Time Series Methods in Hydrosciences (Developments in water science)

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

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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
=

1e x p ( - i > 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
26

Kalman, R.E., 1963. New metClods i n I l l i e n e r f i l t e r i n a t h e o r y . I n P r o c e e d i n n s o f t h e F i r s t Svmposium on E n a i n e e r i n a A w l i c a t i o n s o f Random F u n c t i o n T h e o r y and P r o b a b i l i t ! i . J . L . B a n d a n o f f and F. K o z i n ( E d i t o r s ) . W i l e y , New York. M a c N e i l l , I . B . , 1974. T e s t s f o r chanqe o f n a r a m e t e r a t u n k n o w t i m e and d i s t r i b u t i o n s o f some r e l a t e d f u n c t i o n a l s on b r o w n i a n m o t i o n . Ann. S t a t i s t . 2 : 95C)-962. PacPei11, I . B . , 1978a. P r o p e r t i e s o f seouence o f n a r t i a l sum o f polynomial r e g r e s s i o n r e s i d u a l s w i t h aool i c a t i o n s t o t e s t s f o r chanae o f r e g r e s s i o n a t unknown t i m e . Ann. S t a t i s t . 6 : 422-433. r l a c N e i l l , I.B., 1978b. L i m i t u r o c e s s e s f o r seouences o f a a r t i a l sums o f r e q r e s s i o n r e s i d u a l s . Ann. Prob. 6 : 696-698. F a c N e i l l , I . B . , 1980. D e t e c t i o n o f chanaes i n t h e p a r a m e t e r s o f p e r i o d i c o r p s e u d o - p e r i o d i c systems wllen t h e chanae t i m e s a r e unknown. I n : S. I k e d a ( E d i t o r ) , S t a t i s t i c a l C l i n a t o l o p y . E l s e v i e r , Amsterdam. P l a c k e t t , R.L., 1950. Some theorems i n l e a s t s n u a r e s . B i o m e t r i k a 37: 149-157. P r i e s t l e y , V . R . , 1965. E v o l u t i o n a r y s o e c t r a and n o n - s t a t i o n a r y p r o c e s s e s . J . goy. S t a t i s t . SOC. B 27: 204-237. P r i e s t l e y , P.B. and Subba Rao, T., 1969. A t e s t f o r n o n - s t a t i o n a r i t y o f t i m e s e r i e s . J. Roy. S t a t i s t . SOC. B 31: 140-169. Subba Rao, T. and Tona, H., 1974. L i n e a r t i m e dependent svstems. I . E . E . E . T r a n s . on A u t . Cont., AC-19: 735-737. 1976. R e c u r s i v e aonroaches t o t i m e s e r i e s a n a l y s i s . Younq, P.C., B u l l . I n s t . Flaths. & , 4 o p l i c . 1 0 : 209-224.

27

DISTRIBUTION OF PARTIAL SUMS WITH APPLICATIONS T O DAM CAPACITY AND ACID RAIN A.M. MATHAI McGill University, Montreal, Canada

SUMMARY D i s t r i b u t i o n of t h e t o a l random i n p u t o v e r a p e r i o d o f t i m e i n t o a dam o r s t o r a g e

s c o n s i d e r e d i n t h i s a r t i c l e . The

i n p u t s c o u l d be t h e amounts o f s e d i m e n t s c a r r i e d i n t o a dam o n d i f f e r e n t o c c a s i o n s o v e r a p e r i o d o f t i m e o r t h e amounts o f w a i e r i n e x c e s s o f t h e n o r m a l f l o w due t o r a i n s on d i f f e r e r i t o c c a s i o n s o r t h e amounts o f a c i d d e p o s i t e d i n a l a k e by c l o u d s c a r r y i n g a c i d vapours e t c . E x a c t d i s t r i b u t i o n o f t h e t o t a l IS

0

t Y Pe

t a i n e d when t h e i n d i v i d u a l

input

i n p u t s a r e o f a g e n e r a l gamma

P o s s i b l e a p p l i c a t i o n s o f I-hese r e s u l t s t o v a r i o u s p r a c t c a I

p r o b ems a r e a l s o p o i n t e d o u t .

NTKODUCT I ON

I.

i n t o a s t o r Qe n input over a period of n occasions i s then

C o n s i d e r i n d e p e n d e n t random i n p u t s X 1 7 . . . , X t a n k . The t o t a l

sn

=

X I + ...+ X,

These random i n p u t s may be t h e e x c e s s amounts o f w a t e r f l o w i n g i n - l o a dam due t o r a i n s o f d i f f e r e n t d u r a t i o n s a t t h e w a t e r h e a d , t h e y cou I d b e t h e amounts o f p o l l u t a n t s d i s c h a r g e d i n t o a l a k e f r o m d i f f e r e n t s o u r c e s o r from one s o u r c e on d i f f e r e n t o c c a s i o n s , t h e y c o u l d b e t h e amounts o f s e d i m e n t s c a r r i e d t o a dani o n d i f f e r e n t occasions e t c .

I f X I , X 2,... r e p r e s e n t e x c e s s f l o w due

t o r a i n s t h e n an a p p r o p r i a t e p r a c t i c a l model w o u l d be e x p o n e n t i a l d i s t r i b u t i o n s w i t h d i f f e r e n t mean v a l u e s because t h e d u r a t i o n s

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

0

28 o f t h e r a i n f a l I s a r e u s u a l l y d i f f e r e n t for d f f e r e n t occas i o n s . i s a sum o f i n d e p e n d e n t bu n o t i d e n t i ca I I y n d i s t r i b u t e d e x p o n e n t i a l random v a r i a b l e s . I f w a t e r i s c o l l e c t e d I n such a c a s e S

a t d i f f e r e n t w a t e r h e a d s on d i f f e r e n t o c c a s i o n s b e f o r e f l o w i n g i s t h a t i t i s a sum n o f n i n d e p e n d e n t gamma d i s t r i b u t e d random v a r i a b l e s . I n o r d e r i n t o a dam t h e n a n a p p r o p r i a t e model f o r S

t o answer t h e p r o b l e m s o f o v e r f l o w one needs tp s t u d y t h e

probability that S

n

e x c e e d s a p r e a s s i g n e d number. F o r t h i s and

o t h e r p u r p o s e s one needs t h e e x a c t d i s t r i b u t i o n o f S n . Consider another problem o f a r e f i n e r y e m i t t i n g s u l p h u r d i o x i d e i n t o t h e a i r . The c o n c e n t r a t i o n o f t h e p o l l u t a n t s may be maximum a t a c e r t a i n h e i g h t f r o m t h e chimney and t h e n i t may t h i n o u t i n a l l d i r e c t i o n s . U n d e r some m i l d c o n d i t i o n s i t i s reasonab e t o assume t h a + t h e d i s t r i b u t i o n o f t h e p o l l u t a n t s i n the a i r

s a t h r e e dimensional Gaussian t y p e w i t h t h e o r i g i n

b e i n g a t an o p t i m a l h e i g h t f r o m t h e chimney. Suppose t h a t a p a t c h o f c l o u d i s p a s s i n g t h r o u g h t h a t p l a c e . How much o f t h e p o l l u t a n t s w i I I b e c a r r i e d away by t h a t c l o u d ? I n o t h e r w o r d s how much a c i d r a i n one can e x p e c t f r o m t h a t p a t c h of c l o u d ? Assuming t h a t t h e p a t c h o f c l o u d can b e l o o k e d upon as an e l l i p s o i d , w i t h respect t o t h e water molecule content, w t h a c e n t r e o f i t s own t h e n t h e amount o f a c i d r a i n i s p r o p o r i o n a I 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 a d i s o r i e n t e d e l l i p s o i d i n a 3d i m e n s i o n a l Gaussian d i s t r i b u t i o n . I n t h e g e n e r a l c a s e o f t h i s t y p e o f p r o b l e m one has t h e f o l l o w i n g s i t u a t i o n . C o n s i d e r a p - v a r i a t e v e c t o r random v a r a b l e d i s t r i b u t e d according t o a p-variate normal. That i s

x

%

rd ( u , c ) P

where p = E ( X ) and C

E

i s t h e c o v a r i a n c e m a t - r i x and

denotes t h e

e x p e c t e d va I u e . A d i s o r i e n t e d e l I i p s o i d ,wi I I t h e n be o f -the f o r m A = C ( X - u ) ' C ( X - a ) -<

61

where C i s a s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i x ,

(y.

i s n p-vcctor

o f known c o n s t a n t s , a p r i m e d e n o t e s a t r a n s p o s e and 6 a s s i g n e d number.

i s a pre-

I f v=a t h e n t h e e l I i p s o i d i s c e n t r e d a t t h e

29 e x p e c t e d v a l u e o f X. Then t h e p r o b a b i l i t y c o n t e n t o f A,

denoted

b y P ( A ) i s a v a i l a b l e f r o m t h e m u l t i n o r m a l d e n s i t y as f o l l o w s .

-< 6)

P(A) = P{(X-a)'C(X-a) where X i s a p - v a r i a t e

random v e c t o r n o r m a l l y d i s t r i b u t e d a n d

a,C,6 a r e a l I known. F o r c o n v e n i e n c e we s h a l I r e o r i e n t and r e l o c a t e t h e e l l i p s o i d by making t h e t r a n s f o r m a t i o n Y = C-'/'(x-U) where Z 1 / *

x

or

= 1 ' " ~+ p

= .'/'(y+z-'/'p)

i s t h e symmetric square root o f C.

Then

N

( 0 , I ) and X-a = c 1 / 2 ( Y + E - 1 / 2 ( p - , ) ) P Q (X-a)'C(X-a) = ( Y + B ) ' V ( Y + B )

Y

'L

where

v = c

I /ZCC I /2

,

B = c-1/2(p-a)

S i n c e V i s s y m m e t r i c and p o s i t i v e d e f i n i t e t h e r e e x i s t s an o r t h o g o n a l m a t r i x Q such t h a t Q ' V Q = D = d i a g ( A t h e t r a n s f o r m a t i o n Z = Q ' Y o r QZ=Y and t h e n Z s n

N

D

I

,..., A

).

P (0,k) and

Make

, ,

(1.1) where y=Q'a, y . i s t h e j t h e l e m e n t i n Q ' a and z . i s t h e j t h J J e l e m e n t i n Z. S i n c e z . Q N ( O , I ) , j = l , ...,p and m u t u a l l y i n d e J p e n d e n t (z.+y.)' i s a n o n c e n t r a l c h i s q u a r e w i t h one d e g r e e o f J ? freedom. What i s t h e p r o b a b i I i t y c o n t e n t o f t h e e l I i p s o i d (X-a)'C(X-a)

-i 6

I t i s n o t h i n g b u t t h e p r o b a b i l i t y P { U -<S}

?

where U i s a l i n e a r c o m b i n a t i o n o f i n d e p e n d e n t n o n c e n t r a l c h i s q u a r e v a r i a b l e s w i t h one d e g r e e o f freedom e a c h . . I f we a r e c o n s i d e r i n g t h e amounts of a c i d r a i n t h a t can b e e x p e c t e d f r o m s e v e r a l cloud p a t c h e s p a s s i n g t h r o u g h t h e chimney a r e a t h e n t h e problem reduces t o t h e e v a l u a t i o n of a probabi I i t y of t h e t y p e where X ,..., X a r e i n d e p e n d e n t n I n I n weighted n o n c e n t r a l c h i s q u a r e v a r i a b l e s w i t h d i f f e r e n t deqrees P{Sn ~

6 wj i t h

S =X +...+X

o f f r e e d o m and 6

i s a p r e a s s i q n e d number.

I n t h i s p a p e r we w i I I c o n s i d e r t h e d i s t r i b u t i o n of Y = X

...+X

when X

,. . .,X

a r e m u t u a l l y i n d e p c n d c n t and ( 1 )

I garnmc

P I P d i s t r i b u t e d w i t h d i f f e r e n t parameters ( 2 ) noncentra I ch i s q u a r e d i s t r i D u t e d w i t h d i f f e r e n t d e g r e e s o f freedom.

30 2.

GAMMA I NPUl Cons d e r p i n p u t s i n t o a dam w h i c h a r e m u t u a l y i n d e p e n d e n t l y

gamma d i s r i b u t e d w i t h t h e i t h i n p u t d e n o t e d by X i

and h a v i n g

the densitv

a. c1.- I I - x i /A f.(x.) = ( r ( a . 1 ~ . ) - I x . ' e

,a.>O, A.>O, x.>O,

(2.1)

and z e r o e l s e w h e r e ,

i n p u t i s Y=X + . . . + X

.

I

I

I

t

I

I

I

so t h a t t h e t o t a l

t

I

P S i n c e t h e moment q e n e r a t i n g f u n c t i o n s e x i s t and i n v e r t i b l e i n

t h e f o l l o w i n g p r o b l e m s we w i I l w o r k w i t h t h e moment g e n e r a t i n q f u n c t i o n s f o r c o n v e n i e n c e . The moment g e n e r a t i n g f u n c t i o n o f X

i

IS

and t h a t ~f Y i s (2.2)

The d e n s i t y o f Y i s a v a i l a b l e b y i n v e r t i n g M ( t ) . T h e r e a r e many Y w a y s of doing t h i s b u t we w i l l p r o c e e d as follows.

(1-X.p

(I-A,I)(A/A )(l-(l-A

=

2

Hence

( I - x t)- a I

I

1

-a (l-A,t)

2

=(I-A

I

I

/A,)/(l-A 2

-(a +a I

t)

I

t) )

)

-U

(A2/AI)

2

-cI

(

I - ( I - A I /A2)/(

I - A I f)

)

2

But

f o r -t < I / i 2 ,

...

l / ~ ~where , f o r example ( a ) = (a+l) r d e n o t e d b y g (y), i s I 2' 2 -a* a1+a2-l -y/x 9 2 ( Y ) = (T2/AI) y e I C(a2),( I - A I / X 2 r=O a I tr ! X rI r ( c I l + c 1 2 + r ) )

u t r - I 1.

Hence t h e d e n s i t y o f Y=X +X

I

!Ir/

31 where

F i s a confluent hypergeometric function. Proceeding I I e x a c t l y t h e same way one has t h e f o l l o w i n g r e s u l l - i n t h e

genera I case.

g ( y ) = ( A c1 I . . . A U p r ( a l +. . . + a ) ) - I P I P P

c ...

m

00

r =O 2

c

r =O P

(qr. . . ( aP 2

r

(

a +. . .+a - I P v I -I

-A2

(Al

-I

e-Y/X,

r )y 1

2

...

P

The m u l t i p l e s u m a p p e a r i n g h e r e i s a c o n f l u e n t h y p e r g e o m e t r i c function o f p-l

variables,

namely

0,

( s e e M a t h a i 8 Saxena,1978,

p . 16 3). The d e n s i t y is t h e r e f o r e ,

. . . + a p ; ( X-l I - A 2- I ) y ,

..., ( A -I I - A -PI

)y),

(2.5)

y > 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)

<6 t h e n t h e q u e s t i o n t o be answered

i s what i s t h e p r o b a b i I i t y t h a t ( X - n ) ' C ( X - a ) a s s i g n e d 6 where X i s a 3 - v a r i a t e

<6 f o r a p r e normal w i t h mean v e c t o r p

and c o v a r i a n c e m a t r i x C where p , a

a r e known v e c t o r s and C

is

a known m a t r i x ? I n s e c t i o n I we h a v e t r a n s f o r m e d t h i s p r o b l e m i n t h e g e n e r a I c a s e o f p c o - o r d i n a t e s t o t h e e v a I u a t i o n of t h e P

?

P

where A p r o b a b i I i t y P{U< 6 ) where U = . c A . ( z . + ~ . ) ~ = C A . u J=l J J J j=I j j I' ..., A a r e known c o n s t a n t s and u . i s a n o n c e n t r a l c h i s q u a r e w i t h P J 2 one d e g r e e o f f r e e d o m and n o n c e n t r a l i t y p a r a m e t e r y , / 2 . The J

moment g e n e r t i n

function of u . i s therefore 2 J -Y . / L - 1/2 e ( Y . / 2 ) / ( I - Z t ) ( t )= e J (i-zt) J

5

M

' iJ

2

The moment g e n e r a t i n g f u n c t i o n o f U i s t h e r e f o r e M (t)= e U

-0

P -1/2 B ./( I-2h.t) II(l-2X.t) J e J J j= I

where B . = y 2. / 2 , J J

0 =

1 fi

j- I j'

(4.I )

The i n v e r s i o n o f ( 4 . 1 ) g i v e s t h e

d e n s i t y o f U. T h i s moment g e n e r a t i n g f u n c t i o n i n ( 4 . 1 ) o r t h e c o r r e s p o n d i n g L a p l a c e t r a n s f o r m o f t h e d e n s i t y o f U comes i n a wide v a r i e t y of problems connected w i t h g e o m e t r i c probabi I i t i e s , d i s t r i b u t i o n s o f q u a d r a t i c forms, performance e t c . ,

communication problems,

see f o r example H e l s t r o m (1978), R i c e ( 1 9 8 1 ) ,

G i I I i l a n d and Hansen (19741, Ruben (1962) and t h e many rences t h e r e i n .

radar

refe-

I t a l s o comes i n c e r t a i n t i m e s e r i e s p r o b l e m s ,

see f o r example, MacNei I I (1974). I f t h e r e a r e n i n d e p e n d e n t and i d e n t i c a l o c c a s i o n s o f such

c l o u d f o r m a t i o n s t h e n t h e d e n s i t y f o r t h e t o t a l amount o f a c i d r a i n w i l l have t h e moment g e n e r a t i n g f u n c t i o n -0 -n/2 B . / ( I - 2 ~. t ) Mu(t) = e I II(l-2A.t) e J J j= I J

(4.2)

35 T h i s e x p r e s s i o n a l s o comes i n a v a r i e t y o f p r o b l e m s i n v o l v i n g traces o f noncentral Wishart matrices, (1980).

see f o r example M a t h a i

I n t h i s c a s e i t i s a l s o shown i n M a t h a i ( l 9 8 0 ) t h a t t h e

q u a n t i t y (U-n t r B ) / ( Z n t r B 2 )

H

n goes t o i n f i n i t y where matrix with

A

I

,..., A

P

”*

goes t o a s t a n d a r d normal when

i s a symmetric p o s i t i v e d e f i n i t e

b e i n g i t s e i g e n v a l u e s and t r B d e n o t e s t h e

t r a c e o f €3.

I f t h e r e a r e a number o f i n d e p e n d e n t o c c a s i o n s b u t n o t a l l i d e n t i c a l l y d i s t r i b u t e d then

e w i I I have t h e general s i t u a t i o n

where M ( t ) w i I I have t h e f o l o w i n g f o r m

where n l ,

..., n P

are positive

nt e g e r s

.

We w i I I c o n s i d e r t h e i n v e r s i o n o f

(4.I ) h e r e . O t h e r c a s e s

o f ( 4 . 2 ) and ( 4 . 3 ) c a n b e h a n d l e d i n t h e same f a s h i o n . a,

nP e a J. ( l - Z t h J. ) - l

j=I

=

~

r=O r

( r ! . . . r !I I P

I

c +...+ r

o r1l . . . a P

=r

r

P

P(I-2tA

-I

I

)-rl

(4.4)

Now compare t h i i p r o d u c t c o n t a i n i n g t w i t h t h e e x p r e s s i o n i n

(2.2). Hence t h e d e n s i t y o f I J ,

r ( r +...+ I

where

m2

and

r +p/2))-’ P

,

d e n o t e d by f ( u ) ,

u >

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:

<

p1 = p2

I, i

=

1 , 2.

One wants t o t e s t the n u l l hypothesis.

a n d 0, = 023

(3) I .

t h a t i s , the samples y1 , i , i = 1 , 2 , . . . , n l + l , and y 2 , i , i = 1 , 2 , . . . , n 2 + l , come from identical populations. Write n = min ( n l y n 2 ) a n d define

=

0 Zi-1 + e i i

=

9

Let

fl

2,3,

...,n + l ;

(5)

be the l e a s t squares estimator of 0; then

n+ 1

n+l (6)

which i s consistent a n d asymptotically unbiased (Chatfield, 1975; Christopeit and Helmes, 1980). Define

x1i

= Y1 , i + l

and x2i

=

-

Y2,i+1

$yl,i, i

- 0

y2,i’

1 , 2 ,... J l

=

i

=

1,2,

...,n 2 .

(7)

47

To t e s t Ho, t h e proposed s t a t i s t i c i s ( A l = nl-2rl, T

=

=

=

n2-2r2)

+ (1/m2)1,

(Gl-i+)/GJ{(l/ml)

where ml

A2

nl-2rl+2r1B1

cl , c 2 , G l

(A2-l)c~}/(A1+A2-2);

(8)

and m2 = n 2 - 2 r 2 f 2 r 2 ~ 2 and and

a,

k2

+

= I(A1-l)02 I

a r e T i k u ' s (1967,1978,1980)

MML

( m o d i f i e d maximum 1 ik e l i h o o d e s t i m a t o r s ; these e s t i m a t o r s a r e calculated from equations ( A . l ) xi's

r e p l a c e d by x1 I s and x2i

t o (A.4) g i v e n i n t h e appendix w i t h

' s and r r e p l a c e d by rl = C0.5

f

O.lnl 1

and r2 = C0.5 + 0.1n21, r e s p e c t i v e l y . Theorem:

I f t h e u n d e r l y i n g p o p u l a t i o n s a r e normal

, the asymptotic

(A1 and A2 b o t h t e n d t o i n f i n i t y ) n u l l d i s t r i b u t i o n o f T i s normal

N(O,1). Proof:

Since n i s l a r g e r t h a n min (A1 , A 2 ) ,

t h e r e f o r e , as A1 and A2 4

tend t o i n f i n i t y , n a l s o tends t o i n f i n i t y , i n which case P, converges i = 1,2 ,..., nl, and x2i, i = 1,2 ,n2, a r e t o g; consequently, xli,

,...

i i d normal N(u,a) under Ho and t h e theorem f o l l o w s i m m e d i a t e l y f r o m the f a c t t h a t

E ( a 2 ) = u and J<

=

( ; l - ~ ) / ~ and

i n d e p e n d e n t l y d i s t r i b u t e d as normal N ( 0 , l ) ;

5

(b2-p2)/u

are

see T i k u (1978, Lemmas 1

and 2 ; 1981). Even if t h e u n d e r l y i n g p o p u l a t i o n s a r e non-normal b u t w i t h e x i s t e n t

,.

means and variances, consequently, xli 1,

and

E(u~) =

g2

B

converges t o 0 as n tends t o i n f i n i t y and,

and xZi

are i i d .

I n such s i t u a t i o n s , t h e e s t i m a t o r s

converge ( T i k u , 1980, p.134) t o t h e i r expected values = ko (under H o ) , and s i n c e jl and are l i n e a r

~(o~)

c2

functions o f order s t a t i s t i c s they are asymptotically normally d i s t r i b u t e d under some v e r y general r e g u l a r i t y c o n d i t i o n s ; see S t i g l e r (1974).

For most symmetric p o p u l a t i o n s ( t h e f a m i l y of

S t u d e n t ' s t d i s t r i b u t i o n s w i t h degree o f freedom g r e a t e r t h a n 2, f o r example), T i k u (1980, p.134) showed t h a t V ( f i l a r g e samples; t h e a s y m p t o t i c n u l l d i s t r i b u t i o n o f

h k o ) = 1 for

T f o r such

symmetric non-normal p o p u l a t i o n s i s , t h e r e f o r e , a l s o a p p r o x i m a t e l y Normal N ( 0 , l ) .

TABLE 1

n,I =n,=lO L

Simulated Values of the Probability P(T>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


0.5 which i s o f c o n s i d e r a b l e

p r a c t i c a l i n t e r e s t ( P i e r c e , 1971) and, here, we found t h e normal a p p r o x i m a t i o n adequate. P(T

<

For example, we s i m u l a t e d t h e p r o b a b i l i t i e s

h / H o ) f o r numerous symmetric and skew p o p u l a t i o n s , h b e i n g t h e

upper 100 ( l - S ) % p o i n t o f t h e normal N ( 0 , l ) d i s t r i b u t i o n ; t h e s i m u l a t e d values (based on 80000/nl

Monte C a r l o r u n s ) o f these

p r o b a b i l i t i e s a r e g i v e n i n Table 1.

It i s c l e a r t h a t the n u l l

d i s t r i b u t i o n o f T i s r o b u s t t o p a r e n t p o p u l a t i o n s , and i s c l o s e l y approximated b y t h e normal

N(0,l ) d i s t r i b u t i o n i n t h e range

-0.5 G f~d 0 . 5 . A t t h i s p o i n t , one would perhaps l i k e t o know about t h e robustness p r o p e r t i e s o f t h e analogous S t u d e n t ' s t s t a t i s t i c based on t h e sample means and v a r i a n c e s , t h a t i s , t h e s t a t i s t i c T w i t h rl

=

r2 = 0.

Given

below a r e t h e s i m u l a t e d values (based on 8000 r u n s ) o f t h e p r o b a b i l i t y P(t

<

h l H o ) ; nl = n2 = 10:

la -0.5

0.0

0.1

fl 0.5

0.7

-0.5

Norma 1

5 1

.048 .008

.036 .006

.034 .003

0.0

0.1

0.5

0.7

Dexponerit ia 1 .024 .002

.017 .002

.041 .007

.035 .004

.023

.022

2 x2-2

t2

5 .040 .033 .030 .020 .016 .003 .002 .001 1 .005 .003 -----__-__-----

.033

.004 .002 .002

.041 ,007

.034 .003

.032 .003

.023 .002

.020 .001

51

The r e s u l t s f or ( n l , n 2 ) = (20,lO) and (20,20) are s milar b u t we d o n o t reproduce them f o r conciseness. I t i s c l e a r t h a t the n u l l distribution of t i s very sensitive t o changes in 0 3.

POWER PROPERTIES

The alter natives t o Ho t h a t are o f considerable practical * * i n ter es t are ( E ( x l i ) = u1 a n d E ( x ~ ~= ) p,) H 1 : d = p *l - p *2 > 0 ; large va ues of T lead t o the rejection of Ho in favour of H 1 . t h a t E (y +1 - O Y i ) = u ( 1 - 0 ) .

(9)

Note

Theorem: I f the underlying populations are normal, the asymptotic converges t o 0 as power function of T i s given by (assuming t h a t n tends t o i n f i n i t y ) P[z h-d/u4{(l/ml) + ( l / m 2 ) l l ; z being a standard normal variate.

(10)

Proof: Follows exactly on the same l i n e s as the previous theorem. The corresponding power function f o r the Student's t s t a t i s t i c mentioned above i s given by P l z 2 h-d/uv'i(l/nl) + ( l / n 2 ) l l .

(11 1

I f the underlying populations are normal, the asymptotic values of

the power o f the s t a t i s t i c s T and t can be calculated from equations ( 1 0 ) and ( 1 1 ) and i t i s seen t h a t T i s only s l i g h t l y l e s s powerful t h a n t ; see also the following values of the power. For small samples, we simulated (from 2000 runs) the values o f the power of T a n d t f o r ( n l , n 2 ) = (lO,lO), (20,lO) and (20,20) and for numerous populations a n d found the s t a t i s t i c T considerably more powerful on the whole. For example, we have the following values of the power; n l = n2 = 10, 0 = - 0 . 5 , and the significance level f o r b o t h T and t i s approximately 1 % :

52

d

0.5

Distribution

1.0

1.5

2.0

2.5

3.0

Normal

T t

.13 .13

.47 .48

.84 .85

.98 .99

1.00 1.00

1.00 1.00

Dexponenti a1

T t

.10 .07

.32 .27

.64 .55

.85 .80

.95 .92

.99 .98

T t

.08

.05

.25 .16

.49 .35

.71 .52

.85 .67

.92 .77

T t

.ll .10

.40 .36

.73 .64

.92 .85

.98 .95

1 .OO .99

T t

.19 .16

.61 .53

.89 .84

.97 .96

.99 .99

1 .OO 1.00

(n-l)N(O,l) 1N(0,3) x2-2 2

&

---------Note t h a t t h e above values o f t h e power were o b t a i n e d by adding t h e c o n s t a n t d t o t h e o b s e r v a t i o n s xli,

i = 1,2,.

. . ,nl.

ACKNOWLEDGEMENT Thanks a r e due t o NSERC o f Canada and McMaster U n i v e r s i t y Research Board f o r r e s e a r c h g r a n t s .

