 9(3), 580-585. Vicens, G.J., Rodriguez-Iturbe, I., and Schaake, J.C., Jr., 1975. Bayesian generation of synthetic streamflows, Water Resour. Res. 11(6), 827-838. Wallis, J.R., and O'Connell, P.E., 1973. Firm reservoir yieldhow reliable are historic hydrologic records, Hydrol. Sci. Bull. 18(3), 347-365. Wood, E.F., 1978. Analyzing hydrologic uncertainty and its impact upon decision making in water resources, Advances in Water Resources 1(5), 299-305. Zellner, A., 1971. An Introduction to Bayesian Inference in Econometrics, J. Wiley 6 Sons, Inc., New York.
530
STOCHASTIC FLOOD PREDICTORS: EXPERIENCE I N A SMALL BASIN P. BOLZERN, G. FRONZA AND G. GUARISO Istituto di Elettrotecnica ed Elettronica Centro Teoria dei Sistemi - Politecnico di Milano, P.za Leonard0 da Vinci n. 32 - 20133 Milan - Italy
ABSTRACT - The p a p e r d e s c r i b e s t h e a p p l i c a t i o n o f t h r e e s t o c h a s t i c p r e d i c t o r s o f r i v e r f l o w - r a t e t o a s m a l l b a s i n ( u p p e r bas i n o f Temo, i n S a r d i n i a "180
2 km ) . The f i r s t p r e d i c t o r (ARX,
A u t o R e g r e s s i v e w i t h exogenous i n p u t ) r e q u i r e s q u a n t i t a t i v e r a i n f a l l measurement, t h e second one (ARQX, A u t o R e g r e s s i v e w i t h Qual i t a t i v e exogenous i n p u t ) needs o n l y a g r o s s i n f o r m a t i o n a b o u t r a i n f a l l , t h e t h i r d one (AR, A u t o R e g r e s s i v e ) has no r a i n f a l l i n p u t . The p e r f o r m a n c e o f t h e t h r e e p r e d i c t o r s t u r n s o u t s a t i s f a c t o r y up t o 3-4 h o u r s , p a r t i c u l a r l y b y ARX. However, t h e gap o f q u a l i t y between t h e ARX and t h e o t h e r t w o p r e d i c t o r s i s c o u n t e r b a l a n c e d by t h e c o s t o f t h e r a i n f a l l t e l e m e t e r i n g s y s t e m r e q u i r e d b y ARX.
1. INTRODUCTION A c e r t a i n number o f m a t h e m a t i c a l models have been p r o p o s e d
i n t h e technical l i t e r a t u r e , i n order t o supply r e a l - t i m e f o r e cast o f r i v e r flow-rates, 3f
p a r t i c u l a r l y i n f l o o d s i t u a t i o n s . Most
t h e s e models, such as lumped p a r a m e t e r d e t e r m i n i s t i c ( s e e
f o r i n s t a n c e Dooge, 1977) o r s t o c h a s t i c , a r e s i m p l i f i e d r e p r e -
Reprinted from Time Series Methods i n Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
@
531
s e n t a t i o n s of the r a i n f a l l - r u n o f f phenomenon. As a m a t t e r of f a c t applying t h e c l a s s i c a l p a r t i a l d i f f e r e n t i a l e q u a t i o n s u s u a l l y r a i s e s a number of conspicuous o p e r a t i o n a l d i f f i c u l t i e s , both from d a t a and computational viewpoint. The most common type of s t o c h a s t i c model ( s e e f o r i n s t a n c e C h i u (1978) , Bolzern e t a1 . ( 1 9 8 0 ) ) , proposed f o r d e s c r i b i n g the
r a i nfa 1 1 -f 1 ow-ra t e mec hani sin , i s ARMAX , namely, Au toRegress i ve "loving Average with exogenous i n p u t ( r a i n f a l l ) . I n an ARMAX, the f l o w - r a t e a t each time s t e p i s expressed a s a l i n e a r combination 3f previous f l o w - r a t e s (AR-part) , p l u s a l i n e a r combination o f previous r a i n f a l l volumes (X-part) plus moving average n o i s e . ilatural l y , t h e a c t u a l implementation of an ARMAX f l o w - r a t e 2redictor i n a basin requires t o set u p a r a i n f a l l telemetering system. T h u s , when c o n s i d e r i n g t h e use of ARblAX p r e d i c t i o n , one must compare t e l e m e t e r i n g c o s t s w i t h f o r e c a s t " b e n e f i t s " , a t l e a s t i n terms of f o r e c a s t improvement with r e s p e c t t o more t r i v i a l p r e d i c t o r s , which do n o t use q u a n t i t a t i v e information on rainfall. Such " b e n e f i t s " a r e pointed o u t i n the p r e s e n t p a p e r , which 2 . d e a l s with the upper b a s i n of r i v e r Temo ('L 180 kin ) i n the n o r thwest of S a r d i n i a . S p e c i f i c a l l y , i n the next s e c t i o n an ARX pred i c t o r i s i l l u s t r a t e d , t o g e t h e r with a pure AR and an ARQX ( A u toRegressive with Qua1 i t a t i v e exogenous i n p d t , namely q u a l i t a t i v e information about r a i n f a l l ) . The performance of t h e s e t h r e e f o r e c a s t a l g o r i t h m s , a s well a s the one by t h e p e r s i s t e n t p r e d i c t o r ( P P ) i s d e s c r i b e d i n the l a s t s e c t on, f o r d i f f e r e n t f o r e c a s t ho r i z o n s . For i n s t a n c e , i t turns o u t t h a t .80 of c o r r e l a t i o n b e t ween f o r e c a s t and r e a l i t y i n f l o o d s i t u a t i o n s i s obtained with a p r e d i c t i o n horizon of 4h, 3h, 2.5h 2 h r e s p e c t i v e l y by A R X , A R Q X , AR and P P . 2 . THE PREDICTORS The basin under c o n s i d e r a t i o n i s shown i n F i g . 1 . Hourly a-
verage flow r a t e s were a v a i l a b l e f o r t h e period 1950-1970.
532
Sea
-
0 F i g . 1 - R i v e r Temo B a s i n work)
5
(1s e c t i o n
10
15 2 0 km
considered i n t h e present
Such h i s t o r i c a l r e c o r d i s c h a r a c t e r i z e d b y r a t h e r p r o l o n g e d " f l o o d s " , w i t h peaks o f t e n r e a c h i n g more t h a n t e n t i m e s t h e aver a g e f 1ow-ra t e
.
I n order o f i n c r e a s i n g complexity, t h e f o l l o w i n g r e a l - t i m e p r e d i c t o r s o f f u t u r e f l o w - r a t e have been c o n s i d e r e d ( A t = l h ) . 2.1
The AR p r e d i c t o r The p u r e a u t o r e g r e s s i v e p r e d i c t o r o f o r d e r p ' i s s i m p l y
q(k)
=
average f l o w - r a t e i n the interval
i n t h e s e c t i o n p o i n t e d o u t i n Fig.1, [(k-1) A t , kAt);
533 q(k+flk) PI
=
f o r e c a s t o f q ( k + f ) made a t t i m e k A t ;
{ai
=
p r e d i c t o r parameters.
1
i=l By p r e d i c t o r ( 1 ) f o r e c a s t i s s u p p l i e d o n l y on t h e b a s i s o f r e -
c e n t f l o w - r a t e measurements,
no i n f o r m a t i o n a b o u t r a i n f a l l on t h e
basin i s required. 2.2 The ARQX p r e d i c t o r The AROX p r e d i c t o r makes use o f b o t h r e c e n t f l o w - r a t e mea surements and g r o s s i n f o r m a t i o n a b o u t r a i n f a l l , such a s d i s t i n c t i o n between " r a i n " and " n o r a i n " s i t u a t i o n s . The ARQX p r e d i c t o r o f o r d e r p " i s g i v e n by
P
k+Z/k)
=
61 [-c ( k ) ]
1 B1. r- c ( k - i + 2 ) ]
q(k+l/k)+
q(k-it2)
(2b)
i= 2
Pi
t
[c(k-it3)]
q(k-i+3)
i=3
where c(k)
=
r a i n f a l l "class" i n the i n t e r v a l [(k-l)At, sely a binary variable: c(k) u p p e r Temo b a s i n c ( k )
=
=
kAt), preci-
0 means no r a i n i n t h e
1 means r a i n ( w h a t e v e r i t s v a l u e ) .
N a t u r a l l y , t h e d e f i n i t i o n can be g e n e r a l i z e d t o " c ( k)=O means r a i n b e l o w a c e r t a i n t h r e s h o l d " w h i l e " c ( k ) = l means r a i n above t h a t t h r e s h o l d " ( b u t i n t h e c a s e o f r i v e r Temo, t h e f o r m e r d e f i n i t i o n has p r o v e d more a p p r o priate) ;
{@i
P
I'
= p r e d i c t o r parameters. i=l
534
Eqs. ( 2 ) r e p r e s e n t an a u t o r e g r e s s i v e p r e d i c t o r , where p a s t flowr a t e s have d i f f e r e n t weights, according t o t h e circumstance t h a t they correspond t o a r a i n y o r a non-rainy hour. In p a r t i c u l a r , no information about c ( k t l ) , c ( k t 2 ) ... i s assumed t o be a v a i l a b l e a t time k i l t , t h e r e f o r e i n eqs. ( 2 b ) , ( 2 c ) ... a l l f u t u r e r a i n f a l l c l a s s e s a r e s e t equal t o c ( k ) . 2.3 The ARX p r e d i c t o r The ARX p r e d i c t o r s u p p l i e s f o r e c a s t by using both r e c e n t flow-rate d a t a and r e c e n t q u a n t i t a t i v e r a i n f a l l e v a l u a t i o n on t h e upper Temo b a s i n . I t i s a s l i g h t v a r i a t i o n o f the f o r e c a s t a l gorithm i l l u s t r a t e d by Bolzern e t a l . ( 1 9 8 0 ) . This p r e d i c t o r i s given by
r
r'
I I1
q(ktlIk)=
f g i q ( k - i t l ) t j =1l $j u1 ( k - j t 1 ) t j =11 qj u 2 ( k - j t l ) i=l I a1
q( k t 2 / k ) = glq( ktl 1 k ) t
f oiq( k - i t 2 ) t i =2
r'
1 $ .ul ( k - j t 2 ) t
j=2
J
r ''
...... where PI1 I
pl"
;L1
, {qj} r. ' J=1 , r 1, r " = p r e d i c t o r o r d e r s ;
{gi} i=l ,
{$.)
J
=
p r e d i c t o r parameters;
k-M Y
u1( k )
=
1
t=k-l
a ( t ) 6 as
k-M 5
1
t=k-1
a ( t ) > as
(3a)
535
k-M
, k -M
a(t)
:4
=
r a i n f a l l volume ( m 3s - 1 ) on the basin during the interval [( t-1 )At, t A t ) ;
=
threshold volume;
=
integer representing the "memory of s o i l s t a t u s " . If in the i1 steps before ( k - ] ) A t the overall r a i n f a l l volume has not exceeded the threshold a s ( " s i t u a t i o n of d r y s o i l " ) , the r a i n f a l l a ( k ) in the k - t h time step i s introduced i n t o predictor ( 3 ) as u l . I n the opposite s i t u a t i o n ("wet s o i l " ) , the r a i n f a l l a ( k ) i s introduced as u 2 .
3 . RESULTS
The predictors mentioned in the previous section have been tested on r i v e r Temo f o r the s i x major floods of the period 1950-1970. Orders p ' , r ' , r " , p " , p'", threshold as and index t.1 have been estimated by judgement or t r i a l (see Table 1 ) . TABLE 1
Parameters estimated by judgement or t r i a l
2
2
2
3
3
2
300
Parameters a i , $ j , q j , a i , ai have been reestimated a t each t i me step by the recursive extended l e a s t squares procedure ( P a n u ska, 1969; Young, 1968). Note t h a t flow-rate prediction has been
536
supplied simu taneously with the updating of parameter estimates and n o t a f t e r the convergence of estimates, (as done by Bolzern e t a l . (1980) . I n t h i s sense, predictors ( 1 ) - ( 3 ) a r e "adaptive". The forecast performance, p a r t i c u l a r l y by ARX, has turned o u t s a t i s f a c t o r y . In d e t a i l , Fig. 2 describes the overall perfor rnances (=correlation p between predictions and observations) by A R X , ARQX, AR f o r various forecast horizons f and compares them with persistence prediction PP. The figure accounts for f 6 6: f o r f = 7 a l l q u a l i t i e s f a l l down t o t o t a l l y unacceptable values. Note t h a t for f 6 2 there i s no s i g n i f i c a n t advantage by using r a i n f a l l measurements, while f o r f > 2 such advantage can be ap5
P
1.0 r
0.8
8
0.6 0.4.
Fig. 2
-
ARX ARQX AR PP
Overall forecast performance of the various predict o r s versus prediction horizon f .
537
p r e c i a t e d . For i n s t a n c e , i f p = 0.80 i s considered a s t h e mini-
mum a c c e p t a b l e performance, a t t h a t l e v e l of q u a l i t y ARX horizon i s 30% longer than ARQX, 50% longer than AR and ZOO?; longer than PP. As s t a t e d i n t h e i n t r o d u c t i o n , the value of such e x t r a hour ( o r more) of f o r e c a s t given by ARX must be compared with the c o s t of a r a i n f a l l t e l e m e t e r i n g system.
REFERENCES Bolzern, P.,
F e r r a r i o , M.,
Fronza, G .
, 1980. Adaptive rea:-tirne
f o r e c a s t of r i v e r f l o w - r a t e s from r a i n f a l l d a t a . J . Hydro1 . , 47: 251-267. Chiu, C . L .
( E d i t o r ) , 1978. A p p l i c a t i o n s of Kalman f i l t e r t o hy-
drology, h y d r a u l i c s and water r e s o u r c e s , U n i v e r s i t y of Pi t t s burgh Press, Pittsburgh.
Dooge, J . C . I . ,
1977. Problems and methods of r a i n f a l l - runoff
modelling. In T . A . C i r i a n i .U. Haione, J.R. Wallis ( E d i t o r s ) , Mathematical Plodels f o r Surface Hydrology, John Wiley, New York, 423 p p . Panuska, V . ,
1969. A n a d a p t i v e r e c u r s i v e l e a s t squares i d e n t i -
f i c a t i o n a l g o r i t h m , Proc. 8th I E E E Symp. on Adapt, P r o c e s s e s , Penns. S t a t e Univ. Young, P . C . ,
1968. The use of l i n e a r r e g r e s s i o n a n d r e l a t e d pro-
cedures f o r t h e i d e n t i f i c a t i o n of dynamic p r o c e s s e s , Proc. 7th IEEE Symp. on Adapt. P r o c e s s e s , Los Angeles.
538
TIME SERIES MULTIPLE LINEAR REGRESSION MODELS FOR EVAPORATION FROM A FREE WATER SURFACE JAMES G. SECKLER Associate Professor, Civil Engineering Bradley University, Peoria, Illinois (USA) INTRODUCTION AND SUMMARY The objective of this study was to develop, using time series multiple linear regression techniques, mass transfer models for evaporation from a free water surface.
The basic form of the mass trans-
fer evaporation model may be expressed as Aep Et - CNtUz,t z,t
where E =
U
z,t Ae 2,
t
=
evaporation at time t, Nt
a mass transfer coefficient,
=
wind speed at elevation z above water surface at time t ,
=
vapor pressure difference at elevation z at time t, and
t
C = constant.
The subscript t indicates a time dependent process
whose observations are taken at equal time intervals. The terms E u
t'
, Ae z,t and N t may be considered stochastic processes. After z,t
linearization the above may be analyzed using time series multiple linear regression to obtain estimates for C, Nt, m and p based on exand Ae . z,t z,t Using data collected at Lake Hefner, Oklahoma (Harbeck, et al,
perimental data taken for E
u
t'
1954) and Fort Collins, Colorado (Rohwer, 1931) the following models were obtained: Lake Hefner: Et
0.424~lO-~(k'/ln(z/z~))0.8OU
=
Fort Collins: Et
=
0.00445~
0.50Ae Z' t
ZYt
where k'
=
von Karmen's constant, and z
height in cm.
0
E
t
0.80Ae z,t z,t
is in cm/day, u
ZYt
=
equivalent water roughness
in cm/sec, and Ae
z,t
is in gm/cm
Using the above models the average correlation coefficients Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
@
2
.
539 b e t w e e n o b s e r v e d a n d p r e d i c t e d e v a p o r a t i o n f o r t h e Lake H e f n e r and F o r t C o l l i n s d a t a w e r e c a l c u l a t e d t o b e 0 . 7 7 and 0 . 9 5 r e s p e c t i v e l y . The Lake H e f n e r r e s u l t s compare f a v o r a b l y w i t h r e s u l t s o b t a i n e d i n a p r e v i o u s s t u d y (Harbeck, e t a l , 1 9 5 4 ) .
The Lake H e f n e r model was a l s o
a p p l i e d t o d a t a from t h e E l e p h a n t B u t t e R e s e r v o i r i n New Mexico (Tschantz, 1965).
The c o r r e l a t i o n c o e f f i c i e n t b e t w e e n o b s e r v e d a n d
p r e d i c t e d evaporation w a s c a l c u l a t e d t o be 0.95.
DEVELOPMENT OF MATHEMATICAL MODEL Dimensional Anlaysis The p r i m a r y f a c t o r s a f f e c t i n g e v a p o r a t i o n f r o m a f r e e water s u r f a c e are g i v e n i n Table 1.
Using dimensional a n a l y s i s t h e following
equation w a s developed E = C N u mAe P z z where C i s a c o n s t a n t a n d N i s a m a s s t r , a n s f e r c o e f f i c i e n t g i v e n by 1-a C m
K
m d 1-a/2 ? A
N =
u
[ed k d
'
(2)
i s a wind s p e e d a t e l e v a t i o n z and A e i s a v a p o r p r e s s u r e d i f f e r e n c e
at elevation z.
The s h a p e f a c t o r , Sh, a n d wind d i r e c t i o n , D i , w e r e
e l i m i n a t e d b a s e d upon t h e f o l l o w i n g c o n s i d e r a t i o n s . The s h a p e f a c t o r i s i n t r o d u c e d p r i m a r i l y t o a c c o u n t f o r c h a n g e s i n l a k e o r r e s e r v o i r d e p t h s of g r e a t m a g n i t u d e . t i m e (e.g.,
F o r s h o r t p e r i o d s of
a d a y ) t h e amount o f e v a p o r a t i o n i s o f t h e o r d e r of o n e
i n c h r e s u l t i n g i n a n i n s i g n i f i c a n t c h a n g e i n t h e s h a p e o f a water body. The wind d i r e c t i o n , D i , i s d r o p p e d b a s e d on t h e p r e m i s e i t h a s no a f f e c t on t h e v a p o r f l u x from t h e s u r f a c e of t h e w a t e r body b u t only e f f e c t s t h e d i s t r i b u t i o n of vapor i n t h e atmosphere above t h e
water s u r f a c e . The wind v e l o c i t y u
i s i n t r o d u c e d i n t o e q u a t i o n (2) u s i n g t h e
l o g a r i t h m i c wind l a w g i v e n by U
_ z= -
uj;
l In z/zo k'
(3)
540 TABLE 1 Summary of Variables Affecting Evaporation From a Water Surface Variable
Dimension Force-Length-Time
Symbol
Evaporation per unit area per time
E
LT-l
Area of water surface
A
L2 FL-
Ae=e -e o a
Difference in vapor pressure Equivalent water roughness
Z
L 0
L2f1
V
Kinematic Viscosity of air Coefficient of molecular diffusion
L2f1
Km
Density of air
0
FT2L-4
Shear at water surface
r.
FL-~
Shape factor of water body
Sh
Dimensionless
Di
Dimensionless
Wind direction ~~~
where u*
=
L
0
~
friction velocity = ~ ~ / p .
Recognition of evaporation, wind speed, vapor pressure difference and the mass transfer coefficient as stochastic processes the evaporation model may be written as
(4)
neP Et - CNtUz,t z,t
where the subscript t indicates a time dependent process whose observations are taken at equal time intervals and the subscript z indicates the height of measurement of the mass transfer parameters. Time Series Multiple Linear Regression Model Equation ( 4 ) was logarithmically transformed to obtain
Yt
=
(5)
K + blXl,t + b2X2,t + Et
where Y t
=
log E t , R
=
log C, bl
=
m,Xl,t = l o g u Z,t’
541 b2
=
p, X
= log (Ae
2,t
2, t
) and
E
t
=
log N
t’
Equation (5) may be analyzed using time series multiple linear regression techniques to obtain estimates for K, bl, b
t
E
.
X and X 2,t t’ l,t is assumed to be a stationary random series independent of X
analysis Y E
and
In the t are assumed to be ergodic random processes.
2
1,t is the log
and
This last assumption may be false because (a) E x2,t’ t transformation of N which is related to u and Ae * ( b ) the pret z,t Z,t’ and Ae . and (c) nonsence of errors in observation of u z,t Z,t’ and X . The regression additivity of the transformed variables X l,t 2,t constant K is obtained by
where the bars indicate the means of the respective series. dual series &
t
= Yt
E
t’ - K - b X
The resi-
is determined by (7)
1 l,t - b2X2,t
The use of time series multiple linear regression techniques to determine the best estimates of the regression coefficients has one advantage over traditional regression methods. used to obtain estimates for b
The spectral methods
and b
take into account the variation 2 of the signal to noise ratio with frequency. For the evaporation
1
model there are two signals; i.e., the covariance spectrums between wind speed and evaporation and vapor pressure difference and evaporation.
The noise is represented by the variance spectrum of the resi-
.
The estimates for b and b are weighted for fret 1 2 quency bands which possess high signal to noise ratios. In other dual series,
I
words, the estimates for b
1
and b2 are obtained by giving more weight
to the frequency bands where a large degree of information is provided by wind speed or vapor pressure difference for the prediction of eva-
poration.
DATA ASSEMBLY Mass transfer evaporation data from two investigations were selected for this study.
The first study was performed by the United
States Geological Study (USGS) at Lake Hefner, Oklahoma (Harbeck, et
542
al, 1 9 5 4 ) . The data measured included air temperature, humidity and wind speed at two, four, eight and sixteen meters above the lake surface, and lake surface temperature.
Evaporation was determined using
the water budget method. The second study was conducted by the Colorado Agricultural Experiment Station for the United States Department of Agriculture (USDA) (Rohwer, 1 9 3 1 ) .