Thanks a r e a l s o due t o Mrs. Carmela

C i v i t a r e a l e f o r t y p i n g the manuscript. REFERENCES

C h a t f i e l d , D. (1975). The A n a l y s i s o f Time S e r i e s : Theory and P r a c t i c e . Chapman and H a l l , London. C h r i s t o p e i t , N . and Helmes, K. (1980). S t r o n g c o n s i s t e n c y o f l e a s t squares e s t i m a t o r s i n l i n e a r r e g r e s s i o n models. Ann. S t a t i s t . 8, 778-88. Ljung, G.M. and Box, G.E.P. (1980). A n a l y s i s o f v a r i a n c e w i t h a u t o c o r r e l a t e d o b s e r v a t i o n s . Scandinavian J . S t a t i s t . 7, 172-80. P i e r c e , D.A. (1971). L e a s t squares e s t i m a t i o n i n t h e r e g r e s s i o n model w i t h autoregressive-moving average e r r o r s . B i o m e t r i ka 58, 229-31 2 . S t i g l e r , S.M. (1974). L i n e a r f u n c t i o n s o f o r d e r s t a t i s t i c s w i t h smooth w e i g h t f u n c t i o n s . Ann. S t a t i s t . 2, 676-99. Subrahmaniam, K., Subrahmaniam K a t h l e e n and Messori , J . Y . (1 975). On t h e robustness o f some t e s t s o f s i g n i f i c a n c e i n sampling f r o m a compound normal p o p u l a t i o n . J . Amer. S t a t i s t . Assoc. 70, 435-38. Tiku, M.L. (1967). E s t i m a t i n g t h e mean and s t a n d a r d d e v i a t i o n f r o m censored normal samples. B i o m e t r i ka 54, 155-65.

53

T i k u , M . L . ( 1 9 7 0 ) . Monte C a r l o s t u d y o f some s i m p l e e s t i m a t o r s i n c e n s o r e d normal samples. Biometrika 57, 207-11. T i k u , M . L . ( 1 9 7 1 ) . S t u d e n t ' s t d i s t r i b u t i o n under non-normal s i t u a t i o n s . Aust. J . S t a t i s t . 13, 142-48. T i k u , M . L . ( 1 9 7 8 ) . L i n e a r r e g r e s s i o n model w i t h c e n s o r e d observat i o n s . Commun. S t a t i s t . A 7 ( 1 3 ) , 1219-32. T i k u , M . L . ( 1 9 8 0 ) . Robustness o f MML e s t i m a t o r s based on c e n s o r e d samples and r o b u s t t e s t s t a t i s t i c s . J . S t a t i s t i c a l P l a n n i n g and I n f e r e n c e 4 ( 2 ) , 123-43. T i k u , M . L . ( 1 9 8 1 ) . Testing l i n e a r c o n t r a s t s o f means i n e x p e r i m e n t a l d e s i g n w i t h o u t assuming n o r m a l i t y and homogeneity o f v a r i a n c e s . I n v i t e d Paper: P r e s e n t e d a t the March 22-26, 1981, B i o m e t r i c Colloquium o f the GDR-Region o f the B i o m e t r i c S o c i e t y (paper t o appear i n Biometrical Journal ). T i k u , M . L . and S t e w a r t , D . ( 1 9 7 7 ) . E s t i m a t i n g and t e s t i n g group e f f e c t s from Type 1 c e n s o r e d normal samples i n e x p e r i m e n t a l d e s i g n . Commun. S t a t i s t . A6 ( 1 5 ) , 1485-1501. T i k u , M . L . and S i n g h , M. ( 1 9 8 1 ) . Robust t e s t f o r means when popul a t i o n v a r i a n c e s a r e unequal. Commun. S t a t i s t . A 1 0 ( 2 0 ) , t o appear APPEND1 X

Let

be a Type 1 1 c e n s o r e d sample, o b t a i n e d by a r r a n g i n g n random

o b s e r v a t i o n s x1 x 2 , . . . , x n i n a s c e n d i n g o r d e r o f magnitude and c e n s o r i n g the r s m a l l e s t and r l a r g e s t o b s e r v a t i o n s ; T i k u ' s (1967, Y

1978, 1980) MML e s t i m a t o r s ( d e f i n e d f o r m a l l y by T i k u and S t e w a r t , 1977) o f the l o c a t i o n p a r a m e t e r

and s c a l e p a r a m e t e r u a r e g i v e n

LI

bY

j= c

n-r

xi

2

+ rg ( x ~ ++ ~xn-,)}/m

i =r+l

and

= {B +

where m

c

=

d(B2 + 4AC)1/2JIA(A-l)},

n - 2 r + 2 r ~ , A = n-Zr, B = r n ( X n - r

n-r =

z: xi2

i = r +I

+

2

rB(Xr+l +

2 xn-r) -

-2.

mu

-

X,,,),

and

54

t h e c o e f f i c i e n t s a and 6 a r e g i v e n by T i k u (1967, e q . 6 ) . n 2 10, however,

CY

For

and B a r e o b t a i n e d f r o m t h e f o l l o w i n g s i m p l e r

e q u a t i o n s ( T i k u , 1970): B

=

-f(t){t-f(t)/ql/q

and a = { f ( t ) / q 3 - B t ;

(A.4)

q = r / n , and t i s determined by t h e e q u a t i o n F ( t ) = i t f ( z ) d z = 1 -q -W

and f ( z ) = t1/4(2n)I exp ( - z L / 2 ) , -m and 0 < B < 1 .


m.

Note t h a t 0

<

a

<

1

I n t h e absence o f any knowledge about t h e u n d e r l y i n g

p o p u l a t i o n , r i s always chosen t o be t h e i n t e g e r v a l u e r = [ 0 . 5 + 0 . l n l ; see T i ku (1980) and T i ku and Singh ( 1 981).

a

a r e i n v e s t i g a t e d by The e f f i c i e n c i e s o f t h e e s t i m a t o r s and T i k u (1980); and 0 t u r n o u t t o be remarkably e f f i c i e n t ( j o i n t l y ) . Note t h a t f o r r = 0, ;I and variance

and s

2

.

0’

reduce t o t h e sample mean and

55

INFERENCE ABOUT THE POINT OF CHANGE I N A REGRESSION MODEL WITH A STATIONARY ERROR PROCESS

A.H.

EL-SHAARAWI and S.R.

ESTERBY

N a t i o n a l Water R e s e a r c h I n s t i t u t e , B u r l i n g t o n , O n t a r i o

ABSTRACT 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 model w i t h a n a u t o r e g r e s s i v e e r r o r p r o c e s s of o r d e r one i s c o n s i d e r e d .

Expressions

f o r t h e e x a c t and f o r a n a p p r o x i m a t e 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 t h e p o i n t of change "1 and 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 p are d e r i v e d , and i t e r a t i v e p r o c e d u r e s f o r t h e e s t i m a t i o n of n 1 and p are given.

The two l i k e l i h o o d f u n c t i o n s are shown t o g i v e a p p r o x i m a t e l y

t h e same r e s u l t s when t h e number of o b s e r v a t i o n s i s l a r g e .

For t h e

number of o b s e r v a t i o n s odd, a c o n d i t i o n a l p r o c e d u r e i s used t o e l i m i n a t e t h e dependency between t h e random v a r i a b l e s , and t h u s , t o p e r m i t t h e d e r i v a t i o n of a l i k e l i h o o d f u n c t i o n f o r n 1 a l o n e .

The p e r f o r m a n c e

of t h e above p r o c e d u r e s and t h e l i k e l i h o o d f u n c t i o n o b t a i n e d assuming no a u t o c o r r e l a t i o n are compared i n a Monte C a r l o s t u d y , where t h e samples g e n e r a t e d m i m i c a change i n mean l e v e l of t h e volume of d i s c h a r g e of a r i v e r .

1. INTRODUCTION The s t u d y of t h e c h a r a c t e r i s t i c s of a body of water o v e r a p e r i o d of t i m e f r e q u e n t l y involves t h e q u e s t i o n as t o whether a characteristic

has changed a t some p o i n t .

Although s u c h a s i t u a t i o n c a n o c c u r i n many

ways, an example i s t h e a b r u p t change i n i n d u s t r i a l d i s c h a r g e d u r i n g t h e m o n i t o r i n g of a r i v e r f o r v a r i o u s water q u a l i t y p a r a m e t e r s .

There

e x i s t s a number of d i f f e r e n t s t a t i s t i c a l p r o c e d u r e s d e s i g n e d t o d e t e c t o r e s t i m a t e a p o i n t of change under v a r i o u s s i t u a t i o n s . E s t e r b y and El-Shaarawi

For example,

(1981b) o b t a i n l i k e l i h o o d f u n c t i o n s t o e s t i m a t e

Reprinted from T i m e Series Methods in Hydroseiences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

56

the point of change, Hinkley (1970, 1971) derives the distribution of the maximum likelihood estimate, Pettitt (1979) presents a non-parametric approach to detection and Brown et al. (1975) use cumulative sums and recursive residuals.

Examples of application to river

data, in particular, discharge data of the Nile River, are the use of cumulative sums (Kraus, 1956), an approximation to the conditional distribution of the maximum likelihood estimator (Cobb, 1978) and the relative marginal likelihood function (Esterby and El-Shaarawi, 1981a).

All of the above techniques assume that the observations are

independent. In the present paper, likelihood functions are obtained for the estimation of the point of change in a regression model with an autoregressive error process of order one.

The performance of a likelihood

function assuming independence is also considered.

The likelihood

functions are derived in Section 2 and they are compared in a Monte Carlo study in Section 3 . Consider the model P

...,n1)

1 a e.+ut Yt - j=l tj J q

Yt

1 btj@j -- j=l

+

(t = n 1+1 ,n1+2,.

Ut

where u1,u2,...,un

(1.1)

(t = 1,2,

..,n=nl+n2)

are n successive random variables from a

stationary process.

The coefficients {atj} and {btj} are known but

the parameters {Oj },

{& }

p

and the change point parameter "1, where

n1 and p+q < n , are unkown.

Assume that ul,u2,..

.,un is a sample

from an autoregressive sequence of normal variables, i.e., Ut =

where

P ut-l+et

[

pi

(t =

...,-2,-1,0,1,2,...)

< 1 and the increments {et} are normally and independently

distributed with mean 0 and variance

CI

2

.

57 2.

LIKELIHOOD FUNCTIONS

2.1

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

...,un

The j o i n t d e n s i t y of

U O , U ~ ,

is

n

n f(ut/ut-l) (2.1) t=l 2 4 2 2 " 2 = f(u0) ( 2 1 1 ~ ) exp {-+a ( ~ ~ - p ~} - ~ ) t -1

f ( U O , U l , . * * , U n) = f ( U 0 )

,C

where t h e i n i t i a l v a l u e u o is a n u n o b s e r v a b l e random v a r i a b l e .

Three

main p o s s i b i l i t i e s f o r f ( u 0 ) (Cox and H i n k l e y , 1974) are t o assume t h a t

u o is a known c o n s t a n t , u o = q,, o r f ( u 0 ) h a s t h e form which w i l l

...

make u ~ , u ~ ,,un s t a t i o n a r y .

The a s s u m p t i o n t h a t uO=un w i l l n o t

be c o n s i d e r e d h e r e . 2

Assuming u o , u l , . . . , u

n

stationary leads t o f ( u O )

- N ( 0 ,dy ) . 1- P

P u t t i n g t h i s i n t o e x p r e s s i o n ( 2 . 1 ) and i n t e g r a t i n g o v e r u o g i v e s f(ul,u2

,...,u n )

2 -1112 (1-p2)+ exp{-+a2(u2(1-p2) = (2nd ) 1

The j o i n t l i k e l i h o o d f u n c t i o n f o r

and by yt

-

1

j =1

{ej}, {pj},

btjpj f o r t=nl+l,nl+2,

a2, p and n1 i s

...,n.

The a s s u m p t i o n t h a t u0=0 can be c o n s i d e r e d a n a p p r o x i m a t i o n t o

.. N(O,%)

2

and i t i s used h e r e because i t 1- P p r o v i d e s a s i m p l e r method f o r c a l c u l a t i n g 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 t h e case where u o

f o r nl and p.

For ug=O, t h e j o i n t d e n s i t y of u l , u 2 , . . . , u n is n f(ul,u2, u n ) = ( 2 r 1 1 a ~ ) exp -~~~ {-t 0 2 ( u 2 + ( U ~ - ~ U ~ - ~ )( 2~. 3) )} . 1 t=2 Again 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 f o r { 8 j }, { & }, u 2 , p and

...,

1

E

n1 i s o b t a i n e d by r e p l a c i n g u t by Y t - j = l

a t j 8..I f o r t = 1 , 2 ,

...,nl

58

1

- b . p . f o r t=n1+1,nl+2 t j=1 tJ J

and by y

,...,n. , l

- N(O,-),1- P

lJL

Under e i t h e r t h e a s s u m p t i o n t h a t uo=O o r u o 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 }

d e n o t e d by squares.

},

and { p j

Then u

-

-

t

't

>: 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

<

o2

and H - : Ol

> 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

<

Pr(/sij

c , i - 1,2

,..., n - 1 )

Pr(U G c )

=

n-1 2

‘TT

Pr ( I S i I

c).

i=l Similarly Pr(U

+<

c)

>

n-1 TI

P(Si G c )

i=l A n o t h e r s t a t i s t i c f o r t e s t i n g H can be c o n s t r u c t e d b y t r a n s f o r m i n g t h e v a r i a b l e s S1,. Y19Y2’. Yk

=

.

*

. . ,Sn-l

YYn-l where ( f o r k = 1,2,

(‘k+lSk

02,

..., n - 1 )

- p k s k + l )/‘T(M-T)PkPk+l (‘k+]-‘k)

The d i s t r i b u t i o n o f yi

o1 =

t o t h e s e t o f u n c o r r e l a t e d random v a r i a b l e s

(i

=

1,2,

...,n - 1 )

w i t h mean 0 and u n i t v a r i a n c e .

*

i s a s y r n p o t i c a l l y n o r m a l , when

71

Define the random walk wk

=

wk-l

' YkY

for (k

=

1,2,

...,n-1)

where Wo = 0. The variances and the covariances are given respectively = min ( k , s ) . by Var ( W k ) = k and Cov ( W k , W s ) Approximating {Wk} by the corresponding Gaussian process w i t h the same mean and covariance matrix, Brown e t a1 (1975) have shown the probability t h a t tWkl crosses the l i n e s defined by the two PO nts (01 2 a m),( n - 1 , f 3 a m ) i s ct = .01,.05, .10 f o r a = 1.143, .948, .850 respectively . The empircal l o g i s t i c transform (Cox 1970) i s another method f o r testing H , H+ and H-. To d o t h i s suppose t h a t the l i n e a r l o g i s t i c model specifies t h a t f o r i = 1 , 2 , ..., k hl =

log [ 0 1 / ( 1 - ~ 1 ) 1and f o r i = k+l, ...,n

h 2 = log ~ o 2 / ( 1 - o 2 ) I

The corresponding empircal l o g i s t i c transforms are Zk = log(S,+l/2)/(Mk-Sk+l/2) and Z ' = log (Stk+1/2)/(M'k - S ' k + 1 / 2 ) , where Mk = ml +

.:

+ m k , S t k = T-Sk and

M I k

=

M-MI<.

The large-sample variances of Zk and Z I k are given respectively by (MIk+l)(Mtk+2) . ( Mktl ( Mkf2 'k = M k ( S k + l ) (Mk-Sk+l) and " k - M'k(S'k+l) (Mtk-Snk+l) Under the assumption o1 = 0 2 , Z I k - Z k i s an estimate of the l o g i s t i c difference X 2 - X 1 and has a standard e r r o r approximately For testing H, we use the s t a t i s t i c

Jv,+vk

and f o r t e s t i n g H

t

+

and H- we can similarly define D

and D-. The approximate significance levels associated with D, D , and D- can be calculated in the same manner as t h a t used f o r the s t a t i s t i c U. t

72

E S T I M A T I O N OF THE CHANGE-POINT

P e t t i t t ( 1 9 7 9 ) s u g g e s t e d as an e s t i m a t e f o r k t h e v a l u e

I; w h i c h

1 . T h i s e s t i m a t e o f t h e change p o i n t i s j u s t i f i e d by k,n noting t h a t the conditiona expectation o f U. given under H i s 3 ,n

maximizes / U

x

/n-k

for j for j

< >

k k

f o r r G k.

1x1

c o n s i s t s of two s t r a i g h t l i n e s . T h i s shows t h a t t h e g r a p h o f / U . J ,n The f i r s t , f o r r < k, has t h e p o s i t i v e hope Iu-xl, and t h e second f o r n k > h has t h e n e g a t i v e s l o p e - =lu-?. A n o t h e r way o f e s t i m a t i n g t h e c h a n g e - p o i n t b y t a l k i n g t h e v a l u e k w h i c h m a x i m i z e s t h e s t a t i s t i c D. (i

=

1,2,

. . . ,n - 1 )

F u r t h e r t h e p l o t o f Wi

a g a i n s t i,

can be used t o e s t i m a t e k g r a p h i c a l l y b y t a k i n g k

as t h e p o i n t where t h e p r o c e s s {Wi3

w i l l s t a r t t o follow a systematic

d r i f t from zero. APPLICATIONS I n s e d i m e n t c o r e s , t h e d i s t r i b u t i o n o f f o s s i l s i s a random v a r i a b l e w h i c h i s a f u n c t i o n o f t h e c h e m i c a l , p h y s i c a l , and b i o l o g i c a l c o n t r o l s when t h e b i o l o g i c a l u n i t was a l i v e .

I n c o r e s where a s u b s t a n t i a l

change i n t h e number o f o s t r a c o d e s h e l l s ( s e e d s h r i m p ) has o c c u r r e d a t some p o i n t i n t i m e , o b s e r v a t i o n s o f a c o r r e s p o n d i n g change o f t h e r e l a t i v e c o n c e n t r a t i o n i n a c o r e p r o v i d e s one means o f i d e n t i f y i n g t h e d e p t h , hence t h e t i m e , a t w h i c h a s i g n i f i c a n t change o c c u r r e d .

This

i s i m p o r t a n t because i t a l l o w s one t o d e t e r m i n e t h e t y p e o f change i n t h e h a b i t a t which c o n t r o l l e d t h e faunal element.

A c o r e c o l l e c t e d f r o m Echo Lake i n t h e Q u ' A p p e l l e v a l l e y o f s o u t h e r n Saskatchewan c o n t a i n s s e v e r a l o s t r a c o d e s p e c i e s , one o f w h i c h i s C y c l o c p r i s ampla F u r t o s , 1933.

T h i s s p e c i e s , a l t h o u g h n o t t h e most

abundant, has a c h a r a c t e r i s t i c p r o f i l e w i t h d e p t h down t h e c o r e . s p e c i e s appears t o d i m i n i s h i n t h e t o p t h i r d o f t h e p r o f i l e .

The

73 FIGURE 1 -logistic

difference

0

0

proportion

.09 0

GI a .08.07-

0

$1

.

. -4

Time k

FIGURE 2

c cn '6

74 F i g . 1 p r e s e n t s t h e p l o t of t h e p r o p o r t i o n of C . ampla s h e l l s t o t h e t o t a l number o f o s t r a c o d e s h e l l s found i n each s l i c e o f t h e c o r e . and wk a r e p l o t t e d i n F i g u r e 2 . The maximum o f k,n 61.93 and i s a t d e p t h k = 35 cm. The s i g n i f i c a n c e l e v e l

The values o f U

i s U = IUk,ni cx a s s o c i a t e d w i t h U i s a p p r o x i m a t e l y z e r o i n d i c a t i n g a v e r y h i g h

evidence o f change.

The l i n e s showing t h e 5% and 1% boundaries f o r

t h e process i W k l a r e p l o t t e d on F i g u r e 2 which show t h a t t h e p a t h o f

i W k l crosses these l i n e s a t k

=

Hence, t h e h y p o t h e s i s o f change i s

35.

accepted and as an e s t i m a t e f o r t h e change-point we t a k e 35. 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 (Zlk-Zk)/-

i s shown.

On F i g . 1 This

D i s 6.2563 and occurs a t k = 34. The s i g n i f i c a n c e l e v e l a s s o c i a t e d w i t h D i s almost zero, which i n d i c a t e s t h a t t h e d a t a

indicates that

shows s t r o n g evidence f o r change.

The e s t i m a t e f o r k u s i n g t h e

l o g i s t i c t r a n s f o r m i s k = 34. Based on an i n t e r p r e t a t i o n o f t h e chemical and p h y s i c a l h a b i t a t (Delorme et a l , 1977), from t h e f o s s i l o s t r a c o d e s i n t h e core, a reason f o r t h e change i n t h e s p e c i e s p o p u l a t i o n can be seen f o r t h e d e p t h o f 34 t o 35 cm.

The d e p t h i n t e r v a l 32 t o 37 clil shows an i n t e r p r e t e d

decrease i n m a j o r i o n s and pH. t h e annual p r e c i p i t a t i o n .

Simultaneously, t h e r e i s an i n c r e a s e i n

The a d d i t i o n a l r a i n had t h e e f f e c t o f

d i l u t i n g the ions i n the lake water. am@ Cyclocypris -

Although the count o f s h e l l s f o r

p a r a l l e l s t h e t o t a l she1 1 p o p u l a t i o n , t h e i n t e r p r e t e d

c h e m i s t r y does n o t .

The decrease i n s h e l l p o p u l a t i o n i n t h e t o p t h i r d

o f t h e c o r e r e f l e c t s more t h a n a change i n t h e h a b i t a t .

Species, such

as -C. ampla, which had a m a r g i n a l e x i s t e n c e p r i o r t o t h e change i.n t h e h a b i t a t , d i d n o t r e g a i n t h e i r former s t a t u s .

Some changes may p l a c e

more s t r e s s on t h e s p e c i e s so t h a t i t does n o t f u l l y r e c o v e r .

REFERENCES

Brown, R.L. Durbin, J . and Evans, J.M.(1975). Techniques f o r t e s t i n g t h e constancy o f 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 ( w i t h d i s c u s s i o n ) . J.R. S t a t i s t . SOC. B, 37:149-192.

75

Delorme, L.D., Zoltai, S.C. and Kalas, L.L. (1977 . Freshwater shelled invertebrate ind cators of pa eocl imate n northwestern Canada during late glacial times. Can. J. of Earth Sciences, 14(9):2029-2046. Esterby, S.R. and El-Shaarawi, A.H. (1981a). Likelihood inference about the point of change in a regression regime. J. Hydrology, 53:17-30. Esterby, S.R. and El-Shaarawi, A.H. (1981b). Inference about the point of change in a regression model. Appl. Statist., 30:277-285. Feder, P.L. (1975a). On asymptotic distribution theory in segmented regression problems - Identified case. Annals of Statistics, 3:49-83. Feder, P.L. (1975b). The log likelihood ratio on segmented regression. Annals of statistics, 3:84-97. Ferreira, P.E. (1975). A Bayesian analysis of a switching regression model: known number of regimes. J. of the American Statistical Association, 70:370-374. Hinkley, D.V. (1969). Inference about the intersection in two-phase regression. Biometrika, 56:495-504. Hinkley, D.V. (1970). Inference about the change-point in a sequence of random variables. Biometrika, 57:l-17. McGilchrist, C.A. and Woodyer, K.D. (1975). Note on a distributionfree Cusum technique, Technometrics, 17:321-325. Page, E.S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika, 42:523-527. Page, E.S. (1957). On problems in which a change in a parameter occurs at an unknown point. Biometrika, 44:248-252. Pettitt, A.N. (1979). A non-parametric approach to the change-point problem. Appl. Statist., 28(2):126-135. Quandt, R.E. (1958). The estimation of the parameters of a linear regression system obeying two separate regimes. J. of the American Statistical Association, 53:873-880. Quandt, R.E. (1960). Test of the hypothesis that a linear regression obeys two separate regimes. J. of the American Statistical Assoc., 55 :324: 330. Quandt, R.E. (1972). New approach to estimating switching regressions, J. of the American Statistical Association, 67:306-310.

76

EXPLORATION OF AN EXTREME VALUE PARTIAL TIME SERIES MODEL IN HYDROSCIENCE ASHKAR, F., EL-JABI, N. and ROUSSELLE, J. Ecole Polytechnique de Montr6al

ABSTRACT This paper presents and discusses the characteristics of a vrobabilistic model suitable for the description of the time and snace joint distribution of partial duration series flood flow data. An important functional of the magnitude of exceedances and their time of occurrence, commonly used in the design o f hydraulic structures is the design exceedance corresvonding to a given return period. When building a project based on this design exceedance a certain risk resulting from the estimation of the models’ narameters is encountered.

lhis risk is studied for different periods o f record and

different return periods. In addition to exceedances, the continuous river flow process nroduces two features essential in several areas and flood damage analysis. flood volume.

of

flood control

These are namely flood duration and

The partial duration series amroach is shown to

offer convenience in the definition, calculation and statistical analysis of quantitative aspects of these two features. Relative to the planning of water resources and the management of river banks subject to flooding,a methodology is presented in which technical, economic and physical considerations are integrated. 1.0

INTRODUCTION

Flood flow forecasting is undoubtedly one of the most important considerations in the process of water resources management, given the economic damage and the loss of human lives that may be caused by floods.

According to Gray ( 1 9 7 2 ) , the choice of a forecasting

method depends on several factors:

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

77

i

-

t h e flood f e a t u r e s r e l e v a n t t o t h e p r o j e c t under consideration;

i . e . peak f l o w , f l o o d i n t e n s i t y and f l o o d volume, f l o o d d u r a t i o n , sedimentation problems, e t c . . .

ii

-

t h e a v a i l a b l e d a t a and t h e s e a r c h f o r t h e b e s t method f o r

e x t r a c t i n g i n f o r m a t on from t h e s e d a t a .

...

2.22-

t h e e f f e c t o f t h e b a s i n ’ s t o p o g r a p h i c and h y d r o g r a p h i c c h a r a c -

t e r i s t i c s on r u n o f f

iv

-

t h e i m p o r t a n c e o f t h e p r o j e c t u n d e r c o n s i d e r a t i o n from t h e

economic, as well as from t h e s o c i a l p o i n t s o f view and t h e s p a n o f time c o v e r e d by t h i s p r o j e c t .

The a p p r o a c h w i t h t h e l o n g e s t h i s t o r y , u s e d i n f l o o d f l o w f o r e c a s t i n g i s t h e o n e which u s e s e m p i r i c a l f o r m u l a s t h a t e x p r e s s f l o o d d i s c h a r g e as a f u n c t i o n o f 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 t h e b a s i n , w i t h a s p e c i a l e m p h a s i s on t h e b a s i n a r e a .

The a p p l i c a t i o n o f t h e s e

f o r m u l a s r e q u i r e s a p r o f o u n d knowledge a b o u t t h e t o p o g r a p h i c and hydrographic p r o p e r t i e s o f t h e catchment under c o n s i d e r a t ion. A second approach used t o f o r e s e e t h e f u t u r e behavior o f f l o o d s

i s s t a t i s t i c a l and i s b a s e d on t h e a s s u m p t i o n t h a t a p r o b a b i l i t y measure c a n b e imposed on t h e a v a i l a b l e s t r e a m f l o w d a t a .

The g o a l

here i s t o obtain r e l a t i o n s capable of describing t h e interaction between t h e laws o f randomness g o v e r n i n g t h e b a h a v i o r o f v a r i a b l e s . The c h o i c e among a v a i l a b l e s t a t i s t i c a l t e c h n i q u e s i s g o v e r n e d by 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 t h e phenomenon, t h e m a t h e m a t i c a l p r o p e r t i e s o f t h e d a t a and by i n t u i t i o n and t h e p r e v a l e n t e n g i n e e r i n g practice. Credit f o r the construction of a probabilistic base, suitable f o r

f l o o d a n a l y s i s g o e s t o ‘Todorovic and l’evj c v i c h (1967, 1969) , ‘Todor o v i c ( 1 9 6 7 , 1 9 6 8 , 19701, T o d o r o v i c and Z c l e n h a s i c ( 1 9 7 O ) , ‘Todorovic and R o u s s e l l c ( 1 9 7 1 ) and T o d o r o v i c ( 1 9 7 8 ) . base of t h e c u r r e n t s t u d y .

‘l‘hese iiiodcls a r e a t t h e

A s i n ( t i i r b y , 1909), t h e y t r e a t f l o o d

h y d r o g r a p h s m a i n l y as a s c q u c n c c o f f l o o d p e a k s , s e p c r a t c d by r e l a t i v e l y l o n g p e r i o d s o f low d i s c h a r g e .

78

Both deterministic and stochastic approaches have gained support from several investigators, and a comparison between the two approaches was done by Yevjevich (1974).

The deterministic school is

strong on interpretations and developments of a physical nature, but limited on the regional level. The probabilistic school, on the other hand, is more flexible and offers a variety of readily available techniques, often found useful in desperate situations (Tiercelin, 1973). 2.0 THEORETICAL CONSIDERATIONS

A model based an the theory of extreme values and on the theory of random number of random variables was developed at the Engineering research center of Colorado State University, and was presented by Todorovic (1970). Our flood-plain management methodology, to be p r e sented towards the end of this paper, is based on this stochastic model. 2.1

Analysis of the flood phenomenon

Consider a hydrograph representing the instantaneous river flow at a given station, within an interval of time (O,t], (figure 1). Let us consider as floods only those hydrograph peaks exceeding a certain base level (figure 2).

where

Q

is the base flow (often taken as the discharge when the

water starts to overflow to spread over the banks of the river).

A discrete and non negative stochastic process representing e x ceedances occurring in the interval (O,t] is obtained. Let <, be the vth exceedance occurring at time T ( V ) ( f i g u r e 3). The f l o o d phenomenon may therefore be represented by the following sequence

(5"

; v = 0,1,2

,...,

t

(2)

79

Fl(iLl!
c:xcceJances

80

The e x i s t i n g p r o b a b i l i s t i c models u s e d i n f l o o d f o r e c a s t i n g a r e e s s e n t i a l l y d e v o t e d t o t h e t r e a t m e n t o f t h e two s e q u e n c e s

(tU)and

( ~ ( v ) .) I n t h e case o f a m u l t i p l e - p e a k e d h y d r o g r a p h , s u c h as t h e one o c c u r r i n g a t time

T(V -

1) i n f i g u r e 2 , o n l y t h e h i g h e s t peak i s

T h i s i s done t o j u s t i f y t h e n e e d e d a s s u m p t i o n o f m u t u a l

considered.

independence between exceedances

tV.

The number o f e x c e e d a n c e s i n

( O , t ] , t h e times o f o c c u r r e n c e o f t h e s e e x c e e d a n c e s , as w e l l a s t h e m a g n i t u d e s o f e x c e e d a n c e s a r e random v a r i a b l e s . D i s t r i b u t i o n f u n c t i o n o f t h e number o f e x c e e d a n c e s

2.2

The d i s t r i b u t i o n o f t h e number o f e x c e e d a n c e s i s i m p o r t a n t b e c a u s e design i n t e r e s t usually lies i n t h e multivariate d i s t r i b u t i o n of t h e number o f e x c e e d a n c e s i n ( O , t ] Denote by

n(t)

and t h e magnitudes o f t h e s e exceedances

t h e number o f e x c e e d a n c e s i n ( O , t ] , i . e .

q ( t ) may t h e r e f o r e t a k e any non n e g a t i v e i n t e g e r v a l u e f o r any t > 0 . D e f i n e EL b y

= {q(t) = V)

E:

I n o t h e r terms, E: dances i n (O,t]

=

{ ~ ( u )< t

< ?(V

+ 1))

(4)

i s t h e e v e n t t h a t t h e r e a r e e x a c t l y Li e x c e e -

.

The e v e n t s EL form a p a r t i t i o n o f t h e s a m p l e s p a c e , i . e . E: fl E:

=

8 and ov!

m

EL

=

t h e i m p o s s i b l e e v e n t and

0 for all i f j

=

0,l..

.

where 8 s t a n d s f o r

R stands f o r the certain event.

Denote by A ( t ) t h e e x p e c t e d v a l u e o f q ( t ) , s o t h a t we h a v e A(t)

? v

= u=1

P(E\)

(5)

Mainly d u e t o t h e s e a s o n a l v a r i a t i o n s o f s t r e a m f l o w , i?(t) i s a non l i n e a r f u n c t i o n i n time. Denote by F u ( t ) t h e d i s t r i b u t i o n f u n c t i o n o f t h e time o f o c c u r rence

T(V)

o f t h e vth e x c e e d a n c e ,

Equation (4) c a n , t h e r e f o r e , b e w r i t t e n i n t h e f o l l o w i n g form: P(E:)

= P{T(V)

< t)

- P{T(V

+ 1)

< t)

= F v ( t ) - I:"+,

(t)

(7)

81 By f o l l o w i n g t h e same p r o c e d u r e u s e d i n a r r i v i n g a t e q u a t i o n ( 7 ) we o b t a i n

Summing e q u a t i o n s ( 7 ) , ( 8 ) and ( 9 ) y i e l d s

which may b e p u t i n t h e f o l l o w i n g form m

F"(t)

=

J" P ( E ; )

(11)

Remark t h a t ~ ( v )a r e c o n t i n u o u s random v a r i a b l e s and f o r a l l u = 1,2,.

. .; 0

< ?(V) P(E;)

< ,r(v + 1 ) ; > 0

\d

T(v)

+

m

if

v

t > 0 and v = 0 , l ,

+

and

...

(12)

Suppose t h a t €or a n y " v e r y small" time i n t e r v a l ( t , t + A t ] on t h e time s c a l e ( b y " v e r y small" we mean i n t h e l i m i t when A t + 0) o n l y one o f t h e f o l l o w i n g two e v e n t s c a n o c c u r (Cox and M i l l e r , 1965);

i

- no exceedance o c c u r s i n t h i s i n t e r v a l ;

or

ii

- one

and o n l y o n e e x c e e d a n c e o c c u r s i n t h i s i n t e r v a l . Ilenote t h e s e two e v e n t s , r e s p e c t i v e l y , by E,t + A t

=

{q(t + A t )

- q ( t ) = 0)

E,t + A t

z

{ q ( t + At)

-

q ( t ) = 1)

I g n o r i n g e v e n t s whose p r o b a b i l i t y of o c c u r r e n c e t e n d s t o z e r o as

At

+

0 ( f o r t h e sake o f n o t e n t e r i n g d e e p i n t o complex m a t h e m a t i c a l

d c t a i l s ) w e may w r i t e t h e f o l l o w i n g e q u a l i t y P ( C y ) = P(&)

P ( E ot + n t / E t" - l )

D i v i d i n g t h i s e q u a t i o n by A t y i e l d s

+

P(E;-l)

P ( E tl + A t / Etu - l )

(14)

82

and taking the limit of equation (16) as At

Using the following notation

: A,(t)

=

+

0, we get

I

P(E:+at

lim

,equation

At

At - t o

(17) becomes

with

dw:) dt

=

dP(E'dt )

=

v-1

-

(t) P ( E L - 1 ) - A"(t)

xo(t)

for v

p(Ek)

=

PW:)

(18)

o

(19)

The solution of the system of differential equations (18) and (19) is given by (Zelenhasic, 1970): P(Ei)

=

exp1-d X, (s) d s }

P(&)

=

expC-1; Av(s) ds) .

(20)

r,'

A"-,

(s) ds

.exp{C' [A,(s) - A"-, (s)]ds) .expIt" [A, (s) - A, (s)]ds}

.

.(

(l..

"-l

X0 (tv)

dt, dt,_, . . .dtl

Under the following hypothesis : A,(t)

E A(t)

(21)

equation (21) takes

the form P(E:)

{r,'

=

X(s)ds}"

expi-jd X(s)ds}/v!

When the following expression for A(t):

A(t)

taken along with equation ( 2 2 ) , we obtain

-c

= e

A(

s ) ds

c

v=o

[a' A(s)ds]" (V-l)!

=

,z0

(22)

v

P(E:)

is

83 A(t) A(t)

(

=

depends, therefore, on A(s)

A(s) ds

(24)

which is a non negative function

representing the rate of occurrence of exceedances points in time.

5, at different

'This function is effected by climatic variations

occurring between seasons and/or between years. Equations (22) and (24) finally yield P(E:)

=

{A{tIU

exp{-A(t))

(25)

~

W!

2.3 Distribution function of the largest exceedance

5, within a time interval

Among the set of all exceedances

(O,t],

probably the most important variable for design purposes is the largest exceedance x(t),

defined as

x(t)

non decreasing stochastic process (figure 4).

; x(t)

is a

T ( W ) < t

Let F, (x) be the distribution function of x(t), P{X(t)

5,

Sup

=

i.e. F, (x)

=

G x) . A mathematical expression for F, (x) was derived by

Todorovic (1970) in terms of the following conditional probability

and under the hypothesis of a mutually independent sequence

(c,),

the following equalities were deduced

F,

(XI

K

m

=

=

KgO P[v!o P(E:)

+

15, G

Kz,

XI

fl

&I

(27)

K

15,

G x x ) fl E L ]

(28)

4 schematic representation of the distribution function of X(t)

shown in figure 5. sequences (5,)

and

is

The hypothesis of mutual independance of the two T(V)

along with equation (28) leads to the fol-

lowing expression co

Ft (x) = P ( E i )

+ K=l T:

{[tI(x)IK

.

P(EfK))

fI(x) is the common distribution function of the cxceedances ring in the invertal (O,t].

(29)

c,

occur-

bfaking use of equations (25) and (29)

84

85

we obtain

Ft (x)

=

e

- i Z ( t ) [l-H(x)]

In the foregoing analysis it is supposed that all exceedances 5, occurring during a year are mutually independent and identically distributed, and that the flood process is stationary (i.e. there is no change in the probabilistic behavior of floods from year to year).

If we wish to divide the one-year interval into a certain number k of "seasons" in the following manner: (O,Tl1 , (TI,T2],. . . , (TK-l ,TK]; then the largest exceedance x ( t ) takes the form

x(t)

=

sup {XI (O, T I )

where x ~ ( T ~ ,TK) ~, k

=

9

xz (TI T2), . . .

1,2,.

9

. . , To

Z

3

XK (TK-I,

t) 1

(34)

0 is the largest exceedance

in the interval (TK-l, TK]. In this case, the distribution function of the largest exceedance in the interval of time (O,t] would be given by Ft

K(X)

=

P{X1 (O,T,

x,X2

(TI2Tz)G x , . . . ,xK (TK-, ,t)G x } O
(35)

Following the same procedure that lead to equation ( 3 3 ) , Rousselle (1972) showed that for an interval (O,t], the distribution function

FtK(x) takes the form

where 11, (x) stands €or the distribution function of cxcccdanccs in the ithseason. Rousselle (1972) also found that the cxponcntial distribution function fits adequately well thc data on some rivcrs in the United States,

86

Rousselle and El-Jabi (1976) and Ashkar (1980) showed also that the exponential distribution gives good results when applied to some Canadian rivers. Equations (37), (33) and (36) lead to the distribution function F, (x) of the largest exceedance during a year in the case when

i- a l l exceedances

occurring within a year are independent and

identically distributed

e -B'

F~ (x) = exp{- A.(t)

1

(38)

ii- all exceedances 5, are mutually independent, but are identically distributed only on a seasonal basis. F t 4 (x) = exp-A(T,) e - [fl('l,)

-

[A(T~)- A ( r r l 1 1

- A ( T 2 ) ] e-"'

where A(T, 1 , [ A ( T 2 )

-

A(Tl

- [fl(T4) -

1 I , [A(T3)

-

e-'Zx

fl(T,)]

e-"'

1

(39)

A(T2) I and [ A ( T 4 ) - A(T,)

I

are the mean numbers of exceedances for the four different seasons (assuming that the year is divided into four seasons), B,

, B Z , p3

and P4 are the parameters of the exponential distribution (37) for the four seasons. 3.0 DESIGN CONSIDERATIONS

The return period

3.1

In the process of flood control planning, the choice of the capacity of a hydraulic structure is often hampered by many difficulties.

Whatever this choice may be, a certain risk has to be tole-

rated, and it is desirable in practice to have a quantitative measurc of this rish.

A widely used measure is the return period T defined

as the expected value of N,, the number o f years required to obtain an exceedance greater than the level x for the first time.

The

level x is called the "design flow" associated with the return period 1'. Equations (40) through (44) put the notions of N, and T into mathematical terms.

If

(xV)

is assumed to be a sequence of mutually independent and

identically distributed random variables having the distribution function: Ft ( x )

=

x) , then we obtain

P(xv

where F, (x) is given either by equation (38) or equation (39) depending on whether seasonality is taken into account o r not. Since by definition we have T

we hence obtain

E(N,),

=

3.2 The first passage time

Define T(x) as the time necessary for the process x(t) to exceed the level x (figure 4), i.e. T(x)

=

Inf{t ; x(t)

> x);

T(x) is a

continuous and non decreasing stochastic process, since 0 implies that T(x1 )

T(x2) .

Let

Q,

x1 G x2

(t) be the distribution function

of T(x) (1, (t)

The relation between x(t)

=

P[T(x) G t]

(45)

and T(x) leads to the following equations

{x(t)

G x)

P{X(t)

x)

{T(x) > t) f

P{T(x)

> t>

which in turn yield the following relation between (1, (t) = 1 - Ft

(46)

(XI

(47) Qx

(t) and F,

(x)

(48)

88

4.0 PARAMETER SENSII'IVITY ANALYSIS AND ITS REI,ATION TO DESIGN From a sample of flood data, design flows X, corresponding to different return periods T can be calculated, either by using equations (38) and (44) (in the case when seasonal effects on flood flow are not considered important) and solving for x in terms of T to obtain X,Ex= where C

=

'

-Ln(Ln-

7'-1

);

c+Lnh

(49)

a

A = A(t*);

t* = one-year period

or by using equations (39) and (44) (when seasonal variations shou d be taken into account) and solving numerically for x in terms of T

A, , A,,

A,, th

in the i

A,,

a,

BB and fi4 (A, being the mean number of floods

season).

Because design flows X, are calculated from the available record, which cannot produce all the information containedinthe set of all possible floods, these design flows shouldbe expected to varyifa different flood record was in hand. In other words, a design flow is a random variable whose distribution function can be obtained. This distribution function depends on both the design model used and the length of record. The randomness inherent in the design flow is an essential element in the calculation of the economic risk to be encountered when building a dam of any size. This randomness is due to the random nature h

A

of the parameter estimates A and L3 used in practice to replace h and A

(3

A

in equation (49) (or due to the random nature of A, , B, , i

3,4 in the presence of seasonality).

=

1,2,

It was studied by fiindig (1974)

and Ashkar (1980) for the extreme value model presented in the current study. Parameter estimation was done by Bayesian Techniques (Hindi6, 1974) and by maximum likelihood (Ashkar, 1980).

A comparison between

the two approaches was done by Ashkar (1980). The distribution function of the design flood X, can be taken along with economic information about the flood-prone region, put in the form of an economic "loss function1',to derive the economic risk associated with different possible sizes of the hydraulic structure

89

being planned. 5.0 FLOOD MAGNITUDE, DURATION AND VOLUME

Three important components of flood hydrographs shown in Figure 2 are the exceedance 5, the flood duration D and the flood volume V. Ashkar (1980) showed that on some Canadian rivers, it is possible to make the hypothesis that the flood hydrograph is triangular in shape. Considering the nature of flood flow we may say that the three components

5, D and V are random variables.

Ashkar (1980) divided the flood duration D into two sub-components D, and Df that he called "rise duration" and "fall duration" respectively. lie showed that in practice D, and Df can be considered as statistically independent and that they can be put in the form of a sum of a deterministic component S, and a non-deterministic component

1, i.e. D,

=

S,

+

1, and

Df

= Sf

+ Cf

. He showed that on some

Canadian rivers, the exponential distribution fits adequately well the non deterministic components 1,,c and C f , c associated with a given flood magnitude 5, D and V

=


6.0 FLOOD PLAIN SYSTEM ANALYSIS

This system may be regarded as a complex set containing such elements as streams, floodable lands and the human population living on them. Several decisional constraints complicate the analysis of this system. These constraints may be political, hydrotechnical, environmental, economic, sociological or physical in nature. They form what is called a decisional space which covers the flood phenomenon in its entirety. Some o f these constraints are not amenable to a quantitative treatment due to their piculiar character; nevertheless, they often play a key role in the final implementation of policies, especially when these policies come into conflict with the public's interests. By a hydroeconomic system, we mean that set of elements or components required in the economic quantification process of flood

90

damage or in the determination of the economic value of a given damagereduction measure.

It is this sub-system within which all economic

aspects of a flood occurring on a particular basin at a particular points in time, interact.

These economic aspects are composed of phy-

sical (topography, slope, area, geology, . . . ) and economic (residential damage, reconstruction costs, . . . ) components. El-Jabi (1980) showed that the hydroeconomic system may be described

by the following set of relations usable in a numerical analysis of the systems: ition;

... zzz-

ii- the depth-damage relathe discharge-frequency relation; and iv- the damagethe depth-discharge relation;

frequency relation. The pooling of these relations produces a distribution function p,(d) of flood damage that can be expressed as follows pt (d) = F, {g-’[8-’(K,d)])

d

:

a vector representing different types of physical and non-physical damage ;

k

:

a vector representing the physical capital (residential property, commerces and industries, stock, . . . ) along with the associated activities (production, services,...);

e :

a discharge-damage functional transfer relation;

g I:,

:

a depth-discharge functional transfer relation;

:

a distribution function of hydrologic characteristics considered in an interval of time (O,t] given by equation (44).

This approach to the flood phenomenon and to its impact on flood plains opens the road to the following fundamental objectives

i-

analyzing flood-damage transfer relations to obtain a probability

distribution function of flood damage;

ii- finding an estimate of this damage without hav ng to make use of an after-flood survey;

Lii- comparing the effectiveness of different flood control measures.

91 7.0 CONCLUSIONS The objective of this study was to present an extreme value model designed for flood analysis. Although no numerical applications were given, several of them may be found in publications listed in the references. This model has the following characteristics:

i-

it has no restrictions on the choice of the distribution function

of flood magnitude;

ii-

the introduction of a base flow makes the analysis of flood dura-

tion and flood volume more practical as compared to other flood models in use.

It also increases the amount of information on floods occur-

ring in any interval of time;

iii- when using the model, attention may be restricted to intervals of time as short as one day and as long as desired.

In practice this is

very important for the study of seasonal effects on streamflow in a particular region;

iv- unlike other models in use, this model permits the study of the multivariate distribution of the number of floods occurring in a given interval of time, and the magnitudes of these floods. REFERENCES ASIiKAR, F., (1980). Partial Duration Series Plodels for Flood Analysis, P1i.D. Thesis, Ecole Polytechnique de Montr6a1, 172 p . COX, D.R. and MILLER, H.D., (1965). ses, Chapman and Hall Ltd, 398 p.

The Theory of Stochastic Proces-

EL-JABI, N., (1980). Approche Syst6matique pour l'Am6nagement des Plaines Inondables, Ph.D. Thesis, Ecole Polytechnique de Montr6a1, 263p. GRAY, D . M . , (1972). 13 chapters.

Manuel des Principes d'Hydrologie, CNRC, Canada,

HINDIE, F.S., (1974). Approche Bayesienne pour 1'Estimation de 1'Errcur due 2 1'Echantillonnage dans 1'Evaluation d e s D6bits de Crues, Masters Thesis, Ecole Polytechnique de blontr6a1, 78 p. KIRBY, W., (1969). On the Random Occurrence of Major Floods, Nater Resour. Res., vol. 5, no 4, p p . 778-784. ROUSSELLE, J., (1972). On Some Problems of Flood Analysis, P 1 i . D . Thesis, Colorado State Univ., 226 p.

92

ROUSSELLE, J. and EL-JABI, N. , (1976). Rivisre des Prairies, Repr6sentation Stochastique des Crues, Eau du QuGbec, vol. 9, no 1, p p . 23-28. TIERCELIN, J.R., (1973). Modsles Probabilistes en Hydrologie, La Houille Blanche, no 7, p p . 547-552. TODOROVIC, P., (1967). A Stochastic Process of Monotonous Sample Functions, Math. Inst. of the Republic of Serbia, vol. 4, no 19, pp. 149-158. A Mathematical Study of Precipitation Phenomena, TODOROVIC, P. , (1968) Report no CET67 - 68P 65, Colorado State Univ., 123 p . TODOROVIC, P. , (1970) Random Variables, Ann

On Some Problems Involving Random Number of Math. Statist., vol. 41, no 3, pp. 1059-1063

Stochastic Models of Floods, Water Resour. TODOROVIC, P. , (1978) Res., vol. 14, no 2 , pp. 345-356. TODOROVIC, P. and ROUSSELLE, J., (1971). Some Problems of Flood Analysis, Water Resour. Res., vol. 7, no 5, pp. 1144-1150. TODOROVIC, P. and YEVJEVICH, V., (1967). A Particular Stochastic Process as Applied to Hydrology, Proc. Int. Hydrol. Symposium, Colorado, pp. 298-305. TODOROVIC, P. and YEVJEVICH, V., (1969). Stochastic Process of Precipitation, Colorado State Univ., Hydrol. paper no 35, 61 p . TODOROVIC, P. and ZELENHASIC, E., (1970). A Stochastic Model for Flood Analysis, Water.Resour. Res., vol. 6, no 6 , p p . 1641-1648. YEVJEVICH, V., 1974). Systematization of Flood Control Measures, ASCE, HY11, pp. 1537-1548. ZELENHASIC, E. , (1970). Theoretical Probability Distributions for Flood Peaks, Co orado State Univ.; Hydrol. Paper no 42, 35 p .

93

A COMPARATIVE STUDY ON ESTIMATION OF PARAMETERS OF A MARKOVIAN PROCESS - I A.A. ABD-ALLA AND A.M. ABOUAMMOH University of Riyadh, Saudi Arabia ABSTRACT This paper considers the stationary autoregressive process of order one Xt = aXt-l + ~~~t E I = I ..., -2,-1,0,1,2 ,... 1 where E~ is a sequence of independent normally distributed random variables with zero mean and variance 0:. The conditional maximum likelihood estimates for G: and a are obtained. In addition these estimates are compared with the empirical estimate o f 0:: and the Bayes' estimate for a. The parameter a is taken to be priori distributed as uniform, standard normal and non-standard normal respectively, where the stationarity condition is -1 < a < 1. INTRODUCTION Consider the stationary autoregressive process of order one (AR(1)) Xt

=

aXt-1

+

€t

(1.1)

where { E ~ I ,E ~ I is a sequence of independent normally distributed The process {Xtl,t E I is normally with zero mean and variance oz. E distributed with zero mean and variance G $ = ~:/(l-a~). It is known that such process is stationary if the parameter a satisfies the condition < 1. The main statistical problem in (1.1) is estimating the unknown parameters 0: and a . The problem o f estimating these parameters using different methods such as the Yule-Walker's, the maximum like1 hood and the conditional maximum likelihood ( C . M . L . ) i s discussed by Box and Jenkins (1976). The likelihood function of u z and a is given by Abd-Alla (1980)as n 2: (xi-axi~l)211(l .2) f(a,u:)'=(2. u:) -n'2(1-a2)4 exp c ( 2 0 2 ) - 1 i(l-a2)x:+ E

i =2

Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 8 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

94

T h i s shows t h a t t h e s o l u t i o n s o f t h e l i k e l i h o o d e q u a t i o n s a r e n o t simple. THE CONDITIONAL LIKELIHOOD I n o r d e r t o s i m p l i f y t h e s o l u t i o n s of t h e l i k e l i h o o d f u n c t i o n , we

e q u a t i o n s aL/aa = 0 and aL/au:

By s o l v i n g t h e C.M.L.

L = l o g f(a,aE/xl)

2 E

= (n-1)

-1

zn

t o g e t t h e C.M.L. 2 xi-2{(

n

A

X ~ X ~ - ~ ) /

i=2

i=2

estimators n z

z

= 0, where

2

i=2

n

z

I(

xixi-l

i=2

i=2

xixi-1

n

f o r u2 and a r e s p e c t i v e l y . E

THE BAYESIAN APPROACH I t can be e a s i l y shown t h a t t h e e m p i r i c a l e s t i m a t e

=

1 {n z t=l

i=2

n 2 Xttl)(n

2 1 n-l x t -(-Z n-1

t=l

n 22 t=2

a2E(E)

of u:

is

n Xt-1 t=l

i=l

L e t t h e p r i o r knowledge about a be r e p r e s e n t e d by t h e g ( a ) .

The

o t h e r i n f o r m a t i o n o f sample d a t a a r e expressed i n t h e 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 f(a,u:/xl).

..

The p o s t e r i o r d e n s i t y o f

~1

i s then

I/

95 g(a(xl)

= f(a,aE/xl)

g ( a ) l h f(a.u2E/x1)

and t h e Bayes p o i n t e s t i m a t e

Of

Q

!3(a) da

i s t a k e n as t h e mean o f t h e

posterior distribution a = Jag ( n i x l )

dcl

(3.2)

The f o l l o w i n g t h r e e s p e c i a l cases f o r t h e p r i o r d i s t r i b u t i o n o f a a r e considered :

( i ) the density o f

CY

i s u n i f o r m on ( - 1 , l ) ;

( i i ) t h e d e n s i t y o f a i s normal w i t h mean 0 and u n i t v a r i a n c e ; and ( i i i ) the density o f

~1

i s normal w i t h mean

l-la

and v a r i a n c e o i

I n t h e t h r e e cases, t h e mean o f t h e p o s t e r i o r d i s t r i b u t i o n i s c a l c u l a t e d u s i n g n u m e r i c a l i n t e g r a t i o n methods ( D a v i s and R abinowit z ( 1 9 7 5 ) ) . A

and

G(N)

We s h a l l denote t o t h e Bayes e s t i m a t e s b y

(U)’ f o r t h e u n i fo rm , s t a n d a r d normal and t h e normal p r i o r

YSN) distributions respectively.

The c a l c u l a t i o n s o f

s t r a i g h t forward b u t t h e c a l c u l a t i o n o f o f o:,

M a and

0

&

(N)

& (U)

and “(SN) are r e q u i r e s t h e knowledge

0:.

To f i n d e m p i r i c a l e s t i m a t e s f o r th e s e v a l u e s we n o t e t h e f o l l o w i n g : 2 2 ECXtl = (11: + u2) E Xt-l + oz a E (3.3 a) 2 2 = (11; + 0a2 ) E Xt-2 + . 2 E (3.3 b ) 3 2 E[Xt X t + l l = E a E Xt-l + puoz (3.3 c )

q-,

ECXt Xt-,l

=

E a3 E ! ::-2

(3.3 d)

+ paoE

From ( 3 . 3 a ) and (3.3 b ) one can o b t a i n

From ( 3 . 3 c ) and(3.3 d ) , t h e q u a n t i t y E

Using (3. 4) and ( 3 . 5 ) ’ we o b t a i n

a3

can be o m i t t e d as f o l l o w s :

96

Using (3.3 a ) and ( 3 . 6 ) t h e v a l u e o f u2 = U

0%

a

becomes

C(E X 2t - I E X E - 2 E Xt-(E 2 Xt-l)2}/ 2

Then t h e v a l u e o f a

can be c a l c u l a t e d b y a n u m e r i c a l i n t e g r a t i o n (N) a f t e r r e p l a c i n g t h e values o f n:,ua and u2 b y t h e i r e m p i r i c a l a

estimates.

T h i s method reduces t h e u n d e s i r a b l e e f f e c t o f t h e p r i o r

distribution,

( B a r n e t t (1973), L i n d l e y ( 1 9 6 5 ) ) .

RESULTS The r e s u l t s a r e summarized i n t a b l e s ( a ) and ( b ) . shows t h e c o n d i t i o n a l maximum 1 ik e l ihood e s t i m a t e estimates

""1,

C"(SN)

and n ( N ) .

Table ( a )

0( C L ) ,

the B w s

Table ( b ) g i v e s t h e c o n d i t i o n a l

maximum 1 ik e l ihood e s t i m a t e u2 and t h e e m p i r i c a l e s t i m a t e E(CL) -2 o f 0:.

0~

(E)

From Table ( a ) , i t can be concluded f o r samples o f s i z e 100,150,200, < - a ( which shows t h a t c"(cL) i s c l o s e r t o t h e that I i ( C L ) -

4

IGy

t r u e value a t h a n

( U ) ' '(CL) n = 100 and u = 0.1.0.3,0.4,0.5,

I

' ( N ) - a 1 i s t r u e e x c e p t when n = 150 and a = 0.1,0.3,0.4,0.5.0.7, <

= 200 and u = 0.1,-0.4,0.4,0.5,0.7. Also, - a / < 1 G(CL)-cx i s t r u e when m o s t l y , a and n i n c r e a s e i n p a r t i c u l a r f o r n = 100 and

n

a = 0.1,0.3,0.4,0.5,

n

=

n

=

150 and

200 a = 0.3,0.4,0.5,0.7.

~1

0.1,0.3,0.4,0.5,0.7

and

Generally,

n e g a t i v e values o f a whereas

4(N)

m o s t l y , f o r a l l p o s i t i v e v a l u e s of mean square e r r o r (MSE) o f

=

6'E

(CL)

i s preferable f o r (CL) i s b e t t e r than o t h e r estimates, Table ( b ) shows t h a t t h e

3.

i s l e s s than t h e MSE o f

o2E ( E ) '

ACKNOWLEDGEMENT The a u t h o r s a r e g r a t e f u l t o p r o f .

K.

h i s encouragement and u s e f u l d i s c u s s i o n .