Data measured included (reported as daily averages)
evaporation, water surface temperature, air temperature ahd wind speed at two meters elevation. The data from both investigations was processed to give daily evaporation in cmlday, wind speed in cm/sec and vapor pressure differn
ence in gm/cmL. Using this data best estimates for the regression and the regression constant K in equation (5) 2 were obtained using time series multi-linear regression techniques. coefficients b
1
and b
PRESENTATION AND DISCUSSION OF RESULTS Presented in Table 2 are the results of the regression analyses and tests of validity for the USGS and USDA data. TABLE 2 Summary of Time Series Multilpe Linear Regression Results
USGS, Lake Hefner Validity
Elevation meters Model 2
Results
10%
Et = 0.0038 N u
0.554Ae1.122
t t
t
4
Et
0.912
Invalid
Invalid
6
E = 0.0017 N u 0.545Ae1.431 t t t t
Invalid
Invalid
8
Et
0.0016 N u o*620Ae 1 * 3 3 4
Invalid
=
=
0.0009 N u
1.293
t t
t t
t
I
Invalid
USDA Reservoir Validity Year 1927
Model 0.557Ae1.310 Et = 0.0050 N u
1928
Et = 0.0091 N u
t t
t t
t
Invalid
Results
1
Valid
543 The t e s t s o f v a l i d i t y i n d i c a t e t h a t f o r t h e most p a r t t h e models
are i n v a l i d s t a t i s t i c a l l y .
M a t h e m a t i c a l l y t h e t e s t s of v a l i d i t y a r e
b a s e d o n t h e s t a t i s t i c a l i n d e p e n d e n c e of t h e r e s i d u a l s e r i e s , from X
and X
E
t’ t h e i n d e p e n d e n t t r a n s f o r m e d mass t r a n s f e r v a r i -
2,t’ The i n v a l i d r e s u l t s were n o t e n t i r e l y u n e x p e c t e d s i n c e
1, t
ables.
E
t i s a f u n c t i o n of a t m o s p h e r i c s t a b i l i t y , k i n e m a t i c v i s c o s i t y of t h e a i r ,
a r e a o f t h e water body, wind s p e e d v a r i a t i o n w i t h h e i g h t a b o v e t h e
water s u r f a c e , w a t e r s u r f a c e r o u g h n e s s a n d m o l e c u l a r d i f f u s i o n .
There-
and X and s h o u l d n o t l,t 2,t b e e x p e c t e d t o b e s t a t i s t i c a l l y i n d e p e n d e n t . M o r e o v e r , Hamon a n d fore
E
t
i s n o t p h y s i c a l l y i n d e p e n d e n t of X
Hannon (1963) p o i n t o u t t h a t v e r y r a r e l y w i l l t h e t e s t s of v a l i d i t y i n d i c a t e independence.
They s t a t e t h a t t h e t e s t s o f v a l i d i t y f o r t h e
model w i l l d e t e c t v e r y small d i s c r e p a n c i e s b e t w e e n d a t a and h y p o t h e s i s . T h i s i m p l i e s t h a t a s i g n i f i c a n t s t a t i s t i c a l r e s u l t need n o t correspond t o an operationally l a r g e discrepancy.
The m o J e l s f o r e v a p o r a t i o n
w h i c h a r e v a l i d s t a t i s t i c a l l y a r e , f o r t h e most p a r t , models w h e r e t h e mass t r a n s f e r v a r i a b l e s a r e measured a t t h e two meter l e v e l o r l e s s .
T h i s i s i n t h e r e g i o n where t h e e f f e c t s of s t a b i l i t y a r e s u p p r e s s e d . I t a p p e a r s from T a b l e 2 t h e f o l l o w i n g r e l a t i o n s h i p s b e t w e e n t h e
e x p o n e n t s f o r wind s p e e d ( b ) and v a p o r p r e s s u r e d i f f e r e n c e ( b ) 1 2 e x i s t : b l # 1, b2 # 1 and b l # b 2 . S t u d e n t ’ s t t e s t s w e r e u s e d t o test the hypotheses ( a t the ten p e r c e n t s i g n i f i c a n c e l e v e l )
b l = 1, b 2 = 1 and b
1
= b
2
w i t h t h e r e s u l t s b e i n g summarized i n
T a b l e 3 below. F o r t h e Lake H e f n e r d a t a i t a p p e a r s ( a ) b
is significantly dif1 f e r e n t from u n i t y , and ( b ) b 2 i s n o t s i g n i f i c a n t l y d i f f e r e n t f r o m
unity a t the ten per cent significance level.
F o r b o t h t h e Lake Hef-
n e r and t h e USDA r e s e r v o i r d a t a t h e S t u d e n t t t e s t s i n d i c a t e t h a t , i n general, b
i s s i g n i f i c a n t l y d i f f e r e n t from b F o r t h e two p e r 2 1’ i o d s of a n a l y s i s u s e d f o r t h e USDA r e s e r v o i r d a t a , t h e S t u d e n t t
t e s t s i n d i c a t e t h a t b o t h b l and b 2 a r e s i g n i f i c a n t l y d i f f e r e n t from unity.
544 TABLE 3 Summary of Student t Tests Null Hypotheses: bl = 1, b2 = 1, bl
=
b2
USGS, Lake Hefner Elevation, meters
0.554
1.122
1 = 1 No
4
1.293
0.912
8
0.545
16
0.621
2
bl b2 -
b
Null Hypothesis Acceptance? b2 = 1 bl = b2 Yes
No
Yes
Yes
No
1.431
No
No
No
1.334
No
Yes
No
USDA, Reservoir Year
bl b2 --
1927
0.557
1.310
No
No
No
1928
0.464
1.138
No
No
No
bl
=
b2 = 1
1
bl
=
b2
Development of Evaporation Models A conclusion of the USGS studies (Harbeck, et al, 1954) was the
eight meter level adequately represents the upper limit of the vapor boundary layer.
For this reason, the values of b l and b2 for the two,
four and eight meter results were averaged to give b
1
=
0.797 and
b2
=
1.188. For the USDA data average values of b
b2
=
1.224 for the two years of study were obtained.
bl
=
0.797 for the exponent of wind speed for the Lake Hefner data
agrees very closely with a value of b,
=
1
=
0.510 and The value of
0.78 as reported by Sutton
(1949, 1953). for both Lake Hefner and the USDA reser2 voir is close to unity, the theoretical value. However, the average The average value for b
values for b 2 appear to be approximately 1.20.
These results indicat-
ed Student t tests should be performed to test the following null hypotheses: (1) b2 data; (2) b l
=
Reservoir data.
=
1.20 for both Lake Hefner and U S D A reservoir
0.80 for Lake Hefner data; and (c) b l
=
0.50 for USDA
The above hypotheses were tested at the ten per cent
significance level.
The results are summarized in Table 4 .
545
TABLE 4 Student's t Test Results USGS, Lake Hefner Elevation, meters
Null Hypothesis Accepted? b l = 0.80 b2 = 1.20
bl b2 --
2
0.554
1.122
Yes
Yes
4
1.293
0.912
No
No
8
0.545
1.431
Yes
No
USDA Reservoir bl
0.50
Year
bl b2 --
1927
0.557
1.310
Yes
Yes
1928
0.464
1.138
Yes
Yes
=
b2 = 1.20
The results presented in Tables 3 and 4 suggest the following evaporation model for Lake Hefner = CN 0.80Ae Et t Uz,t 2,t' The mass transfer coefficient, Nt, varies with time and with height above water surface. N
is proportional to (k'/ln(z/z ))m
t
0
where k' is
V. Karman's constant, z is height above water surface, z is a water roughness parameter, and m average values for Nt -
Nt = C'(k'/ln(z/zo)) and setting z
0
=
(N,)
0.80
=
0.80 for the Lake .Hefner data.
Using
and C, letting
,
(9)
0.6 cm (Harbeck et al, 1954) the following model for
the Lake Hefner study was obtained 0.80 0.80Ae u z,t 2,t' Et = 0.424 x 10-2(k'/ln(z/zo)) For the USDA reservoir Table 4 suggests b2 should be 1.20, which is contrary to Dalton's law.
To resolve this discrepancy a prelirri-
nary correlation analysis between observed and predicted evaporation was performed using the following - evaporation relationships: O.5OAe1.20. Et = CN u and t 2,t z,t ' 0.50Ae E =CNu t t z,t 2,t where C was obtained as the antilog of K.
For b
= 1.20 the correla2 tion coefficients between observed and predicted evaporation were
546 calculated to be 0.959 for the 1927 data and 0.934 for the 1928 data. For b2
=
1.00 the calculated correlation coefficients were found to be
0.965 for the 1927 data, and 0.934 for the 1928 data.
Since there was
a small gain in the correlation coefficient between observed and predicted evaporation for the 1927 data where b2 = 1.00, equation (12) was chosen as the evaporation model. For the USDA reservoir the average value of CN of analysis is 0.00445.
t
for the two years
Using this value, the following model for
evaporation is obtained 0 . 50A Et = 0.0045 u z,t ez,t*
(13)
Using the models for evaporation by equations (10) and (13) values of evaporation were predicted for Lake Hefner and the USDA reservoir.
Correlation coefficients between observed and predicted eva-
poration are sumnarized in Table 5.
Shown in Figures 1 and 2 are
example plots of observed versus computed evaporation. TABLE 5 Summary of Correlation Coefficients Lake Hefner July-1, 1950 to August 31, 1951 Two Xeter Four Meter Eight Meter 0.770
0.769
0.762
USDA Reservoir 1928
1927 -
0.965
0.934
Application of Lake Hefner Stochastic Evaporation Prediction
Model to Elephant Butte Reservoir The Lake Hefner model for evaporation, equation (lo), was applied to the mass transfer data reported by Tschantz (1968) for the Elephant Butte Reservoir evaporation study.
The meteorological data was measur-
ed at the two meter level in the Elephant Butte study. Assuming z
0
Et
= =
0.6 cm and z = 200 cm equation (10) becomes -3 0.8OAe 0.500 x 10 u z,t z,t'
Equation (14) was used to predict evaporation from Elephant Butte Reservoir. The results are summarized in Table 6 and Figure 3. Table 6 is a summary of observed and predicted evaporation for the twenty-eight thermal survey periods of the Elephant
1.00
0
0.80 (d
n
\ [I)
&I 0)
u
2
-4
2al
0.60
u c
*d
c
0
-4
0.40 &I 0
a
(d
w?
0
a
? $
0.20
rn P
0
8. 0( (
/
00
I
I
0.13
0.25
F i g u r e 1.
I
I
I
I
I
0.38 0.50 0.63 0.78 0.88 Computed Evaporation, i n Centimeters/Day Computed v e r s u s Observed Evaporation f o r 1927 U . S . D . A . R e s e r v o i r Data
548
0
0
0
0
0
0
I
0
0
N
I
1 \o
0
0
h:
0 0
0
0 0
0
I
0
a l
1 0 0
549 Butte study. tion.
F i g u r e 3 i s a p l o t of o b s e r v e d v e r s u s p r e d i c t e d e v a p o r a -
The c o r r e l a t i o n between o b s e r v e d and p r e d i c t e d e v a p o r a t i o n w a s
d e t e r m i n e d t o b e 0.955.
TABLE 6 Summary of P r e d i c t e d v s . Observed E v a p o r a t i o n f o r Elephant B u t t e Study (T.S.P. = Thermal S u r v e y P e r i o d Number) T.S.P.
Observed Predicted Evaporation, Evaporation, cm/day cm/ d a y
T.S.P.
Observed Predicted Evaporation, Evaporation, cm/day cm/day
1
0.639
0.759
15
0.202
0.289
2
0.229
0.486
16
0.268
0.414
3
0.631
0.587
17
0.365
0.473
4
0.566
0.551
18
0.378
0.414
5
0.411
0.462
19
0.671
0.666
6
0.326
0.425
20
0.757
0.844
7
0.236
0.272
21
0.63;
0.599
8
0.291
0.516
22
0.804
0.716
9
0.185
0.278
23
0.943
0.996
10
0.157
0.278
24
0.886
0.766
11
0.140
0.231
25
0.695
0.768
12
0.118
0.231
26
0.748
0.723
13
0.088
0.211
27
0.686
0.868
14
0.165
0.193
28
0.780
0.883
CONCLUSIONS U s i n g d i m e n s i o n a l a n a l y s i s and r e c o g n i z i n g t h e s t o c h a s t i c p r c p e r . t i e s of e v a p o r a t i o n , wind s p e e d , and v a p o r p r e s s u r e d i f f e r e n c e a lrass t r a n s f e r model f o r e v a p o r a t i o n w a s d e v e l o p e d and i s g i v e n by
(9)
E~ = CN t u z , mAe:,t. t
T h i s model w a s a n a l y z e d u s i n g t i m e s e r i e s m u l t i p l e l i n e a r r e g r e s s i o n t e c h n i q u e s t o o b t a i n e s t i m a t e s of C , N t ,
m and p f o r t h e meteor-
o l o g i c a l d a t a a c q u i r e d d u r i n g t h e Lake H e f n e r S t u d i e s (Harbeck, 1S54) a n d s t u d i e s c o n d u c t e d by t h e USDA (Rohwer,
1931).
Using t h e e i g h t meter l e v e l as t h e u p p e r l i m i t of t h e v a p o r ‘cound a r y l a y e r and t h e l o g a r i t h m i c wind d i s t r i b u t i o n € o r a t m o s p h e r i c f l o w
550
1.0
h
m
a
0.8
a,
a
il)
I-r a,
n/’
u
2
*r(
2
0.6
a,
U
C .d
d 0
;.
0.4
m $4
0
a (d
w>
a aJ
?
I-r
0.2
w
m
P 0
0.0
551 over an aerodynamically rough surface the following model for evaporation was developed for the Lake Hefner data. Et = 0.424 x 10-2(k'/ln(z/zo)) 0 . 8 0 u O.8OAe z,t z,t Analysis of the USDA reservoir data produced the following model for evaporation. Et
=
0.50A 0.00445~ z,t z,t In general the exponent m for both the Lake Hefner and the USDA
data was found to be significantly different from unity at the ten per cent level.
The exponent for vapor pressure difference, p, was found
to be, in general, non-significantly different from unity at the ten per cent significance level.
Also the difference between m and p was
found to be significant at the ten per cent significance level. Equations (10) and (13) were used to predict evaporation froe Lake Hefner and the USDA reservoir respectively.
The average correl-
ation coefficient between observed and predicted evaporation for the Lake Hefner data was 0.767.
For the two periods of analysis for the
USDA Reservoir data the dverage correlation coefficient between observed and predicted evaporation was 0.951. The evaporation model developed for the-Lake Hefner data was applied to the mass transfer data for the Elephant Butte Reservoir. The correlation coefficient between observed and predicted evaporation was 0.955.
REFERENCES
1.
Hamon, B.V. and Hannan, E . , J . , "Estimating Relations Between Time Series", Journal Geophysical Research, 1963, vol. 68, no. 21, p p . 6033-6041
2.
Hdnnan, E.J., Time Series Analysis, 1960, New York, Wiley and Sons
3.
Hannan, E.J., "Regression for lime Series", Symposium _ of _ Time _ Series Analysis, 1963, Wiley and S o n s
4.
Harbeck, et al, "Water Loss tnvrstigation: Lake IIefner Studies",
U.S. Geological Survey Professional P a p e r s 269 and 270, 1954, Washington, D.C.: U.S. Government Printing Office
552
5.
List, R.J., Smithosonian Meteorological Tables, Smithsonian Kiscellaneous Collections, Smithsonian Institution, Washington, D.C.
6.
Rohwer, Carl, "Evaporation From Free Water Surfaces", Technical Bulletin No. 271, U.S. Department o f Agriculture, 1931
7.
Sutton, O.G., "The Application of Micrometerology to the Theory of Turbulent Flow Over Rough Surfaces", Royal Meteorological Society Quarterly Journal, 1949, V o l 74, No. 3 2 6
8.
Sutton, O.G., Micrometerology, 1953, New York: IlcGraw-Hill Book Company, Inc
9.
Tschantz, B.A., Evaporation Investigation at Elephant Butte Resservoir Using Energy Budget and Mass Transfer Techniques, January, 1968, Sc.D. Dissertation, New Mexico State University, Las Cruces, New Mexico
.
553 OPTIMAL MANAGEMENT OF PIULTIRESERVOIR SYSTEb1S USING STREAMFLOW FORECASTS by J o h n W. L a b a d i e Associate Professor
Kogelio C . Lazaro Research Associate and
,Jose D . S a l a s kssociate Professor Department o f C i v i l E n g i n e e r i n g C o l o r a d o S t a t e I l n i v e r s i t y , F t . C o l l i n s , CO
80523
ARSI RACT
I n d e v e l o p i n g a p l a n t o e f f e c t i v e l y manage
complex w a t e r r e s o u r c e
s y s t e m , some form o f water s u p p l y f o r e c a s t i n g i s r e q u i r e d . with
3
In areas
l a r g e s p - i n g snowmelt r u n o f f , a t o t a l s e a s o n a l f o r e c a s t o f
stre:rmflow i s o f t e n a v a i l a b l e from snow s u r v e y s as i n p u t t o some t y p e o f m u l t i p l e l i n e a r r e g r e s s i o n model.
These s e a s o n a l estimates p r o v i d e
a b a s i s f o r an a n n u a l o p e r a t i o n a l p l a n .
rhereafter, the operations
manager m o n i t o r s t h e d a i l y f l o w s and t r i e s t o meet t h e downstream a n d / o r supplementary s u p p l i e s .
A s c o n f l i c t s i n water u s e i n t e n s i f y , t h e n e e d
arises for d i s a g g r e g a t e d m o n t h l y (or even s h o r t e r t e r m ) f o r e c a s t s t o o h t a i n a n o p t i m a l management s t r a t e g y f o r a s y s t e m of r e s e r v o i r s .
This
i s p a r t i c u l a r l y t r u e f o r hydropower p r o d u c t i o n p l a n n i n g i n c o n j u n c t i o n w i t h o t h e r uses.
A s t a t e - s p a c e f o r e c a s t i n g model h a s b e e n d e v e l o p e d
t h a t a t t e m p t s t o r e c o n s t r u c t v i r g i n s t r e a m f l o w s b a s e d on h i s t o r i c a l r e c o r d s decomposed i n t o low f l o w a n d h i g h f l o w s e q u e n c e s .
The Kalman
f i l t e r i s u s e d t o s e p a r a t e model a n d measurement e r r o r s .
The r e s u l t i n g
f o r e c a s t s were found t o be s a t i s f a c t o r y p r i m a r i l y for t h e low f l o w months.
S u b s e q u e n t a d d i t i o n a l u s e o f snow water e q u i v a l e n t d a t a a v a i l -
a b l e p r i o r t o t h e s t a r t o f t h e h i g h f l o w p e r i o d y i e l d e d some improvement
i n f o r e c a s t i n g t h e important high flow period.
Streamflow f o r e c a s t s
f o r up t o a s i x month l e a d time were u s e d t o d e t e r m i n e o p t i m a l water t r a n s f e r s and e x c h a n g e s w i t h i n t h e Cache l a P o u d r e R i v e r , i n n o r t h c e n t r a l Colorado.
A r i v e r b a s i n n e t w o r k o p t i m i z a t i o n model p r o v i d e d
a means o f p r o c e s s i n g f o r e c a s t e d s t r c a m f l o w and o t h e r s y s t e m i n p u t s t o Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
554 i d e n t i f y t h e minimum c o s t f l o w s t o s a t i s f y s t o r a g e and d i v e r s i o n r e q u i r e m e n t s i n a c c o r d a n c e w i t h d e c r e e d water r i g h t s .
Management
y t r a t e g i e s were g e n e r a t e d w i t h t h e o b j e c t i v e o f m i n i m i z i n g w a s t e d o u t f l o w s from t h e b a s i n , i n o r d e r t o i n c r e a s e t h e l i k e l i h o o d o f j u n i o r a p p r o p r i a t o r s b e i n g s e r v e d d u r i n g c r i t i c a l low f l o w p e r i o d s . Rased on a v e r a g e monthly f l o w s , improvement was s i g n i f i c a n t f o r a t y p i c a l d r y y e a r , w i t h o u t f l o w s r e d u c e d from o v e r 4 2 , 0 0 0 a c r e - f t p e r y e a r t o a r o u n d 1000 a c r e - f t t h r o u g h u s e o f t h e combined f o r e c a s t i n g / network o p t i m i z a t i o n m o d e l s . INTRODUCTION E f f e c t i v e o p e r a t i o n a l p l a n n i n g o f water r e s o u r c e s s y s t e m s r e q u i r e s knowledge o f b o t h t h e demand f o r and t h e s u p p l y o f water r e s o u r c e s . The demands t e n d t o b e more p r e d i c t a b l e i n comparison w i t h t h e s u p p l y . f-lowever, t h e water u s e r must o f t e n make i m p o r t a n t p l a n n i n g d e c i s i o n s which a f f e c t h i s u l t i m a t e demand, i n a n t i c i p a t i o n o f f o r t h c o m i n g water s u p p l y . The normal p r a c t i c e i s o f t e n t o make an CduccLtCd g u U b , b a s e d on a v a i l a b l e d a t a , o f what t h e t o t a l incoming s u p p l y f o r t h e s e a s o n w i l l b e , and p a i r i t w i t h t h e Unud demand.
The p l a n implemen-
t a t i o n becomes a matter o f mcLtcking t h e demand w i t h t h e e s t i m a t e d supply.
I n t h e C o l o r a d o f r o n t r a n g e , f o r example, t h e s t r e a m f l o w
f o r e c a s t i s a t o t a l s e a s o n a l a g g r e g a t e b a s e d on snow pack d a t a .
A
g r e a t amount of s k i l l i s n e e d e d by t h e water a d m i n i s t r a t o r i n b r e a k i n g t h i s i n f o r m a t i o n down t o s o m e t h i n g t h a t c a n s e r v e as a b a s i s f o r day t o d a y o r even month t o month d e c i s i o n s .
A system with s t o r a g e
c a p a c i t y o f f e r s some o p e r a t i o n a l f l e x i b i l i t y , d e p e n d i n g upon t h e a v a i l a b l e c a p a c i t y , b u t also g r e a t l y i n c r e a s e s t h e number o f d e c i s i o n a l t e r n a t i v e s , d u e t o t h e need t o d e f i n e r e s e r v o i r o p e r a t i n g r u l e s . Computerized models a r e an e x t r e m e l y v a l u a b l e way o f t e s t i n g o u t v a r i o u s management schemes a t t h e r i v e r b a s i n o r s u b b a s i n l e v e l t o d e t e r m i n e t h e i r i m p a c t s on t h e v a r i o u s water u s e r s and u s e s ; o r f o r r a p i d l y p r e d i c t i n g t h e a f f e c t s o f p h y s i c a l and i n s t i t u t i o n a l c h a n g e s i n t h e system.
T h e s e models can a l s o b e an i m p o r t a n t way o f documenting
t h e e x p e r i e n c e o f t h e system a d m i n i s t r a t o r i n a n t i c i p a t i o n o f t h e day
h e r e t i r e s o r d e c i d e s t h a t t h e p r e s s u r e i s t o o g r e a t and f i n d s a n o t h e r
555 position.
‘l’he new man on t h e j o b w i l l f i n d s u c h a model i n v a l u a b l e ,
u n l e s s h e h a s been e x t e n s i v e l y t r a i n e d b y t h e water manager. A g a i n , t h e c o i n p u t e r i z e d model c a n be a c o n v e n i e n t t o o l f o r h a n d l i n g t h e l a r g e amount o f d a t a and m y r i a d v a r i a b l e s a s s o c i a t e d w i t h a complex
water r e s o u r c e s y s t e m .
Models c a n b e d e v e l o p e d which c a n p r o c e s s
h i s t o r i c a l d a t a a n d p r o d u c e more d e t a i l e d f o r e c a s t s o f f u t u r e streamflows t h a n are g e n e r a l l y a v a i l a b l e .
In a d d i t i o n , a measure o f t h e
confidence a s s o c i a t e d with t h e s e f o r e c a s t s can a l s o b e e s t i m a t e d .
This
i n f o r m a t i o n c a n t h e n b e i n p u t t o a n o t h e r model which c a n p r o c e s s t h e s e d a t a t o produce o p e r a t i o n a l g u i d e l i n e s .
A s new d a t a a r e r e c e i v e d i n
r e a l - t i m e , t h e p a r a m e t e r s o f t h e s e models c a n b e s y s t e m a t i c a l l y u p d a t e d t o p r o d u c e new s t r e a m f l o w f o r e c a s t s and u p d a t e d o p e r a t i o n a l p l a n s . A inodeling s t u d y i s d e s c r i b e d h e r e i n which combines a s t a t e - s p a c e
s t r e a m f l o w f o r e c a s t i n g m o d e l , b a s e d on t h e Kalman f i l t e r , w i t h a r i v e r b a s i n s i m u l a t i o n model w i t h some o p t i m i z i n g c a p a b i l i t y .
‘These m o d e l s
h a v e been t e s t e d on t h e Cache La Poudre R i v e r b a s i n i n n o r t h c e n t r a l C o l o r a d o t o d e t e r m i n e i f i t i s p o s s i b l e t o improve o p e r a t i o n a l e f f i c i e n c y . D e t a i l s o f t h e f o r e c a s t i n g model a r e d e s c r i b e d e l s e w h e r e .
This
p a r t i c u l a r p r e s e n t a t i o n f o c u s e s on how t h e f o r e c a s t s were i n p u t i n t o t h e r i v e r b a s i n model t o d e t e r m i n e r e a l - t i m e o p e r a t i o n a l p l a n s . BRIEF RFVILIV OF LIYDROSYSTI Pl MANAGETENT TECHNIQUES The most w i d e l y u s e d f o r e c a s t i n g t e c h n i q u e i n t h e w e s t e r n U n i t e d S t a t e s is a c o r r e l a t i o n - b a s e d s t r e a m f l o w f o r e c a s t i n g method t h a t y i e l d s o n l y t h e e x p e c t e d t o t a l s e a s o n a l r u n o f f volume ( U S B R ,
1968).
When
hydropower i s i m p o r t a n t , a n a t t e m p t i s made t o d i s a g g r e g a t e t h e t o t a l s e a s o n a l f o r e c a s t i n t o m o n t h l y estimates.
T h i s i s g e n e r a l l y accomplished
by u s i n g t h e m o n t h l y r e c o r d o f f l o w s from a p r e v i o u s y e a r h a v i n g a s i m i l a r t o t a l r u n o f f volume ( B e l l a m y , 1 9 8 0 ) .
A f t e r t h e last snowpack
m e a s u r e m e n t s a r e o b t a i n e d , n o f u r t h e r f o r e c a s t u p d a t e s a r e made.
The
f o r e c a s t s are g e n e r a l l y adequate €or average flow y e a r s , b u t tend t o o v e r e s t i m a t e low flows a n d u n d e r e s t i m a t e h i g h f l o w s (flawley, e t . a l . 1980). I n Colorado, t h e s e monthly f o r e c a s t s are a l s o used €or planning a n n u a l a l l o c a t i o n o f s u p p l e m e n t a l t r a n s b a s i n d i v e r s i o n water, s u c h as
556 done by t h e N o r t h e r n C o l o r a d o Water Conservancy D i s t r i c t (NCWCD) (Barkley, 1974). a r e obtained.
T h e s e a l l o c a t i o n s a r e a d j u s t e d i n time a s new d a t a
Within t h i s same n o r t h e r n C o l o r a d o a r e a , t h e Poudre
R i v e r Commissioner, u n d e r t h e o f f i c e o f t h e S t a t e E n g i n e e r , i n d e p e n d e n t l y p r e p a r e s h i s own w a t e r s u p p l y f o r e c a s t s u s i n g a s l i g h t l y more d e n s e network o f snow c o u r s e s and a s i m p l e a v e r a g i n g p r o c e d u r e (Neutze, 1980).
H i s f o r e c a s t s are u t i l i z e d i n determining t h e percentage o f
normal r e q u i r e m e n t s t h a t c a n b e s e r v e d i n h i s d i s t r i c t .
Day t o day
d e c i s i o n s on how b e s t t o match demand w i t h t h e o b s e r v e d s u p p l y a r e made on t h e b a s i s o f t h e e s t a b l i s h e d w a t e r r i g h t p r i o r i t i e s , g u i d e d by e x p e r i e n c e and i n t u i t i o n . A number o f s t u d i e s i n v o l v i n g u s e o f s t r e a m f l o w f o r e c a s t s i n r i v e r
b a s i n management and o p e r a t i o n s are r e p o r t e d e l s e w h e r e i n t h e literature,
(Hoshi and Burges, 1 9 7 9 ) , (Wilson and K i r d a r , 1 9 7 0 ) ,
(Recker and Yeh, 1974), ( M e j i a , e t . a l . , l 9 7 4 ) , (Wunderluch and G i l e s , 1981) and (Unny, e t . a l . , 1 9 8 1 ) .
DEVELOPMENT 01: STREAMFLOW FORECASTING PIODEL
S e v e r a l s t r e a m f l o w f o r e c a s t i n g methods have a p p e a r e d i n t h e l i t e r a t u r e i n r e c e n t y e a r s , w i t h time s e r i e s a n a l y s i s ( e . g . , P4cKerchar and D e l l e u r (1974)) and s t a t e - s p a c e a p p r o a c h e s (Movarek, e t . al. ( 1 9 7 8 ) ) seeming to b e t h e most p o p u l a r . Graupe (1976) s u g g e s t s a p o s s i b l e rno.hI~&gQ between time s e r i e s a n a l y s i s ( f o r model form and t y p e i d e n t i f i c a t i o n ) and s t a t e - s p a c e a p p r o a c h e s ( f o r c o n s i d e r i n g measurement e r r o r s ) ,
A s is g e n e r a l l y
known, monthly s t r e a m f l o w r e c o r d s e s s e n t i a l l y c o n s i s t of g r o u p s of i n d i v i d u a l d a t a ( i . e . , low f l o w and h i g h flow s e t s , o r t h r e e o r f o u r s e a s o n a l g r o u p s ) p o s s e s s i n g d i f f e r e n t and d i s t i n c t c h a r s c t e r i s t i c s . Panu and Unny (1980) p r o p o s e t h a t two o r more s e p a r a t e m o d e l s , one f o r e a c h group o r c l a s s , b e d e v e l o p e d i n l i e u o f a s i n g l e r e p r e s e n t a t i o n f o r t h e monthly s t r e a m f l o w d a t a . I t i s known t h a t monthly h y d r o l o g i c time s e r i e s u s u a l l y have p e r i o d i c components i n s e v e r a l o f t h e i r s t a t i s t i c a l c h a r a c t e r i s t i c s s u c h as t h e mean, s t a n d a r d d e v i a t i o n s and c o r r e l a t i o n c o e f f i c i e n t s .
This is
p h y s i c a l l y j u s t i f i e d s i n c e i n l a t e summer, f a l l and w i n t e r t h e
557 streamflow is dominated by groundwater or baseflow contributions while in the spring and early summer, direct surface runoff from snowmelt and rainfall is the major contributor.
The stochastic model considered
herein includes these seasonal characteristics, in the form of a low flow sequence, where the measurement errors are relatively small and
a high flow sequence where the measurement errors are usually large. Details on the model structure identification, model parameter and noise statistics estimation as well as diagnostic checks are given in Lazaro, et. al., ( 1981) . The iterative model building procedure of Box and Jenkins (1976) can still be employed in this case.
The exception is that a slightly
different approach in the computation of the autocorrelation and partial autocorrelation functions must be used to account for the discontinuity involved in separating the two groups of flows (Salas, et. al., 1980).
The parameters for each model can be estimated using the
method of maximum likelihood or a sequential least squares regression algorithm.
For Gaussian time series, the latter yields estimates
approaching the maximum likelihood estimates.
blodel adequacy diagnostic
checks via the usual residual analysis for goodness of fit testing then follow. In this modeling approach, the error term allows for the inexactness of the model to represent the true process. reflected in the estimates of the parameters.
Any measurement errors are This state-space
approach to modeling requires complete specification of the predictor parameters and noise terms corrupting the system dynamics and measurement models.
These predictor parameters and noise statistics can be
obtained from the identified time series model under an assumption of an invariant processor.
The time history of measurement errors for
hydrologic events are, however, difficult to obtain or rarely available. Those that are accessible are either incomplete or location and
instrument spec i f i c .
558 RIVER BASIN NETWORK MODEL The T e x a s Water Development Board ( 1 9 7 2 ) h a s d e v e l o p e d a model c a l l e d SIMYLD t h a t i s c a p a b l e of r e p r e s e n t i n g t h e p h y s i c a l c h a r a c t e r i s t i c s o f
l a r g e - s c a l e , complex water r e s o u r c e s y s t e m s a n d s e l e c t i n g t h e o p t i m a l water a l l o c a t i o n i n a minimum c o s t s e n s e .
SIP4YLD i s b a s i c a l l y a s i m u l a -
t i o n model w i t h o p t i m i z i n g c a p a b i l i t y on a month b y month b a s i s .
That
i s , a s e q u e n t i a l , s t a t i c o p t i m i z a t i o n approach i s used r a t h e r than a f u l l y dynamic g l o b a l o p t i m i z a t i o n . The s y n t h e s i z e d model u s e d i n t h i s s t u d y , c a l l e d MODSI?.I, e s s e n t i a l l y r e t a i n s t h e b a s i c s t r u c t u r e o f S I r l Y L D , w h i l e i n c o r p o r a t i n g some additional features.
Conversational, i n t e r a c t i v e coding has been incor-
p o r a t e d t o e n c o u r a g e u s e o f t h e model by s t a t e and l o c a l water r e s o u r c e s p l a n n e r s and managers i n t h e i r t a s k s o f e v a l u a t i n g t h e i m p a c t s o f a l t e r n a t i v e water management p o l i c i e s t h r o u g h o u t a r i v e r b a s i n . The model e s s e n t i a l l y c o n s i d e r s t h e p h y s i c a l f e a t u r e s of t h e s y s t e m i n a c a p a c i t a t e d n e t w o r k form.
'The r e a l s y s t e m components, s u c h as
s t o r a g e and n o n - s t o r a g e p o i n t s ( r e s e r v o i r s , demand p o i n t s , c a n a l d i v e r s i o n s and r i v e r c o n f l u e n c e % ) , a r e r e p r e s e n t e d h y n o d e s .
'The r i v e r
r e a c h e s , c a n a l s and c l o s e d c o n d u i t s a r e d e s i g n a t e d a s node t o node 1i n k a g e s (1 i n k s ) .
M a t h e m a t i c a l D e s c r i n t i o n of t h e Problem The problem o f networh flow a l l o c a t i o n i s s e q u e n t i a l l y s o l v e d through t h e o u t - o f - k i l t e r a l g o r i t h m t o y i e l d t h e minimum " c o s t " f l o w s f o r each month o f o p e r a t i o n , w i t h i n t h e c o n f i n e s o f mass b a l a n c e t h r o u g h o u t t h e networh ( B a z a r r a and . J a r v i s , 1 9 7 7 ) . g i v e n month
t
The o p t i m i z a t i o n probleni f o r any
is:
s u b j e c t t o node c o n s e r v a t i o n o r c o n t i n u i t y of mass
g i v e n bounds on f l o w s i n a 1 1 l i n k s
where Cij
Q. . I!
=
=
t h e flow i n t h e l i n h from nodc
i
t h e c o s t o f t r a n s f e r r i n g e a c h u n i t of f l o w
t o nodc Qij
_i (no r o u t i n g ) , from node
i
to
559 node
j,
and
U . . = u p p e r bound o f l i n k from node
11
i
t o node
j.
C e r t a i n a c t u a l c o s t s , s u c h a s pumping c o s t s , c a n be e a s i l y a s s i g n e d t o appropriate links.
I n many s i t u a t i o n s , i t may b e d e s i r a b l e t o
C.. as w e i g h t i n g o r p r i o r i t y f a c t o r s r a t h e r t h a n a c t u a l 1J S t o r a g e and d i v e r s i o n r i g h t s c a n b e w e i g h t e d i n o r d e r o f
represent the costs.
p r i o r i t y , w i t h s e n i o r r i g h t s g i v e n t h e lower " c o s t s " .
For l i n e a r
o b j e c t i v e s s u c h as t h i s , r e l a t i v e o r d e r i n g o f t h e s e p r i o r i t i e s i s more important t h a n t h e a c t u a l magnitudes. Though t h e model o b j e c t i v e a t t e m p t s t o minimize c o s t s , b e n e f i t s can be r e p r e s e n t e d a s n e g a t i v e c o s t s .
I n f a c t , d e s i r e d end-of-month s t o r a g e
and demand l i n k s a r e a s s i g n e d n e g a t i v e c o s t s . s o l v e d p e r i o d by p e r i o d .
The network i s o n l y
G l o b a l o p t i m a l s o l u t i o n o f t h e e n t i r e networh
f o r s e v e r a l t i m e p e r i o d s a t once i s e x t r e m e l y e x p e n s i v e c o m p u t a t i o n a l l y . More d e t a i l s on t h e model, t h e way l i n k c o s t s a r e a s s i g n e d and t h e e x t e n d e d c a p a b i l i t i e s i n c l u d e d i n MODSIPI can b e found i n S h a f e r , e t . a l . , (1981). PlODSIM model a p p l i c a t i o n s
The r e s e r v o i r o p e r a t i n g r u l e s a s i n p u t t o t h e program may r e f l e c t e i t h e r t h e d e s i r e t o m a i n t a i n s t o r a g e l e v e l s a t some p o i n t beloiv maximum c a p a c i t y d u r i n g c e r t a i n months ( f l o o d c o n t r o l ) , m a i n t a i n l e v e l s a s h i g h as p o s s i b l e t o enhance w a t e r s u p p l y r e l i a b i l i t y ( d r o u g h t c o n t r o l ) , o r s t r i c t l y a b i d e by t h e e s t a b l i s h e d w a t e r r i g h t p r i o r i t i e s d u r i n g t h e normal f l o w y e a r s .
The p r i o r i t i e s p l a c e d on a c h i e v i n g t h e
t a r g e t s t o r a g e l e v e l s a n d / o r d i v e r s i o n demands c a n be m a n i p u l a t e d t o a s s e s s d i f f e r e n t o p e r a t i o n a l schemes f o r a c c o m p l i s h i n g t h e s e g o a l s . The s y n t h e s i z e d model MODSIPI h a s been u s e d t o e v a l u a t e t h e i m p a c t s o f a l t e r n a t i v e management schemes for two c a s e s t u d i e s w i t h i n t h e Cache
l a Poudre Poudre R i v e r b a s i n : (1) d e t e r m i n i n g i f r e c r e a t i o n o p p o r t u n i t i e s c o u l d b e p r o v i d e d i n s e l e c t e d h i g h mountain r e s e r v o i r s by maint a i n i n g s a t i s f a c t o r y monthly s t o r a g e l e v e l s w i t h o u t i n j u r y t o dormstrcam w a t e r u s e r s ( S h a f e r , e t . a l . , ( 1 9 8 0 ) ) , and ( 2 ) d e t e r m i n i n g i f s u f f i c i e n t r e u s a b l e e f f l u e n t from t h e C i t y o f r o r t C o l l i n s , C o l o r a d o i s a v a i l a b l e ( g i v e n an assumed h y d r o l o g i c a l s e q u e n c e ) t o meet monthly w a t e r demands f o r a p r o p o s e d c o a l - f i r e d power g e n e r a t i n g p l a n t ( S h a f c r , c t . a l . , (1981)).
560 CACHE LA POUDRE CASE STUDY
The Cache l a P o u d r e R i v e r s y s t e m i s a f o u r t h o r d e r t r i b u t a r y o f t h e Y i s s i s s i p p i R i v e r , t h e d r a i n a g e s y s t e m o f t h e l o w e r m i d d l e p o r t i o n of t h e N o r t h American C o n t i n e n t
.
The b a s i n c o n s i s t s o f a m o u n t a i n o u s p a r t from which most of t h e w a t e r s u p p l y o r i g i n a t e s and a p l a i n s a r e a i n which t h e water s u p p l y i s
used.
The r i v e r s t a r t s a t P o u d r e L a k e a n d s e v e r a l p l a c e s on t h e
Contixental Divide.
From t h e h e a d w a t e r , t h e r i v e r flows i n a n o r t h e a s t
d i r e c t i o n t o i t s canyon mouth f o r a b o u t 50 m i l e s and t h e n s o u t h e a s t f o r a n o t h e r 35 miles o v e r a n open p l a i n t o w a r d s t h e S o u t h P l a t t e R i v e r . . Ilydrometeorological f e a t u r e s The f a l l and w i n t e r p r e c i p i t a t i o n i n t h e b a s i n i s u s u a l l y i n t h e form of snow, w h i l e t h e s p r i n g a n d summer p r e c i p i t a t i o n g e n e r a l l y o c c u r s as thunderstorms.
Snow a c c u m u l a t i o n s i n t h e w a t e r s h e d a r e m o n i t o r e d from
some e i g h t snow c o u r s e s from November t h r o u g h A p r i l .
There i s one
m e t e o r o l o g i c a l s t a t i o n t h a t m e a s u r e s m o u n t a i n p r e c i p i t a t i o n y e a r round and t h r e e s t a t i o n s a r e l o c a t e d on t h e p l a i n s a t F o r t C o l l i n s , G r e e l e y and Windsor. During t h e p e r i o d from May t o . J u l y , t h e s p r i n g r u n o f f b e g i n s .
About
SO t o 7 0 p e r c e n t of t h e s u r f a c e water r u n o f f o c c u r s d u r i n g A p r i l t h r o u g h ,July as snowmelt from t h e m o u n t a i n t r i b u t a r i e s .
Groundwater
b a s e flow s u s t a i n s t h e s t r e a m f l o w d u r i n g t h e months from J u l y t h r o u g h October.
'I'he f a l l and w i n t e r f l o w s a r e c o n s t a n t l y low.
Two USGS
s t r e a m f l o w g a g i n g s t a t i o n s w i t h l o n g and c o n t i n u o u s r e c o r d s a r e l o c a t e d
a t t h e mouth o f tlie canyon a t F o r t C o l l i n s and a t G r e e l e y n e a r t h e confluence w i t h t h e South f'latte River. Water s u p p l y d i s t r i b u t i o n and u s e s y s t e m
Development o f t h e water r e s o u r c e s i n t h e a r e a p a r a l l e l s t h a t i n most w e s t e r n s t a t e s o f tlie U.S.
There are s e v e r a l import s o u r c e s
d i v e r t i n g water t o t h c b a s i n t o augment t o t a l s u p p l y .
These a r e t h e
S h y l i n e I)i t c h , Grand R i v e r I)i t c h , Cameron Pass D i t c h , Laramie-Poudre ?'unnel, M i c h i g a n D i t c h , Wilson S u p p l y D i t c h , and t h e C o l o r a d o Big Thompson p r o j e c t .
561 The snow-fed mountain streams p r o v i d e e x c e s s w a t e r i n t h e s p r i n g and i n a d e q u a t e amounts l a t e r i n t h e s e a s o n .
A s y s t e m o f r e s e r v o i r s was
t h e r e f o r e developed t o c o r r e c t t h i s s i t u a t i o n .
T h e r e a r e now some 200
r e s e r v o i r s i n t h e b a s i n , w i t h e l e v e n major o n e s i n mountain p o r t i o n s
of t h e b a s i n ( i . e . , Long D r a w , P e t e r s o n , J o e W r i g h t , Chambers L a k e , Barnes Meadow, Big Beaver, Twin Lakes, Commanche, W o r s t e r , l l a l i g a n and Seaman).
The r e s t a r e s i t u a t e d on t h e p l a i n s .
The p r i m a r y u s e o f t h e b a s i n w a t e r s u p p l y i s f o r i r r i g a t i n g some 2 4 0 , 0 0 0 acres o f l a n d on t h e p l a i n s .
Water i s d i s t r i b u t e d by a b o u t 3 3
i r r i g a t i o n companies which o p e r a t e and m a i n t a i n t h e d i s t r i b u t i o n and r e s e r v o i r systems.
The a r e a i s w i t h i n t h e N o r t h e r n C o l o r a d o Water
Conservancy D i s t r i c t , (NCWCD) and i s p a r t o f t h e p r i m a r y s e r v i c e area o f t h e Colorado-Big 'Thompson P r o j e c t .
Supplemental w a t e r from t h i s
p r o j e c t i s d e l i v e r e d v i a l l o r s e t o o t h R e s e r v o i r t o t h e Cache l a Poudre R i v e r n e a r t h e p o i n t where i t emerges from t h e mountains o n t o t h e p l a i n s M u n i c i p a l and i n d u s t r i a l w a t e r s u p p l y i s p r o v i d e d t o t h e c i t i e s of G r e e l e y and F o r t C o l l i n s w i t h d i r e c t f l o w r i g h t s t o t a l i n g 1 2 . 5 c f s and 19.93 cfs, respectively.
Both c i t i e s a l s o own r i g h t s which allow them
t o i m p o r t water i n t o t h e b a s i n .
They a l s o own 13,125 and 1 0 , 2 5 0 s h a r e s ,
r e s p e c t i v e l y , o f water from t h e Colorado-Big Thompson (CBT) P r o j e c t ( R e i t a n o and H e n d r i c k s , 1 9 7 8 ) .
Water r i g h t s : D o c t r i n e o f p r i o r a p p r o p r i a t i o n I n C o l o r a d o , t h e r i g h t s t o t h e u s e o f w a t e r a r e b a s e d f i r m l y on t h e d o c t r i n e o f p r i o r a p p r o p r i a t i o n which s i m p l y means t h a t t h e 6 h A Z i n
&'mc u s e r s a r e &&&
i n high;t.
'The o r d e r o f p r i o r i t y o f w a t e r r i g h t
h o l d e r s a r e s p e c i f i e d such t h a t a t v a r i o u s l e v e l s o f streamflow, t h o s e who a r e p e r m i t t e d t o d i v e r t w a t e r can be d e t e r m i n e d (Anderson, (1975)). The r i g h t s a r e a c q u i r e d by a p p r o p r i a t i o n and a r e d e f i n e d t h r o u g h t h e a d j u d i c a t i o n p r o c e d u r e c o n d u c t e d by t h e c o u r t s . 'The d e c r e e d o r a d j u d i c a t e d water r i g h t s a r e a d m i n i s t e r e d by t h e S t a t e I n g i n e e r and h i s s u b o r d i n a t e s , t h e d i v i s i o n e n g i n e e r s and t h e water commissioners.
A water commissioner i s r e s p o n s i b l e f o r e a c h
major stream w i t h i n a d i v i s i o n .