Alam o f Clemson U n i v e r s i t y f o r

-0.5

n n

yCL) “(U)

100

“(SN)

f(N]

“(CL)

I

-0.755 -0.760

-0.543 -0.557

-0.636 -0.644 -0.735

-0.462 -0.552 -0.521

-0.738

-0.527

-0.422

-0.674

-0.440

-0.720 -0.728 -0.732

-0.523 -0.540 -0.545

-0.648

-0.420

h

-

-0.4 -0.436

-0.3

I

-0.328

-0.1

0.1

0.3

I -0.113 I 0.189 I

0.4

0.219

I 0.284 ~

~~~

-0.446

-0.337

-0.116

0.193

0.215

0.278

-0.447

-0.333

0.438

-0.333

0.101 0.108

0.312

-0.441

-0.174 -0.115

-0.416

-0.312

-0.103

0.017

0.335 0.225

0.448 0.373

-0.317

-0.105

0.017

0.366

-0.416

-0220

-0.163

0.101

0.220 0.302

0.404

-0.419 -0.422 -0.447

-0.316 -0.346 -0.350

-0.105 -0.149 -0.151

0.106 0.081

0.315 0.306

0.419 0.293

0.880

0.306

0.297

h

CI

150

(U)

h

:(SN)

“(N)

,.

“(CL) CI(U)

200

CI

(SN)

A

“(N)

I

-0.416

1

-0.445

I

-0.736

-0.512

1

I

-0.368

I

-0.348

I

-0.103

I

-0.150

I

I

I

0.055

I

0.056

I

I

0.261

I

0.271

I

I

0.369

I

0.381

I

I 0.579

0.743

+ 0.424

0.745

0.418

0.763

98

Table ( b )

0.1

0.3

0.4

0.5

0.7

0.94074

1 .lo103

1 .I7386

1.25807

1.54349

1.01 633

1.01548

1 .01451

1 . 0 1 308

1.00921

~

7

1 .08237

1 .22856

7 .29405

1.36664

1 .59721

1.04618

1.04601

1 .04582

1 .04550

1.04446

1.01127

1.12622

1.7537

1.23152

1.427

1 .00936

1.01033

1.01089

1.01 130

1.01103

1 .86904

1.18252

2.32612

2.4955

3.05623

2.03257

2.03070

2.02874

2.02591

2.01847

2.15476

2.44326

2.57290

2.71 689

3.01 733

2.09231

2.09191

2.09152

2.09089

3.08903

2.01 540

2.24116

2.33960

2.4509

2.83880

2.01892

2.02073

2.02177

2.02256

2.02213

2.801 23

3.26742

3.48178

3.73048

4.57235

3.04873

3.04584

3.04288

3.03869

3.02787

3.22945

3.6601 1 -

3.8541 1

4.06932

4.75156

3.1 3840

3.13781

3.13722

3.13634

3.13363

3.02137

3.35998

3.50550

3.67207

4.2521 8

3.02854

3.031 1 9

3.03272

3.03392

3.03326

3.73543

4.3541 2

4.63924

4.97021

6.0901 8

4.06492

4.06097

4.05704

4.05152

4.03720

4.30535

4.87809

5.13635

5.42298

6.33088

4.18451

4.18371

4.18298

4.181 79

4.17820

4.02829

4.47858

4.67229

4.8941 8

6.66631

4.03820

4.041 60

4.04370

4.04525

4.04435

150

I1 N

W

0

100

I I

':(E)

%L)

c u I1

N W

5

0-l

I1 N U

b

100

d

1

02

E(E)

I1 N

W

0

-

99

REFERENCES :

Abd-Alla, A . A . ( 1 9 8 0 ) ; The e s t i m a t i o n of the parameters o f the f i r s t and t h e second o r d e r a u t o r e g r e s s i v e process with z e r o l e v e l . In the proc. of the 6th Saudi National comp. conf. P . P . 247-259. B a r n e t t , V . (1973): Comparative S t a t i s t i c a l I n f e r e n c e . Wiley. Box, G . E . P . and J e n k i n s , G . M . (1976): Time s e r i e s a n a l y s i s , Forecasting and C o n t r o l . Holden-Day. Davis, P.J. and Rabinowitz, P . ( 1 9 7 5 ) : Numerical I n t e g r a t i o n . B1 a i s d e l l . Lindley, D . V . (1965): I n t r o d u c t i o n 'to P r o b a b i l i t y and S t a t i s t i c s from a Bayesian Viewpoint. P a r t 2 I n f e r e n c e . Cambridge U n i v . Press.

100

GENERALIZED LEAST SQUARES PROCEDURE FOR REGRESSION WITH AUTOCORR E LATED ERRORS U.L. GOURANGA RAO

Dal housie U n i v e r s i t y , H a l i f a x , Canada

ABSTRACT

In this paper the generalized least squares procedure is used to estimate the parameters of the standard linear regression model where the errors follow a first-order autoregressive process. This procedure is similar to the full maximum likelihood procedure of Beach and MacKinnon (1978). Asymptotically the two procedures are equivalent. Results of sampling experiments suggest that the generalized least squares procedure is marginally superior to the full maximum likelihood procedure.

1. INTRODUCTION

Consider the standard linear regression model where the errors follow a first-order autoregressive process, y = X B + u t=

PU

(1)

t-1 +

I P I < 1,

Et,

E ( E ) = 0 and Var(& ) =

t

t

u2

t

=

1, 2,

. . .,

T

t

=

1, 2,

. . .,

T.

Given the assumptions regarding

E

t’

the following results are valid

for the error vector u: E(u)

=

0 and Cov(u) =

cr2 V -

(1-P2)

where P

P2

1

P

.

P

T-1

PT-2

v= 0

T-2

For p c (-1,+l), V is positive definite. The parameters of the model to be estimated are: B,

I-,

and u’.

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

101 Given the above set-up, V V-l

-1

can be written as:

= Q'Q

(3)

where 0

0

1

0

0

0

Q= -P

The typical Cochrane and Orcutt

1

(1949) estimator is a generalized

least squares (GLS) estimator which ignores the information contained in u

1

and minimizes the generalized sum of squares of residuals

u'V-~Uwhere V 1

1

= QiQ, and that Q

1

is of order (T-1)xT which is obtair

ed by deleting the first row of Q. Gurland (1954), Kadiyala (1968), and Rao and Grilliches (1969) have studied the relative efficiency of the GLS estimators based on V

and V and concluded that using V 1 1 instead of V could, in certain cases, result in a substantial loss

of efficiency. In this paper, the GLS procedure is used to obtain estimates of 8, p and

2.

0'.

THE GENERALIZED LEAST SQUARES PROCEDURE A

The generalized least squares estimator 8 of 8 is obtained by minimizing the generalized sum of squares of residuals S ( B , p ) = (y

-

where y*

- XP) 'V-1(Y

- XB)

(4)

(y*-X*B)'v-l(y*-X*a) =

Qy and X*

=

QX.

If p is known, the efficient Aitken's (1935) estimator is given by B(p) =

(x*'x*)-lx*'y*

=

-1

(X'V X)

-1

x'v-ly

(5)

If p is not known, p ( p ) represents partial minimization of S ( 8 , p ) with respect to 8. Substituting (5) in (4), we have S(p)

=

(p)v-%' ( p )

(6)

A

where u(p)

=

y - XB(p) and that S ( p ) has to be minimized with respect A

to p . Noting that S, u, and 8

are all functions of p , the argument

p in these functions will henceforth be dropped for convenience.

102 It has been pointed by many researchers including Dhrymes (1971, p. 68) that the expression in (6) is highly nonlinear in p which precludes a direct approach to the solution of dS/dp

=

0. The non-

linearity of the expression (6) led Hildreth and Lu (1960) to suggest a grid-search procedure for estimating p . Fortunately, the solution

-

of dS/dp = 0 is not difficult as we will demonstrate below. -1Consider S = u'V u.

The expression in (7) is obtained using the following results:

,.,

-1

A

X'V

u

=

X'V

-1

(y

- X4)

=

0.

yields the unique solution T,.. T-1 2 p = c u u t t-1 2 The second derivative of S with respect to p is

The derivative of (8.2) with respect to p implies

and using (111, we can express (10) as

(8.2)

103

The second derivative of V

-1

with respect to p in ( 1 2 ) is

dp where I* is modified unit matrix with zero in the left and right hand corner elements. Using (13), we can express (12) as

2 Thus d S/dp2 is the difference between two quadratic forms which are

positive definite involving the GLS estimated residuals ut and hence data-relative. Assuming that the second order condition is satisfied, we have a minimum for S. We may now summarize the GLS estimators of the parameters of the model as follows:

T h A T-1 c u u U2 2 tt-11 t I\

p

=

A

h

;

a2(p)

=

^-1

where V

S(P)/

(15.2)

(15.3)

T

is the matrix V

-1

with p substituted for p .

The consistency of p was proved by Hildreth and Lu (1960) and A

h

A

h

Dhrymes (1971, pp. 91-96) and hence a ( p ) and a 2 ( p ) are also consisA

A

tent. An estimate of the asymptotic covariance matrix of B ( p ) is given by A

h-1

h

Cov(B(p))

=

(X'V X)

-1^

A

U2(P)

-

(16)

So far we made no assumptions regarding the probability distribu-

tion of

except that the first and second moments exist. If we t assume that E 's are NID(0,u2), then we can apply the maximum likelit hood procedure. Following Beach and MacKinnon (1978) and Dhrymes E

(1971, p. 7 0 ) , the concentrated log-likelihood function can be written as

104 It is clear from (17) that maximizing L"(p) with respect to p is equivalent to minimizing 2 b 'T 0 2 ( P ) / (1 - P ) which yields the full maximum likelihood (FML) estimates of B, p and

u 2 . In GLS we minimize 02(p)

a2(p)

whereas in FML we minimize

/ (1 - p2)liT. Asymptotically the two procedures are equivalent

because the limit of (1 - p2)llT is unity as T

-f

00.

3. AN ALGORITHM FOR MINIMIZING S ( B , p )

Equations (5) and (9) constitute the two components of an algorithm for computing the GLS estimates of the parameters of the model by The procedure consists in alternatively minimizing

minimizing S ( B , p ) .

with respect to 6,

S(B,p)

p

held constant, and minimizing

respect to p , 6 held constant. Usually, we start with p A

A

=

S(6,p)

with

0 and compute

A

B and p . Since S reaches its minimum value at p

= p,

it follows that

A

S is an increasing function of p for p > p , and a decreasing function h

S

T - -

< 0, then t t-1 decreases for values of p < 0. Since our objective is to minimize

of p for p < p . Accordingly, if p

=

0 > p , i.e., C u u 2

h

S, we choose p = p for the next iteration. If, on the other hand, n T - p = 0 < p , i-e., C u u > 0, then S decreases for values of p > 0. 2 t t-1 Since our objective is to minimize S, we choose p

= p

> 0 for the

next iteration. Thus, in either case, the computed p will be the value to be used for the next iteration. When two p-values in successive iterations are close, the iterative procedure is stopped. Iterative procedures of the type described above raise two questions. The first is concerned with the convergence of

S

to a

stationary value and the second is concerned with the attainment of a global minimum as opposed to a local mimimum. p being an estimator of the correlation coefficient, it follows from equation (9) that the A

stationarity condition I p I < 1 may not always be satisfied and the convergence of S may be jeopardized. But the results of sampling experiments we have conducted reveal that this is not a serious drawback of the algorithm. The question of the attainment of global minimum was i-nvestigated by Sargan (1964) who concluded that there was

105 no indication of the occurrence of multiple minima.

4. MONTE CARL0 RESULTS The GLS procedure is similar to the FML procedure of Beach and MacKinnon (1978). The advantage of GLS is that it is distributionfree. In order to compare the relative performances of these two estimators in small sample situations, we conducted three sampling experiments. The model-structure used for purposes of experimentation is the same as of Beach and MacKinnon (1978): Yt=

8,

+ P2Xt +

Utl

Ut =

Put-l

+

Et l

E

'L

NID(0,.0036).

The x's contain a large trend component'and are computed from the t w 'L NID (0,.0009). The values of 8 equation x = exp( .04t) + w 1 t t' t and 6, are set equal to unity. In each of the three experiments, three values of p were considered and they are p

=

.60, .80, and -99.

Experiments 1 and 2 differ only in the sample size used; in experiment 1 the sample size is 20 and in experiment 2 the sample size is 50. In experiment 3 the sample size is 20, and the

on a

x2

are realizations t random variable with one degree of freedom adjusted for zero

mean. The variance of

E~

E'S

is, of course, 2.0. Thus experiment 3 permits

us to examine the effect of a specification error (nonnormal errors) on the FML estimator and this might shed light on the robustness of the FML estimator. Each experiment involved one thousand replications. On each replication, FML and GLS estimators are used to compute point

6, and p for a given realization of the E ' S . It may 1' be mentioned here that in order to ensure that the error process was estimates of

L

Since FML and GLS point 1' estimates have to be computed by iterative procedures, we started off stationary, u

1

was generated as (l/(l-pz) ')

in each case with the initial value of p

E

= 0

and stopped iterating -5

when two successive estimated values of p differed by less than 10

.

In Table 1, the mean biases, root mean square errors (RMSE) and the number of GLS estimates that were closer to the corresponding true values than the FML estimates (in the 'NO' column) are presented by experiments. In Table 2, the ratics of the mean square error of FML to the mean square error of GLS are presented by experiments.

106 The first element in each cell represents the FML estimator and is followed by the corresponding statistic for the GLS estimator. Since experiments 1 and 2 are identical to those of Beach and MacKinnon (1978) except for the number of replications, their format is borrowed

for presenting the results to facilitate easy comparisons of our results with theirs. The biases of both estimators for

and are not significant 1 2 at the .05 level. The biases of both estimators for p are negative and significant at the .O1 level. Each estimator produced biases of

6, and 0, which are opposite in sign. The biases are invariably smaller in the case of the GLS estimator for p . This is also evident from the frequency of the GLS estimates of p which came closer to the true value of p than the FML estimator. In experiments 1 and 2, the biases of the GLS estimator are very close to the corresponding biases of the FML estimator. Though the assumption of normal errors is violated, yet, the FML estimator proved to be slightly superior to the GLS and 0, in experiment 3. 1 Most often, the frequency of the GLS estimates which came closer to estimator by generating smaller biases for B

the true values of the parameters is significantly larger at the .05 level. The large sample bionomial test is used for this purpose. See the frequencies marked with an asterisk in Table 1. The RMSE's of the GLS estimator are most often smaller than the corresponding RMSE's of the FML estimator. This is expected of the GLS estimator in view of its minimum variance property. The relative efficiencies of the GLS estimator with respect to the FML estimator are most often greater than unity. The number of iterations in each replication are either equal or the GLS estimator required just one more iteration. Computational efficiency is the same for both estimators. The average number of iterations in each experiment for the two estimators are displayed in Table 3. In this context one remark is in order. Equation (9) which defines the p for each iteration may not satisfy the stationarity

I I

condition p

< 1. In such an event, that sample was discarded and

a new sample was added. The number of samples thus discarded for want

BIAS

Parameter

TABLE 1 AND ROOT MEAN SQUARE ERROR OF FML AND GLS ESTIMATORS~

True p

Experiment 1 Bias RMSE NO.

.60 -.2385

P

-.2171 -80 -.3050 -.2751 -99 -.4177 -.3816 .60

62

a

.3235 792* -.0985 .3177 -.0818 -3736 910" -.1154 .3582 -.lo06 .4674 988* -.1704 .4425 -.1516

-0006 -1269 520 -0006 .1271

.0014 -0014 .99 .0001 -.0002 .80

Experiment 2 Bias RMSE NO.

.1940 537* .1932 .5293 592* .5259

.60 -.0002 .0771 508 -.0003 .0773 .80 -.0004 .1155 560* -.0004 .1150 -99 .0016 .2095 631* .0019 .2061

-0004 -0004 .0010 -0010 .0057 .0056

-.OOOO -.OOOO -.0001 -.0002 -.0005 -.0005

Experiment 3 Bias RMSE NO.

.1588 732* .1552 -1644 816* .1565 .2006 1000* .1886

-.2254 -.2028 -.2905 -.2593 -.4137 -.3770

.3005 811* -2933 .3507 914* -3334 -4618 990* .4372

.0410 499 -0410 -0743 524 -0743 .4422 528 -4412

-0201 2.9127 -0221 2.9315 -.0303 4.3450 -.0339 4.3380 -.0135 9.4827 -.0353 9.4440

.0106 499 -.0041 1.7657 .0106 -.0017 1.7741 .0184 513 -0301 2.5911 .0185 .0390 2.5803 .0566 609" .0575 4.9334 .0557 .0795 4.8583

574* 583* 596*

547* 568* 613*

The f i r s t f i g u r e i n each c e l l r e f e r s t o t h e FML e s t i m a t o r . The second f i g u r e i n each c e l l r e f e r s t o t h e GLS e s t i m a t o r . The frequency of GLS estimates which came c l o s e r t o t h e t r u e parameter value than t h e corresponding FML e s t i m a t e s and marked with an a s t e r i s k a r e s i g n i f i c a n t a t t h e .05 l e v e l .

108

TABLE 2 R A T I O S O F ROOT MEAN SOUARE ERRORS

Parameter

Experiment 1 Experiment 2

True p

Experiment 3

P

.60 -80 -99

1.0185 1.0432 1.0562

1.0234 1.0505 1.0753

1.0245 1.0517 1.0562

%

-60 .80 .99

.9981 1.0046 1.0064

.9999 .9996 1.0023

-9936 1.0016 1.0040

.60 .80 -99

-9983 1.0038 1.0166

1.0001 .9999 1.0167

.9953 1.0042 1.0155

B2

TABLE 3 AVERAGE NUMBER OF I T E R A T I O N S R E W I R E D

True p

FML

GLS

E x p e r i m e n t 1:

.60 .80 .99

4.557 5.036 5.509

4.735 5.365 6.211

E x p e r i m e n t 2:

.60 .80 -99

3.597 3.963 4.728

3.636 4.064 5.233

E x p e r i m e n t 3:

.60 .80 .99

4.617 5.129 5.664

4.832 5.525 6.396

TABLE 4 NUMBER O F SAMPLES DISCARDED

True p

Experiment 1

- 60

0

.80 -99

2 10

Experiment 2

0 0

27

Experiment 3

9

12 16

109 of convergence constitutes a very small percentage as is indicated by the figures in Table 4. A similar problem could very well arise in the case of the Cochrane and Orcutt (1949) estimator. See Rao and Grilliches (1969, footnote 6, p. 256).

5. CONCLUSION The GLS estimator considered in this paper is slightly more efficient than the FML estimator as revealed by the results of the sampling experimets. Except for a few cases wherein convergence is not achieved, the performance of the GLS estimator compares very favourably with the FML estimator. In empirical work it is recommended that both the estimators be used so that the merits of one over the other could be exploited to benefit the researcher.

REFERENCES Aitken, A. C., 1935. On least squares and linear combinations of observations. Proceedings of the Royal Society of Edinburg, 55: 42-48. Beach, C.M. and MacKinnon, J. G., 1978. A maximum likelihood procedure for regression with autocorrelated errors. Econometrica, 46: 51-58. Cochrane, D. and Orcutt, G. H., 1949. Applications of least squares regression to relationships containing autocorrelated error terms. Journal of the American Statistical Association, 44: 32-61. Dhrymes, P. J., 1971. Distributed Lags. Holden Day, San Francisco. Gurland, J., 1954. An example of autocorrelated disturbances in linear regression. Econometrica, 22: 218-227. Hildreth, C. and Lu, J. Y., 1960. Demand relations with autocorrelated disturbances. Research bulletin 276, Michigan State University Agricultural Experiment Station. Kadiyala, K. R., 1968. A transformation used to circumvent the problem of autocorrelation. Econometrica, 36: 93-96. Rao, P. and Grilliches, Z., 1969. Small-sample properties of several two-stage regression methods in the context of auto-correlated errors. Journal of the American Statistical Association, 64: 253-272. Sargan, J. D., 1964. Wages and prices in the United Kingdom: A study in Econometric methodology. In: P.E.Hart, G.Mills and J.K.Whitaker (Editors), Econometric Analysis for National Economic Planning. Butterworth and Co. Ltd., London, pp. 25-54.

110 FITTING DYNAMIC MODELS TO HYDROLOGICAL TIME SERIES KEITH W .

HIPEL*, A .

I A N MCLEOD:

and DONALD J . NOAKES*

ABSTRACT Based upon t h e p h y s i c a l p r o p e r t i e s o f t h e phenomena b e i n g mode l l e d and v a l i d s t a t i s t i c a l p r i n c i p l e s , t e c h n i q u e s a r e p r e s e n t e d f o r 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 .

Procedures

a r e d e v i s e d f o r p r o p e r l y i n c o r p o r a t i n g one o r more c o v a r i a t e s e r i e s i n t o a dynamic model.

F u r t h e r m o r e , when t h e r e a r e m i s s i n g d a t a

p o i n t s i n t h e s e r i e s , t h e s e can b e e s t i m a t e d by i n c l u d i n g a s p e c i a l type o f i n t e r v e n t i o n component i n t h e dynamic model.

The e f f i c a c y

of t h e model b u i l d i n g t e c h n i q u e s i s c l e a r l y d e m o n s t r a t e d by d e s i g n i n g a t r a n s f e r f u n c t i o n - n o i s e model t o d e s c r i b e t h e dynamic r e l a t i o n s h i p s c o n n e c t i n g a monthly r i v e r flow s e r i e s i n Canada t o p r e c i p i t a t i o n and t e m p e r a t u r e c o v a r i a t e s e r i e s .

1.1

INTRODUCTION

I n o r d e r t o p r o p e r l y d e s i g n and o p e r a t e w a t e r r e s o u r c e s p r o j e c t s ,

i t i s n e c e s s a r y t o measure o v e r t i m e and s p a c e p e r t i n e n t h y d r o l o g i c a l phenomena which may i n c l u d e r i v e r f l o w s , p r e c i p i t a t i o n , and temperature.

T i m e s e r i e s a n a l y s i s t h e r e f o r e c o n s t i t u t e s a very

i m p o r t a n t t o o l f o r u s e i n s o l v i n g w a t e r r e s o u r c e s problems.

In par-

t i c u l a r , h y d r o l o g i s t s o f t e n r e q u i r e s t o c h a s t i c models which r e a l i s t i c a l l y d e s c r i b e t h e dynamic r e l a t i o n s h i p s c o n n e c t i n g a s i n g l e o u t - . p u t s e r i e s , such a s s e a s o n a l r i v e r f l o w s , t o one o r more c o v a r i a t e o r i n p u t s e r i e s such a s p r e c i p i t a t i o n and t e m p e r a t u r e .

Consequently,

t h e p u r p o s e o f t h i s p a p e r i s t o c l e a r l y d e m o n s t r a t e how b o t h a sound 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 and comprehensive s t a t i s t i c a l p r o c e d u r e s can b e employed f o r d e v e l o p i n g a comprehensive dynamic model t o f i t t o a s e t o f t i m e s e r i e s .

*

Department o f Systems Design E n g i n e e r i n g , U n i v e r s i t y o f W a t e r l o o , W a t e r l o o , O n t a r i o , Canada -1 Department o f S t a t i s t i c a l and A c t u r i a l S c i e n c e s , 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

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

111 The s p e c i f i c dynamic model employed i n t h i s p a p e r i s 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 which i s d e s c r i b e d by Box and J e n k i n s ( 1 9 7 0 ) . By a d h e r i n g t o t h e 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 d i a g n o s t i c c h e c k i n g s t a g e s o f model c o n s t r u c t i o n , a t r a n s f e r f u n c t i o n - n o i s e model i s developed f o r l i n k i n g a monthly r i v e r flow s e r i e s t o p r e c i p i t a t i o n and t e m p e r a t u r e d a t a s e t s .

The b e s t dynamic model i s t h e n s e l e c t e d

u s i n g t h e Akaike I n f o r m a t i o n c r i t e r i o n ( A I C )

(Akaike, 1 9 7 4 ) .

The

A I C p r o v i d e s a combined measure o f model parsimony and good s t a t i s -

tical fit.

The model which h a s t h e minimum A I C v a l u e s h o u l d b e

s e l e c t e d when t h e r e a r e s e v e r a l competing models. I n t h e p r o c e s s o f s e l e c t i n g t h e most a p p r o p r i a t e model t o f i t t o t h e s e r i e s , a number o f u s e f u l m o d e l l i n g p r o c e d u r e s a r e s u g g e s t e d . Often t h e r e a r e more t h a n one p r e c i p i t a t i o n and t e m p e r a t u r e s e r i e s and a s t a t i s t i c a l p r o c e d u r e i s p r e s e n t e d f o r c r e a t i n g a s i n g l e s e quence t o r e p r e s e n t t h e p r e c i p i t a t i o n o r t e m p e r a t u r e s e r i e s .

This

approach can b e u t i l i z e d i n p l a c e o f t h e r a t h e r a d hoc methods such a s t h e I s o h y e t a l and T h i e s s e n polygon t e c h n i q u e s ( s e e Viessman e t al.

( 1 9 7 7 ) f o r a d e s c r i p t i o n o f t h e I s o h y e t a l and T h i e s s e n polygon

methods).

To d e c i d e upon which s e r i e s t o i n c l u d e i n t h e dynamic

model and a l s o d e s i g n t h e form o f t h e t r a n s f e r f u n c t i o n c o n n e c t i n g a c o v a r i a t e s e r i e s t o t h e o u t p u t , c r o s s - c o r r e l a t i o n analyses of t h e r e s i d u a l s of models f i t t e d t o t h e s e r i e s can b e employed.

It is

a l s o e x p l a i n e d how a dynamic model can be used f o r e s t i m a t i n g m i s s i n g d a t a p o i n t s i n t h e o u t p u t o r c o v a r i a t e s e r i e s and f o r m o d e l l i n g t h e e f f e c t s of one o r more e x t e r n a l i n t e r v e n t i o n s upon t h e mean l e v e l o f the output s e r i e s .

The t r a n s f e r f u n c t i o n - n o i s e model which i s u l -

t i m a t e l y chosen by f o l l o w i n g contemporary m o d e l l i n g p r o c e d u r e s can be used f o r a p p l i c a t i o n s such a s f o r e c a s t i n g and s i m u l a t i o n and a l s o p r o v i d i n g i n s i g h t i n t o 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 t h e phenomena b e i n g examined. 1.2

THE GENERAL INTERVENTION TRANSFER FUNCTION-NOISE MODEL

D e t a i l s of t h e mathematical theory underlying t r a n s f e r functionn o i s e and i n t e r v e n t i o n models a r e g i v e n i n a number o f p a p e r s

(see

112 f o r example Box and T i a o ( 1 9 7 5 ) , Hipel e t a l .

(1975), Hipel e t a l .

(1977b) and H i p e l and McLeod ( 1 9 8 2 ) ) . To g i v e t h e r e a d e r some p e r s p e c t i v e o f t h e model, t h e b a s i c development i s now p r e s e n t e d . The 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 may b e w r i t t e n i n t h e form