562 I r r i g a t i o n u s e r s ( e i t h e r i n d i v i d u a l or c o m p a n i e s ) o f t e n own a number o f w a t e r r i g h t s t h a t v a r y b o t h i n d a t e o f p r i o r i t y and i n q u a n t i t y o f water c l a i m e d .
D u r i n g t h e s e a s o n a water r i g h t h o l d e r may h a v e some
r i g h t s u n d e r which h e c a n draw w a t e r , w h i l e o t h e r r i g h t s c a n n o t b e e x e r c i s e d h e c a u s e someone e l s e has a h i g h e r p r i o r i t y .
A t t i m e s , water
r i g h t h o l d e r s will g i v e ul) u s i n g water when o n l y some of t h e i r r i g h t s
a r e a c t i v e , t h e r e b y p e r m i t t i n g o t h e r water r i g h t h o l d e r s on o t h e r d i t c h e s t o d i v e r t i n e x c h a n g e f o r water a t some l a t e r t i m e . T h i s m u l t i p l e water r i g h t mahes t h e t a s k o f d e l i v e r i n g water t o i n d i v i d u a l s a n d / o r companies complex f o r t h e water c o m m i s s i o n e r .
lle
must heep a l l d i t c h e s i n f o r m e d o f which o f t h e i r s e v e r a l r i g h t s e n t i t l e them t o d i v e r t water and which do n o t .
kle h a s t o k e e p t r a c k o f which
r i g h t h o l d e r s a r e n o t d i v e r t i n g water, even t h o u g h t h e y a r e e n t i t l e d t o . fie i s a l s o r e s p o n s i b l e for a l l o c a t i n g r e t u r n f l o w t h a t i s a v a i l a b l e f o r
diversions. 1)istrict operational planning practices I n p r e p a r i n g f o r t h e District o p e r a t i o n p l a n , t h e r i v e r c o m m i s s i o n e r makes a f o r e c a s t o r e s t i m a t e of e x p e c t e d water s u p p l y a s e a r l y as t h e first snowfall starts.
He m o n i t o r s t h e r e s o u r c e s by o b s e r v i n g t h e
snow c o u r s e s e s t a b l i s h e d by t h e Snow S u r v e y U n i t of t h e S o i l C o n s e r v a t i o n S e r v i c e and m a i n t a i n s o t h e r c o u r s e s t h a t p r o v i d e him a d d i t i o n a l c o n f i d e n c e i n e s t i m a t i n g t h e snowpack a c c u m u l a t i o n s i n t h e w a t e r s h e d . T h i s i s done t h e f i r s t of e v e r y month from November t h r o u g h A p r i l of each year.
By May l s t , t h e o n s e t of t h e i r r i g a t i o n s e a s o n , h e h a s a f i n a l
a s s e s s m e n t o f t h e snowpack a c c u m u l a t i o n s .
He d e t e r m i n e s t h e a v e r a g e
snow water d e p t h e q u i v a l e n t on some 2 4 7 , 0 0 0 acres o f snow f i e l d , which
i s t h e a p p r o x i m a t e a r e a o f t h e s n o w l i n e a t 9000 f e e t e l e v a t i o n .
With
t h i s i n f o r m a t i o n , and a l l o w i n g f o r a b o u t 8% s h r i n A a g e l o s s , h e o b t a i n s an e s t i m a t e o f t h e t o t a l volume o f water t h a t would b e made a v a i l a b l e to
t h e d i s t r i c t f o r t h e c u r r e n t y e a r o p e r a t i o n s t o f i l l i r r i g a t i o n water n e e d s and o t h e r ( d o m e s t i c a n d i n d u s t r i a l ) u s e s w i t h i n t h e D i s t r i c t . r ' i t i o of c u r r e n t estimates with p r e v i o u s long-term r e c o r d s is denoted .is a p e r c e n t a g e o f normal w a t e r s u p p l y a v a i l a b l e .
The l o n g - t e r m
, i v e r a g e demands 5y t h e d i s t r i c t i s e s t i m a t e d a t 2 8 0 , 0 0 0 a c r e - f t .
'The
563
The estimated supply is then related with this demand to yield the percentage of demand that could possibly be satisfied. This apLioLi knowledge would subsequently indicate the approximate number of appropriators that could be served. This figure is of course a relative one and is highly dependent on the future weather conditions. If the temperature is cooler, the snowpack may hold and snowmelt will be slow.
Conversely, if the weather is hot and dry, snowmelt occurs
rapidly. The snowpack could therefore be considered as a nCLtWLdf i e . h & f ~ v u hwith
a certain capacity, except that the fieLe~~5e.h are
beyond man’s control.
If the expected supply is estimated to be less than the normal demand, junior appropriators who may not be served make their own assessment of the situation and may plan to buy reservoir water early in the season to augment their rights later. They may opt to run their groundwater pumps more than usual when needed. Those unsure of what their status will be may delay making decisions and wait and see what the weather conditions will be.
Others may gamble and hope that
localized cloudbursts will fill their needs when they require them or that surplus reservoir water will still be available when needed. On the other hand, if the expected supply is greater than the average demand, appropriators who could not be served during the average years would have a chance to exercise their rights. However, if the weather conditions are such that early and/or rapid snowmelt occurs, the low lying areas downstream are threatened by flood damages. The forecast of the water supply is considered important in planning the operation of the reservoir in the District, but the day to day management is highly reactive to the weather conditions. If the snowpack is less than normal, reservoirs at the headwaters would be filled first to minimize transmission losses and severe shortages later in the season. The plains reservoirs generally have the higher priorities and therefore are entitled to fill first. A provision in the law, however, allows the mountain reservoirs or any other reservoirs to be filled outof-priority to capture water that might otherwise be lost, particularly when the plains reservoir are already full (Radosevich, et. al., (1975).
564 R e s e r v o i r r e l e a s e s s t a r t a t t h e l o w e s t r e s e r v o i r s which c a n b e exchanged w i t h r i v e r water u p s t r e a m ( s i n c e t h e y a r e n o t d i r e c t f l o w s ) and p r o g r e s s i v e l y work u p s t r e a m .
The p l a i n s r e s e r v o i r s a r e drawndown f i r s t
t o a l l o w water e x c h a n g e s w h i l e t h e r e i s r i v e r w a t e r .
The m o u n t a i n
r e s e r v o i r s a r e drawndown l a s t , b e c a u s e t h e i r water c a n be t a k e n d i r e c t l y and i s on
cdee
a t any time.
D i s t r i c t c o m m i s s i o n e r ' s manaeement a c t i v i t i e s
The f t i v e r Commissioner h a s a m u l t i f a c e t e d t a s k o f : (1) s a t i s f y i n g
as many a p p r o p r i a t o r s as h e c a n , g i v e n t h e c u r r e n t water s u p p l y ; ( 2 ) m i n i m i z i n g t h e c o n s e q u e n c e s of low s u p p l y from d r o u g h t , as well as
m i n i m i z i n g damages r e s u l t i n g from r a p i d r e l e a s e of snow a c c u m u l a t i o n s , thereby causing flooding i n t h e p l a i n s ;
( 3 ) r e d u c i n g o u t f l o w s from t h e
s y s t e m , g i v e n a v a i l a b l e s t o r a g e and c a n a l c a p a c i t i e s ;
(4) m i n i m i z i n g
w a s t a g e and (5) maximizing u t i l i z a t i o n o f r i v e r w a t e r b e f o r e t h e supplementary supply sources a r e a p p l i e d . A t y p i c a l day f o r t h e r i v e r commissioner involves t h e determination
and a s s e s s m e n t o f t h e USGS g a g e f l o w a t t h e mouth o f t h e canyon i n F o r t Collins.
Given t h e c u r r e n t f l o w r a t e and a i d e d by l o n g y e a r s of
e x p e r i e n c e , t h e number o f a p p r o p r i a t o r s t h a t can be s e r v e d i s o b t a i n e d . Knowing who i s c u r r e n t l y b e i n g s e r v e d , h e knows who n e e d s t o c o n t i n u e d i v e r t i n g , who has t o s t o p , who h a s t o r e d u c e t h e amount b e i n g d i v e r t e d and who n e e d s t o be a d d i t i o n a l l y s e r v e d . t o r s f o r a d j u s t m e n t s o r t h e y c a l l him.
E i t h e r he c a l l s t h e appropriaWhen r i v e r d i v e r s i o n and s t o r a g e
r i g h t s h a v e been a c c o u n t e d for, t h e n s u p p l e m e n t a l water s o u r c e s a r e considered next.
I f h i s t r a n s b a s i n s u p p l y i s n o t enough t o f i l l t h e
n e e d s , he t h e n c a l l s , as r e p r e s e n t a t i v e o f t h e D i s t r i c t water u s c r s , f o r q u o t a a l l o c a t i o n r e l e a s e s from t h e NCIVCI), a n d / o r a l l o w s pumping from groundwater r e s o u r c e s . Twice a wcek, a f i e l d i n s p e c t i o n i s d o n e . t h e r i v e r a p p r o x i m a t e l y i n IiAlf.
Ile and his d e p u t y d i v i d e
Each o n e m o n i t o r s and e v a l u a t e s t h e
r i v e r f l o w s and d i v e r s i o n s a t v a r i o u s t a k e - o f f p o i n t s i n some t h i r t y three ditches.
They b o t h check i f t h e p r o p e r amounts a r e i n t h e r i v e r
and i f d i v e r s i o n s a r e w i t h i n t h e s p e c i f i e d r i g h t s .
Wasteways o r
s l u i c e w a y s a r e l i k e w i s e o h s e r v e d t o clicck i f w a t c r i s b e i n g w a s t e d .
56 5 Water l o s t d u e t o l a c k of s t o r a g e i s a l s o d e t e r m i n e d .
T h i s i s water
t h a t c o u l d h a v e b e e n s t o r e d b u t t h e r e was no room f o r i t s i n c e e x i s t i n g r e s e r v o i r s were a l r e a d y f u l l . i s a l s o r e a d and r e c o r d e d .
The USGS g a g e n e a r t h e C i t y o f G r e e l e y
T h i s i n d i c a t e s t h e amount o f water l e a v i n g
t h e system. The r e s e r v o i r l e v e l s a r e r e p o r t e d t o him by t h e owners and a r e g e n e r a l l y v e r i f i e d o n c e a month.
Any d a y i s open i f t h e r e a r e c o m p l a i n t s ,
or a d j u s t m e n t s are n e e d e d , anywhere w i t h i n t h e d i s t r i c t , t w e n t y - f o u r h o u r s a d a y , s e v e n d a y s a week.
Weekly r e p o r t s a r e p r e p a r e d f o r t h e
S t a t e E:ngineer on t h e f l o w l e v e l o f t h e r i v e r , t h e r i g h t h o l d e r s e n t i t l e d t o d i v e r t water, and t h e r i g h t h o l d e r s a c t u a l l y d i v e r t i n g .
Monthly
summaries of o p e r a t i o n s a r e t r a n s m i t t e d t o t h e S t a t e E n g i n e e r t h r o u g h t h e Division engineers.
A t t h e end o f t h e c a l e n d a r y e a r , a summary o f t h e
annual o p e r a t i o n i s prepared.
The D i s t r i c t w a t e r s u p p l y s o u r c e and
a p p l i c a t i o n s are i d e n t i f i e d a l o n g w i t h s e e p a g e i n f l o w s and s y s t e m o u t floivs t o p r o v i d e i n d i c a t i o n s o f t h e e n d - o f - t h e - y e a r
available storage.
I t i s e v i d e n t t h a t t h e t a s k s o f t h e R i v e r Commissioner a r e e x t r e m e l y complex and c h a l l e n g i n g .
I t i s berieved t h a t the forecasting/network
o p t i m i z a t i o n models p r e s e n t e d h e r e h a v e t h e p o t e n t i a l o f b e i n g e x t r e m e l y u s e f u l f o r him, and t h o s e i n s i m i l a r p o s i t i o n s o f r e s p o n s i h i l i t y ,
in
p r o v i d i n g s e a s o n a l r e a l - t i m e mariagement g u i d e l i n e s . RESULTS OF Tl1E MANAGEIENT S'TIJDY
The p r i m a r y o b j e c t i v e o f t h i s management s t u d y was t o d e t e r m i n e i f t h e comhined u s e o f f o r e c a s t i n g / n e t w o r k o p t i m i z a t i o n models c o u l d h a v e r e d u c e d w a s t e d f l o w s ( i . e . , f l o w s a b c v e r e q u i r e d downstream r e l e a s e s ) during a dry period.
C a p t u r i n g t h i s water would i m p l y t h a t more
a p p r o p r i a t o r s could be s e r v e d . 'The f o r e c a s t r e s u l t s €or t h e low flow g r o u p were v e r y c l o s e t o t h e
a c t u a l s t r e a m f l o w v a l u e s , a s shown i n F i g u r e 1 .
h i g h f l o w g r o u p , however, t e n d e d t o t h e mean.
The forecast f o r t h e 'I'he model a l s o o v e r -
e s t i m a t e d t h e low f l o w y e a r and u n d e r e s t i m a t e d t h e h i g h f l o w year.
Some
o t h e r m e a s u r a b l e i n p u t c o n t r i b u t i n g t o the s t r e a m f l o w needed t o b e c o n s i d e r e d t o m o d i f y t h e model f o r e c a s t .
llence, t h e l a s t (Apri 1 .3Oth)
snoic w a t e r c q u i v a l e n t m c n s u r e m c n t s , c o n s i d e r e d a s an n c c e s s i h l e i n p u t , \ \ s ~ r cr e g r e s s e d t \ i t h t h c t o t a l s e a s o n a 1 s t r e a m f l o w . '['he iclcnt i f i e d
566
I'
Actual monthly flow Model forecast 6 months lead time - Revised forecast (Iinnear regression predicted runoff) Revised forecast (multiple correlation predicted 150,000 runoff WPRS 1 RMSE= root mean square error
Figure
1.
R e v i s e d m o n t h l y s t r e a n i f l o w f o r e c a s t u s i n g t h e predic:ed t o t a l s e a s o n a l r u n o f f from l i n e a r r e g r e s s i o n and m u l t i p l e c o r r e l a t ~ o nt e c h n i q u e f o r t h e w a t e r y e a r s 197; and 1 9 7 8 , h i g h flow g r o u p s e q u e n c e .
567 r e l a t i o n s h i p was t h e n u t i l i z e d t o p r o v i d e i n d i c a t i o n s o f e x p e c t e d s e a s o n a l r u n o f f volume, which i n t u r n was d i s a g g r e g a t e d u s i n g t h e i n f o r m a t i o n a v a i l a b l e from t h e model r e p r e s e n t i n g t h e h i g h f l o w s e r i e s . A l t e r n a t i v e l y , t h e t o t a l s e a s o n a l s t r e a m f l o w f o r e c a s t from t h e m u l t i p l e c o r r e l a t i o n - b a s e d model of t h e Bureau o f R e c l a m a t i o n was a l s o u s e d . The r e v i s e d m o n t h l y f o r e c a s t s f o r t h e h i g h f l o w p e r i o d were c l o s e r t o t h e a c t u a l strcamflow v a l u e s . System c o n f i g u r a t i o n and network r e p r e s e n t a t i o n ‘The p h y s i c a l f e a t u r e s o f t h e b a s i n was t r a n s f o r m e d i n t o a c a p a c i t a t e d f l o w networA, a s shown i n r i g u r c 2 . The s t o r a g e ( r e s e r v o i r s ) and nons t o r a g e ( r i v e r c o n f l u e n c e , c a n a l d i v e r s i o n s , and demands) p o i n t s a r c r e p r e s e n t e d as n o d e s w h i l e t h c r i v e r r e a c h e s and c a n a l s a r c d e s i g n a t e d as l i n k s connecting t h e nodes.
D e s c r i p t i o n o f and d a t a f o r t h e n e t w o r k
components a r e g i v e n i n ‘Tables 1 and 2 . The e l e v c n m a j o r h i g h c o u n t r y r e s e r v o i r s were lumped t o g e t h e r .
Some
o f t h c r e s e r v o i r s on t h e p l a i n s w e r e s i m i l a r l y a g g r e g a t e d by o w n e r s h i p and d i v e r s i o n c a n a l s s e r v i n g them.
T h i s approach reduced t h e dimension
o r s i z e o f t h e nctworA w i t h o u t s a c r i f i c i n g management f l e x i b i l i t y , defined earlier. 41!
t h e d i v e r s i o n demands a r e i n d i v i d u a l l y c o n s i d e r e d .
have h i g h e r p r i o r i t i e s compared t o i d e a l s t o r a g e l e v e l s .
They a l s o Imports t o
t h e b a s i n t o augment t h e n a t u r a l s u p p l y were a l s o lumped t o g e t h e r . S u p p l e m e n t a l water s u p p l y d e l i v c r i e s from t h e C o l o r a d o - B i g Thompson p r o j e c t v i a l l o r s e t o o t h R e s e r v o i r was c o n s i d e r e d s e p a r a t e l y . The two 1JSGS g a g e s a t t h e mouth o f t h e canyon n e a r t h e C i t y of r‘ort C o l l i n s and a t t h e c o n f l u e n c e o f t h e Cache l a P o u d r e R i v e r w i t h t h e South P l a t t c R i v e r n e a r t h e C i t y of Greeley s e r v e d as an a d d i t i o n a l check i n t h c c a l i b r a t i o n r u n s and an i n d i c a t i o n o f t h e amount o f water leaving t h e system. Model calibration ____ C a l i b r a t i o n r u n s o f t h e networA model were made t o s i m u l a t e t h e e x t r e m e l y d r y ’ 7 6 - ’ 7 7 water y e a r . assigned.
To do t h i s , ” c o s t s ”
C . . must b e 1J ‘I’hese were p r o v i d e d by t h e I’oudrc R i v e r Commissioner, w i t h
some a d j u s t m e n t s .
Some m i n o r d i s c r e p a n c y between t h e h i s t o r i c a l d e s i r e d
Figure 2 .
Network representation of the Cache la Poudre r i v e r b a s l n ’ s physicdl features.
569 Table 1. Network node components, Cache la Poudre River Basin. Node
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21
Description Mountain Reservoirs (9 aggregated) North Poudre Company (20 aggregated) Windsor Reservoir and Canal ( 3 aggregated) Horsetooth Reservoir Water Supply and Storage (9 aggregated) Claymore Terry Lake Warren Lake Windsor Reservoir Fossil Creek Timnath Reservoir Windsor and Seeley Lake (2 aggregated ) City of Greeley Gage (terminal node) Demand Junction Ogilvy Demand Boyd and Freeman Demand Greeley No. 3 Demand Demand Junction Jones Demand Whitney Demand Eaton Demand
Node #
Description
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Greeley No. 2 Demand Demand Junction Demand Junction Boxelder Demand Chaffee Demand Coy Demand Lake Demand Demand Junction Arthur Demand Larimer and Weld Demand Demand Junction Larimer County No. 2 Taylor and Gill New Mercer Demand Little Cache La Poudre Demand Junction Jackson Demand Pleasant Valley and Lake Larimer County Demand City of Fort Collins Gage City of Greeley Diversion Demand Junction Demand Junction Poudre Valley Demand Fort Collins Diversion Mountain Ditch Demand North Poudre Canal Demand
Table 2.
Network link components. Cache la Poudre system
Link Number
From To Capacity Link Node Node Acre-Feet Number
~
1 2 3 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30' 31 32 33 34 35 36
1 43 43 43 43 2 2 43 44 44 44 3 3 43 4 4 41 41 37 5 37 37 5 5 37 37 6 39 6 37 32 32 32
7 7 32
43 46 47 48 2 48 5 44 2 3 45 45 5 41 44 41 42 37 40 40 5 38 38 9 39 6 39 10 29 32 35 36 7 36 9 34
300,000 1,680 3,240 23,100 23,100 23,100 23,100 21,000 21,000 21,000 21,000 21,000 45,000 300,000 91,000 91,000 2,500 300,000 42,000 42,000 45,000 3,000 3,000 60,000 7,500 7,500 7,500 7,500 7,500 300,000 3,600 7,500 7,500 7,500 60,000 720
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
70 71
From To Capacity Node Node Acre-Feet 32 32 18 33 32 29 29 29 9 29 29 24 24 24 24 11 24 23 10 23 11 23 18 18 18 18 18 12 12 18 14 14 14 14 3
8 33 33 10 29 30 31 9 31 22 24 26 25 27 28 28 23 10 18 11
12 18 21 19 12 22 20 14 22 14 17 16 16 13 9
9,000 9,000 9,000 9,000 300,000 3,600 60,000 60,000 60,000 31,500 300,000 1,500 3,300 600 9,600 9,600 300,000 15,000 15,000 15,000 15,000 300,000 1.800 960 36,000 36,000 4,200 36,000 36,000 300,000
9.000 600 4,800 300,000 60,000
570
storage levels and those obtained from the simulation was expected due to the adjustments made to account for reservoir evaporation and approximations of channel conveyance losses, seepage inflows and leakage outflows into and from the reservoirs. The difference between the computed and desired storage levels never exceeded 20% and were generally lower than 10%. This was deemed acceptable. The observed '76-'77total flow at the USGS gage near Fort Collins was miscalculated by only 3.8% on the high side. The calculated high flow months differ by 15% on the average while the low flow months are in error by 30%. The latter was not considered serious since the flow magnitudes are extremely low. Considering that some unmeasurable variables such as the losses and seepage inflows could only be roughly estimated, this calibration run was accepted without further adjustment of the priority factors. The final rankings of priorities for the various storage and diversion demands obtained from the calibration runs are shown in Tables 4 and 5. Management objective-minimize wasted outflows The estimate of the total seasonal streamflow based on the snowpack accumulations only allows the commissioner to determine whether that available supply is below, above or even with the normal u s c .
This
information is of benefit to appropriators who feel that they could not be served at all or that continuing service to fill up their needs could not be assured. Day to day water allocation activities of the river commissioner have been modeled by Thaemart (1976).
In this model, the
daily flows are simply apportioned by a direct search of users with the highest priority and deducting the amount entitled to divert in succession. Establishing monthly targets for reservoir storage and diversions, based on forecasted monthly streamflows for the duration of the season that can be updated in real-time, is one possible additional step to take. This would maintain the efficient and effective operation of the system
in anticipation of future streamflows. Through this approach, a basic framework could also be provided €or real-time operations over a shorter time interval, such as daily or weekly.
571 Table 3. Reservoir storage target levels as percent of total (full) capacity, Cache la Poudre River Basin (water year 1977-1978). ~
Reservoirs
C ~ D . Nov.
Dec.
Jan.
Feb.
Mar.
Am.
Mav
Jun.
Mountain Reservoir (11) NPC Reservoir (20) WRC (3) WSS Reservoir (9) Claymore Terry Lake Warren Lake Windsor Fossil Timnath Windsor and Seeley Reservoirs
42511 0.27 41121 0.62 41960 0.34 25553 0.69 954 0.64 8028 0.67 2089 0.66 16786 0.41 11100 0.64 1007C 0.00 2587 0.71
0.28 0.61 0.33 0.68 0.62 0.70 0.61 0.45 0.48 0.00 0.70
0.27 0.63 0.33 0.67 0.63 0.70 0.59 0.49 0.58 0.00 0.73
0.27 0.63 0.33 0.65 0.67 0.70 0.56 0.54 0.66 0.00 0.71
0.28 0.61 0.32 0.64 0.76 0.69 0.54 0.58 0.79 0.00 0.64
0.38 0.61 0.32 0.64 0.85 0.73 0.50 0.62 0.86 0.00 0.93
0.53 0.58 0.28 0.60 0.71 0.97 0.58 0.75 0.82 0.00 0.81
0.59 0.54 0.56 0.54 0.21 0.17 0.42 0.52 0.34 0.24 0.64 0.56 0.42 0.68 0.59 0.36 0.74 0.74 0.00 0.04 0.83 0.86
Table 4.
Jul.
AUE.
Seo.
Oct. -
0.30 0.30 0.10 0.31 0.43 0.48 0.83 0.11 0.37 0.15 0.72
0.15 0.28 0.18 0.37 0.02 0.28 0.57 0.11 0.25 0.28 0.65
0.14 0.29 0.18 0.26 0.00 0.36 0.51 0.14 0.36 0.31 0.69
Annual rankings for the ditch diversion demand nodes, Cache la Poudre river basin (lower numbers represent higher priority)
NODENO. 2 NODENO. 3 NODENO. 5 NODENO. 8 NODENO. 9 NODE NO. 12 NODE NO. 13 NODE NO. 15 NODE NO. 16 NODE NO. 17 NODE NO. 19 NODE NO. 20 NODE NO. 21 NODE NO. 22 NODE NO. 25 NODE NO. 26
YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1
RANK = 66 RANK = 33 RANK = 64 RANK = 4 0 RANK = 51 RANK = 4 6 RANK = 7 5 RANK = 71 RANK = 15 RANK = 38 RANK = 30 RANK = 17 RANK = 20 RANK = 4 6 RANK = 25 RANK = 5 6
NODE NO. 27 NODE NO. 28 NODE NO. 30 NODE NO. 31 NODE NO. 33 NODE NO. 34 NODE NO. 35 NODE NO. 36 NODE NO. 38 NODE NO. 39 NODE NO. 40 NODE NO. 42 NODE NO. 45 NODE NO. 46 NODE NO. 47 NODE NO. 48
YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1 YEAR 1
RANK = 22 RANK = 61 RANK = 4 3 RANK = 51 RANK = 4 0 RANK = 28 RANK = 53 RANK = 35 RANK = 69 RANK = 59 RANK = 64 RANK = 1 2 RANK = 33 RANK = 10 RANK = 48 RANK = 66
572 Table 5
Reservoir characteristics, Cache l a P o u d r e River Basin (lower n u m b e r s represent higher priority).
CAPACITIES I N ACRE-FEET
Name of Reservoirs
Minimum
Startine '
1. 2. 3. 4. 6. 7. 8. 9. 10. 11. 12.
11412 24406 15029 17074 448 4721 1441 6019 5782 0 1779
Mountain Reservoir (11) NPC Reservoir (20) WRC Reservoir ( 3 ) WSS Reservoir (9) Claymore Terry Lake Warren L a k e Windsor Reservoir Fossil Creek Timnath Windsor a n d Seeley ( 2 ) TOTAL
CASES
Maximum
147 155 134 166 100 141 113 163 152 164 151
42511 41121 41960 25553 954 8028 2089 16786 11100 10070 2475
0
0 0 0 0 0 0 0 0 0
0
202647
88111
I
System Outflows and Computed Reductions in Acre-Feet NOV-API Irrigation Season Subtotal Jun Jul May
Historical Gaged System Outflow Gage # 6752500 la/ Management Study Run ,=1 Computed Model Outflow Ib/ Computed Outflow Reduction (Stored) 2a/ Management Study Run +2 Target Minimum Outflow 2b/ Computed Outflow Reduction (Stored)
Rank Priorities
31,855 (66%)
May-Oct Annual Subtotal Total
AUK
Sep
Oct
1,701 556 6,584 1,721 1,273 4,372
30,971 (73%)
0
0
823 (15%)
1,701 556
646
0
360 (34%)
1 3 0 130
130
130
31,495 (67%)
1,571 426
5,938 1,721
16,207 (34%)
48,062 (100%)
914 2,92C 11,483 (27%)
42,468 (100%)
359 1,452
5,598 (100%)
130
60
4,714 (85%)
710 (66%)
6,454 1,591 1,143 4,312 15,497 (33%)
1,070 (100%) 46,992 (100%)
l a / MODSIM model computed system outflow, under the policy that the outflow is allowed t o fluctuate within the minimum and maximum capacity of the river reach and assigned the lowest priority demand. l b / Amount of water that was historically wasted but is available for storage on a month-to-month basis under the policy stated in l a above. 2a/ Target minimum system outflow set equal t o the recommended minimum streamflow for the Poudre River with assigned priority in between the lowest priority diversion and highest priority storage. 2b/ Amount of water that was historically wasted but is available for storage on a month-to-month basis under the policy stated in 2a above.
573 The h i s t o r i c a l summary o f t h e D i s t r i c t ' s a n n u a l o p e r a t i o n as c o m p i l e d by t h e P o u d r e R i v e r Commissioner, shows t h a t , on a n a n n u a l b a s i s , an amount r a n g i n g from 7 , 0 0 0 t o 1 5 1 , 0 0 0 a c r e - f t . o f water ( f o r 23 y e a r s o u t o f 29) was l o s t from t h e D i s t r i c t b e c a u s e o f i n a b i l i t y t o convey t o s t o r a g e .
RctCk 0 6 c i t 0 k U g Q
or
E v e r y y e a r f o r t h e l a s t 30 Years of t h e
D i s t r i c t o p e r a t i o n , a n a v e r a g e amount of 9 6 , 0 0 0 a c r e - f t . o f water d r a i n e d t o t h e S o u t h P l a t t e R i v e r , as o b s e r v e d a t t h e USGS g a g e n e a r t h e C i t y of G r e e l e y .
T h i s amount i s r o u g h l y 35% of t h e a v e r a g e a n n u a l
v i r g i n f l o w and r e p r e s e n t s 20% o f t h e a v e r a g e t o t a l water s u p p l y from a l l sources.
I f a f r a c t i o n o f t h i s amount c o u l d b e s a v e d , or s t o r e d
and u s e d , a d d i t i o n a l a p p r o p r i a t o r s c o u l d b e s e r v e d . To d e t e r m i n e t h e w o r t h o f t h e m o n t h l y s t r e a m f l o w f o r e c a s t , t h e model PlODSIM was u s e d t o o b t a i n m o n t h l y o p e r a t i o n p o l i c i e s , t h a t would r e d u c e
o u t f l o w s from t h e Cache l a P o u d r e R i v e r s y s t e m , a s s u m i n g h i s t o r i c a l d i v e r s i o n demands must b e s a t i s f i e d .
Water y e a r ' 7 6 - ' 7 7 was a g a i n c h o s e n ,
b u t f o r e c a s t e d streamflows were u s e d r a t h e r t h a n a c t u a l o b s e r v e d f l o w s . The o b s e r v e d m o n t h l y s t r e a m f l o w s f o r t h e s i x (6) months p r i o r t o May
were u s e d t o f o r e c a s t t h e s u c c e e d i n g s i x (6) months s t r e a m f l o w .
The
f o r e c a s t was m o d i f i e d somewhat u s i n g t h e i n f o r m a t i o n o b t a i n e d from t h e r e s u l t s of t h e l a s t ( A p r i l ) snow c o u r s e s u r v e y and i n p u t i n t o bIOI)SI?~l. The h i s t o r i c a l r e s e r v o i r s t o r a g e l e v e l s were s p e c i f i e d a s i d e a l t a r g e t l e v e l s a n d h i s t o r i c a l d i v e r s i o n demands were u s e d .
However, t h e
C.. l d
p r i o r i t i e s from t h e c a l i b r a t i o n r e s u l t s were m o d i f i e d t o d i s c o u r a g e wasted outflows.
T h i s i m p l i e s some f o r e k n o w l e d g e u s e d i n t h e e x p e r i m e n t .
However, a s p e c i f i c minimum b a s i n o u t f l o w t a r g e t was s e t w i t h a h i g h e r p r i o r i t y t h a n t h e r e s e r v o i r s , so u s e o f t h e h i s t o r i c a l s t o r a g e l e v e l s
as t a r g e t l e v e l s d i d n o t a p p r e c i a b l y e f f e c t t h e r e s u l t s .
As
for u s e
o f t h e h i s t o r i c a l d i v e r s i o n demands, we a r e a s s u m i n g t h a t t h e s e c a n be f o r e c a s t e d w i t h r e a s o n a b l e a c c u r a c y on a month t o month lxisi s . Cornptational llesults
R e s u l t s a r e summarized i n T a h l e 6 .
Flanagement r u n ? I a l l o w e d t h e
b a s i n o u t f l o w t o f l u c t u a t e f r e e l y , e s s e n t i a l l y u s i n g t h e same p r i o r i t i e s as from t h e c a l i b r a t i o n r u n s .
C.. 'I 'I'he s p e c i f i e d d i v e r s i o n demands
h a v i n g a h i g h e r p r i o r i t y were s a t i s f i e d ;is e x p e c t e d .
'['he t o t a l s y s t e m
574 computed o u t f l o w s were l o w e r by 1 2 % compared t o t h e o b s e r v e d o u t f l o w s
a t t h e gage n e a r G reeley.
The t o t a l d i f f e r e n c e , 5 , 5 9 8 a c r e - f t . , was
s t o r e d i n r e s e r v o i r s w i t h h i g h p r i o r i t y and s u f f i c i e n t u n u s e d c a p a c i t y . Notice t h a t t h e observed system outflow during t h e i r r i g a t i o n season (May-Oct)
i s o n l y a b o u t 34% of t h e a n n u a l t o t a l r e c o r d e d w a s t e d water
l e a v i n g t h e system.
T h i s i s an i n d i c a t i o n o f t h e r e l a t i v e l y e f f i c i e n t
manner o f water use and s y s t e m management i n t h e a r e a , p r i m a r i l y b a s e d
on t h e d o c t r i n e of p r i o r a p p r o p r i a t i o n .
The o p p o r t u n i t y t o s a v e a
g r e a t e r amount o f t h e w a s t e d water a p p e a r s t o b e d u r i n g t h e f a l l and w i n t e r months, when r o u g h l y t w o - t h i r d s o f t h e a n n u a l t o t a l h a s o c c u r r e d . T h e o r e t i c a l l y , t h e amount l e a v i n g t h e s y s t e m c a n be r e d u c e d t o z e r o ( t h i s has a l r e a d y o c c u r r e d i n some s e c t i o n s o f t h e r i v e r , C o l o r a d o a n , 1 9 7 4 ) s i n c e t h e r e i s no l e g a l l y b i n d i n g c o n t r a c t or a g r e e m e n t f o r t h e
Poutlre R i v c r t o p a s s a c e r t a i n amount o f water t o t h e S o u t h P l a t t e River.
I n r e a l i t y , e s p e c i a l l y i n t h e lower r e a c h e s , t h e r e would be canal
conveyance l o s s e s and farm w a s t e s from b o t h s i d e s o f t h e r i v e r .
For
management r u n # 2 , t h e amount o f r e t u r n f l o w was a p p r o x i m a t e d b y t h e recommended minimum m o n t h l y s t r e a m f l o w , which amounts t o 60 a c r e - f t . from O c t o b e r t h r o u g h A p r i l arid 130 a c r e - f t . from May t o September (Rhinehart, 1975).
T h e s e minimum monthly f l o w v a l u e s were t h e r e f o r e
d e s i g n a t e d as t h e t a r g e t minimum o u t f l o w s from t h e Poudre R i v e r b a s i n . The p r i o r i t y a s s i g n e d t o t h i s f l o w was made lower t h a n t h e l o w e s t p r i o r i t y f o r d i v e r s i o n demand b u t h i g h e r t h a n t h e h i g h e s t p r i o r i t y f o r t h e s t o r a g e demand Table 6.
T h e s e r e s u l t s a r e shown i n t h e l a s t two rows o f
The improvement i s s i g n i f i c a n t .
The amount o f water t h a t was
wasted and c o u l d h a v e been s t o r e d a p p e a r s i n t h e r e s e r v o i r s which h a v e h i g h e r p r i o r i t y and s u f f i c i e n t c a p a c i t y . The c a l c u l a t e d end o f month s t o r a g e , i n some i n s t a n c e s , i s h i g h e r than t h e desired o r t a r g e t storage l e v e l s .
T h i s t a r g e t s t o r a g e i s being
u t i l i z e d i n t h e MODSIM model as a g u i d e ; h e n c e , t h e water l e v e l s c a n f l u c t u a t e between t h e d e s i r e d l e v e l and maximum s a f e l e v e l f o r e a c h of the reservoirs.
The amount t h a t i s s t o r e d d e p e n d s on t h e p r i o r i t y
assigned t o thc reservoir.
F i g u r e 3 shows t h a t Management r u n i f 2
r e s u l t e d i n t h e c a p t u r e and s t o r a g e o f c o n s i d e r a b l y more water t h a n
575
occurred historically for Fossil Creek Reservoir. Note that the maximum reservoir storage levels were never actually reached in a l l months. As an illustration of the kind of improved storage strategies the
model produces, consider a reach of the Poudre River shown in Figure 4. Fossil Creek reservoir is located south of the river, but its
releases to the river could serve several ditches downstream. Greeley No. 2 (or Cache la Poudre No. 2) canal is the last ditch where it is possible to divert and store river flows at either Windsor, Neff or Seeley Lakes. Any flows passing by this ditch could satisfy demands at Whitney, B. H. Eaton, Jones, Greeley No. 3, Boyd and Freeman, and Ogilvy ditches. Any excess flows beyond the last ditch, i.e., passing by Ogilvy canal, drain into the South Platte river and are lost downstream. Out-of-priority diversion of the flows that would otherwise be lost from the system could be stored at Fossil Creek, Windsor Lake or Seeley Lake. Any shortfalls at Ogilvy could be served by Windsor or Seeley Lakes.
Shortages at any of the other five (5) ditches could be easily
filled by releases from Fossil Creek Reservoir. MODSIM could therefore be useful in devising voluntary exchange and trade schemes that will greatly benefit a l l the users in the basin. FINAL, NOTES The results presented here need to be qualified according to the assumptions made.
First, since all streamflows are in average monthly
values, there is no guarantee that computed average monthly diversions could actually be realized during the month. This is because an intense thunderstorm, €or example, could produce a large flow in a short period of time which could not be captured because of insufficient canal capacity; even if there is sufficient offstream storage capacity. A large onstream reservoir could, of course, capture a significant
portion of this flow. This study area does not currently have such a reservoir. This problem could be indirectly considered by reducing the effective canal capacity by an amount based on an analysis of average daily flows within the month in relation to daily available canal
576
1.0-
-
---_
--__
-
c
. X_
dd
Usable storage = 11,100 acre-feet _ -- Storage level alternative MODSIM run
01 0
2
Fossil Creek r e s e r voir capacity= 11,508 acre-feet
2
- Desired storage level water year 7976 -77 Historical storage level water year 1965- 66
0.5-
E"
+
.'
c C
u
z
3provided by the Poundre River commissioner
.' ..,.'
a 1
I
NOV.
I
I
I
I
DEC. .JAN. FEE. MAR. APR.
Fiwre 3 .
I
MAY
I
I
JUN. JUL.
I
I
AUG. SEP
I
OCT. Months
:.. s t o l q e ievels under a reduced system outflow case and relationship with the historical storage levels for the lowest f l o w year (1976-1977) and comparable low flow year *I--^_.._.
nTi_l-..yIL
(1965-1966), Fossil Creek Reservoir, Cache la I'oudre River Basin
Greeley No.2 ditch (also Cache la Poudre No.2 ditch)
7 t Seeley lake
South Platte River ',Creek
Cache la Poudre River 1
Fossil
,' Creek Fossil Creek Reservoir
Figure 4 .
' Jones ditch
USGS gage near Greeley
E.H Eaton ditch
Schematic diagram to illustrate the out o f priority diversion for storage and subsequelit release strategy for ditch diversion demand satisfaction t o minimize outflows from n water resource system, using the Cache la Poudre basin as an example.
577 capacity.
An i t e r a t i v e p r o c e d u r e c o u l d b e d e v i s e d between t h e MODSIM
model and a d a i l y f l o w a c c o u n t i n g model i n o r d e r t o d e t e r m i n e how much t h e e f f e c i v e c a n a l c a p a c i t y shou d b e r e d u c e d . A s mentioned e a r l i e r , t h e r e i s a q u e s t i o n a b o u t t h e s a f e t y o f many o f
the offstream reservoirs. filled.
T h i s p r e v e n t s s e v e r a l o f them from b e i n g
Though t h e t a r g e t maximum s t o r a g e l e v e l s used i n t h i s s t u d y a r e
b e l i e v e d t o b e r e a s o n a b l e , f u r t h e r work i s needed t o r e f i n e them.
They
may a l s o need t o b e m o d i f i e d i n a c c o r d a n c e w i t h e x p e c t e d i c i n g c o n d i t i o n s d u r i n g w i n t e r t h a t would r e d u c e e f f e c t i v e s t o r a g e c a p a c i t y . A l s o , t h e model d o e s n o t g u a r a n t e e t h a t s t o r a g e r i g h t s w i l l b e e x e r c i s e d i n o r d e r of p r i o r i t y .
The w e i g h t i n g f a c t o r s
C..
on c a r r y o v e r s t o r a g e
11
p r i m a r i l y c o n t r o l t h e r e l e a s e o f water r a t h e r t h a n p r i o r i t i e s i n initially f i l l i n g the reservoirs.
F u r t h e r work i s needed t o more
a c c u r a t e l y i n c l u d e f i l l i n g p r i o r i t i e s i n t h e model. F i n a l l y , s i n c e t h i s model u s e s a s e q u e n t i a l s t a t i c a p p r o a c h t o t h e o p t i m i z a t i o n , t h e n it c a n n o t f u l l y u t i l i z e a n e x t e n d e d , m u l t i p e r i o d f o r e cast.
R a t h e r , a one-month ahead f o r e c a s t i s t h e most i n f o r m a t i o n i t
can a c t u a l l y u s e . a g i v e n month.
The model t h e n o p t i m i z e s a l l o c a t i o n o f w a t e r w i t h i n
However, u s i n g , s a y , a s i x month f o r e c a s t , t h e
weighting f a c t o r s
C.. 1J
c a n be a d j u s t e d t o r e f l e c t l a t e r problems
a n t i c i p a t e d by t h e s t r e a m f l o w f o r e c a s t .
F o r example, i f a d r o u g h t
period i s forecasted, weighting f a c t o r s
C.
4
on s t o r a g e c o u l d b e
a l t e r e d t o t a k e s h o r t a g e s e a r l i e r and r e t a i n more w a t e r i n t h e r e s e r v o i r s as a hedge on t h e coming d r y p e r i o d .
F u r t h e r work i s needed
on f u l l y i n c o r p o r a t i n g i n t o MODSIM i n f o r m a t i o n o b t a i n e d from t h e f o r e c a s t model, p a r t i c u l a r l y t h e s t r e a m f l o w f o r e c a s t c o v a r i a n c e e s t i m a t e s ,
i n o r d e r t o p r o p e r l y a n a l y z e t h e r i s k a s s o c i a t e d w i t h v a r i o u s management s c e n a r i o s . ACKNOWLEDGEMENTS T h i s r e s e a r c h was p a r t i a l l y s u p p o r t e d by f u n d i n g p r o v i d e d by t h e Office o f Water R e s e a r c h and Technology, U . S. Department o f I n t e r i o r
( a u t h o r i z e d u n d e r P . L. 95-467) as a u t h o r i z e d by t h e Water R e s e a r c h and Development Act of 1978 and t h e U . S . - S p a n i s h P r o j e c t : C o n j u n c t i v e Water Uses o f Complex S u r f a c e and Groundwater S y s t e m s .
578 REFERENCES Anderson, R . L . , "The E f f e c t s of S t r e a m f l o w V a r i a t i o n on P r o d u c t i o n and Income o f I r r i g a t e d Farms O p e r a t i n g Under t h e D o c t r i n e o f p r i o r A p p r o p r i a t i o n , " Water R e s o u r c e s R e s e a r c h , V o l . 11, No. 1, 1975. B a r k l e y , J . R . "The N o r t h e r n C o l o r a d o Water C o n s e r v a n c y D i s t r i c t , " Loveland, C o l o r a d o , 1 9 7 4 .
Bazaraa, M . S . and J . J . J a r v i s , " L i n e a r Programming and Network Flows," J o h n Wiley and S o n s , I n c . , 1977. Becker, L and W-G Yeh, " O p t i m i z a t i o n of Real-Time O p e r a t i o n o f a M u l t i p l e R e s e r v o i r System," Water R e s o u r c e s R e s e a r c h , V o l . 1 0 , No. 6 , December, 1974.
Be1 1amy, R . , " P e r s o n a l communication,
I'
1980.
Box, G. E . P . and G . M . J e n k i n s , "Time S e r i e s A n a l y s i s , F o r e c a s t i n g and C o n t r o l , " Second E d i t i o n , Holden-Day, 1976. C o l o r a d o a n , "140 T r o u t K i l l e d i n S e c t i o n of P o u d r e R i v e r , " Newspaper a r t i c l e on t h e D r y i n g up o f t h e Poudre R i v e r , F o r t C o l l i n s , September 11, 1974. Graupe, D . , " I d e n t i f i c a t i o n o f S y s t e m s , " Second E d i t i o n , R o b e r t F . K r e i g e r P u b l i s h i n g C o . , H u n t i n g t o n , N e w York, 1 9 7 6 . Ilawley, b f . E . , R . H . McCuen and A . Rango, "CompariTons o f Models f o r F o r e c a s t i n g Snowmelt Runoff Volumes," Water R e s o u r c e s B u l l e t i n , Vol. 1 6 , No. 5, 1980. Hoshi, K . and S . J . Burges, " I n c o r p o r a t i o n o f F o r e c a s t e d T o t a l S e a s o n a l Runoff Volumes i n t o R e s e r v o i r Management S t r a t e g i e s , " R e l i a b i l i t y i n Water R e s o u r c e s Management, McBean, E . A . and T. F . Unny ( e d i t o r s ) , Water R e s o u r c e s P u b l i c a t i o n s , F o r t C o l l i n s , C o l o r a d o , 1979. L a b a d i e , J . W . , J . M . S h a f e r and R . Aukerman, " R e c r e a t i o n a l Enhancement o f High C o u n t r y Water S u p p l y R e s e r v o i r s , " Water R e s o u r c e s B u l l e t i n , V o l . 1 6 , No. 3 (June 1 9 8 0 ) . L a z a r o , R . C . , L a b a d i e , J . In;. and J . D . Salas, " S t a t e - S p a c e S t r e a m f l o w F o r e c a s t i n g Model f o r Optimal R i v e r B a s i n Management," p r e s e n t e d a t t h e I n t e r n a t i o n a l Symposium on Real-Time O p e r a t i o n of Hydrosystems, J u n e 24-26, U n i v e r s i t y o f W a t e r l o o , IVaterloo, O n t a r i o , Canada, 1 9 8 1 . bicKerchar, A . I . and J . IV. D e l l e u r , " A p p l i c a t i o n o f S e a s o n a l Parametric L i n e a r S t o c h a s t i c blodels t o Monthly Flow Data," Water R e s o u r c e s R e s e a r c h , Vol. 1 0 , No. 3, 1974. bfejia, J . M . , P . E g l i and A. L e c l e r c , "Evaluating M u l t i r e s e r v o i r O p e r a t i n g R u l e s , 1 1 Water R e s o u r c e s R e s e a r c h , Vol. 1 0 , No. 6 , December, 1 9 7 4 .