where t i s d i s c r e t e t i m e ; y t i s t h e r e s p o n s e v a r i a b l e ; N

is the t s t o c h a s t i c n o i s e component which may b e a u t o c o r r e l a t e d ; and f ( _ k ,_x , i , t )

. The dynamic t e r m i n c l u d e s a s e t o f t k , a group of c o v a r i a t e s e r i e s 5 , and t h e s e t o f i n t e r parameters i s t h e dynamic component o f y

vention s e r i e s

E, which a r e r e q u i r e d when model1 ng t h e e f f e c t s o f

external interventions.

When r e q u i r e d , b o t h y

t

formed u s i n g a s u i t a b l e Box-Cox t r a n s f o r m a t i o n

and x

may b e t r a n s t Box and Cox, 1 9 6 4 ) .

The r e a s o n s f o r t r a n s f o r m i n g t h e s e r i e s i n c l u d e s t a b i l i z i n g t h e v a r i a n c e and improving t h e n o r m a l i t y assumption o f t h e w h i t e n o i s e s e r i e s which i s i n c l u d e d i n N

I n h y d r o l o g i c a p p l i c a t i o n s , Box-Cox t’ transformations, i n p a r t i c u l a r the logarithmic transformation, o f t e n r e c t i f y problems a s s o c i a t e d w i t h t h e u n t r a n s f o r m e d s e r i e s .

I t should

a l s o be n o t e d t h a t t h e same Box-Cox t r a n s f o r m a t i o n need n o t b e applied t o a l l of t h e s e r i e s . Once a g i v e n s e r i e s h a s been t r a n s f o r m e d , i t i s t h e n n e c e s s a r y t o remove any t r e n d s o r s e a s o n a l i t y i n t h e d a t a .

A common p r o c e d u r e

employed i n hydrology f o r monthly sequences i s t o d e s e a s o n a l i z e t h e s e r i e s by s u b t r a c t i n g t h e e s t i m a t e d monthly mean and d i v i d i n g by t h e e s t i m a t e d monthly s t a n d a r d d e v i a t i o n f o r each d a t a p o i n t . I n c l u d e d i n t h e dynamic component of t h e model a r e t h e e f f e c t s of a l l i n p u t c o v a r i a t e s e r i e s and a l l e x t e r n a l i n t e r v e n t i o n s . general, i f there are I

In

i n p u t c o v a r i a t e s e r i e s and I i n t e r v e n t i o n s , 1 2 t h e dynamic component o f t h e model i s g i v e n by T

T

i i

113 where x

,

tl

i s t h e ith t r a n s f o r m e d and d e s e a s o n a l i z e d i n p u t c o v a r i a t e

i s t h e jth i n t e r v e n t i o n s e r i e s c o n s i s t i n g o f 0 ' s and 1's 'tj t o i n d i c a t e t h e nonoccurrence and o c c u r r e n c e , r e s p e c t i v e l y , of t h e

series;

jth

i n t e r v e n t i o n ; and B i s t h e backward s h i f t o p e r a t o r such t h a t

Bx

= x

t

p

l

and BSx = x where s i s a p o s i t i v e i n t e g e r . t t-s

The t e r m

( B I B 7/6.(B) = v . ( B ) i s t h e t r a n s f e r f u n c t i o n of t h e j t h i n t e r v e n j 7 3 t i o n o r i n p u t s e r i e s and b . i s t h e d e l a y t i m e f o r . or x to 3 t 7 t j effect y The t r a n s f e r f u n c t i o n , v . ( B ) , h a s t h e form t' 3

(fi

<

(W il,( B )

3

=

. B - U

. - - W

03

11

( 1 - 6 . B - 6

11

21

2j

B

B

2

2

-

-

... ...

U

- w

- 6

.B j )

Uj7

b. B 7

(3)

.BrJ)

rjl

I n p r a c t i c e , u . and r . a r e u s u a l l y 0 o r 1 ( s e e f o r example H i p e l et al.

7 3 (1975), Hipel e t a l .

Baracos e t a l .

( 1 9 7 7 b ) , D'Astous and H i p e l ( 1 9 7 9 ) ,

(1981)). When t h e r e a r e no i n p u t c o v a r i a t e s e r i e s

i n ( 2 ) and hence I1 i n t e r v e n t i o n model.

=

0 , t h e model i n (1) i s r e f e r r e d t o as an

On t h e o t h e r h a n d , i f t h e r e a r e no e x t e r n a l

i n t e r v e n t i o n s i n ( 2 ) and t h e r e f o r e I2 = 0 , t h e model i n (1) i s c a l l e d a t r a n s f e r f u n c t i o n - n o i s e model. The n o i s e component 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 i s d e f i n e d by

That i s , t h e n o i s e t e r m o f t h e model i s simply t h e d i f f e r e n c e between the response v a r i a b l e , y t ,

and t h e dynamic component.

The form of

the noise t e r m , N

i s n o t r e s t r i c t e d t o any p a r t i c u l a r form, b u t t' a commonly employed f a m i l y o f models i s t h e a u t o r e g r e s s i v e movingaverage (ARMA) models. MODEL CONSTRUCTION

2.1

INTRODUCTION

No m a t t e r what t y p e s o f s t o c h a s t i c models a r e b e i n q c o n s i d e r e d f o r f i t t i n g t o a s p e c i f i e d d a t a s e t , i t i s recommended t o f o l l o w t h e

114 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 d i a g n o s t i c check s t a g e s o f model construction.

A t the identification stage,

t e n t a t i v e model forms

a r e i d e n t i f i e d by employing v a r i o u s s t a t i s t i c a l t e c h n i q u e s which a r e u s u a l l y e a s i e s t t o i n t e r p r e t g r a p h i c a l l y (Box and J e n k i n s , 1970; H i p e l e t a l . , 1977a; McLeod e t a l . , 1977; H i p e l e t al., 1 9 8 1 ; H i p e l and McLeod, 1 9 8 2 ) .

For t h e c a s e of t h e 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 i n (l), b o t h t h e dynamic and n o i s e components must be identified.

F o l l o w i n g t h i s , maximum l i k e l i h o o d e s t i m a t e s

of t h e model p a r a m e t e r s a r e o b t a i n e d (McLeod, 1 9 7 7 ) .

(MLE’s)

Finally,

d i a g n o s t i c checks a r e employed t o i n s u r e t h a t t h e key m o d e l l i n g assumptions a r e s a t i s f i e d f o r a g i v e n model. 2.2

IDENTIFYING THE DYNAMIC COMPONENT O f t e n i t i s known i n advance w h e t h e r o r n o t a c o v a r i a t e s e r i e s ,

x

causes t h e o u t p u t s e r i e s , y t . For i n s t a n c e , p r e c i p i t a t i o n t’ obviously causes r i v e r flow. I f , however, i t i s n o t c e r t a i n w h e t h e r one s e r i e s c a u s e s a n o t h e r , p r o c e d u r e s a r e a v a i l a b l e f o r d e t e r m i n i n g t h e t y p e o f r e l a t i o n s h i p which may e x i s t between t h e two s e r i e s

(see

f o r example Granger ( 1 9 6 9 ) , P i e r c e and Haugh ( 1 9 7 7 ) , H i p e l e t a l . ( 1 9 8 1 ) , and H i p e l and McLeod ( 1 9 8 2 ) ) . the cross-correlation

function

In particular,

t h e form o f

(CCF) o f t h e r e s i d u a l s from s t o c h a s t i c

models f i t t e d t o t h e two t r a n s f o r m e d and d e s e a s o n a l i z e d s e r i e s , x and y

t’

t

r a n be examiried.

I n g e n e r a l , t h e ARMA models f i t t e d t o t h e two s e r i e s , x

t

and y

t’

may be symbol.ically w r i t t e n a s

and

i s NID(0, r L ) ; v i s N I D ( O , C T ~ ) ;and d ( B ) and O ( B ) a r e t h e u t AR and MA o p e r a t o r s r e s p e c t i v e l y . When t h e r e a r e n o b s e r v a t i o n s i n

where u

t

e a c h s e r i e s t h e r e s i d u a l CCF a t l a g k between t h e r e s i d u a l s f o r t h e two models c a n t h e n be e s t i m a t e d by

115

(7)

i s t h e estimated cross-covariance

estimated r e s i d u a l series

6t and

f u n c t i o n a t l a g k between t h e

ct ;

c,(O) i s t h e e s t i m a t e d v a r i a n c e u of t h e 6 s e q u e n c e ; and c , ( 0 ) i s t h e e s t i m a t e d v a r i a n c e o f t h e 6t t. V series. Approximate c o n f i d e n c e i n t e r v a l s f o r t h e CCF may b e o b t a i n e d by and y a r e i n d e p e n d e n t (Haugh, 1972, 1 9 7 6 ) . However, t t McLeod (1979) o b t a i n e d t h e a s y m p t o t i c d i s t r i b u t i o n of t h e r e s i d u a l

assuming x

and y s e r i e s do n o t have t o t t be independent of e a c h o t h e r , and c o n s e q u e n t l y more a c c u r a t e c o n f i CCF f o r t h e g e n e r a l c a s e where t h e x

dence l i m i t s can b e o b t a i n e d by u t i l i z i n g h i s r e s u l t s . Causal r e l a t i o n s h i p s between x

and y can b e d e t e c t e d o r cont t firmed by examining t h e form o f t h e r e s i d u a l CCF ( P i e r c e and Haugh, F o r example, i f x

causes y which i s t h e c a s e when t h e x t t t' series i s p r e c i p i t a t i o n and t h e y sequence i s made up of r i v e r f l o w s , t t h e n f o r k ?: 0 t h e r e w i l l b e a t l e a s t one v a l u e of r A , , ( k ) which is uv 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 . However, a l l v a l u e s o f t h e 1977).

= ~ . ~ + x < l , x ~ x \< ~ 1 < - L .

zero.

LS>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

,


otherwise i n (10) can be i d e n t i f i e d by f i t t i n g an ARMA t' s e r i e s w i t h o u t t h e i n t e r v e n t i o n component. Once t h e

The n o i s e t e r m , N model t o t h e y

t form o f t h e model i s i d e n t i f i e d , t h e p a r a m e t e r e s t i m a t e s o f t h e n o i s e

model a r e e s t i m a t e d s i m u l t a n e o u s l y w i t h t h e e s t i m a t e o f w

0'

A t time

t = T t h e model r e d u c e s t o

Thus, t h e MLE o f --u t i o n a t t i m e T.

0

c o n s t i t u t e s an e s t i m a t e of t h e m i s s i n g o b s e r v a -

By c o n s i d e r i n g t h e s t a n d a r d e r r o r o f e s t i m a t e f o r

oo, a c o n f i d e n c e i n t e r v a l c a n be o b t a i n e d f o r t h e e s t i m a t e d m i s s i n g o b s e r v a t i o n r e p r e s e n t e d by

-0

0'

Furthermore, because t h e e s t i m a t e

c f t h e m i s s i n g d a t a depends on t h e n o i s e component, N c o r r e l a t i o n s t r u c t u r e of t h e s e r i e s i s p r e s e r v e d .

the t' To obtain the

e s t i m a t e of t h e m i s s i n g o b s e r v a t i o n f o r t h e o r i g i n a l s e r i e s , t h e i n v e r s e d e s e a s o n a l i z a t i o n t r a n s f o r m a t i o n f o l l o w e d by t h e i n v e r s e Box-Cox t r a n s f o r m a t i o n can b e i n v o k e d . I f more t h a n one m i s s i n g o b s e r v a t i o n a r e t o b e e s t i m a t e d , t h e model i s simply e x t e n d e d by adding t h e a p p r o p r i a t e number of i n t e r vention t e r m s .

F o r example, i f I

3

missing d a t a a r e t o be e s t i m a t e d ,

t h e model i s g i v e n by

where y

t

i s t h e t r a n s f o r m e d and d e s e a s o n a l i z e d s e r i e s , and

118 1

,

if t = T.

0

,

otherwise

The t e r m T . i n t h i s c a s e r e f e r s t o t h e t i m e o r l o c a t i o n of t h e i

th

missing d a t a i n t h e series. 2.5

COMBINING MULTIPLE TIME SERIES O f t e n more t h a n one c o v a r i a t e s e r i e s i s a v a i l a b l e t o t h e a n a l y s t .

I n h y d r o l o g i c a l s t u d i e s , d a t a from s e v e r a l p r e c i p i t a t i o n and t e m p e r a t u r e s t a t i o n s w i t h i n o r n e a r t h e b a s i n may b e a v a i l a b l e .

A

common p r o c e d u r e employed by h y d r o l o g i s t s t o r e d u c e model c o m p l e x i t y

i s t o combine s i m i l a r t y p e s o f s e r i e s t o form a s i n g l e i n p u t c o v a r i a t e series.

I n t h e c a s e of p r e c i p i t a t i o n d a t a , t h e r e c o r d s from t h e

v a r i o u s s t a t i o n s a r e o f t e n combined t o p r o v i d e a s i n g l e s e r i e s of mean a r e a p r e c i p i t a t i o n .

TWO common methods of combining p r e c i p i -

t a t i o n s e r i e s a r e t h e I s o h y e t a l and t h e T h i e s s e n polygon t e c h n i q u e s (Viessman e t a l . ,

1977).

These p r o c e d u r e s a r e e s s e n t i a l l y g r a p h i c a l

methods and r e q u i r e a s k i l l e d a n a l y s t t o o b t a i n r e a s o n a b l e and consistent results.

I n an e f f o r t t o automate p r o c e d u r e s f o r combining

s i m i l a r t y p e s of s e r i e s and p r o v i d e more c o n s i s t e n t r e s u l t s , a t e c h nique based on combining t r a n s f e r f u n c t i o n c o e f f i c i e n t s i s p r e s e n t e d . Consider t h e c a s e where two i n p u t c o v a r i a t e s e r i e s , x

and x t2' tl a r e t o be combined t o form a s i n g l e i n p u t c o v a r i a t e s e r i e s x If t' x , causes y i n s t a n t a n e o u s l y , then t h e t r a n s f e r function-noise tl t models f o r t h e two s e r i e s would be -

Yt

*?OIXtl

+ N

tl

and -

Y t- 302Xt2 where

+ N

t 2

(16)

and w a r e the t r a n s f e r function parameters f o r t h e s e r i e s 01 02 and x I n t h i s c a s e t h e two s e r i e s x xtl and x t 2 , r e s p e c t i v e l y . tl t 2 would be combined u s i n g t h e r e l a t i v e r a t i o o f t h e t r a n s f e r f u n c t i o n ri)

119 c o e f f i c i e n t s such t h a t

I f more t h a n two i n p u t s e r i e s a r e a v a i l a b l e , t h i s p r o c e d u r e c o u l d simply be e x t e n d e d t o combine a l l of t h e a v a i l a b l e d a t a i n t o one input covariate series. M O D E L L I N G HYDROMETEOROLOGICAL DATA

3.1

INTRODUCTION

The g e n e r a l p r o c e d u r e i n any m o d e l l i n g problem i s t o s t a r t w i t h a s i m p l e model and t h e n i n c r e a s e t h e model c o m p l e x i t y u n t i l an a c c e p t a b l e d e s c r i p t i o n o f t h e phenomenon i s a c h i e v e d o r u n t i l f u r t h e r improvements i n t h e model c a n n o t b e o b t a i n e d by i n c r e a s i n g t h e model complexity.

I n o r d e r t o d e m o n s t r a t e t h e model b u i l d i n g t e c h n i q u e s

discussed i n the previous s e c t i o n s , d i f f e r e n t t r a n s f e r functionn o i s e models a r e c o n s i d e r e d and t h e most a p p r o p r i a t e model i s eventually selected.

The o u t p u t f o r each t r a n s f e r f u n c t i o n - n o i s e model

always r e p r e s e n t s t h e a v e r a g e monthly flows of t h e Saugeen R i v e r a t Walkerton, O n t a r i o , w h i l e t h e c o v a r i a t e s e r i e s c o n s i s t o f e i t h e r p r e c i p i t a t i o n o r temperature d a t a s e t s , o r both types of s e r i e s . The t y p e , l e n g t h and l o c a t i o n o f measurement f o r t h e d a t a s e t s ent e r t a i n e d a r e shown i n Table 1 f o r t h e s i n g l e r i v e r flow s e q u e n c e , t h e two p r e c i p i t a t i o n and t h e two t e m p e r a t u r e s e r i e s .

The r i v e r

flow d a t a a r e o b t a i n e d from Environment Canada (1980a) and t h e p r e c i p i t a t i o n and t e m p e r a t u r e d a t a a r e p r o v i d e d by t h e Atmospheric Environment S e r v i c e i n Downsview O n t a r i o (Environment Canada, 1 9 8 0 b ) . 3.2

E S T I M A T I N G M I S S I N G DATA

P r i o r t o performing t h e c r o s s - c o r r e l a t i o n

a n a l y s e s and f i t t i n g

t h e t r a n s f e r f u n c t i o n - n o i s e models, any m i s s i n g d a t a must b e e s t i m a t e d . The o n l y m i s s i n g d a t a i n t h i s s t u d y a r e t e n p r e c i p i t a t i o n and c o r r e s ponding t e m p e r a t u r e d a t a p o i n t s f o r t h e Lucknow s t a t i o n where t h e d a t e s of t h e s e m i s s i n g d a t a a r e g i v e n i n Table 2 .

When c o n s i d e r i n g d a t a

120 TABLE 1.

A v a i l a b l e Monthly Data Location

Period

Riverflows

Saugeen R i v e r a t Walkerton, Ontario.

196 3-1979

Precipitation

Paisley, Ontario.

1963-1979

Precipitation

Lucknow, O n t a r i o .

1950-1979

Temperature

Paisley, Ontario.

1963-1979

T e m p e rat u r e

Lucknow, O n t a r i o .

1950-1979

sets i n t h e r e s i d u a l cross-correlation

analyses o r f o r use an i n p u t

o r o u t p u t s e r i e s i n t h e dynamic m o d e l , t h e d a t a a r e o n l y c o n s i d e r e d f o r t h e t i m e p e r i o d d u r i n g which a l l o f t h e s e r i e s o v e r l a p .

How-

e v e r , when e s t i m a t i n g m i s s i n g v a l u e s i n a s i n g l e s e q u e n c e , t h e e n -

t i r e t i m e series i s u t i l i z e d i n o r d e r t o t a k e f u l l advantage o f a l l t h e a v a i l a b l e i n f o r m a t i o n a n d t h e r e b y o b t a i n b e t t e r e s t i m a t e s of the missing d a t a . TABLE 2.

E s t i m a t e s of M i s s i n g T e m p e r a t u r e D a t a a t Lucknow Monthly Mean

Date

(C")

E s t i m a t e (CO) (Standard Error)

February

1953

-6.48

-6.57 (2.32)

May

1968

11 -99

11.81 (1.78)

15.17

15.34 (1.16)

S e p t e m b e r 1968 October

1973

9.60

9.78 (1.67)

August

1975

18.89

18.59 (1.20)

S e p t e m b e r 1975

15.17

15.29 (1.16)

July

1976

19.68

19.47 (1.09)

S e p t e m b e r 1978

15.17

15.30 (1.16)

October

1978

9.60

8.30 (1.70)

August

1979

18.89

18.56 (1.18)

F o r t h e case o f t h e p r e c i p i t a t i o n and t e m p e r a t u r e s e r i e s a t Lucknow a s e p a r a t e dynamic model i s f i t t e d t o e a c h o f t h e s e r i e s

121 where t h e r e a r e t e n i n t e r v e n t i o n components f o r e s t i m a t i n g t h e t e n missing observations plus a c o r r e l a t e d noise t e r m .

The most a p p r o -

p r i a t e n o i s e model f o r t h e Lucknow t e m p e r a t u r e d a t a i s found t o b e a n ARMA(0,4) model w i t h t h e s e c o n d and t h i r d MA p a r a m e t e r s c o n s t r a i n e d t o zero.

U s i n g t h i s form of t h e n o i s e m o d e l , M L E ' s of t h e m i s s i n g

d a t a a r e o b t a i n e d and a r e shown i n T a b l e 2 .

I n a l l cases, t h e e s t i -

m a t e s a r e w i t l i i n o n e s t a n d a r d d e v i a t i o n o f t h e m o n t h l y mean.

The

b e s t n o i s e model f o r t h e Lucknow p r e c i p i t a t i o n d a t a i s f o u n d t o b e a w h i t e n o i s e model.

Thus, t h e estimates of t h e m i s s i n g d a t a a r e

s i m p l y t a k e n a s t h e a p p r o p r i a t e e s t i m a t e d mean monthly v a l u e s . 3.3

IDENTIFYING THE DYNAMIC COMPONENT Once e s t i m a t e s o f t h e m i s s i n g d a t a a r e o b t a i n e d , a r e s i d u a l c r o s s -

c o r r e l a t i o n a n a l y s i s c a n b e p e r f o r m e d t o i d e n t i f y t h e forms o f possible t r a n s f e r functions f o r l i n k i n g p r e c i p i t a t i o n o r temperature t o t h e r i v e r flow o u t p u t .

A p p r o p r i a t e ARMA models a r e f i r s t f i t t e d

t o e a c h of t h e f i v e t r a n s f o r m e d , a n d d e s e a s o n a l i z e d s e r i e s a n d t h e model r e s i d u a l s a r e e s t i m a t e d .

The CCF b e t w e e n t h e r e s i d u a l s from

t h e model f i t t e d t o t h e Saugeen R i v e r f l o w s a n d e a c h o f t h e o t h e r f o u r r e s i d u a l s e r i e s are t h e n c a l c u l a t e d . The r e s u l t s o f t h e c r o s s - c o r r e l a t i o n

a n a l y s e s shown a p o s i t i v e

s i g n i f i c a n t r e l a t i o n s h i p a t l a g z e r o f o r e a c h o f t h e two p r e c i p i t a t i o n series.

F o r i n s t a n c e , t h e p l o t o f t h e CCF f o r t h e Lucknow

p r e c i p i t a t i o n and Saugeen R i v e r f l o w s i s d i s p l a y e d i n F i g u r e 1 along w i t h t h e e s t i m a t e d 95% c o n f i d e n c e i n t e r v a l .

The v a l u e o f t h e

CCF a t l a g z e r o i n F i g u r e 1 i s 0.448 w h e r e a s f o r t h e P a i s l e y p r e c i p i t a t i o n t h e e s t i m a t e d v a l u e of 0.365 i s s l i g h t l y s m a l l e r .

Although

t h e CCF p l o t f o r t h e P a i s l e y p r e c i p i t a t i o n i s n o t shown, i t i s i n d e e d s i m i l a r i n form t o F i g u r e 1.

The c h a r a c t e r i s t i c s o f t h e CCF's f o r

t h e two p r e c i p i t a t i o n s e r i e s makes i n t u i t i v e s e n s e from a p h y s i c a l p o i n t o f view s i n c e f o r monthly d a t a , most o f t h e p r e c i p i t a t i o n f o r a p a r t i c u l a r month w i l l r e s u l t i n r u n o f f i n t h e same month. The r e s u l t s o f t h e c r o s s - c o r r e l a t i o n

a n a l y s e s f o r t h e two t e m -

p e r a t u r e s e r i e s and t h e S a u g e e n R i v e r f l o w s a r e somewhat d i f f e r e n t .

122

amQ25

-

0

It

* I 1

1

Ir. 0

.

I I

1

,I

-.

I

" -0.25-

4

-a7

F i g . 1.

CCF f o r t h e Lucknow P r e c i p i t a t i o n and Saugeen River Flows.

I n t h e s e cases t h e r e a r e no s i g n i f i c a n t c r o s s - c o r r e l a t i o n s a t any lag.

However, from a p h y s i c a l view p o i n t , one might expect t h a t

above average temperatures during t h e w i n t e r season would i n c r e a s e snow melt and thus r i v e r flow.

For t h i s reason, t h e temperature

s e r i e s a r e considered i n t h e model b u i l d i n g .

The temperature s e r i e s

a r e assumed t o have a s i g n i f i c a n t c o n t r i b u t i o n a t l a g zero and a s

w i l l be shown l a t e r , t h i s assumption i s found t o be j u s t i f i a b l e . 3.4

THE TRANSFER FUNCTION-NOISE MODELS

The various t r a n s f e r function-noise models examined and t h e i r a s s o c i a t e d A I C values a r e presented i n Table 3 .

A decrease i n t h e

value of t h e A I C i n d i c a t e s t h a t t h e a d d i t i o n a l model complexity i s probably warranted s i n c e a b e t t e r s t a t i s t i c a l f i t i s obtained.

As

expected, each i n c r e a s e i n model complexity l e a d s t o a corresponding decrease i n t h e value of t h e A I C .

Thus a ' b e t t e r ' d e s c r i p t i o n of t h e

phenomena r e s u l t s with each a d d i t i o n of a v a i l a b l e information. D e t a i l s of each of t h e models a r e now d i s c u s s e d .

123 TABLE 3.

T r a n s f e r Function-Noise

Models F i t t e d t o t h e Data

Output and C o v a r i a t e T i m e Series

Parameter E s t i m a t e s (Standard Errors)

Saugeen Flows

91 91

AIC

= 0.407 (0.064)

963-121

= 0.405 (0.064)

936.129

925.606

Lucknow P r e c i p i t a t i o n (w,)

& = 0.310 (0.055) 0 9 1 = 0.429 (0.063) 0, = 0.350 (0.053)

Saugeen Flows

8,

=

0.418 (0.064)

926.340

Combined P r e c i p i t a t i o n ( w0) Saugeen Flows

6

=

0.350 (0.054)

8,

= 0.407 (0.064)

Summer P r e c i p i t a t i o n (a

& = 0.444 (0.065) 01 2 = 0.183 (0.092) 02 8, = 0.408 (0.064)

Saugeen Flows P a i s l e y P r e c i p i t a t i o n (w ) 0 Saugeen Flows

01

Winter P r e c i p i t a t i o n ( w

)

02

Saugeen Flows Summer P r e c i p i t a t i o n ( w

)

01

02

Saugeen Flows P a i s l e y T e m p e r a t u r e (w ) 0 Saugeen Flows Lucknow T e m p e r a t u r e ( w ) 0 Saugeen Flows )

01 (w )

02

Lucknow T e m p e r a t u r e ( w

)

03

Saugeen Flows Summer P r e c i p i t a t i o n ( w

)

01 W i n t e r P r e c i p i t a t i o n (a ) 02 Combined T e m p e r a t u r e

(w

924.037

& = 0.448 (0.066) 01 = -0.162 (0.109) 02 8, = 0.414 (0.064) 963.100 =

0.073 (0.061)

81

=

0.419 (0.064)

00

= 0.112 (0.062)

8,

=

0

Summer P r e c i p i t a t i o n ( w

922.270

a

Accumulated Snow (w )

Winter P r e c i p i t a t i o n

0

)

03

0.422 (0.064)

& = 0.453 (0.064) 01 & = 0.194 (0.090) 02 6 = 0.144 (0.055) 03 9, = 0.420 (0.064)

601 602 6 03

=

961.860

917.558

918.586

0.453 (0.064)

= 0.194 (0.090) =

0.133 (0.055)

The f i r s t model c o n s i d e r s o n l y t h e Saugeen r i v e r f l o w s .

The t i m e

series i s f i r s t t r a n s f o r m e d b y t a k i n g n a t u r a l l o g a r i t h m s o f t h e d a t a . T h i s s e r i e s i s t h e n 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 the e s t i m a t e d monthly mean a n d d i v i d i n g by t h e e s t i m a t e d monthly s t a n d a r d d e v i a t i o n

124

f o r each o b s e r v a t i o n .

An A R M A ( 1 , O ) model i s t h e n f i t t e d t o t h e s e

deseasonalized flows.

The v a l u e o f t h e A I C i s 9 6 3 . 1 2 1 and t h i s v a l u e

i s used a s a b a s i s f o r comparing improvements i n each of t h e models.

3.4.1

P r e c i p i t a t i o n S e r i e s a s Inputs

A s s u g g e s t e d by t h e r e s u l t s o f t h e c r o s s - c o r r e l a t i o n a n a l y s e s ,

each o f t h e p r e c i p i t a t i o n s e r i e s i s used a s an i n p u t c o v a r i a t e s e r i e s . P r i o r t o f i t t i n g 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, each o f the s e r i e s

i s f i r s t deseasonalized.

Each s e r i e s i s t h e n u s e d i n d e p e n d e n t l y a s an

i n p u t c o v a r i a t e s e r i e s i n a t r a n s f e r f u n c t i o n - n o i s e model. i n T a b l e 3 , t h e t r a n s f e r function parameter, w

0'

A s shown

f o r P a i s l e y and

Lucknow a r e e s t i m a t e d a s 0.310 and 0 . 3 5 0 , r e s p e c t i v e l y .

Note t h a t

t h e A I C v a l u e f o r t h e Lucknow p r e c i p i t a t i o n s e r i e s i s s i g n i f i c a n t l y l e s s than t h e A I C value f o r Paisley.

T h i s may s u g g e s t t h a t t h e p a t t e r n

o f t h e o v e r a l l p r e c i p i t a t i o n which f a l l s on t h e Saugeen R i v e r b a s i n upstream from Walkerton, i s more s i m i l a r t o t h e p r e c i p i t a t i o n a t Lucknow t h a n t h e p r e c i p i t a t i o n a t P a i s l e y , even though P a i s l e y i s c l o s e r t o Walkerton t h a n Lucknow. Using t h e p r o c e d u r e o u t l i n e d i n t h e s e c t i o n e n t i t l e d "Combining I n p u t C o v a r i a t e S e r i e s " , t h e two p r e c i p i t a t i o n s e r i e s a r e combined t o form a s i n g l e i n p u t c o v a r i a t e s e r i e s .

I n t h i s s t u d y , t h e Lucknow

and P a i s l e y p r e c i p i t a t i o n s e r i e s a r e combined i n t h e r a t i o 5 3 : 4 7 , respectively.

T h i s combined p r e c i p i t a t i o n s e r i e s i s t h e n deseason-

a l i z e d and used a s an i n p u t s e r i e s f o r 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.

The r e s u l t i n g A I C v a l u e i s o n l y s l i g h t l y l a r g e r t h a n t h e A I C

v a l u e o b t a i n e d when o n l y t h e Lucknow p r e c i p i t a t i o n s e r i e s i s employed. S i n c e t h e d i f f e r e n c e i s s m a l l , e i t h e r model would b e s a t i s f a c t o r y and f o r the b a l a n c e o f t h i s p a p e r t h e combined p r e c i p i t a t i o n s e r i e s i s employed. I n t h e p r e v i o u s models t h e p r e c i p i t a t i o n series are i n p u t as a s i n g l e series having a s i n g l e t r a n s f e r function parameter.

In these

cases i t i s t h e r e f o r e assumed t h a t t h e c o n t r i b u t i o n o f p r e c i p i t a t i o n

i s t h e same t h r o u g h o u t the y e a r .

P h y s i c a l l y , however, i t makes s e n s e

t h a t t h e c o n t r i b u t i o n o f p r e c i p i t a t i o n d u r i n g t h e w i n t e r months would

125 be less t h a n t h e c o n t r i b u t i o n d u r i n g t h e warmer p e r i o d s o f t h e y e a r s i n c e t h e p r e c i p i t a t i o n a c c u m u l a t e s on t h e ground i n t h e form o f snow d u r i n g t h e c o l d s e a s o n .

I n an e f f o r t t o b e t t e r r e f l e c t r e a l i t y ,

t h e s i n g l e p r e c i p i t a t i o n s e r i e s formed by combining t h e Lucknow and P a i s l e y d a t a , i s d i v i d e d i n t o two s e p a r a t e s e a s o n s . I n d i v i d i n g t h e p r e c i p i t a t i o n i n t o two s e a s o n s , t h e w i n t e r s e a s o n

i s t a k e n a s t h o s e months where t h e mean monthly t e m p e r a t u r e i s below zero degrees c e l c i u s .

For b o t h t h e P a i s l e y and t h e Lucknow tempera-

t u r e s e r i e s , December, J a n u a r y t e m p e r a t u r e s below f r e e z i n g .

F e b r u a r y and March have mean monthly Therefore, the winter p r e c i p i t a t i o n

s e r i e s c o n s i s t s of t h e deseasonalized p r e c i p i t a t i o n s f o r t h e s e f o u r months and z e r o s f o r the o t h e r e i g h t months o f t h e y e a r .

Conversely,

t h e s u m m e r p r e c i p i t a t i o n s e r i e s h a s z e r o s f o r t h e f o u r w i n t e r months and t h e d e s e a s o n a l i z e d p r e c i p i t a t i o n s f o r t h e r e m a i n i n g e n t r i e s . These two s e r i e s a r e i n p u t a s s e p a r a t e c o v a r i a t e s e r i e s w i t h s e p a r a t e t r a n s f e r function parameters.

The r e s u l t i n g t r a n s f e r f u n c t i o n - n o i s e

model i s yt = 0 . 4 4 4 ~ tl

+

0.183~

t2

+

a

(1

t

(18)

- 0.407B)

i s t h e d e s e a s o n a l i z e d l o g a r i t h m i c flow a t t i m e t ; x is t tl t h e combined d e s e a s o n a l i z e d summer p r e c i p i t a t i o n s e r i e s ; and x

where y

t2

i s t h e combined d e s e a s o n a l i z e d w i n t e r p r e c i p i t a t i o n s e r i e s .

As

e x p e c t e d , t h e t r a n s f e r f u n c t i o n c o e f f i c i e n t f o r t h e summer p r e c i p i t a t i o n is l a r g e r than t h e t r a n s f e r f u n c t i o n parameter f o r t h e w i n t e r precipitation.

I t i s a l s o reassuring t o note t h a t t h e b e t t e r represen-

t a t i o n of t h e p h y s i c s o f t h e system a l s o l e a d s t o an improved s t a t i s t i c a l f i t a s i n d i c a t e d by a lower A I C v a l u e . A second t y p e o f dynamic model aimed a t m o d e l l i n g t h e s p r i n g

r u n o f f r e s u l t i n g from snowmelt i s a l s o c o n s i d e r e d .

I n t h i s model,

t h e summer p r e c i p i t a t i o n i s t h e same a s t h e model i n ( 1 8 ) .

However,

t h e snow accumulated d u r i n g t h e w i n t e r months from December t o March

i s r e p r e s e n t e d a s a s i n g l e p u l s e i n p u t i n A p r i l where t h e t e m p e r a t u r e i s above z e r o f o r t h e f i r s t t i m e and hence s p r i n g r u n o f f o c c u r s . the other

e l e v e n months o f t h e y e a r t h i s s e r i e s h a s z e r o e n t r i e s .

For

126 T h i s t y p e o f dynamic model h a s been shown t o work w e l l f o r r i v e r systems l o c a t e d i n a r e a s t h a t e x p e r i e n c e A r c t i c c l i m a t e (Baracos e t a l . , 1981) and t h e r e a r e r a r e l y any thaws d u r i n g t h e w i n t e r months. However, t h e c l i m a t i c c o n d i t i o n s i n t h e Saugeen R i v e r b a s i n d u r i n g t h e w i n t e r a r e n o t e x t r e m e l y c o l d and s e v e r a l midwinter melts r e s u l t i n a s i g n i f i c a n t r e d u c t i o n i n t h e accumulated snow c o v e r on t h e ground.

For t h i s r e a s o n , t h e t r a n s f e r f u n c t i o n p a r a m e t e r f o r t h e

accumulated w i n t e r p r e c i p i t a t i o n 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 zero. 3.4.2

Temperature S e r i e s a s I n p u t s

Although t h e c r o s s - c o r r e l a t i o n a n a l y s e s i n d i c a t e no s i g n i f i c a n t r e l a t i o n s h i p s between t e m p e r a t u r e and r i v e r f l o w , t h e two t e m p e r a t u r e s e r i e s a r e used a s i n p u t c o v a r i a t e s e r i e s i n t r a n s f e r f u n c t i o n - n o i s e models.

A s b e f o r e , b o t h s e r i e s a r e f i r s t d e s e a s o n a l i z e d by sub-

t r a c t i n g o u t t h e e s t i m a t e d monthly means and d i v i d i n g by t h e e s t i m a t e d monthly s t a n d a r d d e v i a t i o n s f o r each o b s e r v a t i o n .

These s e r i e s a r e

then input independently a s covariate series i n t r a n s f e r functionn o i s e models.

The r e s u l t i n g models and t h e i r a s s o c i a t e d A I C v a l u e s

a r e shown i n Table 3 .

Because 1 . 9 6 t i m e s t h e s t a n d a r d e r r o r f o r each

parameter i s l a r g e r than t h e parameter e s t i m a t e , n e i t h e r t r a n s f e r function parameter i s s i g n i f i c a n t a t t h e f i v e p e r c e n t s i g n i f i c a n c e level.

R e c a l l t h a t t h e CCF f o r each t e m p e r a t u r e s e r i e s and t h e

Saugeen River f l o w s , a l s o s u g g e s t s t h a t t h e r e may n o t be a marked r e l a t i o n s h i p between t h e t e m p e r a t u r e s and r i v e r f l o w s .

However, t h e

Lucknow t e m p e r a t u r e p a r a m e t e r 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 z e r o a t the ten percent significance level.

A s a r e s u l t , t h e Lucknow

t e m p e r a t u r e s e r i e s i s i n c l u d e d i n t h e t r a n s f e r f u n c t i o n - n o i s e models where b o t h t h e t e m p e r a t u r e and p r e c i - p i t a t i o n s e r i e s a r e i n c l u d e d . 3.4.3

P r e c i p i t a t i o n and Temperature S e r i e s a s I n p u t s

I n an e f f o r t t o combine a l l t h e a v a i l a b l e i n f o r m a t i o n , b o t h t h e t e m p e r a t u r e and t h e p r e c i p i t a t i o n d a t a a r e used a s i n p u t c o v a r i a t e s e r i e s t o 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 t h e f i r s t model o f

t h i s t y p e i n T a b l e 3 t h e combined p r e c i p i t a t i o n i s d e s e a s o n a l i z e d and

127 s p l i t i n t o two s e a s o n s as i s done i n ( 1 8 ) .

The d e s e a s o n a l i z e d Lucknow

t e m p e r a t u r e d a t a i s used a s a n o t h e r i n p u t c o v a r i a t e series.

The

r e s u l t i n g model i s g i v e n by a

yt = 0 . 4 5 3 ~ tl

+

0.194~ t 2

+

0.144~ t3

+ (1 -

t 0.422B)

(19)

i s t h e d e s e a s o n a l i z e d l o g a r i t h m i c flow a t t i m e t ; x is t h e t tl d e s e a s o n a l i z e d summer p r e c i p i t a t i o n ; xt2 i s t h e d e s e a s o n a l i z e d w i n t e r

where y

p r e c i p i t a t i o n ; xt3 i s t h e d e s e a s o n a l i z e d Lucknow t e m p e r a t u r e d a t a ; and

is t h e white n o i s e t e r m . The model and i t s a s s o c i a t e d A I C a r e a l s o t shown i n T a b l e 3 . T h i s model p r o v i d e s a s i g n i f i c a n t improvementover any

a

o f t h e models p r e v i o u s l y employed w i t h a d e c r e a s e o f a l m o s t f i v e i n t h e A I C when compared t o t h e model i n ( 1 8 ) .

Also, t h e t r a n s f e r f u n c t i o n

parameter f o r t h e temperature series i s s i g n i f i c a n t l y d i f f e r e n t fromzero i n t h i s case.

R e c a l l from b e f o r e t h a t t h e t r a n s f e r f u n c t i o n p a r a m e t e r

f o r e i t h e r t e m p e r a t u r e series 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 z e r o a t the f i v e percent significance level.

However, when t h e p r e c i p i t a t i o n

series i s i n c l u d e d i n t h e model t h e t e m p e r a t u r e s e r i e s p r o v i d e s a s i g nificant contribution.

T h i s p o i n t i l l u s t r a t e s t h e need f o r more r e s e a r c h

i n i d e n t i f y i n g t h e dynamic component of t r a n s f e r f u n c t i o n - n o i s e models when more t h a n one i n p u t c o v a r i a t e s e r i e s i s a v a i l a b l e . The l a s t model f i t t e d t o t h e d a t a employs t h e combined p r e c i p i t a t i o n and t h e combined t e m p e r a t u r e d a t a .

The t e m p e r a t u r e s e r i e s a r e

combined i n t h e same f a s h i o n a s t h e p r e c i p i t a t i o n s e r i e s b u t i s n o t d i v i d e d i n t o two s e p a r a t e s e a s o n s . a s s o c i a t e d A I C a r e shown i n T a b l e 3 .

The r e s u l t i n g model and i t s Note t h a t t h e AIC v a l u e i s o n l y

m a r g i n a l l y l a r g e r t h a n t h a t of t h e p r e v i o u s model where t h e Lucknow t e m p e r a t u r e d a t a i s employed i n s t e a d o f t h e combined t e m p e r a t u r e series.

I n t h i s c a s e e i t h e r o f t h e s e l a s t two models c o u l d b e employed

a s t h e most a p p r o p r i a t e model f o r t h e a v a i l a b l e d a t a . CONCLUSIONS

As e x e m p l i f i e d by t h e a p p l i c a t i o n i n t h i s p a p e r , b o t h p h y s i c a l j u s t i f i c a t i o n s and f l e x i b l e s t a t i s t i c a l methods can be employed t o d e s i g n a s u i t a b l e t r a n s f e r f u n c t i o n - n o i s e model t o f i t t o h y d r o l o g i c a l

t i m e series.

When t h e c o v a r i a t e s e r i e s p o s s e s s m i s s i n g d a t a , t h e s e

128 c a n b e e s t i m a t e d by u s i n g a s e p a r a t e i n t e r v e n t i o n model f o r e a c h

series.

The c o v a r i a t e s e r i e s c a n t h e n b e l i n k e d w i t h t h e o u t p u t r i v e r

f l o w s e r i e s i n a n o v e r a l l t r a n s f e r f u n c t i o n - n o i s e model.

I f there

are m o r e t h a n one p r e c i p i t a t i o n o r t e m p e r a t u r e series, a p r o c e d u r e i s a v a i l a b l e f o r o b t a i n i n g a s i n g l e p r e c i p i t a t i o n or temperature series. F o r t h e case w h e r e snow a c c u m u l a t e s d u r i n g t h e w i n t e r t i m e , t h e p r e c i p i t a t i o n s e r i e s c a n be i n c o r p o r a t e d i n t o t h e dynamic model i n s p e c i f i e d manners s o t h a t t h e model makes s e n s e from a p h y s i c a l p o i n t o f view.

F o r t h e case o f t h e b e s t S a u g e e n R i v e r dynamic m o d e l , t h e

p r e c i p i t a t i o n s e r i e s w a s s p l i t i n t o a w i n t e r a n d summer s e r i e s , a n d

a separate t r a n s f e r f u n c t i o n w a s designed f o r each o f t h e series. The r e s i d u a l CCF i s u s e f u l f o r s t a t i s t i c a l l y i d e n t i f y i n g a t r a n s f e r f u n c t i o n t o l i n k a c o v a r i a t e series w i t h t h e o u t p u t w h i l e s t a n d a r d i d e n t i f i c a t i o n p r o c e d u r e s c a n be u s e d t o d e s i g n t h e f o r m o f t h e n o i s e

t e r m i n ( 1 ) . I f t h e o u t p u t s e r i e s h a s b e e n a f f e c t e d by n a t u r a l or man i n d u c e d i n t e r v e n t i o n s , a n i n t e r v e n t i o n component c o u l d be b u i l t i n t o t h e model a s shown i n ( 2 ) .

L i k e w i s e , i n t e r v e n t i a n components

c a n b e b u i l t i n t o t h e dynamic model c o n t a i n i n g t h e c o v a r i a t e s e r i e s

to estimate missing d a t a i n t h e o u t p u t series.

An a u t o m a t i c s e l e c t i o n

p r o c e d u r e s u c h as t h e A I C p l u s d i a g n o s t i c c h e c k s c a n b e employed f o r choosing t h e b e s t o v e r a l l t r a n s f e r function-noise

model w h i c h c a n

t h e n b e used f o r p r a c t i c a l a p p l i c a t i o n s . REFERENCES A k a i k e , H . , 1 9 7 4 . A new l o o k a t t h e 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 o n A u t o m a t i c C o n t r o l , 1 9 : 716-723. B a r a c o s , P.C., H i p e l , K . W . a n d McLeod, A . I . , 1 9 8 1 . M o d e l l i n g h y d r o l o g i c t i m e s e r i z s f r o m t h e A r c t i c . Water R e s o u r c e s B u l l e t i n , 1 7 ( 3 ) . Box, G.E.P. a n d Cox, D . R . , 1 9 6 4 . An 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 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 , S e r i e s B , 2 6 : 211-252. Box, G . E . P . a n d J e n k i n s , G . M . , 1 9 7 0 . T i m e S e r i e s A n a l y s i s : Forec a s t i n g and C o n t r o l . Holden-Day, S a n F r a n c i s c o . Box, G.E.P. a n d T i a o , G . C . , 1 9 7 5 . Intervention analysis with a p p l i c a t i o n s t o economic a n d e n v i r o n m e n t a l p r o b l e m s . J o u r n a l o f t h e American S t a t i s t i c a l A s s o c i a t i o n , 70 ( 3 4 9 ) : 70-79. D'Astous, F . and H i p e l , K.W., 1979. Analyzing environmental t i m e s e r i e s . J o u r n a l o f t h e Environmental E n g i n e e r i n g D i v i s i o n , American S o c i e t y o f C i v i l E n g i n e e r s , 1 0 5 ( E E S ) : 979-992.

129 Environment Canada, 1980a. H i s t o r i c a l S t r e m a f l o w Summary, O n t a r i o t o 1979. I n l a n d Waters D i r e c t o r a t e , Water R e s o u r c e s B r a n c h , Water Survey o f Canada, O t t a w a , O n t a r i o , Canada. Environment Canada, 1980b. Monthly M e t e o r o l o g i c a l Summary , O n t a r i o t o 1979. M e t e o r o l o g i c a l B r a n c h , Environment Canada, O t t a w a , O n t a r i o , Canada. 1 9 6 9 . I n v e s t i g a t i n g c a u s a l r e l a t i o n s by e c o n o m e t r i c Granger, C.W.J., models and c r o s s - s p e c t r a l methods. E c o n o m e t r i c a , 3 7 ( 3 ) : 424-438. Haugh, L . D . , 1972. The I d e n t i f i c a t i o n of T i m e S e r i e s I n t e r r e l a t i o n Ph.D. T h e s i s , s h i p s w i t h S p e c i a l R e f e r e n c e t o Dynamic R e g r e s s i o n . Department o f S t a t i s t i c s , U n i v e r s i t y of W i s c o n s i n , Madison, W i s c o n s i n , U.S.A. Haugh. L.D. , 1 9 7 6 . Checking t h e i n d e p e n d e n c e o f t w o c o v a r i a n c e s t a t i o n a r y t i m e series: a u n i v a r i a t e r e s i d u a l c r o s s - c o r r e l a t i o n a p p r o a c h . J o u r n a l of t h e American S t a t i s t i c a l A s s o c i a t i o n , 7 1 ( 3 5 4 ) : 378-385. Haugh, L . D . a n d Box, G.E.P., 1 9 7 7 . I d e n t i f i c a t i o n of dynamic r e g r e s s i o n ( d i s t r i b u t e d l a g ) models c o n n e c t i n g t w o t i m e s e r i e s . J o u r n a l o f t h e American S t a t i s t i c a l A s s o c i a t i o n , 7 2 ( 3 5 7 ) : 121-130 H i p e l , K . W . , Lennox, W.C., Unny, T . E . and McLeod, A . I . , 1 9 7 5 . I n t e r v e n t i o n a n a l y s i s i n water r e s o u r c e s . Water R e s o u r c e s R e s e a r c h , 11( 6 ) : 855-861. H i p e l , K . W . , L i , W . K . a n d McLeod, A . I . , 1 9 8 1 . C a u s a l a n d Dynamic R e l a t i o n s h i p s b e t w e e n N a t u r a l Phenomena. T e c h n i c a l Report N o . TR-81-05, D e p t . o f S t a t i s t i c a l and A c t u a r i a l S c i e n c e s , 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. H i p e l , K.W. a n d McLeod, A . I . , 1 9 8 2 . T i m e S e r i e s M o d e l l i n g f o r Water Resources and Environmental Engineers. E l s e v i e r , Amsterdam, i n press. H i p e l , K . W . , McLeod, A . I . a n d Lennox, W . C . , 1977a. Advances i n BoxJ e n k i n s m o d e l l i n g , 1, model c o n s t r u c t i o n . Water R e s o u r c e s R e s e a r c h , 1 3 ( 3 ) : 567-575. H i p e l , K . W . , McLeod, A . I . a n d McBean, E . A . , 1977b. S t o c h a s t i c m o d e l l i n g of t h e e f f e c t s o f r e s e r v o i r o p e r a t i o n . J o u r n a l of Hydrology, 32: 97-113. McLeod, A . I . , 1977. Improved Box-Jenkins e s t i m a t o r s . B i o m e t r i k a , 64 ( 3 ) : 531-534. McLeod, A . I . , 1 9 7 9 . D i s t r i b u t i o n o f t h e r e s i d u a l c r o s s - c o r r e l a t i o n i n u n i v a r i a t e ARMA t i m e s e r i e s m o d e l s . J o u r n a l o f t h e American S t a t i s t i c a l A s s o c i a t i o n , 74 ( 3 6 8 ) : 849-855. McLeod, A . I . , H i p e l , K.W. and Lennox, W . C . , 1 9 7 7 . Advances i n BoxJenkins modelling, 2 , a p p l i c a t i o n s . Water R e s o u r c e s R e s e a r c h , 1 3 ( 3 ) : 577-586. P e i r c e , D . A . and Haugh, L . D . , 1 9 7 7 . C a u s a l i t y i n t e m p o r a l s y s t e m s . J o u r n a l o f E c o n o m e t r i c s , 5 : 265-293. L e w i s , G . L . a n d Harbaugh, T . E . , Viessman, W . , J r . , Knapp, J . W . , 1 9 7 7 . I n t r o d u c t i o n t o Hydrology, 2nd E d i t i o n , Harper a n d Row, N e w York.

130

SOME ASPECTS OF NON-STATIONARY BEHAVIOUR I N HYDROLOGY

N.T. KOTTEGODA Department o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f Birmingham, Birmingham, B15 2TT, England, U.K.

ABSTRACT Non-stationary behaviour i n hydrology i s i n v e s t i g a t e d by paying p a r t i c u l a r a t t e n t i o n t o modelling.

A p p l i c a t i o n i s made t o 12 s e r i e s

o f a n n u a l r a i n f a l l and r i v e r f l o w s f r o m N o r t h e r n U t a h and t o t h e N i l e f l o w s a t Aswan Dam.

D i f f e r e n t a s p e c t s o f non-randomness a r e

i n v e s t i g a t e d b y f i t t i n g 1 i n e a r a u t o r e g r e s s i v e t y p e o f models and e x a m i n i n g t h e r e s i d u a l s b y means o f p a r a m e t r i c and n o n - p a r a m e t r i c tests.

The a l t e r n a t i v e a p p r o a c h a d o p t e d h e r e i n i s t o i n v e s t i g a t e

e v o l u t i o n a r y changes t h r o u g h e s t i m a t e d s p e c t r a l f u n c t i o n s o f o v e r l a p p i n g sequences.

Non-stationarities are then quantified by r e l a t i v e

mean v a l u e s o f c h i - s q u a r e d i n p a r t i c u l a r f r e q u e n c y bands. GENERAL APPROACH TO NON-STATIONARITY I n h y d r o l o g i c t i m e s e r i e s n o n - s t a t i o n a r y b e h a v i o u r may be seen t o o c c u r i n a number o f d i f f e r e n t f o r m s .

Random s h i f t s i n t h e mean

c o n s t i t u t e a b a s i c example o f d e p a r t u r e s f r o m s t a t i o n a r i t y .

Compara-

t i v e l y d e t e r m i n i s t i c i n t y p e a r e c y c l i c a l changes such as t h e a n n u a l seasons and t h e d i u r n a l p e r i o d i c i t i e s , b r o u g h t a b o u t b y e x t e r n a l stimuli.

More c o m p l i c a t e d s e r i e s have n o n - s t a t i o n a r i t y i n t h e mean

and v a r i a n c e and p o s s i b l y i n t h e h i g h e r moments, caused b y c l i m a t i c and e n v i r o n m e n t a l changes.

I n g e n e r a l , we can v i e w n o n - s t a t i o n a r y

s e r i e s as ones t h a t a r e b e i n g d i s t u r b e d s t o c h a s t i c a l l y b y t r a n s i e n t e f f e c t s o f one o r more k i n d s w i t h d i f f e r e n t c a u s a t i v e f a c t o r s . I n t h e a n a l y s i s o f t i m e s e r i e s , s h i f t s i n t h e mean may be i n v e s t i g a t e d by d i f f e r e n c i n g o f t h e f i r s t order. d i f f e r e n c i n g can be u s e d t o s e p a r a t e t r e n d s .

Second o r d e r

A l t e r n a t i v e l y , we may

t a k e moving a v e r a g e s o r s i m p l e (%,%) w e i g h t s o f a d j a c e n t v a l u e s . Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

0 1982

131

These techniques b e l o n g t o t h e c a t e g o r y o f methods c a l l e d l o c a l filtering.

F i t t i n g o f a polynomial o r t h e use o f a general g l o b a l

model c o n s t i t u t e a n o t h e r approach.

An o r i g i q a l s t u d y o f t r e n d s and

jumps o f v a r i o u s k i n d s a p p l i c a b l e t o h y d r o l o g y was made by Y e v j e v i c h and Jeng (1969).

I f t h e r e i s s t r o n g p h y s i c a l r e a s o n i n g f o r a change

i n t h e mean and i f t h e p o i n t i n t i m e when t h i s change occurs can be a p p r o x i m a t e l y i d e n t i f i e d , t h e n a n o n - s t a t i o n a r y s e r i e s may be modelled by i n t e r v e n t i o n a n a l y s i s (Box and Tiao, 1965). O f g r e a t e r concern and more d i f f i c u l t t o assess p r a c t i c a l l y a r e e v o l u t i o n a r y changes i n t h e s t r u c t u r e o f a t i m e s e r i e s .

For i n s t a n c e ,

slow movements may be induced by t h e g e n e r a t i n g process, which can be s i m u l a t e d by changing t h e parameters o f a model o v e r d i f f e r e n t t i m e spans.

These m o d i f i c a t i o n s a r e caused p r i m a r i l y b y gradual movements

i n c l i m a t e such as i n c r e a s i n g o r d e c r e a s i n g r a i n f a l l s o v e r f i n i t e time horizons.

U r b a n i z a t i o n , v a r i a t i o n s i n catchment c h a r a c t e r i s t i c s ,

which may be n a t u r a l o r man-made, gradual s i l t i n g o f r e s e r v o i r s and changes i n e v a p o r a t i o n can a l s o have s i g n i f i c a n t e f f e c t s . P r i o r t o i n v e s t i g a t i o n , e r r o r s o f measurement and i n c o n s i s t e n c i e s i n t h e d a t a should be e l i m i n a t e d , i f t h e y e x i s t .

Then i f a l o n g

s e r i e s i s a v a i l a b l e , a s i m p l i s t i c approach i s t o d i v i d e t h e s e r i e s i n t o n o n - o v e r l a p p i n g segments, compute s u i t a b l e s t a t i s t i c s o r e m p i r i c a l f u n c t i o n s such as s e r i a l c o r r e l o g r a m s from each segment and t e s t these f o r significant differences.

There i s however s u b j e c t i v i t y i n t h e

c h o i c e o f sub-samples and hence some b i a s i s o f t e n i n t r o d u c e d . OBJECTIVES AND SCOPE OF STUDY T h i s s t u d y aims t o f i n d o b j e c t i v e procedures f o r i n v e s t i g a t i n g nons t a t i o n a r y behaviour.

Because n e a r l y a l l h y d r o l o g i c s e r i e s possess

some degree o f non-randomness we commence by i n i t i a l l y f i t t i n g l i n e a r s t o c h a s t i c models (Box and Jenkins, 1976). analysed by d i f f e r e n t methods.

Then t h e r e s i d u a l s a r e

These i n c l u d e i n a d d i t i o n t o t h e Box-

P i e r c e portmanteau l a c k - o f - f i t t e s t , t h r e e non-parametric t e s t s , t h e Von Neumann r a t i o t e s t f o r normal d a t a and r e s c a l e d range.

More

i m p o r t a n t l y , a l i k e l i h o o d r a t i o t e s t i s used t o t e s t f o r a s h i f t i n

F

Table 1

0

T e s t s on Independent R e s i d u a l s

to

T

I

I

T e s t S t a t i s t i c f o r Coillparisoo Standard Normal V a r i a t e s ParaNon-parametric

[

-with

Station

INo.

iI

'Type' Portmanteau S t a t i s t i c ears

1

~

-

Jordan R i v e r , L e h i R 1914 t o 1976 2 Bear R i v e r , C o l l i n g s t o n R 1890 t o 1977 I 3 Weber R i v e r , P l a i n City i R 1906 t o 1977 4 B l a c k s m i t h Fork, Hyrum R 1914 t o 1977 5 Logan R i v e r , Logan R 1901 t o 1976 6 Weber R i v e r , Oakley R 1905 t o 1977 7 Farmi ngton ' P 1890 t o 1977 8 Kelton I P 1897 t o 1929 9 Eel P P 1897 t o 1929 0 Corrine P 1871 t o 1972 1 Ogden P 1871 t o 1977 2 Tree Ring Index T Rex Peak, Bear Lake Drainage, 1698 t o 1977

'\I

Ratio

AR(1)

4.83

7.69

-0.89

-0.89

-0.62

88

AR(2) 15.03

24.20

-0.43

-0.43

-0.40

72

AR(1) 10.56

11.25

-0.12

-0.83

-0.83

-1.32

64

AR(1)

4.93

8.26

0.13

1.14

1.14

-1.22

76

AR(1)

9.35

10.69

1.28

1.65

1.65

-1.46

73

AR(1)

7.05

10.15

-0.24

0.47

1.40

-1.07

88

AR(1)

9.83