579 Movarek, I. E., M. H. Salem and H. T. Dorrah, "Hydrological Studies on the River Nile - I. Forecasting," Research Report, Cairo University/MIT, Technological Planning Program, Cairo, 1978. Neutze, J., 'Tersonal communication," 1980. Panu, U. J. and T. F. Unny, "Extension and Applications of Feature Prediction Model for Synthesis of Hydrological Records," IVater Resources Research, Vol. 16, No. 4, 1980. Radosevich, G. E., D. H. Hamburg and L. L. Swick, "Colorado Water Laws, A Compilation of Statutes, Regulations, Compacts and Selected Cases," Information Series No. 17, Center for Economjc Education and Environmental Resources Center, Colorado State University, 1975. Reitano, B. M. and D. W. Hendricks, "Input-Output Modeling of the Cache la Poudre Water System," Environmental Engineering Technical Report 78-1683-01, Dept. of Civil Engineering, Colorado State University, Ft. Collins, Colorado, Nov., 1978. Rhinehart, C. G . , "Minimum Streamflows and Lake Levels in Colorado," Environmental Resources Center, Information Series No. 18, Colorado State University, August, 1975. Salas, J. D., Delleur, J. IY. Yevjevich, V. and Lane, IV., 1980, "Applied blodeling of Hydrologic Time Series," Water Resources Publications, Littleton, Colorado. Shafer, J. b l . , "An Interactive River Basin Water Management Model: Synthesis and Applications," Technical Report No. 18, Colorado Water Resources Research Institute, Colorado State University, 1979. Shafer, J. b l . , Labadie, J. 1V. and E. Bruce Jones, "Analysis of Firm Water Supply Under Complex Institutional Constraints," Water Resources Bulletin, June, 1981. Texas Water Development Board, "Economic Optimization and Simulation Techniques for blanagement of Regional Water Resource Systems, River Basin Simulation Model SIMYLD-II --Program Description," Prepared by the Systems Engineering Division, Austin, Texas, July, 1972. Thaemart, R. L., "Vathematical Model of Water Allocation blethods," Ph.D. Dissertation, Colorado State University, 1976. United States Bureau of Reclamation, "Western Division h'ater Supply Forecasting: Electronic Computer Program Description," Loveland, Colorado, 1968. IJnny, T. C . , Divi, R., fiinton, B. and A. Robert, Proceedings of the International Symposium on Real -'I ime Operation of Hydrosystems," Volumes I and 11, University of IVatcrloo, Il'aterloo, Ontario, Canada, June 24-26, 1981. Wilson, .J. R. and E. Kirdar, "llse of RunofF Forecasting in Reserv o i r Operations," ,Journal of the Trrigation and Ilrninnge Division,
580
Proceedings of the ASCE, Vol. 96, IR3, September, 1970. Wunderlich, W. 0. and J. E. Giles, Proceedings of the International Symposium on Real-Time Operation of Hydrosystems," Volumes I and 11, University of Waterloo, Waterloo, Ontario, Canada, June 24-26, 1981.
581
APPROPRIATE SAMPLING PROCEDURES FOR ESTUARINE AND COASTAL ZONE WATER- Q UAL ITY MEAS UREMENTS
M. HILLg’
and JOHN
DONALD STEVEN GRAHAd’
Introduction Many
environmental
problems
in
e s t u a r es
involve
relating
changes i n w a t e r q u a l i t y t o some h y p o t h e s i z e d b i o l o g i c a l e f f e c t ; o r r e l a t i n g ecosystem r e s p o n s e t o f o r c i n g f u n c t ons, t a n t o f which i s u s u a l l y water q u a l i t y . traditional
sampling
techniques
often
t h e most i m p o r -
However, used do n o t
t h e associated i n c l u d e con-
s i d e r a t i o n o f d e t e r m i n i s t i c and p r e d i c t a b l e hydrodynamic and w a t e r quality variations. U n t i l r e c e n t l y e c o l o g y was p r i m a r i l y a q u a l i t a t i v e s c i e n c e and s u r v e y s t e n d e d t o be used t o a s s i s t d e s c r i p t i v e s t u d i e s . past
few
years
the
emphasis
has
been
placed
on
I n the
statistically
d e s i g n e d s t u d i e s [ s e e S m i t h (1978)] w h i c h a r e b e i n g used t o t e s t hypotheses
r e g a r d i n g model
v a r i e s g r e a t l y however, i) ii)
1/ -?./
validity.
The d e g r e e o f
complexity
b u t f o u r g e n e r a l c a t e g o r i e s can be d e f i n e d :
deterministic statistical
a)
b i o l o g i c a l and w a t e r - q u a l i t y o n l y
b)
b i o l o g i c a l and w a t e r - q u a 1 i t y c o u p l e d t o h y d r o d y n a m i c s .
Tudor E n g i n e e r i n g Company, San F r a n c i s c o , C a l i f o r n i a , 94105 C i v i l E n g i n e e r i n g Department, L o u i s i a n a S t a t e U n i v e r s i t y , B a t o n Rouge, L o u i s i a n a , 70803
Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors)
o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
582
The
simplest
difficult the
of
type
ecological
An example of
i s (ib).
l a t t e r N a j a r i a n and Harleman
model
is
(iia),
the
most
t h e f o r m e r i s TECO (1980), (1977).
and
The advantage o f t h e
l a t t e r i s t h a t i t i s p r e d i c t i v e if c i r c u m s t a n c e s a r e a l t e r e d .
Use
o f models o f t y p e ( i b ) r e q u i r e s c o n s i d e r a t i o n o f t i m e s c a l e s i n t h e sampling
and
e ngineering, ticians.
analyses.
Such
are
standard
i n fluids
b u t appear t o be q u i t e novel t o b i o l o g i s t s and s t a t i s It
is
the
purpose
concepts i n a g e n e ra l way, great
concepts
advantages
of
o f t h i s paper
w i t h examples,
to
i n t r o d u c e these
and t o i l l u s t r a t e t h e
u s i n g 1 i n k e d hydrodynamic
and water-qua1 i t y
models i n a s s o c i a t i o n w i t h e c o l o g i c a l s t u d i e s i n dynamic e n v i r o n ments.
!-lowever
t h e r e l e v a n c e o f d i s c u s s i o n i s n o t l i k e l y t o be
limited t o biologists, pointed out, tion
for,
as N a j a r i a n and Harleman (1977) have
t h e t r a d i t i o n a l methods o f water-qua1 i t y d a t a c o l l e c -
i n transient
environments
by f i e l d
sampling
teams may be
o b s o l e t e i n s o f a r as th e s e d a t a a re now o f t e n used as i n p u t t o r e a l time
numerica l
models.
Because
the
e n g i n e e r i n g works must now be s t u d i e d , gists
and eng i n e e rs
has
environmental
impacts
of
i n t e r a c t i o n between b i o l o -
i n c re a s e d m a r k e d l y .
bwever,
f o r many
p r a c t i c a l e n g i n e e r i n g a p p l i c a t i o n s where b i o l o g i c a l i n f o r m a t i o n i s needed,
such as e n v i ro n m e n ta l impact statements, t h e d a t a p r o v i d e d
a r e n o t c o m p a t i b l e w i t h t h e re q u i re m e n t s o f water q u a l i t y models.
If t hese data, which are e s s e n t i a l l y t i m e s e r i e s , a r e t o be u s e f u l , th e n c e r t a i n sampling c o n s i d e r a t i o n s must be t a k e n i n t o account. The purpose o f t h i s paper specific
examples
for
i s t o o u t l i n e i n general terms,
illustration,
some
with
considerations
a c q u i r i n g usef u l d a t a i n e s t u a r i n e ‘ i d c o a s t a l environments.
for It i s
i n t e n d e d t o be p r a c t i c a l and i l l u s t r a t i v e , r a t h e r t h a n r i g o r o u s . A d vect ion and D i s p e r s i o n Unfortunately, for
environme n ta l
t h e t r a d i t i o n a l method o f s u rv e y s
has
biological
been t o measure b i o t a
sampling and water
583
SCALE MILES ~
FIGURE
1
YEROPLANKTON SAMPLING STATIONS 1976-1977
qua1 i t y v a r i a b l e s s i m u l t a n e o u s l y a t
rather
long time
intervals.
vs. A f r o m B ( t ) and The two are sometimes t h e n c o r r e l a t e d as B A ( t ) , a l t h o u g h even t h i s i s o f t e n n o t done. An exarnple o f a t y p i c a l sampling scheme i s shown i n F i g u r e 1 i n which t e n s t a t i o n s
were sampled a t two-week i n t e r v a l s i n o r d e r t o p r e d i c t t h e e f f e c t s
584
o f an i n c r e a s e i n t h e h e a t d i s c h a r g e f r o m t h e Bend Bend S t a t i o n i n Tampa Bay,
A t y p i c a l t i m e s e r i e s o f l a r v a e numbers i s
Florida.
shown i n F i g u r e 2 (TECO,
1980).
T h i s graph f u r t h e r demonstrates
t h e need t o change t r a d i t i o n a l sampling schemes t o t a k e i n t o cons i d e r a t i o n t h e n a t u r e o f an ever-changing p o p u l a t i o n i n a dynamic water body.
A l s o shown i n F i g u r e 1 i s t h e o u t p u t o f a s i m p l e temfrom TECO
p e r a t u r e plume model
(1980)
showing t h e temperature
g r a d i e n t a t 6 4 % p l a n t l o a d under t h e assumption t h a t t h e r e are no t i d e s i n Tampa Bay.
Obviously,
f o r t h with the t i d e i n r e a l i t y .
O
t h e plume i s advected back and !-!owever, n o t e t h e f o l l o w i n g f a c t s :
t h e temperature d i f f e r e n c e along t h e plume i s o f t h e o r d e r o f 6OC,
which
i s a l s o t h e o r d e r o f t h e annual
ambient
temperature v a r i a t i o n t h e temperature g r a d i e n t along t h e plume i s l i k e l y g r e a t e r t h a n t h e seasonal temperature g r a d i e n t '
t h e temperature g r a d i e n t across t h e plume i s v e r y high, r e l a t i v e t o t h e along-plume o r seasonal change
O
because
a
plume
instantaneous
is
gradient
a
separated-flow would
be
much
phenomenon, larger
than
the a
vs. b e l l - s h a p e d ) t i d a l l y - a v e r a g e d one ( t o p - h a t -
O
t h e plume v e l o c i t y would be o f t h e same o r d e r as, o r l e s s than,
t h e t i d a l rms v e l o c i t y a t a d i s t a n c e n o t f a r f r o m
t h e source
O
if t h e plume i s o f w i d t h B and t h e rms t i d a l v e l o c i t y i s o f o r d e r Ut,
then
585
FIGURE 2 D e n s i t y of Menippe Mercenaria Larvae a t Inshore S t a t i o n s 1976-1977
JAN
FEB
M A R C H APRIL
MAY
JUNE
JULY
Day
-
i
!
o JAN
Night
Legend ~~~~
_
_
Source
0 8 I 6 I ~111 13 TCCO, 1980
-
I 16
.. . . . . . . I
3
- - - - ~ - BC 8
AUG
SEPT
OCT
NOV
DEC
586
I t i s e v i d e n t t h e n t h a t b o t h t h e temporal and s p a t i a l s c a l e s o f t h e p h y s i c a l and b i o l o g i c a l sampling are n o t a p p r o p r i a t e t o t h e s c a l e s
o f t h e hydrodynamic phenomena.
Furthermore, s i n c e t h e y cannot be
r e l a t e d , t h e e f f e c t o f a change i n t h e l a t t e r upon t h e f o r m e r cannot predicted. Using t h e TECO example t o p r o v i d e a general theme,
-diffusive
s i o n o f a d v e c t i v e vs. that
a water
quality or
a discus-
e f f e c t s i s now i n o r d e r .
biological
variable
is
Assume
a function
of
another ( D O as te m p e ra tu re f o r i n s t a n c e ) :
(2)
DO = f n ( T )
Now, t emperat ure i s q u a s i - c o n s e r v a t i v e and can be approximated i n a two-dimensional E u l e r i a n C a r t e s i a n frame i n t h e x - d i r e c t i o n b y
T - t e mp e ra tu re
where
U - velocity i n x-direction
-
V
velocity i n y-direction
-
Ki
dispersion coefficients
Te
-
e q u i l i b r i u m te mp e ra tu re
Ti
-
sources o f heat
To
-
sinks o f heat
Q
h e a t exchange c o e f f i c i e n t
Note t h a t t h i s e q u a t i o n i s l i n e a r i n
T.
Further, note t h a t i f
T were c o m p l e t e l y c o n s e r v a t i v e and steady, t h e n
-U+a Tax
UaT ay
-
a K - aT+ ax
x ax
a aY
K -
aT
y aY
(4)
where t h e terms on t h e r h s r e p r e s e n t a d v e c t i v e t r a n s p o r t and on t h e
587
l e f t dispersive transport.
@.
same t h i n g ,
Both are d i f f e r e n t expressions o f t h e
motion.
Most b i o l o g i c a l
(and a s s o c i a t e d water-
qua1 i t y ) sampl i n g schemes assume d i s p e r s i v e processes t o c o m p l e t e l y dominate.
T h i s i s o n l y c o r r e c t f o r c e r t a i n t i m e s c a l e s i f U, V # 0
i n an E u l e r i a n frame however; sion,
the
entire
indeed, except f o r m o l e c u l a r d i f f u -
d i s p e r s i v e term
r e s u l t s f r o m averaging.
is
inherently f i c t i t i o u s
and
Using t h e usual conventions f o r disag-
gregating the j o i n t time-series:
U = U + U '
T = T + T '
- -
-
UT = UT + UT'
+ U'T + U ' T '
Averaging UT t h e n r e s u l t s i n
- - UT = UT + U ' T ' so t h a t t h e t r u e a d v e c t i v e t e r m a/ax(UT)
i n equation 3 i s rep-
r e s e n t e d i n an averaged f o r m a t by
- -- U- aT+
( -a KaT)
ax
ax
ax
where t h e d i f f u s i o n analogy i s used f o r t h e second t e r m on t h e r h s of
e q u a t i o n 9 by assumption o f
magnitude o f
a lengthscale.
Therefore,
the
t h e d i s p e r s i v e term m e r e l y r e p r e s e n t s t h e a l i a s i n g
e f f e c t s o f t h e l e n g t h s c a l e s and t i m e s c a l e s by t h e sampling scheme used.
Sampling procedures suggested by Smith (1978) do n o t attempt
t o m i n i m i z e t h i s term.
588
FICUEE 3 Water-Quality Data A t Apalachicola, Florida
~~
(./OO)
SALINITY
APALACHICOLA CITY
( ' / o )
.20
40
-0
DISSOLVED OXYGEN
''"9
I6
S o u r c e : Graham e t a l . ,
1978
589
The e f f e c t s due t o a d v e c t i o n o f t e n dominate a t t h e s m a l l t i m e and l e n g t h s c a l e s o f c o l l e c t i o n o f t h e i n d i v i d u a l sample.
During
t h e sampling t i m e a t a s t a t i o n t h e t i d e can be e i t h e r i n o r out,
A
f o r instance.
sampling s t a t i o n i n F i g u r e 1 can e i t h e r be i n s i d e
o r o u t s i d e t h e plume w h i l e t h e sample i s b e i n g t a k e n .
Therefore,
f o r t h e d a t a t o have meaning i n r e g i o n s w i t h sharp g r a d i e n t s a t s h o r t sampling t i m e s ( b u t n o t n e c e s s a r i l y i n t e r v a l s ) , one must know t h e r e l a t i o n between t h e d a t a and t h e v e l o c i t y f i e l d . Another F i g u r e 3,
example
is
p r o v i d e d f r o m Graham e t
which d e p i c t s w a t e r - q u a l i t y
the Apalachicola River,
(1978)
al.
as
d a t a t a k e n a t t h e mouth o f
F l o r i d a over
a t e n hour t i m e i n t e r v a l .
Note t h e r a p i d decrease i n s a l i n i t y f r o m 20 t o 0 p p t ,
and t h e
a s s o c i a t e d r i s e i n DO f r o m 0 t o 5 mg/l near t h e b o t t o m as a r e s u l t of
tidal
adv e c ti o n .
Obviously,
an animal
with
a short-period
t h r e s h o l d DO t o l e r a n c e o f 2.5 mg/l would s u r v i v e a t t h i s l o c a t i o n on t h e average,
b u t be dead i n r e a l i t y .
A r a p i d sample t a k e n h e r e
a t a b i w e e k l y i n t e r v a l c o u l d have a v e r y l a r g e v a r i a t i o n , b u t t h i s would
be q u i t e p r e d i c t a b l e . they
were
Finall.,
ta k e n
at
t h e d a t a would
the
bottom,
top,
be q u i t e
different
if
or
depth-
averaged.
The p o i n t o f t h i s example i s t h a t a t t h e t i m e s c a l e s and
l e n g t h s c a l e s we are o f t e n most i n t e r e s t e d i n , o r t e n d t o sample i n , advection i s important r e l a t i v e t o diffusion-dispersion. The e f f e c t o f t h e a d v e c t i v e t e r m e n t e r s i n t h e l h s o f e q u a t i o n
3.
I n a 2-dimensional h o r i z o n t a l E u l e r i a n frame t h e u s u a l f o r m o f
t h e momentum e q u a t i o n f o r t i d a l f l o w i s :
-a+u - + -uau at
ax
uav ay
fV
--
+
gH- a q
-C,(U
C D p l O l ulox
ax 2
2 1/2
+ V )
u H2
590
-
a ( E xy (-au + -)) av -a (Exx -)au - ax
aY
ay
ax
-Mx=O where
U - vertically-averaged velocity in x-direction V - vertically-averaged velocity in y-direction f - Coriolis parameter, = 2 w sin (latitude) D - depth below MLW rl - height above MLW H = D + q
- density of air p - density of water Ul0 - air velocity at 3m Ulox - air velocity at 3m in x-direction CD - air-water momentum transfer coefficient pair
EXy - eddy viscosity terms M,
- momentum addition rate per unit area
Note that equation (11) is quadradic in velocity and hence is nonlinear. The advantage o f a dispersive assumption to remove the nonlinearity can certainly be appreciated, but this would lead to a nonpredictive capability in almost all circumstances.
Assuming then that it is desirable to know the velocity field in almost all practical water quality problems in coastal areas, how then can it be predicted and how well? Use of equation (11) along with an analogous equation in the y-direction and a continuity equation closes the problem. Several good computer models exist to predict the velocity field. An example showing the
591
quality o f a simulation to measured data in Apalachicola Bay from Daniels and Graham (1981) is presented as Figure 4. This illustrates that the velocity field can be represented quite nicely in real-time (l-minute increments) even in a case in which windimparted momentum is significant. The advective terms in equation (2) are deterministic therefore and need not be considered random in either the sampling scheme or the time-series analysis of the data. The water-qua1 ity field can then be subsequently calculated with known boundary and initial conditions, with the velocity field input. For most cases the effects of water-quality on the momentum field can be neglected.
FIGURE 4 Computer S i m u l a t i o n of T i d a l K a v e , Apalachicola Bay, Florida
S o u r c e : D a n i e l s a n d Graham (1981)
592 Continuous t r a c e s o f water q u a l i t y v a r i a b l e s a t a p o i n t are o f t e n dominated b y a d v e c t i v e e f f e c t s , t a b l e and,
therefore,
3,
Figure
see
and t h i s v a r i a n c e i s p r e d i c -
removable from t h e s i g n a l .
Figure
5
of
Gunnerson
(1966)
N a j a r i a n and Harleman (1977) f o r examples. t h i s approach i s t h a t e f f e c t s of velocity, the
and hence w a t e r q u a l i t y ,
extent
processes.
advection Methods
coefficient nonrigorous,
to
the
dominates of
relating
velocity
field
I n addition t o
and
the
paper
by
An o b v i o u s advantage o f
changed c i r c u m s t a n c e s upon t h e are r e l a t i v e l y p r e d i c t a b l e t o diffusion
in
the
transport
the
of
the
dispersion
value
are
generally empirical
and
so t h e p r o b l e m c a n n o t be c l o s e d c o m p l e t e l y a t t h i s
time. The n e x t t h r e e q u e s t i o n s t o l o g i c a l l y a r i s e a r e :
1.
What i s t h e a p p r o p r i a t e s a m p l i n g i n t e r v a l t e m p o r a l l y ?
2.
What i s t h e a p p r o p r i a t e s a m p l i n g i n t e r v a l s p a t i a l l y ?
3.
What i s t h e a p p r o p r i a t e d u r a t i o n o f s a m p l i n g ?
and
Because t h e hydrodynamics p e r i o d i c t i d a l wave phenomena,
o f coastal
areas are dominated by
t h e s a m p l i n g i n t e r v a l must be a b l e
t o r e s o l v e t i d a l e f f e c t s b o t h s p a t i a l l y and t e m p o r a l l y . methodology f o r periodicities
s a m p l i n g and a n a l y z i n g t i m e - s e r i e s
i s t h e well-known
power
needs no f u r t h e r e l a b o r a t i o n h e r e .
The u s u a l
with inherent
spectrum technique,
!-lowever,
which
t h e b e s t methods t o
o b t a i n d a t a w h i c h can be p r o p e r l y i n p u t i n t o d e t e r m i n i s t i c models needs
some
Harleman
and
re-evaluation Najarian
i n the
( 1 9 7 7 ) were
light
of
the f i r s t
current
technology.
t o point
out
this
requirement: Large temporal v a r i a t i o n s o f n u t r i e n t c o n c e n t r a t i o n s have been shown t o o c c u r w i t h i n a t i d a l p e r i o d a t f i x e d
593 locations. T h i s suggests t h a t a r e e v a l u a t i o n o f t r a d i t i o n a l e s t u a r i n e f i e l d data c o l l e c t i o n techniques i s n e c e s s a r y . More a t t e n t i o n must be g i v e n t o d e t e r m i n i n g t h e t e m p o r a l as w e l l as t h e s p a t i a l v a r i a t i o n s o f c o n s t i t u e n t s i n e s t u a r i n e water q u a l i t y surveys. R e l a t i v e l y l i t t l e u s e f u l i n f o r m a t i o n can be e x p e c t e d f r o m t h e random c o l l e c t i o n o f samples f r o m a moving b o a t . Probably t h e greatest 1 i m i t a t i o n t o t h e c o n t inued development and improvement o f p r e d i c t i v e w a t e r q u a l i t y models i s i n t h e g e n e r a l l a c k o f c o o r d i n a t i o n between d a t a needs f o r model v a r i a t i o n and t h e d a t a a c t u a l l y o b t a i n e d i n f i e l d surveys. ( p . 537) The t y p i c a l p r a c t i c a l s a m p l i n g p r o b l e m can be o u t l i n e d b y t h e f o l l o w i n g example.