16.26

0.75

1.40

-0.57

-1.73

53

AR(1)

5.84

10.06

-0.57

-0.57

0.45

-1.04

81

AR(1)

2.92

3.24

0.45

0.10

-2.17

7.74

102

'

precipitation

AR(1)

6.34

0.10

0.99

-1.86

107

AR(3)

7.83

9.65

0.99

1.25

-0.82

280

4R(1), 9.10

12.54

1.25

0.47

R : Flow Recording

T

rN

Likeli-

range

Ratio Test

1

64

-

P

'15

'l?ldlolfowitz

**

1

1

* **

QIO

Run5 Above nannr Below 'Yhitne) DAve,.age

Other P a r a m e t r i c T e s t Statistics

13.35

-1.07

__

: Tree R i n g I n d e x

95 p e r c e n t v a l u e s o f c h i - s q u a r e d a r e 16 f o r , degrees o f freedom,

v =

10; 25 f o r v

=

15; 31.4 f o r v

=

30 and 67.5 f o r v

=

50.

2.55

1

133

the mean without 2 p r i o r i knowledge of the point in time a t which i t occurs. I n the second phase of the study evolutionary spectral d e n s i t i e s a r e estimated from overlapping sections and mean chi-squared values are compared over c e r t a i n frequency bands. This i s applied t o h i s t o r i c a l d a t a , and repeated using synthetic data i n i t i a l l y with model parameters estimated from a f u l l record and then from d i f f e r e n t sections of i t . APPLICATION TO NORTHERN U T A H . Application i s made t o s e r i e s of annual riverflows and p r e c i p i t a t i o n in Northern Utah some of which extenh in time over 100 years.

A tree r i n g index s e r i e s from the same region i s a l s o investigated. O p t i m u m use of scarce water resources i n t h i s area has necessitated the use of r e a l i t i c and adequate models f o r time s e r i e s , so t h a t proper assessment can be made o f f u t u r e supplies. The hydrological s e r i e s studied a r e l i s t e d in Table 1 . Comparison i s made with a 1 7 2 year segment o f the Nile flows a t Aswan. RESIDUALS FROM L I N E A R STOCHASTIC MODELS

Figure 1 shows t r a c e s of the s i x time s e r i e s of r i v e r flows and the p r e c i p i t a t i o n s e r i e s are shown in Figure 2 with the t r e e ring

I n general, no s i g n i f i c a n t trends o r p e r i o d i c i t i e s are evident. There are varying degrees of s e r i a l c o r r e l a t i o n in nearly a l l the s e r i e s . Figure 3 shows the s e r i a l correlograms, r k , and estimated index.

p a r t i a l autocorrelograms ,

f i v e typical s e r i e s . The s e r i a l correlation r k f o r lag k i s computed from observations ( x , ' x 2 , . . . ' x N ) with mean u a n d estimated

'k,k'

x, as

Of

follows.

(1)

I n i t i a l l y on the assumption of s t a t i o n a r i t y , appropriate Box-Jenkins models are f i t t e d from the general ARMA ( p , q ) type given by

134

Bear River at Collingston, 6267ml

2 000 000 1 000000

0 1940 Logan River near Logan, 21 8 mI2

1950

200 0 1900

1930

Blacksmith Fork River near Hyrum, 130 m l P

1960

1920

I I

1950

Weber River at Plain City, 2060 m12 1910

Jordon River at Lehi , 3000 mi2

1940

1970

1940

1970

400 000 200 000

0 3000007

1

Weber River near Oakley, 163 m l

L

J

'

1910

Fig.1. Six River Flow Series From Northern Utah.

135

Tree Ring Index at Rex Peak for Bear Lake Drainage

10

0

00

1900

1950

Delp Prec ip i t a t ion

10

Kelt on Prec i p i t a t i on

O 1900

1920

1940

20 Cor I nne Prec i p i t at i on

10 0

1870

1900

1930

Farmington Precipitation

Ogden Prec I p i t a t ion

Fig 2 .

Annual Tree Ring Index and 5 From Northern Utah.

Precipitation

Series

1960

136

Bear River at Col lings ton, 88 years 1890 - 1977

r

"bk

O.7

t

'k,k

0.2

00

10

20

Logan River near Logan, 76year s 1901-1976

0.4 Corrinne Precipitat ion, 107 years 1871-1977

@den Precipitat ion 107 years 1871 -1977

,-

0.4

r

~

Tree Ring rk Index Rex Feok 280 years 1698 -1977

"I\ 0.2

- 0.2 1

Fig. 3. Serial Correlograms and P a r t i a l Autocorrelograms of 5 Annual Hydrological Series From Northern Utah.

137

P Xt

=

u

+

9

- u)

6i ( X t - i

2

+

Zt

i=l

s

+

e i Zt-i

i =1

i n which t h e $i and e i a r e parameters and Z t i s an independent normal v a r i a t e with z e r o mean. The f i t t e d models a r e of the a u t o r e g r e s s i v e t y p e : models AR(1), AR(2) and AR(3). S p e c t r a l d e n s i t i e s a r e c a l c u l a t e d , f o r a truncation point M, a s follows M- 1 S(o)

=

{ro + 2

rkCOS(wk/M)

(3)

rMCOS(w))/n

k=l

where w = frequency i n r a d i a n s per u n i t time ( y e a r ) , and ;(w) i s c a l c u l a t e d a t hi = TIM, 2n/M, ...,n . The a r e smoothed u s i n g t h e Tukey window which has d e g r e e s of freedom given by v = 2.67 N / M , i f N

sk

o b s e r v a t i o n s a r e used f o r e s t i m a t i o n . Figure 4 shows on the l e f t e s t i m a t e d s p e c t r a l d e n s i t i e s f o r 5 o f the s e r i e s using two v a l u e s of M i n each c a s e . These a r e p l o t t e d on semi-logarithmic paper. Also shown a r e the t h e o r e t i c a l s p e c t r a f o r t h e a p p r o p r i a t e model from e q u a t i o n ( 2 ) a s c a l c u l a t e d from 1 + el e - j W + e 2 e - 2 j ~

+

r(w)

=

o2 2

1 -

1

e-ju

-

$ 2 e - j ~-

... + e

e-qju -pjo

2

(4)

138

El17

5x117

\ *

0.7

?y--

13TU17

TI;

n/17

5TCl17

TL/20

n;/4

9x117

13TI;/17

TI;

Bear R. Collingstonx M=17 M=12

3 f j X

1.

9TW17

x

k.

-

X

2 .

Corinne Precipitation

-x

M=20

3n/4

Tc/2

Tc

Ogden Precipitationx M=2l

TI; 01

K/3

.

2TV3

Tc

Tree Ring Index __

.;--"I 0-5

0.3 Tc/50

,

$9 TC/5

2lT/5

JT1;/5

Raw Data

4TC/5

,

,

,

,

2TI;/5

3TC/5

4TV5

TI;

0. TL

TI;/50

Tc/5

Residuals From AR Model

Fig. 4 . Estimated Spectra of 5 Annual Hydrological Series From Northern Utah. Curves Show Theoretical Spectra of AR Models. Also Estimated Spectra of Independent Residuals are Shown on the Right

139

t h a t some bias i s introduced because the estimated r e s i d u a l s are investigated and n o t the t r u e r e s i d u a l s . NON-PARAMETRIC TESTS Runs Test I n t h i s basic one sample non-parametric t e s t , runs above and below the estimated median of the s e r i e s a r e counted. The t o t a l number of runs i s standardised using the mean and standard deviation

from the normal d i s t r i b u t i o n of the number o f runs of a random sequence as given f o r instance by Conover (1971 ) and Siege1 (1956). Values are well within the range - 1 . 9 6 t o 1 . 9 6 , so they are n o t s i g n i f i c a n t a t the 5-percent level of s i g n i f i c a n c e . If they are s i g n i f i c a n t i t imp1 i e s non-randomness on account of s e r i a l c o r r e l a t i o n , periodic movement o r trend. The t e s t i s n o t a powerful one b u t i s nevertheless useful f o r prel iminary purposes. Man-Whitney and Wald-Wolfowitz Tests F o r these well-known two sample t e s t s (Conover, 1 9 7 1 ) , the f u l l sequence of residuals i s divided i n t o two samples of equal length. The Mann-Whitney t e s t i s t o determine whether there i s a s i g n i f i c a n t difference in the locations of the two samples. On the other hand significance in the Wald-Wolfowitz t e s t can mean differences in general d i s t r i b u t i o n properties such as locat i o n , dispersion and skewness. The t e s t s t a t i s t i c s a r e , a s i n the runs t e s t , standardised by using the respective means and standard deviations of the l a r g e sample normal d i s t r i b u t i o n s of the s t a t i s t i c s expected i n a random population. Again the r e s u l t s do n o t show any significance. All of the aforementioned t e s t s a r e d i s t r i bution-free. Kolmogorov-Smirnov Goodness o f F i t Test The d i s t r i b u t i o n s of the residuals are then t e s t e d using the Kolmogorov-Smirnov Goodness of F i t Test. As shown in Table 1 the s t a t i s t i c s are much l e s s t h a n the c r i t i c a l value of 1.36 which i s the 95 percent value f o r large samples. On account of the normality of the r e s i d u a l s , i t i s possible t o carry

o u t three additional t e s t s of importance as follows.

140 PARAMETRIC TESTS Von-Neumann R a t i o T e s t

T h i s t e s t o f randomness can be used a g a i n s t

I t c o v e r s i n c o n s i s t e n c i e s such as jumps.

unspecified alternatives.

One needs t o c a l c u l a t e t h e r a t i o o f t h e mean s q u a r e s u c c e s s i v e d i f f e r e n c e s t o t h e v a r i a n c e o f t h e o b s e r v a t i o n s , g i v e n by

V

(N/N - 1 ) )

=

N

N-1 2 (xi -

(5)

i=l

i=l

F o r N > 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
=

i n which

x

IN

- min

(x1+x2+ ...+ x i - i i ) / s 1GiGN i s t h e e s t i m a t e d mean and sN i s t h e e s t i m a t e d s t a n d a r d

(6)

d e v i a t i o n from the observations. I n o r d e r t o a s c e r t a i n t h e s m a l l sample 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 , RN xi,

i

=

, 100,000 s e t s o f i n d e p e n d e n t s t a n d a r d n o r m a l v a r i a t e s

1,2,...,N,

**

were g e n e r a t e d , t h e rN were t h e n c a l c u l a t e d f r o m

each s e t and t h e e m p i r i c a l p e r c e n t a g e p o i n t s o b t a i n e d f o r N = 50,60,

...,110.

These a r e shown b y t h e c u r v e s on t h e l e f t o f F i g u r e 5 .

**

g i v e n a r e t h e e s t i m a t e s rN f o r each o f t h e r e s i d u a l s e r i e s .

Also

Two o f

t h e s e r i e s show s i g n i f i c a n t d e p a r t u r e s f r o m randomness. Likelihood Ratio Test

F o l l o w i n g t h e work o f W o r s l e y ( 1 9 7 9 ) a l i k e l i -

hood t e s t can be used t o t e s t f o r a s h i f t i n l o c a t i o n a t unknown t i m e i n a normal p o p u l a t i o n ; t o t e s t f o r a change i n v a r i a n c e one may a p p l y t h e B a y e s i a n method o f M e n z e f r i c k e (1981 ) and, a1 t e r n a t i v e l y ,

t h e non-

p a r a m e t r i c approach o f P e t i t t (1979) determines whether t h e r e i s a s i g n i f i c a n t change i n d i s t r i b u t i o n a t an unknown p o i n t .

The W o r s e l y

t e s t i s based on a s t a t i s t i c W w h i c h i s c a l c u l a t e d as f o l l o w s .

141

...,XN

L e t Xl,X2,

d i s t r i b u t i o n s Xi

Ho:

pi

=

I

be a sequence o f 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

N(ui,a2) i

1-1, i = 1,2

=

1,2,..

.,N.

Consider t h e hypothesis

,...,N

against the a l t e r n a t i v e hypothesis H1:

pi

=

LI,i = 1,2

= P I ,

,..., k

i = k + l , k + 2 ,..., N

I f we d e n o t e t h e mean o f t h e f i r s t

where u2, LI, P ' and k a r e unknown. k observations by

-

xk

and t h e mean o f t h e r e m a i n i n g N - K o b s e r v a t i o n s b y

x i , t h e w i t h i n g r o u p s sum o f s q u a r e s o f t h e o b s e r v a t i o n s i s

(7) i = l

i=k+l

A l s o , t h e n o r m a l i s e d b e t w e e n - g r o u p s sum o f s q u a r e s i s g i v e n b y Ti

=

- i i )2 , k

(k(N-k)/N)(xk

=

1,

..., N-1

(8)

Then t h e l i k e l i h o o d r a t i o t e s t i s based on t h e s t a t i s t i c

**

As i n t h e c a s e o f RN

, t h e e m p i r i c a l d i s t r i b u t i o n o f W was o b t a i n e d b y

s i m u l a t i o n u s i n g independent standard normal v a r i a t e s . I t i s f o u n d t h a t f o r sample s i z e s g r e a t e r t h a n N = 50 t h e f o r m o f

t h e d i s t r i b u t i o n can be a p p r o x i m a t e d b y a s i n g l e c u r v e as shown on t h e r i g h t o f F i g u r e 5.

The e s t i m a t e s o f W f o r each o f t h e r e s i d u a l s e r i e s

a r e g i v e n on t h e same f i g u r e .

About 4 of t h e s e r i e s a r e s i g n i f i c a n t

a t t h e f i v e p e r c e n t l e v e l showing s h i f t s i n l o c a t i o n .

A l s o shown on

t h e r i g h t i s t h e s t e p f u n c t i o n f o r t h e 12 d a t a s e r i e s .

I t i s seen

t h a t t h e r e i s a s i g n i f i c a n t d i f f e r e n c e i n d i s t r i b u t i o n i n terms o f t h e Kolmogorov-Smirnov s t a t i s t i c between t h i s and t h e smooth c u r v e . TESTS BASED ON THE SPECTRUM The l o n g e r s e r i e s o f d a t a were t e s t e d f o r e v o l u t i o n a r y changes b y means of t h e s p e c t r u m and o v e r l a p p i n g samples.

The t h e o r y o f t h e

s p e c t r u m i s g i v e n b y J e n k i n s and W a t t s ( 1 9 6 7 ) .

There i s e m p i r i c a l

work i n r e l a t i o n t o c l i m a t e b y W.M.O.

(1966).

P r i e s t l e y (1965)

1.00r

*-

0.9 0

0.90 -

0 80

0.80.

0 70

0.70 -

0 60

*c= 0-50 0 40 0 300 200 10 0

4

8 12 16 Rescaled Range R Y

20

24

0

1 W

2 3 Statistic

4

Fig 5 Emperical Probability D i s t r i b u t i o n of Rescaled Range a n d W S t a t i s t i c of Independent Standard Normal Variates and Estimates From Samples of Residuals of Hydrological Data, With Step Function of W Statistic for 12 Ser:es

143

originated analytical work on evolutionary spectra and non-stationary processes. Results f o r the 107 year record of precipitation a t Ogden are given i n Figure 6. Against the background of the theoretical spectrum based on equation 4 f o r an AR(3) model , the estimated spectral density functions of ten overlapping sequences are shown here. The f i r s t o f these span the i n i t i a l 53 years. Using the same sample s i z e , a second sample i s formed s t a r t i n g s i x years a f t e r the commencement of the f i r s t . Then the t h i r d i s lagged by a f u r t h e r s i x years and the procedure i s repeated t o cover the f u l l 107 year period. Figure 7 shows a repetition o f the method b u t in t h i s case i t i s applied t o a generated 107 sample o f data based on the AR(3) model applicable t o the s e r i e s . Although r e s u l t s are affected by the choice of the truncation p o i n t M for which a value of M = 1 7 i s adopted, the dispersion i n the h i s t o r i c a l spectra i n Figure 6 seems t o be greater. I n order t o evaluate the v a r i a b i l i t y of the spectra i n time, a theoretical spectrum S ( O ) based on equation 4 was f i t t e d t o the f i r s t 53 years a f t e r estimating the best model and i t s estimates. Then the spectra ;(w) estimated from each h i s t o r i c a l sequence of 53 years and shown i n Figure 6 were used t o calculate values of V ; ( W ) / S ( W ) which have the chi-squared d i s t r i b u t i o n . Results given in Table 2 show the spectral variations i n time a t d i f f e r e n t frequencies. Rapidly increasing o r decreasing values of chi-squared a t some frequencies indicate non-stationarities of some form. For purposes of comparison SOW percentage p o i n t s from tables of the chi-squared d i s t r i b u t i o n are given below Table 2 . The following procedure i s used t o quantify the non-stationary behaviour. I n i t i a l l y a frequency band was selected. For the Ogden precipitation frequencies 10 t o 1 7 seem t o be the ones having cons i s t e n t l y higher than the median value of chi-squared and showing the greatest changes i n time. I t i s seen t h a t chi-squared values tend t o increase in time; t h i s implies t h a t the Ogden precipitation has become more random in recent years, although differences may not be s i g n i f i c a n t . For each sequence these chi-squared values are averaged

144

Table 2 j

z"/w

34

1 2 3 4 5 6 7 8 9 10

17

x:,")

6.8

5.7

4.9

4.3

3.8

6.5

11.1 6.3 5.5 4.3 3.2 5.2 3.3 3.8 2.7 3.2

11.8 8.7 7.3 4.0 3.4 4.6 3.3 4.9 4.3 4.4

6.9 6.8 7.0 3.7 3.5 4.1

4.1 5.2

4.1

6.5

6.2 5.5 5.4

7.3 6.0 7.1 8.3 6.0 5.4 7.1 6.5 7.8 7.9

4.7 4.R 6.4 5.0 6.4 5.2

5.6 4.1 4.6

x?7,.oo5 5.7

=

r Table 3

5.6 7.2 6.9 7.2

x?7,.w 7.6

5.0

4.1 4.2 4.4 6.7 7.0 6.3 5.9

5.0 6.9 5.6 5.8 5.1

I 4 7 , .o5 8.7

Ogden P r e c i p i t a t i o n

- 4 3 -

"/17

/w

34

1 2 3 4

11.3

8.5

9.0 7.6 8.0 9.5 9.8 9.9 8.9 10.2 10.7 11.4 -

2.2 3.3 6.n 3.1 7.0 3.8 3.7 7.0 5.8 3.6 3.2 4.7 8.2 8.8 4.2 6.2 6.2 4.6 - -

8.7 11.1 7.9 11.7 10.0 15.5 9.9 14.1 7.8 15.7 8.2 17.0 9.0 17.7 11.1 19.5 12.9 21.8 9.9114.5

~ ? 7 , . i o 4 7 , . 50 10.1 16.3

2.8

2.6

2.4

2.3

2.1

17.4 16.8 21.4 20.9 20.4 21.2 21.8 18.2 19.4 19.0

16.1 14.1 19.5 19.5 17.4 18.0 18.0 12.7 13.5 17.8

7.3 6.0 11.0 11.8 12.2 14.6 14.7 12.0 15.6 12.3

3.3 5.7 7.1 8.8 8.0 10.7 11.3 10.2 12.2 7.8

10.8 8.4 10.6 9.6 12.3 16.2 15.4 14.3 14.2

x;7,.90 24.8

8

55 --

6.8 - - 5.6 5.5

3.1

4:9

4.3 -

3.2 5.4 6.0 6.6 6.9 7.5 6.4 6.3 6.2 3.4

6.7 8.0 9.8 11.1 11.4 11.0 8.8 10.2 8.9 6.2 -

1.2 2.8 2.0 2.6 2.7 4.0 3.7 5.9 4.1 3.1

footnote'

3.5

I

~ ? 7 , . 9 5 ~ ? 7 , . 9 7 5 G 7 , .995 27.6 30.2 35.7

Values o f vi(o)/s(w) from a 107 y e a r s y n t h e t i c sequence u s i n g an AR(3) model w i t h parameters e s t i n a t e d from h i s t o r i c a l d a t a . 7

- -

17

3.8 12.0 5.4 12.0 5.7 12.2 7.9 14.2 5 7.7 14.3 6 7.1 14.6 7 7.3 14.5 8 3.712.7 9 4.9 14.3 10N o t a7t i. o5 n 16.8

*

n/l In/l

3.4

--

-

5 -

5

n/l;

14 15 16 '4n /17 15n/17 16n,'17

-

8.5

~

w

9 10 11 12 13 9n/17 10n/17 ' l T / 1 7 12n/17 '"/17

11.3

5.9 7.3 6.5 9.0 9.3 7.5 7.2 8.2 7.4 8.2

6.5 8.7 7.0 9.2 11.0 7.4 7.6 8.7 8.9 10.5

W x 2

) H i s t o r i c a l Data Values of v i ( t ~ ) / s ( ~from

2 3 4 5 6 7 8 2n/17 3 n / i 7 4 n / i 7 5 n / i 7 6"/17 7 n / i 7 "/17

1 '/17

L

Ogden P r e c i p i t a t i o n : 107 y e a r s

--

12

38134131 6.5 8.9 6.6 6.8

4.6 5.5 4.3

13

14

'"/17

13'/17

14'/17

2.8

i.6

2.4

2.4 5.9 6.8 7.2

The v a l u e s o f K a r e c a l c u l a t e d as i n T a b l e 2 e x c e p t t h a t s y n t h e t i c d a t a a r e used. I n t h e l a s t column K i s averaged o v e r f i v e randomly s e l e c t e d sequences o f 107 y e a r s

15

16

15n/17 16'/17 2.3

2.1

' 17 r

K

9

from one

from f i v e

2

sequence*

sequences

0.37 0.34 0.30 0.22 0.22 0.22 0.26 0.22 0.30 0.31

0.34 0.31 0.32 0.29 0.32

Table 4

Values v;(u)/s(u) from a 107 y e a r s y n t h e t i c sequence u s i n g an AR(1) model f o r t h e f i r s t h a l f w i t h oarameter e s t i m a t e d f r o m h i s t o r i c a l d a t a and a random normal sequence f o r t h e

@den P r e c i p i t a t i o n

4

4rr/l

8 9 ' 1 0 ~ 1 1 i 1 2 / 1 3 6 7 6n/17 7 n / 1 7 8"/17 9 n / 1 7 l o n / 1 7 1 ' n / 1 7 1 ' 2 n / 1 7 13'/1

5 "/17

I _

8.5

5.7

6.8

4.9

4.3

3.8

3.4

3.1

2.8 I

6 7

8 9

9.6

0.9 6.8 6.5

1.4 1.2 1.6 1.5 8.3

\ 12.5

1

12.7 12.4 9.9 9.7

10.2 9.3 9.4 9.8 10.0 8.1 6.1 10.2 7.1 7.5

4.6 5.8 5.0 5.6 5.8 5.5 5.0

1

1

2.6 4.7 3.4 4.0 4.5 6.3 5.8

'::? f:; 6.9 5.3

5.0 7.8 8.9 9.6 10.4 10.6 9.6 9.5 9.7 6.2

9.6 6.5 1 1 . 8 6.1 14.8 8.2 1 7 . 0 10.0 1 7 . 3 9.6 16.3 10.3 15.1 11.1 15.1 12.3 13.5 10.0 10.3 10.0 I

Notation p r ( x Z ,c x : , ~ )

=

5.6 5.3 4.2

L

I

16

14

6n

14"/li -

/1

-

K

K

from one

from f i v e sequences

2.6

2.4

2.3

2.1

2

sequence*

6.4 5.3 5.6

10.9 8.5 8.1 7.5 6.1 6.6 5.1 5.1 9.7 __ 11.5

16.4 13.3 9.3

14.4 14.4 13.9 10.2 10.8 11.0 12.0 11.5 20.4 19.0 __

10.7 13.4 18.2 16.7 17.0 15.4 16.0 13.6 21 .0 17.3

0.33 0.31 0.30 0.74 0.24 0.24 0.29 0.27 0.42 0.43

5.1 4.8 6.4 6.2 1.2 16.2 17.7

0.38 0.39 0.39 0.47

l

1

I

147 and d i v i d e d b y 27.6 ( w h i c h i s t h e c o r r e s p o n d i n g 95% v a l u e o f c h i s q u a r e d ) and e n t e r e d a s t h e v a r i a b l e r a t e a t which

K

K

on t h e r i g h t o f T a b l e 2.

The

approaches and exceeds u n i t y i s t a k e n as an i n d e x o f

the lack o f r e l i a b i l i t y o f a hydrological record f o r modelling future events.

I n another s i t u a t i o n i n which s e r i a l c o r r e l a t i o n tends t o

i n c r e a s e w i t h t i m e , o r i n g e n e r a l when c h i - s q u a r e d v a l u e s become v e r y low,

K

o u g h t t o be o b t a i n e d b y d i v i d i n g t h e 5% v a l u e o f c h i - s q u a r e d

b y t h e e s t i m a t e d mean v a l u e o v e r t h e f r e q u e n c y band.

There i s a l s o

the p o s s i b i l i t y o f seeing both types o f behaviour over a l o n g period, i n w h i c h c a s e t h e 97.5% and 2.5% v a l u e s o f c h i - s q u a r e d s h o u l d be used. The s h o r t n e s s o f r e c o r d s i n a d d i t i o n t o t h e common d i v i s o r o f

S(W)

l e a d s t o t h e h i g h l y c o r r e l a t e d c h i - s q u a r e d v a l u e s i n t h e t a b l e and t h i s p r e c l u d e s more s t r i n g e n t s t a t i s t i c a l t e s t s .

F o r p u r p o s e s o f com-

p a r i s o n T a b l e 3 shows t h e c o r r e s p o n d i n g c k i - s q u a r e d v a l u e s o b t a i n e d

from a s y n t h e t i c sequence u s i n g an AR(3) model. t i m a t e column a r e

K

v a l u e s f o r t h i s sequence.

d i s t r i b u t e d as shown b y t h e

Given i n t h e p e n u l These a r e u n i f o r m l y

K v a l u e s a v e r a g e d o v e r 5 sequences w h i c h

a r e g i v e n i n t h e l a s t column. To c o m p l e t e t h i s a s p e c t o f t h e s t u d y , ARMA models were f i t t e d s e p a r a t e l y t o t h e f i r s t and second h a l v e s o f t h e Ogden p r e c i p i t a t i o n r e c o r d o f 107 y e a r s .

T h i s meant, t h a t t h e f i r s t h a l f was g e n e r a t e d

u s i n g an AR(1) model w i t h

i=

sequence f o r t h e second h a l f .

0.30 f o l l o w e d b y a c o m p l e t e l y random

Then t h e same p r o c e d u r e as i n T a b l e 3

was r e p e a t e d and e v o l u t i o n a r y s p e c t r a were o b t a i n e d f r o m t h e r e s u l t i n g s y n t h e t i c n o n - s t a t i o n a r y sequence. v a l u e s w i t h mean v a l u e s o f

T a b l e 4 shows t h e c h i - s q u a r e d

K i n t h e p e n u l t i m a t e column and t h e K v a l u e s

averaged o v e r 5 sequences a t t h e e x t r e m e r i g h t .

The d i f f e r e n c e s i n

t h e s e mean v a l u e s o f K f r o m T a b l e s 3 and 4 i s a measure o f t h e nonstationarity.

R e s u l t s shown i n T a b l e 4 a r e c o m p a r a b l e w i t h t h o s e f r o m

t h e h i s t o r i c a l d a t a , shown i n T a b l e 2. NILE FLOWS By way o f c o m p a r i s o n t h e method i s a p p l i e d t o a 172 y e a r r e c o r d o f

t h e N i l e a n n u a l f l o w s a t Aswan Darn f r o m 1701 t o 1872.

Some a s p e c t s o f

-~ Table

5

2 __

2n/ll

n/ll

O'

"/w

I

7n/ll

an/l 1 -

9n/ll

On / 11

3.1

2.8

2.4

2.2

12.9 19.2 19.7 41.1 40.9 25.5 20. 0 21.7 20.1 25.2 21.9 24.2 28.5 23.6 32.3 27.8 25.0 43.0 56.1 70.7

25.6 32.1 36.6 62.0 60.9 34.8 28.9 33.4 33.4 32.4

23.3 25.2 32.6 51.8 49.3 32.4 29.2 35.0 41.4 35.4 41.7 36.5 36.1 60.4 70.9 64.9 89.3 103.5 102.4 78.0

15.3 13.5 17.4 40.1 37.3 28.4 22.1 26.2 32.4 30.9 42.6 43.1 25.9 37.8 44.4 40.9 77.8

y l

__ 5.5

5.7 5.7 7.3 4.4 4.0 7.3 5.8 6.2 6.4 7.6 8.9 8.3 7.7 6.4 6.1 6.7 5.4 5.3 3.5 3.4

8.0 7.7 10.5 7.8 8.2 14.2 13.2 13.4 13.8 16.7 18.4 18.4 17.0 14.9 11.1 11.5 9.4 9.6 7.6 8.1

8.8 8.2 10.5 16.5 19.6 22.3 23.7 22.6 21.6 23.8 23.7 25.0 24.2 23.9 15.7 16.3 15.4 16.4 17.3 16.6

16.0

i -

10 __

7

7.3

15.3 17.7 34.2 40.8 29.6 28.2 26.8 23.4 20.7 21.4 20.1 20.1 20.7 20.1 22.0 21.3 20.3 25.1 22.3

__

__ __ 13.8 14.7 20.4 35.5 46.1 27.5 23.2 22.1 21.9 19.6 22.6 17.9 18.1 17.4 21.8 21.5 17.8 15.4 2q. 3 24.7

6.3 11.1 15.7 27.6 33.7 22.0 17.6 17.2 19.5 21.7 21.2 18.3 19.7 17.8 23.1 20.1 16.0 20.2 21.2 35.7

f r o m H i s t o r i c a l Data

9

4

11

V;(U)/S(U)

8 __

3 ~

22

- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Values o f

N i l e R i v e r Flows : 172 years

___ __ __

30.6 29.6 39.0 50.2 60.0 56.7 62.4 93.7 11.6 03.2

__

~

46.6 49.2 38.7

14.2 13.4 16.2 46.3 42.9 34.1 22.2 25.2 27.4 30.0 41.5 44.5 21.0 25.3 26.8 30.1 67.5 18.3 28.7 28.3

--

0.45 0.52 0.61 1.20 1.15 0.77 0.61 0.71 0.77 0.77 0.89 0.89 0.75 0.98 1.17 1.10 1.61 1.52 1.74 1.59

Notation

*

Value o f K o b t a i n e d i s t h e mean v a l u e o f v ~ ( w ) / ~ ( wfrom ) columns 7 t o 11 d i v i d e d by x $ ~ , .( =~ 4~Q . l ) .

149 t h i s r e c o r d have been s t u d i e d b y R e i h l and M e i t i n ( 1 9 7 9 ) ; i t i s t h o u g h t t h a t t h e r e a r e no e r r o r s o f measurement and t h a t nons t a t i o n a r i t i e s a r e l e s s t h a n i n t h e p e r i o d a f t e r 1872.

However, i t i s

found t h a t t h r e e n o n - o v e r l a p p i n g sub-samples f r o m t h e chosen r e c o r d have c o n t r a s t i n g f e a t u r e s w i t h some e v i d e n c e o f n o n - s t a t i o n a r i t y p a r t i c u l a r l y i n t h e p e r i o d a f t e r 1766.

I n v e s t i g a t i o n s a l s o show t h a t

f i r s t and second d i f f e r e n c i n g and w e i g h t i n g do n o t a p p e a r t o be h e l p f u l f o r m o d e l l i n g purposes. F i g u r e 8 w h i c h shows h i s t o r i c a l e v o l u t i o n a r y s p e c t r a f o r t h e N i l e f l o w s c o r r e s p o n d s t o F i g u r e 6 f o r t h e Ogden p r e c i p i t a t i o n .

The base

p e r i o d i s t a k e n a s 58 y e a r s and t h e l a g g i n g o f c o n s e c u t i v e o v e r l a p p i n g samples i s s i x y e a r s as b e f o r e g i v i n g 20 sequences.

Then, t h e p r o -