A p a l a c h i c o l a Bay,
km w i d e and 30 km l o n g . km/h.
F l o r i d a , i s a p p r o x i m a t e l y 10
A b o a t i s a v a i l a b l e w h i c h can go about 10
A t r a v e r s e o f t h e e s t u a r y t h e n t a k e s about 8 h o u r s , w h i c h i s
o f t h e t i m e s c a l e o f t h e dominant M2 t i d e .
Therefore,
intratidal
v a r i a t i o n s a t any s a m p l i n g p o i n t c a n n o t be d e t e r m i n e d .
Regarding
F i g u r e 3, i t i s known t h a t t h e s e i n t r a t i d a l v a r i a t i o n s can d o m i n a t e i n t e r t i d a l ones Further,
if
at
a p o i n t , .and
t h e r e f o r e c a n n o t be n e g l e c t e d .
DO i s a p a r a m e t e r o_f i n t e r e s t , t h e n
i t i s a l s o known
t h a t DO i s dependent on s u n l i g h t ( t h r o u g h p h y t o p l a n k t o n a c t i v i t y ) , w i n d and wave a c t i o n , and t e m p e r a t u r e .
Temperature v a r i a t i o n s a t a
p o i n t are influenced by t h e v e l o c i t y f i e l d over t h e period o f a tidal
cycle
Mexico o f t e n
i n this
a r e a because t h e
have d i f f e r e n t
dominant
periodicity
daylight
at
the
of
equinox
temperatures.
about is
river
12.42
hours,
12 h o u r s .
i n f l o w and G u l f o f t i d e has
The
M2
and
the
period
a of
To s e p a r a t e t i d a l
from
DO,
it i s
d i u r n a l t e m p e r a t u r e and p h y t o p l a n k t o n - i n d u c e d changes i n necessary t o devise a temporal sampling i n t e r v a l .
I n t h i s case,
t h e N y q u i s t c r i t e r i o n on t h e more f r e q u e n t 12 h o u r s i g n a l h o l d s , so t h e s a m p l i n g i n t e r v a l must be a t l e a s t 6 h o u r s .
O f course,
for
practical
r e a s o n s s a m p l i n g s h o u l d be done a t a r a t e r o u g h l y f i v e
times
dense
as
therefore
as
the
appropriate.
Nyquist However,
frequency.
Yourly
then
record
the
sampling length,
is AT,
n e c e s s a r y t o s e p a r a t e t h e two s i g n a l s w i t h f r e q u e n c i e s f l = 1/12 h
5 94 and f 2 = U 1 2 . 4 2 h r e m a i n s u n a f f e c t e d b y t h e s a m p l i n g i n t e r v a l ,
and
is:
AT
AT
1
>
- f2 -
-> 355
fl
h r s = 14.8 days
(16)
assuming a r e c t a n g u l a r d a t a window.
This duration i s f o r t u i t o u s l y
v e r y c l o s e t o t h e p e r i o d o f t h e f o r t n i g h t l y MI
so
that
a
Therefore, weeks
is
sampling as
period o f
a general
a reasonable
r u l e o f thumb, minimum f o r variables,
varying
water-qua1 i t y
activity,
i n t h e c o a s t a l zone.
!iourly
a 15-day
t i d e (327.9 h o u r s ) ,
length
should
h o u r l y sampling f o r two
determination and
suffice.
of
associated
diurnallybiological
s a m p l i n g f o r a 15-day p e r i o d t h e n e n t a i l s 360 samples a t
each p o i n t , e x c l u s i v e of t h e s a m p l i n g t a k e n b e f o r e h a n d t o e s t a b l i s h the i n i t i a l conditions.
Obviously,
i f p e r i o d i c i t i e s i n the data
w h i c h a r e p u r e l y d e t e r m i n i s t i c can be removed f i r s t , t h e l e n g t h o f the
required
sample
l e n g t h can be c o n s i d e r a b l y r e d u c e d i n most
I f fli s removed i n e q u a t i o n 14, f o r i n s t a n c e , t h e n AT need
cases.
o n l y be g r e a t e r t h a n 12 h o u r s . lost
in this
analysis, degree, describe,
so
procedure, that
predictable.
the
as
Further,
phase i n f o r m a t i o n i s n o t
i s t h e case w i t h
series
is
variance
reconstitutable
T h i s i s an e s s e n t i a l p o i n t :
b u t n o t be used t o p r e d i c t ,
and,
spectrum to
some
s t a t i s t i c s can
i f t h e f o r c i n g functions are
changed. I t s h o u l d be p o i n t e d o u t t h a t many o f t h e s e same c o n s i d e r a t i o n s
595
a r i s e w i t h r e s p e c t t o man-induced p e r i o d i c i t i e s . o f the generating s t a t i o n i n Figure 1 f o r p e r i o d i c i t y which w i l l
have a 24-hour
The p l a n t f a c t o r
instance w i l l tend t o
be o u t o f phase w i t h t h e
and change phase w i t h r e s p e c t t o t h e M2 t i d e .
s o l a r day, i s true of
a STP,
The same
o r r e l e a s e s th r o u g h a h y d r o e l e c t r i c p r o j e c t .
Occasional sampling a t t h e same t i m e o f day can r e s u l t i n a h i g h l y a l i a s e d r e c o r d o f t h e s e e f f e c t s on w a t e r q u a l i t y . The number o f p o i n t s r e q u i r e d t o be sampled w i l l v a r y w i t h t h e particular
location
and
problem,
but
t h e r e should
be a t
least
enough t o p r o v i d e t h e necessary s p a t i a l r e s o l u t i o n . I n coastal areas, a t i d a l wave i s a s h a l l o w one moving a t c e l e r i t y (gH) 1/2
.
Hence i f .cI = 2m, t h e wave moves a t 4.4 m/s and t r a v e l s about 100 km p e r o n e - h a l f M2 t i d a l p e r i o d . km l e n g t h o f
T h i s i s n o t l o n g r e l a t i v e t o t h e 30
A p a l a c h i c o l a Bay.
The s p a t i a l
resolution of
the
phenomenon sh o u l d be t h e analog o f t h e t e m p o r a l N y q u i s t frequency, o r 50 km a t l e a s t and 10 km o r so i d e a l l y .
Obviously,
the time
t a k e n t o t r a v e r s e 10 km on t h e water (1 h o u r ) i s so g r e a t t h a t no F u r t h e r , a 10 km
approximat ion of c o i n c i d e n t sampling can be made. g r i d c o u l d miss many water
q u a l i t y phenomena o f
i n t e r e s t whose
l e n g t h s c a l e i s much s m a l l e r t h a n t h i s ( t h e plume i n F i g u r e 1, f o r instance,
which
resolution of
the
sampling s t a t i o n s i n t h i s experiment, b u t a p p a r e n t l y does n o t ) .
In
o t h e r words,
should
d e te rmi n e
the
spatial
numerous water mass b o u n d a r i e s a r e o f t e n missed due t o
l a c k o f an a p p r o p r i a t e sampling g r i d .
As an a t t e m p t t o address
t h i s problem we have been e x p e r i m e n t i n g w i t h use o f remotely-sensed Landsat
satellite
spatial field.
images t o
provide synoptic resolution o f
the
An example of such an image f o r A p a l a c h i c o l a Bay i s
present ed as F i g u r e 5.’
F i e l d sampling i s necessary t o r e l a t e t h e
A c o l o u r v e r s i o n o f t h i s image can a l s o be f o u n d on t h e cover o f t h e J o u r n a l Water P o l l u t i o n C o n t r o l F e d e r a t i o n , Volume 53, No. 4, A p r i l , 1981.
5 96
FIGURE 5 Enhanced L a n d s a t Image of A p a l a c h i c o l a
Bay, F l o r i d a , 26 F e b r u a r y 1975
597 image
patterns
color),
but
the
resolution sampler.
to
w a te r
quality
image
of
the
allows
parameters greatly
data-collecting
satellite
(1976).
turbidity,
expanded
capability
spatial
of
a
ground
A t i m e and c o s t - e f f i c i e n t water sampling scheme which can
r a p i d l y sample l a r g e b o d i e s o f water, with
( i .e.,
overpasses,
is
p a r t i c u l a r l y i n conjunction
described
in
Hill
and
Dillon
A d e s c r i p t i o n o f an experiment t o use s a t e l l i t e t e c h n o l o g y
t o p r o v i d e s p a t i a l d a t a f o r model v e r i f i c a t i o n appears i n Graham and H i l l (1980).
SUMMARY AND CONCLUSIONS Biological
and
water
quality
studies
in
the
coastal
zone
t y p i c a l l y consist o f gathering series o f data a t selected points. These
sets
of
time-series
c o n c l u s i o n s which, problem(s).
data
are
then
f o r engineering studies,
The u t i l i t y o f th e s e data,
analyzed
to
yield
a r e used t o s o l v e t h e
and hence,
the a b i l i t y o f
t h e a n a l y s t t o s o l v e t h e problem, depends g r e a t l y upon t h e sampling method.
Many processes i n t h e c o a s t a l environment a r e p e r i o d i c and
are r e l a t i v e l y d e t e r m i n i s t i c .
Sampling must be conducted i n such a
manner t h a t a1 ia s i n g o f p e r i o d i c processes does n o t occur, t h e r e b y masking t h e
u n d e r l y i n g randomness o r
parameters.
T h i s r e q u i r e s t h e t i m e s c a l e s and l e n g t h s c a l e s o f a l l
the
hydrodynamic
water
t a k e n i n t o account.
qua1 i t y
interrelationships
and b i o l o g i c a l
S i n c e t h e most u s e f u l
often l i e s i n the residuals,
parameters t o be
portion o f the data
t h e s e can be made most m e a n i n g f u l b y
removing s i g n i f i c a n t known d e t e r m i n i s t i c v a r i a t i o n s , and d i u r n a l e f f e c t s . which,
i n turn,
between
such as t i d a l
T h i s can be b e s t done u s i n g n u m e r i c a l models
r e q u i r e d a t a t o be c o l l e c t e d i n a manner d i f f e r e n t
f r o m usual p a s t p r a c t i c e .
I n p a r t i c u l a r h i g h - f r e q u e n c y sampling a t
a few p o i n t s and measurement o f boundary c o n d i t i o n s a r e r e q u i r e d . S y n o p t i c s p a t i a l d a t a may have t o be a c q u i r e d b y remotely-sensed
598
S i m i l a r l y b i o l o g i c a l surveys may be much more meaningful if
means. they
are
taken
i n context
of
the
hydrodynamics
which,
again,
r e q u i r e s sampling on d i f f e r e n t t i m e and space s c a l e s than has been
A discussion o f c h a r a c t e r i s t i c b i o l o g i c a l
t h e custom i n t h e past.
and chemical t i m e s c a l e s i s presented b y Ford and Thorton (1979). Acknowledgements The a s s i s t a n c e o f acknowledged.
John D a n i e l s w i t h F i g u r e 4 i s g r a t e f u l l y
T h i s research was
p r o v i d e d by t h e U.S.
supported,
i n part,
by funds
Department o f Commerce, N a t i o n a l Oceanographic
and Atmospheric A d m i n i s t r a t i o n ,
O f f i c e o f Sea Grant,
under Grant
NO. 04-158-44046.
References
1. Daniels, J.P.
and Graham, D.S.,
" A p p l i c a t i o n and c a l i b r a t i o n o f
t h e CAFE-1 model t o A p a l a c h i c o l a Bay, F l o r i d a " , the
5th
Canadian
F r e d e r i c t o n , N.B., 2.
Ford, D.E.
Hydrotechnical
Conference,
Proceedings of CSCE,
held at
26-27 May 1981, Vol. 2, pp. 515-536.
and Thorton, K.W.,
"Time l e n g t h s c a l e s f o r t h e one-
dimensional assumption and i t s r e l a t i o n t o e c o l o g i c a l models", 15, ( l ) , 1979, pp. 113-120. Water Resources Research, 3.
Graham,
D.S. and H i l l ,
Qua1i t y V e r i f i c a t i o n " ,
J.M.,
" F i e l d Study f o r Landsat Water
i n Proceedings o f ASCE Symposium " C i v i l
Engineering A p p l i c a t i o n s o f Remote Sensing",
h e l d a t Madison,
Wisconsin, August 13-14, 1980, pp. 101-117. 4.
Graham,
D.S.,
e t al.,
Stormwater Runoff i n t h e A p a l a c h i c o l a
Estuary, F l o r i d a Sea Grant Report R/EM-11, p l u s Appendices.
March 1978, 76 pp.
599 5.
Gunnersen,
C.G.
,
"Optimizing sampling i n t e r v a l s i n estuaries",
Journal o f t h e S a n i t a r y Engineering D i v i s i o n ,
ASCE,
92,
(SA2),
Proc. Paper 4799, A p r i l 1966, pp. 103-125. 6.
J.M.,
Hill,
and
Dillon,
T.M.,
"A
unique
and
effective
o c e a n o g r a p h i c s u r f a c e t r u t h m o n i t o r i n g program f o r c o r r e l a t i o n s w i t h r e m o t e l y sensed s a t e l l i t e and a i r c r a f t imagery," B u l l e t i n No. 76-2, Texas A.
Technical
Texas E n g i n e e r i n g and E x p e r i m e n t S t a t i o n ,
and M. U n i v e r s i t y , C o l l e g e S t a t i o n , Texas, A p r i l 1976,
209 pp. 7.
Najarian,
and Harleman, D.R.F.,
T.O.,
n i t r o g e n c y c l e i n an e s t u a r y " ,
"Real-time s i m u l a t i o n o f
Journal
Engi n e e r i ng D i v i s i on , ASCE , 104, (EE4) Aug 8.
Smith,
"Environmental
W.,
approach",
survey
o f t h e Environmental
.
design:
1977, pp. 523-538. a
E s t u a r i n e and C o a s t a l M a r i n e Science,
2 17 - 224. 9.
TECO
(Tampa
Station 10. Thomann,
-
Electric
Company),
316
time 6, -
Demonstration,
series
1978,
pp.
B i g Bend
U n i t 4, Aug. 1, 1980. R.V.
,
"Time-series
analysis o f water-quality
J o u r n a l o f t h e S a n i t a r y E n g i n e e r i n g D i v i s i o n , ASCE, Proc. Paper 5108, Feb. 1967, pp. 1-23. Copyright,
1982, D.S.
G r a h a m a n d J.M. H i l l
93, -
data", (SAl),
600
TIME SERIES ANALYSIS OF SOIL MOISTURE DATA SHAW L. YU AND JAMES F. CRUISE Department of Civil Engineering, University of Virginia, Charlottesville, Virginia 22901
INTRODUCTION Recent studies have shown the feasibility of statistically based investigations of infiltration and soil moisture regimes. Cordova and Bras (1981) and others have utilized probabilistic models in the analysis of soil moisture and infiltration. It has been well recognized that the soil moisture regime is a stochastic variable, consisting of both deterministic and random components. However, until now sufficient data has not been generally available to allow detailed probabilistic description of the soil moisture process. Therefore, an analysis of the variability of soil moisture based on a sufficient data sample is highly desirable. In this study, time series arTalysis techniques and a linear autoregressive prediction model are employed in an effort to examine the internal structure of the soil moisture process. The data were generated from a study of soil moisture fluctuations under various vegetative covers which was conducted during 1950-55 at the Calhoun Experimental Forest near Union, South Carolina (Kent, et al, 1981). There were a total of 5 years of data on rainfall amounts and total soil moisture to a depth of 1.68 meters. However, only for two of the years continuous daily observations were available without any missing observations. Consequently these 730 daily records were utilized in this study. The vegetation cover was a forest of Loblolly pine. HARMONIC ANALYSIS Initially, the monthly average and the standard deviations o f the soil moisture data were computed and are listed in Table. 1. The results indicated generally high soil moisture in the winter months Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S . R . Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
o
601 TABLE 1 MONTHLY AVERAGES AND STANDARD D E V I A T I O N S OF SOIL MOISTURE DATA
Month
Mean cm
Standard D e v i a t ion cm
January February March April May June July August September October November December
45.37 47.63 51.9 49.89 45.33 42.10 40.93 40.07 39.32 38.69 40.02 42.30
.356 .813 1.825 1.648 1.349 1.325 1.655 1.039 1.072 .335 .548 1.609
and l o w s o i l m o i s t u r e i n t h e summer months.
H i s t o g r a m s o f t h e summer
and w i n t e r month s o i l m o i s t u r e were p l o t t e d i n F i g u r e 1, w h i c h shows t h a t s o i l m o i s t u r e i s h i g h e r i n t h e w i n t e r months and a l s o has h i g h v a r i a b i l it y .
A h a r m o n i c a n a l y s i s was t h e n p e r f o r m e d on t h e s o i l m o i s t u r e d a t a . The r e s u l t s o f t h i s a n a l y s i s a r e shown i n F i g u r e 2.
A t f i r s t glance
t h e r e a p p e a r s t o be an annual c y c l e p r e s e n t i n t h e d a t a .
Visual
i n s p e c t i o n o f t h e d a t a a l s o s u g g e s t s t h e p r e s e n c e o f t h e annual c y c l e . T h e r e i s , however, r e a s o n t o s u s p e c t a c e r t a i n d e g r e e o f c o r r e l a t i o n i n t h e d a t a which would t e n d t o d i s t o r t t h e frequency spectrum.
This
d i s t o r t i o n i s r e f e r r e d t o as " r e d n o i s e " ( G i l m a n , e t a l , 1963; M i t c h e l l , 1964).
The p r e s e n c e o f " r e d n o i s e " t e n d s t o s u p p r e s s t h e r e l a t i v e
v a r i a n c e a t h i g h e r f r e q u e n c i e s and c o n s e q u e n t l y t o i n f l a t e t h e r e l a t i v e variance a t the lower frequencies. The " r e d n o i s e " s p e c t r u m i s a f u n c t i o n o f t h e a u t o r e g r e s s i v e c o e f f i c i e n t o f the data.
F i g u r e 3 shows a f a m i l y o f " r e d n o i s e "
spectra f o r various autoregressive c o e f f i c i e n t s .
( G i l m a n , e t a1 , 1 9 6 3 ) .
I t can be seen t h a t f o r a s u b s t a n t i a l l y l a r g e a u t o r e g r e s s i v e
coefficient,
t h e r e l a t i v e importance o f t h e f i r s t two harmonics i n t h e
602 d a t a i s s i g n i f i c a n t l y reduced.
A periodogram a n a l y s i s o f t h e spectrum i s g i v e n h e r e r a t h e r t h a n the Blackman-Tukey approach because i n t h i s i n s t a n c e t h e spectrum i s e a s i l y smoothed and a degree o f a u t o c o r r e l a t i o n i s p r e s e n t . The advantages o f t h e periodogram i n t h e s e i n s t a n c e s have been p o i n t e d o u t by Jones (1 9 6 4 ).
Among t h e s e advantages a r e t h a t t h e
periodogram spectrum i s always p o s i t i v e and t h a t t h e c r o s s s p e c t r a l d e n s i t y i s e a s i l y c a l c u l a t e d fro m i t . An a n a l y s i s o f t h e s i g n i f i c a n c e o f t h e peaks i n v o l v e d i n t h e f i r s t two harmonics showed, as e x p e c t e d - - t h a t t h e y c o n t r i b u t e d l i t t l e s i g n i f i c a n c e when compared w i t h t h e " r e d n o i s e " spectrum.
Despite the
a p parent l a k o f s i g n i f i c a n c e i n t h e annual c y c l e i t was s t i l l cons i d e r e d adv s a b l e t o remove i t f r o m t h e d a t a b e f o r e p r o c e e d i n g w i t h t h e r e s t o f the analysis.
T h i s was done because, g i v e n t h e presence o f t h e
" r e d n o i s e " d i s t o r t i o n , i t was n o t p o s s i b l e t o p o s i t i v e l y d e t e r m i n e t h e r e a l s i g n i f cance o f t h i s component o f t h e d a t a .
The c y c l e was
removed by computing t h e mo n th l y averages, month b y month, f o r t h e f i v e y e a r s o f a v a i l a b l e d a t a and t h e n s u b t r a c t i n g t h e a p p r o p r i a t e average f r o m each d a i l y o b s e r v a t i o n .
Mo n th l y averages were used i n t h i s case
i n s t e a d o f t h e recommended d a i l y means (Jones, 1964) due t o t h e s m a l l sample s i z e .
I t was f e l t t h a t d a i l y means would be u n d u l y b i a s e d i n
t h i s case. CROSS-CORRELATION ANALYSIS
A c r o s s - c o r r e l a t i o n and c ro s s -s p e c tr u m a n a l y s i s was t h e n a t t e m p t e d on t h e anomalies f r o m t h e above o p e r a t i o n and t h e d a i l y r a i n f a l l observations.
A c ro s s -s p e c tru m a n a l y s i s i s concerned w i t h t h e c o n t r i -
b u t i o n o f each f r e q u e n c y t o t h e t o t a l c o v a r i a n c e o f t h e two s e r i e s ( J e n k i n s , 1961).
The a n a l y s i s was p e r f o r m e d b y methods d e s c r i b e d b y
Panofsky and B r i e r (1 9 5 8 ). The r e s u l t s o f t h i s a n a l y s i s d i d n o t r e v e a l any s i g n i f i c a n t r e l a t i o n s h i p between t h e two s e r i e s a t any fre q u e n c y .
I n o r d e r t o account f o r
any d e l a y i n t h e measurement o f s o i l m o i s t u r e v a l u e s a f t e r a r a i n f a l l e v e nt , t h e s o i l m o i s t u r e was th e n l a g g e d one day, b u t s t i l l no
603
significant correlation o r coherence was obtained. I n order t o t e s t the hypothesis t h a t the lack o f correlation i n the two s e r i e s was due t o very short duration perturbation, the soil moisture data was f i l t e r e d by taking 7-day moving averages. This s e r i e s was then correlated w i t h the 7-day r a i n f a l l s e r i e s . I n t h i s analysis a correlation c o e f f i c i e n t of only about 0.2 was computed. AUTOCORRELATION ANALYSIS An autocorrelation analysis was next performed on the soil moisture
d a t a f o r various lags. The r e s u l t s are shown in Figure 4. As can be seen from the figure, an extraordinarily h i g h degree of correlation e x i s t s in the data. A very large correlation coefficient was obtained a t a l-day lag and a large degree of persistence was observed. As can be seen, the data showed s i g n i f i c a n t correlation a t lags even greater than 150 days. The r e s u l t s suggest a very strong "carry-over" nature of the soil moisture data. AUTOREGRESSIVE MODEL I t seemed reasonable t o assume t h a t the strong autocorrelation evidenced by the s o i l moisture data was one o f the reasons f o r the lack of correlation between i t and the r a i n f a l l s e r i e s . Therefore, i t
was decided t o f i t an autoregressive type model t o the data. I n i t i a l l y , a f i r s t - o r d e r Markov type model was used. This model i s of the form:
x ( t ) = ux(t-1) + q where : x ( t ) = soil moisture a t time t x ( t - l ) = s o i l moisture a t time t-1 u = autoregressive c o e f f i c i e n t ri = random residual component The method used t o f i t t h i s model i s outlined by Jones (1964) and will n o t be repeated here. I n t h i s analysis an autoregression or predictor constant (a) of .9856 was obtained with a standard e r r o r o f prediction
604
o f .0061 and a n R 2 o f 97%. T h i s a u t o r e g r e s s i o n c o e f f i c i e n t c o u l d h a v e been o b t a i n e d b y d i r e c t comparison o f t h e s p e c t r a l a n a l y s i s w i t h curves o f t h e " r e d n o i s e " spectrum f o r d i f f e r e n t values o f
CY
g i v e n b y Gilman, e t a1 ( 1 9 6 3 ) and
M i t c h e l 1 ( 1964).
CROSS-CORRELATION BETWEEN RAINFALL AND SOIL MOISTURE The random component, o r r e s i d u a l s f r o m t h e above a u t o r e g r e s s i v e model, r e p r e s e n t t h e t r u e random f l u c t u a t i o n i n t h e o b s e r v e d s o i l m o i s t u r e d a t a because t h e d e t e r m i n i s t i c c o m p o n e n t s - - c y c l e s and s h o r t t e r m a u t o r e g r e s s i v e e f f e c t s - - h a v e a l r e a d y been e l i m i n a t e d .
Therefore,
i t w o u l d seem r e a s o n a b l e t o assume t h a t t h e r e s i d u a l s w o u l d be b e t t e r c o r r e l a t e d t o t h e o b s e r v e d r a i n f a l l s e r i e s t h a n was t h e o r i g i n a l d a t a . T h i s s u p p o s i t i o n was b o r n o u t b y c r o s s - s p e c t r a l a n a l y s i s .
5 and 6 show t h e r e s u l t s o f t h i s a n a l y s i s .
The t o t a l Pearson c o r r e -
l a t i o n c o e f f i c i e n t due t o a l l f r e q u e n c i e s was 0.46, s i g n i f cant a t the appear t h a t t h e r e
5% l e v e l .
Figures
w h i c h was
From F i g u r e 6a and 6b i t does n o t
s a n y s i g n f i c a n t phase l a g between t h e t w o d a t a
series From t h e r e s u l t s o f t h i s s t u d y i t w o u l d a p p e a r t h a t once t h e d e t e r m i n i s t i c components a r e removed f r o m t h e s o i l m o i s t u r e d a t a , t h e v a r i a n c e i n t h a t s e r i e s i s w e l l e x p l a i n e d b y t h e o c c u r r e n c e o r nonoccurrence o f p r e c i p i t a t i o n over t h e watershed.
I n t h i s analysis i t
i s assumed t h a t s o i l m o i s t u r e f l u c t u a t i o n s a r e due p r i m a r i l y t o t h e e v a p o t r a n s p i r a t i o n p r o c e s s and t o p r e c i p i t a t i o n .
The e v a p o t r a n s p i r a t i o n
process i s e x p l a i n e d by t h e a u t o r e g r e s s i v e e f f e c t s present i n t h e data. T h i s seems t o s u g g e s t t h a t on t h e a v e r a g e a b o u t 1.5% o f t h e s o i l m o i s t u r e i s l o s t each d a y due t o e v a p o t r a n s p i r a t i o n .
The r e m a i n i n g
p a r t o f t h e u n e x p l a i n e d v a r i a n c e i n t h e s o i l m o i s t u r e i s a p p a r e n t l y due t o measurement e r r o r s o r some p h y s i c a l p r o c e s s w h i c h c a n n o t be accounted f o r i n t h i s t y p e o f a n a l y s i s .
605
CONCLUSIONS From t h e r e s u l t s o f t h i s s t u d y i t i s p o s s i b l e t o make s e v e r a l u s e f u l conclusions.
( 1 ) The presence o f an over-powering a u t o c o r r e l a t i o n i n
s o i l m o i s t u r e d a t a makes i t i m p o s s i b l e t o p e r f o r m a F o u r i e r a n a l y s i s on t h i s s e r i e s because o f t h e " r e d n o i s e " d i s t o r t i o n o f t h e frequency spectrum.
F o r t h e same reason, no d i r e c t c o r r e l a t i o n can be o b t a i n e d
between t h e raw s o i l m o i s t u r e d a t a and t h e d a i l y r a i n f a l l s e r i e s .
( 2 ) T h i s a u t o r e g r e s s i v e e f f e c t o r c o r r e l a t i o n i n t h e d a t a shows a v e r y h i g h p e r s i s t e n c e o u t t o a t l e a s t 150 days.
Thus, c a r e should be t a k e n
when p e r f o r m i n g s t a t i s t i c a l a n a l y s i s on s o i l m o i s t u r e d a t a where independence assumptions a r e necessary.
(3) A f i r s t order l i n e a r
a u t o r e g r e s s i v e model a d e q u a t e l y d e s c r i b e s t h e s o i l m o i s t u r e d a t a . T h i s model f i t s t h e d a t a v e r y w e l l and accounts f o r o v e r 90% o f t h e variance i n the data.
( 4 ) Once t h e d e t e r m i n i s t i c components o f t h e
s o i l m o i s t u r e s e r i e s have been removed, a s i g n i f i c a n t c o r r e l a t i o n i s o b t a i n e d between t h e r e s i d u a l s and t h e r a i n f a l l s e r i e s .
( 5 ) The auto-
r e g r e s s i o n a n a l y s i s i n d i c a t e d t h a t on t h e average about 1.5% o f t h e t o t a l s o i l m o i s t u r e i s l o s t each day, p o s s i b l y due t o e v a p o t r a n s p i r a t i o n , when t h e v e g e t a t i o n c o v e r i s L o b l o l l y p i n e . ACKNOWLEDGEMENTS The w r i t e r s would l i k e t o t h a n k J.E.
Douglass o f t h e U n i t e d S t a t e s
Department o f A g r i c u l t u r e , F o r e s t S e r v i c e , Southeastern F o r e s t Experiment S t a t i o n , who p r o v i d e d t h e h y d r o l o g i c d a t a f r o m t h e Calhoun Experimental F o r e s t .
The w r i t e r s assume f u l l r e s p o n s i b i l it y f o r a1 1
c o n c l u s i o n s drawn f r o m these d a t a . REFERENCES Cordova, Jose R., and Rafael L. Bras, " P h y s i c a l l y Based P r o b a b i l i s t i c Models o f I n f i l t r a t i o n , S o i l M o i s t u r e , and A c t u a l E v a p o t r a n s p i r a t i o n , " Water Resources Research, V o l . 17, No. 1 (Februarv 1981) D P . 93-106. Gilman, D.L., F r e g l i s t e r , J., and J.M. M i t c h e l l , Jr., "On t h e Power Spectrum o f 'Red N o i s e ' , " Journal of t h e Atmospheric Sciences, V o l . 20 (March 1963) pp. 182-184. Jenkins, G.M., "General C o n s i d e r a t i o n s i n t h e A n a l y s i s o f Spectra," Technometrics, V o l . 3, No. 2 (May 1961) pp. 133-143.
606
Jones, Richard H . , " S p e c t r a l Analysis and Linear P r e d i c t i o n o f Meteor o l o g i c a l Time S e r i e s , " Journal of Applied Meteorology, V o l . 3 (February 1964) p p . 45-52. Kent, Edward J . , Roy Burke 111, and Shaw L . Yu, "Some Control of Stormwater Through J o i n t Use of Constructed Storage and S o i l Management," Proceedings, I n t e r n a t i o n a l Symposium on Urban Hydrology, Hydraulics and Sediment C o n t r o l , U n i v e r s i t y of Kentucky, Lexington, KY ( J u l y 1981) p p . 453-464. M i t c h e l l . J . Murray. J r . , "A C r i t i c a l Appraisal o f P e r i o d i c i t i e s i n Climate," CAED Report No. 20, Weather'and Our Food Supply, Iowa S t a t e U n i v e r s i t y (1964) p p . 189-227. Panofsky, H . A . , and G.W. B r i e r , Some A p p l i c a t i o n s o f S t a t i s t i c s t o Meteorolo , The Pennsylvania S t a t e U n i v e r s i t y , U n i v e r s i t y Park, d p . 224.
607
0
26
60
m
m
126
1M
116
203
LAG (days)
FC*RE 4 AUTCCCWELATKN VS LAG
FREQUENCY (CPO) FIGW(E 6a COHERENCE (P,SM)
FREQUENCY (CW) FKilREBb PMASE ANGLE (P-SM)
608
FORE CAST I N G UNDE R LINEAR PART1 AL IN FORMAT1 ON M. BEHARA AND E . K O F L E R McMaster University a n d University of Zurich
AB ST RA CT I n a classical forecasting procedure, using regression models, the c r e d i b i l i t y of a forecast i s based on the ( i ) c r e d i b i l i t y of the fixed exogene and endogene variables in the observation period ( i i ) extrapolation of the regression l i n e s into the forecast period ( i i i ) credib i l i t y of the fixed exogene variables in the forecast period e t c . There are also rigid conditions imposed on the residual variables. I n t h i s paper, w i t h the help of linear-partial-information ( L P I ) method, we modify the classical forecasting by assuming fuzziness f o r the exogene and endogene variables i n observation and exogene variables in
the forecast-spaces which w o u l d r e s u l t i n greater c r e d i b i l i t y of the respective variables. Similarly, the extrapolation i s assumed t o be fuzzy. Further, the LPI-fuzziness of the residual variables i s also considered. The b e t t e r quality of the LPI-forecast i s then tested i n a decision-theoretic way. 1.1
INTRODUCTION
2 Let ( X t ’ Y t ) t = ,. l . . , n R , where xt and yt denote the exogenous and the endogenous variables respectively. The method of ordinary l e a s t squares yields the regression 1 ine y t = m + b ( x t - mx) + u t , t = l , . . . , n Y where m and m are the mean values of x and y respectively; b i s the X Y regression c o e f f i c i e n t a n d u t , t = l ,. . . , n are the white noise e r r o r variables. Based on the principle o f extrapolation, the t r a n s i t i o n from the estimation-domain t = 1 ,. . . , n t o the forecast-domain (without the consideration of u t ’ a t f i r s t ) i s given by the regression l i n e f o r t = n + l , n + 2 , . . . and yields the corresponding point-forecast. This i s the so called classical regression problem. Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
609 i s given Se dom, i n f o r e c a s t i n g problems, an o b s e r v a t i o n (xt,yt) by an e x a c t p o i n t . We g e t a more c r e d b l e process i f t h e v a r i a b l e s
a r e a lowed t o assume wide range o f va ues r a t h e r t h a n s i n g l e p o i n t s . For example, i t i s more c r e d i b l e t o assume t h e r a t e o f i n f l a t i o n f o r a c e r t a i n y e a r t o be between 10 and 1 3 p e r c e n t r a t h e r than e x a c t l y 11 percent.
We may d e f i n e " c r e d i b i l i t y " o f (xt,yt)
as an open o r c l o s e d
R2 a c c o r d i n g t o a p r i o r s p e c i f i e d m e t r i c . The c r e d i b i l i t y o f an o b s e r v a t i o n i s , t h e r e f o r e , d i r e c t e d , t o some neighbourhood o f (xt,yt)
in
t = 1
degree, by t h e assumption o f f u z z i n e s s f o r ( x t ( x t , y t ) , and f o r xntl,
x ~ + ~. ., .
.
,. . . ,n)
We s h a l l use t h e LPI-method o f K o f l e r 1,2
t o f o r e c a s t i n g problems where c r e d i b i l i t y o f o b s e r v a t i o n ( x t , y t ) , t = 1 ,. . . ,n as w e l l as c r e d i b i l i t y o f xntl,
x ~ + ~... , g i v e n a t n have
been assumed.
1.2
THE LPI-FUZZINESS
L e t us c o n s i d e r t h e case o f d i s c r e t e endogenous v a r i a b l e s where n o n - s t o c h a s t i c f u z z i n e s s i s assumed. L e t xt be a s s o c i a t e d w i t h t h e 1 2 L e t S = *st. Then S f i n i t e s e t st = (y, ,... ,yt), t = 1 ,... ,n. consists o f k
=
klk 2...kn
n - t u p l e s PW1, PW2 ,... ,PWk where each PW
j = 1 ,..., k i s a s s o c i a t e d w i t h a r e g r e s s i o n l i n e
Hence, we g e t k d i f f e r e n t f o r e c a s t s f o r
RG
j
=
j'
1 ,..., k.
j' by e v a l u a t i n g k
r e g r e s s i o n l i n e s i n x ~ + ~ C. l e a r l y , f o r kl = k2 =
...
= kn = 1 , t h e
above problem reduces t o c l a s s i c a l r e g r e s s i o n problem. I n t h e case o f s t o c h a s t i c LPI f o r t h e d i s c r e t e endogenous v a r i a b l e s , on t h e o t h e r hand, t h e LPI f o r each st, known.
I f LPI ( s t )
t = 1 , ...,n i s assumed t o be
i s g i v e n f o r t = 1 , ...,n, then, f o r e c a s t regard-
i s determined t o be an e x p e c t a t i o n i n t e r v a l as f o l l o w s :
i n g Yn+l E(Y,+~ )
L(Yn+l)
where E and
9
E(Yn+1)
denote t h e minimal and t h e maximal expected values
respectively.
T h i s i s e a s i l y seen, as t h e r e s u l t i n g LPI ( S ) ,
due t o
t h e theorem o f t h e a g g r e g a t i o n o f t h e L P I ' s , o v e r t h e k s t a t e s i s obtained. LPI
(s)
Thus, we have
= LPI (PW,
,. . . ,
PWk).
610 S i m i l a r l y , t h e L P I - f u z z i n e s s o f o t h e r v a r i a b l e s may be s t u d i e d .
2.1
INTERVAL-UNCERTAINTY O f t e n t h e experimental r e s u l t s are found t o l i e i n an i n t e r v a l
r a t h e r than t a k i n g d i s c r e t e values as discussed above.
L e t us considel
an endogenous v a r i a b l e Y t o assume an i n t e r v a l o f u n c e r t a i n t y w i t h r e s p e c t t o an exogenous v a r i a b l e an i n t e r v a l f o r yt.
L e t at,
bt,
t = 1
,. . . ,n
be such
A t f i r s t , we assume no p r i o r i n f o r m a t i o n on t h e
d i s t r i b u t i o n o f t h e y t l s i n at, uncertainty-interval a interval-forecast f o r
X.
t’
bt.
Obviously, f o r a g i v e n x ~ + ~an,
bt o f y t l e a d s t o t h e d e t e r m i n a t i o n o f an as
-
E- [&+I Yn+1’
Yn+l
3
y and where -
denote t h e minimum and maximum values o f y .
Assuming now t h e d e n s i t y f ( y t ) g i v e n f o r t h e i n t e r v a l at, t = 1 ,. . . ,n,
yt,
bt,
of
i t may be e a s i l y proved t h a t f o r a g i v e n x ~ + ~ ,
t h e r e e x i s t s a f o r e c a s t f o r E(Y,+~).
This i s the perfect stochastic
case. F i n a l l y , f o r t h e case o f l i n e a r p a r t i a l i n f o r m a t i o n f o r each yt
E
Cat,
btl,
t h e r e e x i s t s an LPI(Cat,
a p a r t i t i o n o f t h e i n t e r v a l Cat,
t = 1 ,... ,n.
btl),
And, f o r
b t l , t h e r e e x i s t LPI-statements on
the p a r t i t i o n s . For LPI (Cat
, b t l ) , we have -
E(Yn+1) E [E(Y,+~ 9 E(Yn+1)’. T h i s i s o b t a i n e d by t h e f o l l o w i n g procedure: Cat, b t l ,
t = 1,
... ,n,
For t h e i n t e r v a l
a f i n i t e s e t o f s t a t e s Cz,)
To each s e t I z t l t h e r e corresponds an L P I ( i z t l ) , t D e f i n i n g the Cartesian p r o d u c t L x L z t , o f t h i s product.
l e t w,,
i s assigned. =
. . . ,wm
1,
... ,n.
be t h e elements
According t o t h e theorem on t h e c o m p o s i t i o n o f t h e
independent L P I ’ s , t h e components o f L P I ( I z t l ) determine t h e LPI(w . ) , J j = 1 ,..., m. Each w j = 1 ,...,m i s t h e n a s s o c i a t e d w i t h an i n t e r v a l j’ fo r e c a s t .
611
3.1
EXAMPLE O F AN LPI-FUZZY ENDOGENOUS VARIABLE In a given sample
( x , , ~ , ) , . . . , ( x j , y j ) ,. .. , ( x k , y k ) , . . . , ( x n , y n ) , y j and y k a r e c o n s i d e r e d t o be LPI-fuzzy. Considering only two values f o r each of y j and y k , t h e LPI-assignments a r e a s f o l l o w s : 1 2 1 y j E { y j , YJ. } , Prob ( yJ. ) = p l y P O 1 2 yk E { y’k , y2k ? , Prob (yk) = 9 1 ’ Prob ( Y k )
r
, L P I ( yJ . ) : =
= p1 2 p 2 .
4 2 , L p I ( Y k ) : = 41
42’
In t h i s c a s e , t h e r e a r e 2 x 2 = 4 d i f f e r e n t s e t s of r e g r e s s i o n p o i n t s : PW,, PW2, PW3, PW4 and hence 4 r e g r e s s i o n l i n e s corresponding t o the r e g r e s s i o n p o i n t s . From t h e above LPI-assignment, we o b t a i n t h e r e s u l t i n g LPI ( r l , r 2 , r 3 , r 4 ) by composition o f t h e d i f f e r e n t LPI-components, where rl = p l q l , r2 = p1q2, r3 = p 2 q 1 , r4 = p2q2. C l e a r l y , we have
rl 2 r2 2 r 4 , rl 2 r3 2 r4, rl
- r2 2
r3
-
r4
with t h e m a t r i x o f t h e e x t r e t w p o i n t s given by
Hence we have f o u r f o r e c a s t values f o r yn+, given by: RG2(xn+l), RG3(xn+11, RGq(xn+l)* the m a t r i x o f the extreme p o i n t s g i v e n above, we determine t h e In minimal and the maximal e x p e c t a t i o n values E ( y n t l ) and E ( y n + l ) . Therefore, RG1
1 9
E(Yn+1) E
[ E ( Y ~ + ~ E(Yn+1 ) I 9
4.1
DECISION-THEORETIC EVALUATION OF FORECAST The e v a l u a t i o n o f f o r e c a s t i n g problems belongs t o t h e f i e l d o f semantic information t h e o r y . The p r i n c i p l e o f e v a l u a t i o n g i v e n a s f o l 1 ows : For every m u l t i s t a g e d e c i s i o n s i t u a t i o n , each dynamic and c o n t r o l problem, an optimal s t r a t e g y i s gained by u s i n g the maxEmin-principle C31 with t h e use o f L P I ’ s .
Every new f o r e c a s t g e n e r a l l y changes the
612
i n f o r m a t i o n - s t a t e o f t h e d e c i s i o n s i t u a t i o n and, t h e r e f o r e , t h e maxEmin-optimal
strategy.
Hence, erroneous f o r e c a s t s a r e a s s o c i a t e d
w i t h erroneous maxEmin-optimal
strategy.
L e t us c o n s i d e r t h e d e c i s i o n scheme P1
P2
z1
z2
dl
9 1
u1 2
d2
9 1
9 2
4
dm
where di,
m2 : .
i = l,...,m
a r e m d i f f e r e n t money-deposits i n a fund, z1 and
z2 denote a low and a h i g h i n f l a t i o n r a t e r e s p e c t i v e l y .
The d i s t r i b u -
t i o n o f z1 and z2 i s g i v e n by p1 and p 2 r e s p e c t i v e l y . According t o t h e i n f o r m a t i o n o f t h e p a s t few months on p1 ( t h e r e l a t i v e frequency o f z,)
we have observed, say
( X l , P1( 1 1 ),... Y(Xn,P1( q .
A c e r t a i n o u t p u t may be L P I - f u z z y . output.
L e t us s u pose p i s ) i s such an
I f we do n o t t a k e L P I - f u z z i n e s s o f p
i n t o c o n s i d e r a t i o n we
o b t a i n a f a l s e f o r e c a s t F2, say and determine p i n + ' ) each o f which i s s i n g l e - v a l u e d .
f r o m t h e p1 ,. . . ,pn
On t h e o t h e r hand, i f we t a k e LPIleads t o the P1 where t h e t r u e f o r e c a s t F,, say l i e s .
f u z z i n e s s i n t o account, t h e n t h e m u l t i - v a l u e d i n t e r v a l CE(pl( " ' I ) ,
'E(pintl))l
Now, t h e r e a r e two d e c i s i o n s i t u a t i o n s : (1)
The c o r r e c t d e c i s i o n , denoted by D1, w i t h g i v e n i n t e r v a l ~ ~ ( p j n + ' ,) )i ( p { n + 1 ) ) 1 f o r p1( n + l )
-
(2)
The i n c o r r e c t d e c i s i o n , denoted by D2, w i t h t h e f a l s e f o r e c a s t pin+').
L e t d*'
D2 r e s p e c t i v e l y .
and d** denote t h e o p t i m a l s t r a t e g i e s i n D1 and The e v a l u a t i o n o f t h e f o r e c a s t i s then g i v e n by
613
where V D denotes t h e v a l u e o f t h e s t r a t e g y i n t h e d e c i s i o n s i t u a t i o n
D.
Thus,
V(F1) = V
- VD ( d * 2 )
(d*') Dl
1
w h i c h i s n o n n e g a t i v e s i n c e d*2 i s n o t t h e maxEmin-optimal
strategy.
V(F1) i s t h e loss due t o t h e e x a c t b u t u n f o r t u n a t e l y f a l s e f o r e c a s t I t f o l l o w s t h a t by u s i n g t h e LPI-formulation associated w i t h t h e
F2.
L P I - f o r e c a s t F1 t h i s loss may be reduced. 4.2
EXAMPLES Example 1. z1
Two-person zero-sum game.
z2
z3
4
8
24
Solution: d* = (0,1/3,2/3)
*
value ( d ) LPI:
dl
p1
p2
G
G
p3
22
z3
5.75 6.00
6.00 5.66
7.00 7.00
6.25
7.00
7.50
G
5
p4 24 d
d2 d3
=
7.00
Solution:
* =
d3
*
v a l u e ( d ) = 6.25
We o b s e r v e t h a t u s i n g t h e L P I , t h e v a l u e o f t h e game has i n c r e a s e d f r o m 5 t o 6.25. Example 2. L e t t h e i n f o r m a t i o n on t h e s t a t e s be g i v e n b y an e x a c t p r o b a b i l i t y d i s t r i b u t i o n I1= ( 0 . 3 , 0 . 3 , 0 . 2 , 0 . 2 ) . i n f o r m a t i o n F1. j = 1,2,3,4.
The L P I on t h e s t a t e s i s g i v e n b y
For the u t i l i t y matrix: z1
T h i s we c o n s i d e r as t h e f a l s e
z2
z3
z4
I2
= 0.2