cedure i s r e p e a t e d f o r a r a n d o m l y s e l e c t e d 172 y e a r s y n t h e t i c sequence u s i n g an A R ( 5 ) model and t h e r e s u l t i n g s p e c t r a a r e shown i n F i g u r e 9 . Table 5 g i v e s t h e chi-squared values f o r t h e N i l e annual f l o w s f o l l o w i n g t h e same p r o c e d u r e as f o r T a b l e 2.

The f r e q u e n c y band

s t u d i e d i s t h e h i g h f r e q u e n c y s e c t i o n numbered 6 t o 11.

Here t h e 95%

v a l u e o f c h i - s q u a r e d i s 40.1 and v a r i a b l e K commences w i t h 0.45 f o r t h e f i r s t sequence and i s seen t o exceed 1 a f t e r a f e w l a g s o f s i x y e a r s . T h i s i n d i c a t e s a l a r g e r d e g r e e of n o n - s t a t i o n a r i t y t h a n f o r t h e Ogden precipitation.

I t a l s o means t h a t i n a p p l i c a t i o n s such a s t h i s

model1 i n g becomes a l e s s r e l i a b l e p r o c e d u r e . CONCLUSION The r e s u l t s o f t h i s s t u d y show t h a t a v a r i e t y o f t e c h n i q u e s s h o u l d be a d o p t e d when t e s t i n g t h e v a l i d i t y o f s t o c h a s t i c models i n h y d r o l o g y . T h i s i s p a r t l y on a c c o u n t o f t h e l o w power o f some s t a t i s t i c a l t e s t s . Some f o r m o f n o n - s t a t i o n a r y b e h a v i o u r i s e v i d e n t i n s e v e r a l s e r i e s as seen h e r e .

I t i s i m p o r t a n t t o gauge t h e n a t u r e , m a g n i t u d e and

extent o f the variations.

The outcomes c a n be c o n s i d e r e d t o be a

measure o f t h e r e s i l i e n c e one needs t o b u i l d i n t o p l a n n e d w a t e r resource p r o j e c t s t o account f o r f u t u r e u n c e r t a i n t i e s .

150

AC KNOWL E DGEMENTS The work commenced w h i l s t t h e a u t h o r was a V i s i t i n g P r o f e s s o r a t Utah S t a t e U n i v e r s i t y . D.S.

Thanks a r e due t o Professors L.D. James and

Bowles and t o D r . W.R.

Logan.

James o f t h e Water Research L a b o r a t o r y ,

The N i l e d a t a was r e c e i v e d f r o m D r . J. M e i t i n o f t h e U n i v e r s i t y

o f Colorado a t B o u l d e r .

REFERENCES Time S e r i e s : F o r e c a s t i n g and Box, G . E . P . and J e n k i n s , G.M. C o n t r o l . r e v i s e d edn., Holden Day, San F r a n c i s c o , 1976. Box, G.E.P. and T i a o , G.C. A change i n l e v e l o f n o n - s t a t i o n a r y t i m e s e r i e s . B i o m e t r i k a , 52, pp. 181-192, 1965. Conover, W.J. P r a c t i c a l Nonparametric S t a t i s t i c s . Wiley, New York, 1971. Jenkins, G.M. and Watts, D.G. 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 Francisco, 1968. Menzefricke, U.A. Bayesian a n a l y s i s o f a change i n t h e p r e c i s i o n o f a sequence o f independent normal random v a r i a b l e s a t an unknown t i m e p o i n t . A p p l i e d S t a t i s t i c s , v o l . 30, pp. 141-146, 1981. P e t t i t t , A.N. A non-parametric approach t o t h e change-point problem. A p p l i e d S t a t i s t i c s , v o l . 28, pp. 126-135, 1979. P r i e s t l e y , M.B. E v o l u t i o n a r y s p e c t r a and n o n - s t a t i o n a r y processes. J.R. S t a t i s t . SOC., B, 19, 1-12, 1965. R i e h l , H. and M e i t i n , J. Discharge o f t h e N i l e R i v e r . A Barometer o f S h o r t P e r i o d C l i m a t e V a r i a t i o n . Science, v o l . 206, pp. 1178-1179, 1979. S i e g e l , S. Nonparametric s t a t i s t i c s f o r t h e b e h a v i o u r a l sciences. McGraw H i 1 1, New York, 1956. Worsley, K.J. On t h e L i k e l i h o o d R a t i o T e s t f o r a S h i f t i n L o c a t i o n o f Normal P o p u l a t i o n s . J o u r . Am. S t a t . Assoc., v o l . 74, pp. 365-367, 1979. World M e t e o r o l o g i c a l O r g a n i s a t i o n . C l i m a t i c Change. World Meteor. Organ., Geneva, Tech. Note No. 79, 1966. Yevjevich, V. and Jeng, R . I . P r o p e r t i e s o f Non-Homogeneous H y d r o l o g i c S e r i e s . Hydrology Papers, no. 32, 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 , 1969.

151

PERSISTENCE ESTIMATION FROM A TIME-SERIES CONTAINING OCCASIONAL MISSING DATA DANIEL A. CLUIS A N D PIERRE BOUCHER U n i v e r s i t e du Quebec, Quebec, Canada

ARSTRACT I n a r e a l l i f e s i t u a t i o n , such as t h a t encountered w i t h weekly sampled w a t e r - q u a l i t y parameters, measured t i m e - s e r i e s a r e o f t e n i ncomplet e. The m i s s i n q data c o m p l i c a t e t h e e s t i m a t i o n of t h 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 , a most i m p o r t a n t parameter i n t h e e v a l u a t i o n of s e r i a l p e r s i s t e n c e necessary f o r e f f i c i e n t Samplinq strateqies. I n t h e case of o c c a s i o n a l m i s s i n q data, we have i n v e s t i q a t e d t h e h i ases and samplinq v a r i a n c e s r e s r i l t i n q f r o m t h e i n t r o d u c t i o n of s i mp l e h u t n o n -o p ti ma l e s t i m a t o r s (e.q. t h e "qen e r a l mean" e s t i m a t o r , t h e " l o c a l mean" e s t i m a t o r and t h e "comhi ned mean" e s t i m a t o r ) or f r o m compressinq t h e t i m e - s e r i e s by s i m p l y n e q l e c t i n q t h e m i s s i n q data. These s t u d i e s were c a r r i e d o u t i n two ways :

1) v i a t h e o r e t i c a l developments, i n t h e case of l a r q e samples and 2 ) v i a s i m u l a t i o n of Markovian s e r i e s i n t h e case of s m a l l e r samp l e s where a c l a s s i c a l s h o r t sample c o r r e c t i o n was combined w i t h t h e t h e o r e t i c a l h i ases d e r i v e d f r o m t h e l a r g e samples case. The e f f i c i e n c i e s of t h e e s t i m a t o r s ( r e s i d u a l h i ases and sampling v a r i a n c e s ) , h i c h depend on t h e sample l e n g t h s and on t h e p e r s i st ence parameter of t h e p a re n t popul a t i on , have been e v a l u a t e d h e n 5 % , l o % , 15% and 20% of t h e d a t a were m i s s i n g . The proposed approach (combined mean e s t i m a t o r ) 1eads, i n many cases , t o b e t t e r r e s u l t s than would he o b t a i n e d u s i n q c l a s s i c a l te chniques d e r i v e d f r o m F i s h e r ' s Z t r a n s f o r m a t i o n .

INTRODUCTION A N D PREVIOUS W O R K The e s t i m a t i o n of t h e l a g 1 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 of a s t a t i o n a r y t i m e - s e r i e s c o n s t i t u t e s an i m p o r t a n t s t e p i n t h e e v a l u a t i m of t h e s h o r t - t e r m p e r s i s t e n c e d t h i s s e r i e s . Th i s knowledReprinted 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

152 ge i s u s e f u l h o t h f o r t h e u n d e r s t a n d i n q of t h e phenomenon r e p r e s e n t e d b.y t h e s e r i e s and a l s o t o t h e d e t e r m i n a t i o n of t h e o p t i m a l s a m p l i n q r a t e of f u t u r e d a t a a c q u i s i t i o n proqrams. It happens q u i t e o f t e n t h a t i n o r d e r t o assess t h i s 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 one has a t h i s d i s p o s a l a sample of equispaced, s y s t e m a t i c a l l y sampled d a t a , hilt c o n t a i n i n g e i t h e r o c c a s i o n n a l m i s s i n q d a t a o r ahnormal v a l u e s which c o u l d d i s t o r t c o n s i d e r a b l y t h e r e s u l t s and h i a s any i n t e r p r e t a t i o n . I n t h i s s i t u a t i o n , two t y p e s of t e c h n i q u e s h a v e been d e s c r i b e d t o assess t h e v a l u e s of t h e a u t o c o r r e l a t i on. The f i r s t one c o n s i s t s , on each c o n t i n u o u s s e c t i o n of d a t a , i n t h e c a l c u l a t i o n of 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 t h a t may he h i g h l y b i a s e d . Given t h e r e l a t i v e l y s h o r t l e n g t h of each s e c t i o n , t h e F i s h e r - z - t r a n s f o r m a t i o n i s performed t o n o r m a l i z e t h e i r d i s t r i b u t i o n and t h e t r a n s f o r m e d v a l u e s a r e combined a c c o r d i n q t o t h e r e l a t i v e l e n g t h of t h e d a t a . F i n a l l y , t h e i n v e r s e z - t r a n s f o r m a t i m on t h e c o m p o s i t e v a l u e s i s o h t a i n e d , q i v i n q an e s t i m a t i o n o f t h e q l o h a l a u t o c o r r e l a t i on c o e f f i c i e n t . The second t e c h n i q u e c o n s i s t s i n t h e e v a l u a t i o n of m i s s i n q d a t a u s i n q a model of v a r i a b l e c o m p l e x i t y , e s t a h l i s h e d with a l l t h e v a l i d d a t a . T h i s p r o c e d u r e q i ves t h e m i s s i n q d a t a v a l u e s m a i n t a i n i n q t h e b a s i c c h a r a c t e r i s t i c s of t h e sample: mean, v a r i a n c e and p e r s i s t e n c e . Given a s u f f i c i e n t number of d a t a , t h e RnX and J E N K I N S approach, f o l l o w e d b,y a f o r w a r d and backward r e s i d u a l q e n e r a t i o n p r o c e d u r e , p r o v i d e s an e s t i m a t i o n of each m i s s i n q v a l u e . Such a t e c h n i q u e which s h o u l d he a p p l i e d i n a s t e p w i s e and i n t e r a c t i v e manner f o r each m i s s i n q d a t a i s q u i t e time-consuminq; t h e p r o c e d u r e can even e v e n t u a l l y d i v e r g e i f t h e number of m i s s i n q d a t a i s l a r q e and i f t h e i r d i s t a n c e i s t o o s m a l l t o e l i m i n a t e t h e possi b l e i n t e r a c t i o n s . I n t h e r e c e n t l i t e r a t u r e , numerous papers d e a l w i t h t h e p r o b l e m o f e s t i m a t i n q m i s s i n g d a t a i n u n i v a r i a t e and m u l t i v a r i a t e cases; i t i s e s s e n t i a l l y a p r o b l e m of d a t a i n t e r p o l a t i o n u s i n q t h e l o c a l t r e n d of t h e d a t a . WILKINSON (1958) and PREECE (1971) h a v e app l i e d t h i s k i n d of t e c h n i q u e t o e x p e r i m e n t a l d a t a , h u t i n our case, i t s v a l u e i s l i m i t e d as n o a t t e m p t i s made t o p r e s e r v e t h e autocorrelati m structure. More r e c e n t l y RRIIRACHER and WILSON (1976) h a v e d e v e l o p e d a p r e d i c t i v e approach based on t h e l e a s t squares t e c h n i q u e t o e s t i m a t e t h e e f f e c t of a n a t i o n a l h o l i d a y on t h e e l e c t r i c i t y demand u s i n q p r e v i o u s and p o s t e r i o r n o r m a l con s u m p t i o n s ; t h e n i n t e r p o l a t e d v a l u e s a r e o b t a i n e d v i a a method o f f o r e c a s t i n q and backf o r e c a s t i n q which minimi zes t h e r e s i d u a l s e r i a l s ; t h e r a t i o of t h e a c t u a l demand t o t h e h i s t o r i c a l demand d u r i n q h o l i d a y s a l l o w s t h e e v a l u a t i o n of t h e f u t u r e demand f o r s i m i l a r h o l i d a y s . The t e c h n i q u e seems w e l l a d a p t e d t o t h e p r o blem, even if numerous m i s s i n q d a t a a r e t o be r e c r e a t e d s i m u l t a n e o u s l y . The v e r y n a t u r e of e l e c t r i c i t y demand e x p l a i n s why such an approach was s i i c c e s f u l l and why such qood r e s u l t s a r e n o t t o be

153 e x p e c t e d f o r any t i m e - s e r i e s . A model of t h e ROX and J E N K I N S (1970) t y p e can he i d e n t i f i e d f o r t h e whole s e r i e s f r o m a s h o r t sequence of d a t a ; w e e k l y f l u c t u a t i o n s p o s s e s s i n q a much l a r g e r v a r i a b i l i t y t h a n y e a r l y f l u c t u a t i o n s , a few weeks of d a t a a l l o w s u f f i c i e n t i d e n t i f i c a t i o n f o r t h e g l o b a l model and i s n o t s i g n i f i c a n t l y d i s t u r b e d by t h e h o l i d a y s . D'ASTOIIS and HIPEL (1979) h a v e p r e s e n t e d an I n t e r v e n t i o n A n a l y s i s model d e r i v e d f r o m t h e ROX and JENKINS approach which p e r m i t s , on a v e r y q e n e r a l way, t o r e c o n s t i t u t e missing-data. The s i m u l a t i o n s p e r f o r m e d by t h e s e a u t h o r s on homoqeneous and non-homoqeneous s e r i e s of y e a r l y s t r e a m f l o w and m o n t h l y phosphorous d a t a seem ve y c o n c l u s i v e - .

PRESENT

APPROACH

I n t h i s paper, we s u g g e s t t h e u s e of some n o n - o p t i m a l , hut v e r y s i m p l e e s t i m a t o r s of t h e m i s s i n g d a t a ; t h e y i n t r o d u c e b i a s e s i n t h e c a l c u l a t i o n of t h e a u t o c o r r e a t i o n c o e f f i c i e n t s which a r e e a s v t o e s t i m a t e and t h u s can be c o r r e c t e d . The f i r s t e s t i m a t o r r o i s h u i l t i n r e p l a c i n g t h e m i s s i n g v a l u e s o r z e r o if t h e sample h a s heen c e n t e r e d : such an e s t i m a t o r g e n e r a l l y u n d e r e s t i m a t e s t h e t r u e v a l u e r nf t h e a u t o c o r r e l a t i o n of t h e sample. The second e s t i m a t o r ri i s h u i l t

hy t h e g e n e r a l mean,

i n r e p l a c i n q each m i s s i n q v a l u e hy t h e l o c a l mean, i .e. t h e a r i t h m e t i c a l mean between t h e p r e c e e d i n g and t h e f o l l o w i n g v a l u e s : such an e s t i m a t o r g e n e r a l l y o v e r e s t i m a t e s t h e t r u e v a l u e r of t h e a u t o c o r r e l a t i o n of t h e sample. A combined v a l u e rc = f ( r o , ri) The p o s s e s s i n g a l a r g e sample u n b i a s e d n e s s p r o p e r t y i s d i s c u s s e d . t h i r d e s t i m a t o r s t u d i e d , rn, i s o b t a i n e d as t h e s e r i e s i s s i m p l y compressed of i t s m i s s i n q v a l u e s , s h o r t e n i n g i t s l e n g t h ; t h i s method of d e a l i n q w i t h m i s s i n q d a t a i s u s e d i n p r a c t i c e q u i t e o f t e n and i s n o t w i t h o u t danqer. F o r t h e s e cases, t h e h i a s e s and t h e amp1 i f ic a t i on f a c t o r s of t h e sampl i n q v a r i ance a r e d e r i ved t h e o r e t i c a l l y and compared i n t h e l a r g e sample case. I n t h e case of a s h o r t sample, WALLIS and O'CONNELL (1972) h a v e d e m o n s t r a t e d t h e e f f i c i e n c y of t h e KENDALL (1954) c i r c u l a r l e n g t h c o r r e c t i o n t o r e l a t e t h e sampled c o r r e l a t i o n v a l u e w i t h t h a t of t h e p a r e n t p o p u l a t i o n which i s t h e p a r a m e t e r t o he assessed. G i ven t h e two c o r r e c t i orls, a 1 arge-sample e s t i m a t o r c o r r e c t i on and a l e n g t h c o r r e c t i o n , a s i m u l a t i o n programm u s i n g t h e Monte-Carlo t e c h n i q u e was d e v i s e d t o assess t h e o r d e r i n which t h q y s h o u l d he a p p l i e d and t o compare t h e e f f i c i e n c i e s of t h e e s t i m a t o r s w i t h t h a t of t h e c l a s s i c a l F I S H E R ' S z - t r a n s f o r m a t i o n t e c h n i q u e . nne s h o u l d n o t e t h a t t h e p r e v i o u s e s t i m a t o r s a r e making n o h y p o t h e s i s about t h e q e n e r a t i n q p r o c e s s e s of t h e samples which i s c o n s i s t e n t w i t h t h e f a c t t h a t a same s h o r t t r a c e may o r i g i n a t e f r o m d i f f e r e n t p a r e n t p o p u l a t i ons.

154

HYPOTHESIS AND NOTATIONS

...

(i= 1, N) f r o m a s t a t i o n a r y t i m e - s e r i e s , o f l e n g t h N i n which n values a r e m i s s i n g ; we w i l l develop here o n l y t h e case o f occasional m i s s i n g data i.e. t h e i r number i s r e l a t i v e l y small n <( N; each m i s s i n g d a t a i s i s o l a t e d and f o l l o w e d by n o n - i s o l a t e d s e c t i o n Zi o f v a l i d data, so t h a t t h e sample may be w r i t t e n : z ? Z ?....?Z ? Z 1 2 n n+l L e t ' s c o n s i d e r a sample Xi

.

Evidence presented PARZEN (1964) and RODRIGUEZ-ITURBE (1971) has l e a d us t o c o n s i d e r t h e unbiased d e f i n i t i o n o f v a r i a n c e and covar i a n c e t o be s p e c i a l l y w e l l s u i t e d f o r r e l - a t i v e l y s h o r t samples:

and

m

=

N

c

i=1

X i / N

The corresponding expression o f t h e l a g 1 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 i s t h a t o f JENKINS and WATTS (1968) and o f BOX and J E N K I N S (1970) : var

and

=

N

c Xi2

i=1

/ (N-1) r

=

cov =

N- 1

c

i=1

N- 1 C XiXi+l i=1

/ (N-1)

N

/ .c

XiXi+l

1 =1

Xi2

The expected values o f t h e products i n v o l v i n g m i s s i n g values are:

EIXiXi+l] E[Xi2]

= cov

= [(N-1)

/ N] var

THE LARGE SAMPLE CASE T h i s s e c t i o n deals w i t h t h e case where, w i t h r e g a r d t o t h e samp l i n g e r r o r , t h e KENDALL (1954) c i r c u l a r l e n g t h c o r r e c t i o n i s n e g l i g i b l e , i.e. when t h e l e n g t h o f t h e sample c o n t a i n i n g m i s s i n g values i s l a r g e r than 100.

155

B i a s r e l a t e d t o t h e e s t i m a t o r ro Using t h e general mean as an e s t i m a t i o n o f m i s s i n g d a t a l e a d s t o t h e disappearance i n t h e numerator o f e q u a t i o n (1) o f two p roduct s and o f one square a t t h e denominator f o r each o f t h e n m i s s i n g data. T a k i n g t h e expected values, one g e t s :

E[ro]

-- -N-2n-1 N N-n N-1

which can be i n v e r t e d as:

T h i s expressi o n can be expanded t o o r d e r N-2:

The replacement o f E [ r o ] by i t s sample e s t i m a t e r o y i e l d s t h e adequate b i a s c o r r e c t i o n : r

-

ro = N I L ( ~ + ~N ~ A L U ) ~ ~

(4)

i f we suppose a s u f f i c i e n t number o f m i s s i n g d a t a ( n w 1) and by

p u t t i n g -?. = a << 1, t h e n t h e p re c e e d i n g e q u a t i o n s reduce t o :

N

r = (1 + a + 2 a 2 ) ro

r-ro =

a (1+2a) r o

(5) (6)

One can n o t e t h a t f o r p o s i t i v e v a l u e s o f ro, which i s t h e most common case i n hydrosciences, t h e b i a s i s p o s i t i v e ; t h e e s t i m a t o r underest imate s t h e t r u e values o f r. Bias r e l a t e d t o t h e e s t i m a t o r ri Using t h e l o c a l mean as an e s t i m a t i o n of y i s s i n g data, any m i s s i n g value Ximis e s t i m a t e d by: Xim = X i +1 i-1so t h a t each product and each square c o n t a i n i n g Xim

2

a r e e s t i m a t e d by:

156

I n t r o d u c i n g th o s e e x p re s s i o n s i n e q u a t i o n (1) and a p p r o x i m a t i n g Xi-1] by EIXi+l Xi] g i v e s ; t h e expected values EIXi+l

E[ri]

=

2N(N-n-l)r t 2n(N-1) nNr + (2N-n) (N-1)

(7)

which can be i n v e r t e d as:

-- -Nn LLL ri

1 ) E [ri] (1- L,(1- 2N N ( 1 - n- +) l- 1 1 N N

r =

1 ) (1- N

(8)

2

T h i s expre s s i o n can be expanded t o o r d e r N-2:

Eb.1

- ( 1 +1) (1- E [ r i ] ) [ 1+

r = E[ri]

N

2

2

N2

E[r.] 1 ( 2+1+- ) ] 2

n

by i t s sample e s t i m a t e ri y i e ds an

The replacement o f E [ r i ] adequate b i a s c o r r e c t i o n :

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 d a t a ( n xa 1 ) and - = a << 1, t h e n t h e p re c e e d i n g e q u a t i o n s reduce t o :

by p u t t i n g

R

N

r = ri

-

a (1

ri - 2

ri2 -

2

)

-

a2

(1

-

3

ri2

- -1 r i 3 ) 4

(10)

One can n o t e t h a t f o r a l l v a l u e s o f ri, t h e b i a s i s always n egat ive; t h e e s t i m a t o r o v e r e s t i m a t e s t h e t r u e v a l u e o f r. D e f i n i t i o n o f an almost unbiased combined e s t i m a t o r I n t h e prec e e d i n g s e c t i o n s , we have developed t h e c a l c u l a t i o n o f t h e biases o f two v e ry s i mp l e e s t i m a t o r s as a f u n c t i o n o f t h e percentage o f m i s s i n g v a l u e s a, one o v e r e s t i m a t i n g , t h e o t h e r u n d e r e s t i m a t i n g t h e t r u e v a l u e of r. I n t h i s s e c t i o n we d e f i n e a

157

combined estimator r as a combination of r and ri which reduces the bias t o order a2, making bias correction unnecessary. Taking the linear p a r t of equations ( 5 ) and (10) and eliminating a between them gives the definition of rc:

r~

=

ro(l+ri) ( 2-ri ) 2ro + (l-ri ) (2+ri)

Using the same technique as previously, the bias can be a p pr ox i mat ed by :

nc =

2a2

2r

( 2 - rC2) l+rc

which i s a negative and small value of order a2 and vanishes for rc = 0 and rc = 1. The Table 1 presents the estimator rc as a function of the biased sample estimators ro and ri. The Figure 1, which can be used as a nomograph, shows the range of application of equation (12) and the relative location of the three estimations:

l > r i > O > 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/<) 3 s i n 2D exp [i(R-RG)I, =

=

3 2 (6n/5)’i h 2 ( E / [ ) cos D exp [2i(R-RG)I,

.....

187 where h2 = 0.16458m (Sun), 0.35838m (Moon), and RG is the Right Ascension of the Greenwich meridian, namely

RG

I

h -

?T

f

2~r (Universal Time in days).

The radiationa potential elements are formed similarly, from the Sun's elements alone. REFERENCES Alcock, G.A. & Cartwright D.E., 1978. Some experiments with 'orthotides', Geophys.J.R.astr.Soc. 54: 681-696. Amin, M., 1976. The fine resolution of tidal harmonics. Ge0phys.J.R. astr.Soc. 44: 293-310. Cartwright, D.E., 1968. A unified analysis of tides and surges. Phil.Trans.R.Soc. A, 263: 1-55. Cartwright, D.E., 1975. A subharmonic lunar tide in the seas off Western Europe. Nature, 257: 5524, 277-280. Cartwright, D.E., 1976. Anomalous M1 tide at Lagos. Nature, 263: 5574, 217-218. C a r G i g h t , D.E. , 1977. Ocean tides. Rep.Prog.Phys. 40: 665-708. Cartwright, D.E. & Edden, A.C., 1973. Corrected tablesof tidal harmonics. Geophys.J.R.astr.Soc. 33: 253-264. Cartwright, D.E. & Edden, A.C., 1977. Spectroscopy of the tidegenerating potentials. Ann.Geophys. 33: 179-182. Cartwright D.E. & Tayler, R.J., 1971. Newcomputations of the tide generating potential. Geophys.J.R.astr.Soc. 23: 45-74. Corkan, R.H., 1934. An annual perturbation in therange of the t iG,e Proc .R. SOC .London A, 100: 305-329. Desnoes, M.Y., 1977. Le bruitdans les analyses de marge. Ann.Hydrog. 5(2): 31-46. Franco, A.S. & Rock, N.J., 1971. The FFT and its application to tidal oscillations. Bol.Inst.Oceanog.Univ.S.Paulo, 20: 1-56. Godin, G., 1972. The analysis of tides. Univ. Toronto 264pp. Graff, J. & Karunaratne, A., 1980. Accurate reduction of sea level records. 1nt.Hydrog.Rev. 57: 2, 151-166. Groves, G.W. & Reynolds, R.W., 1975. An orthogonalised convolution method of tide prediction. J.Geophys.Rev. 80: 30, 4131-4138. Horn, W . , 1960. Some recent approaches to tidalproblems. Int. Hydrog.Rev. 37: 2, 65-84. Karunaratne, D.A, 1980. An improved method for smoothing and interpolating hourly sea level data. 1nt.Hydrog.Rev. 57: 1 , 135-148. M u n E W.H. & Cartwright, D.E., 1966. Tidal spectroscopy and prediction. Phil.Trans.R.Soc. A, __ 259: 533-581.

188

Munk, W.H., Zetler, B. & Groves, G., 1965. Tidal Cusps. Geophys. J.R.astr.Soc. 10: 211-219. Pugh, D.T. & V a s s i c J.M., 1976. Tide and surge propagation offshore in the Dowsing region of the North Sea. Deut.Hydrog.Zeitsch. 29: 163-213. Pugh, D.T. & Vassie, J.M., 1978. Extreme sea levels from tide and surge probability. Proc. 16th Conf.Coasta1 Eng., Hamburg, - 911-930. Ch.52: Analyse de 19 ans d'observations de marge Simon, B., 1977. a' Brest. Ann.Hydrog. 8(1): 5-17. Zetler, B., Cartwright, D, & Berkman, S. , 1979. Some comparisons o f response and harmonic tide predictions. 1nt.Hydrog.Rev. 56: 2, 105-115. Z e t E r , B.D. i Munk, W.H., 1975. The optimum wiggliness of tidal admittances. J .Mar.Res.Supp. 33: 1-1 3.

189 IDENTIFICATION OF INTERNAL TIDES IN TIDAL CURRENT RECORDS FROM THE MIDDLE ESTUARY OF THE ST. LAWRENCE

LANGLEY R . MUIR Bayfield Laboratory f o r Marine Science and Surveys, Department o f Fisheries and Oceans, Canada Centre f o r Inland Waters, Burl i n g t o n , O n t a r i o , Canada

Abstract Observations of t i d a l c u r r e n t s a r e t r a d i t i o n a l l y analysed by means of s p e c t r a l a n a l y s i s , t o ensure t h a t only t i d a l f r e q u e n c i e s are p r e s e n t and by harmonic a n a l y s i s , t o o b t a i n t h e amplitude and phases o f the t i d a l c o n s t i t u e n t s and t o allow t h e p r e d i c t i o n of f u t u r e c u r r e n t s . When t h i s was done f o r a s e r i e s o f records obtained i n the S t . Lawrence i t was found t h a t 95% of t h e energy was a t t h e t i d a l f r e q u e n c i e s , and t h a t the c a l c u l a t e d " t i d a l c o n s t i t u e n t s " allowed hindcasting the t i d a l c u r r e n t s with g r e a t accuracy. However, n e i t h e r t h e amplitudes nor t h e phases of t h e " t i d a l c o n s t i t u e n t s " could be i n t e r p r e t e d as s t a n d i n g o r p r o g r e s s i v e waves over t h e whole Estuary a n d t h e s e amplitudes and p h a s e s seemed t o be e n t i r e l y u n c o r r e l a t e d , even though t h e o b s e r v a t i o n s were s e p a r a t e d by re1 a t i v e l y s h o r t d i s t a n c e s when compared t o t h e expected wave l e n g t h s of t h e b a r o t r o p i c waves. The topography and d e n s i t y f i e l d of t h e Estuary allow t h e g e n e r a t i o n and propagation of i n t e r n a l g r a v i t y waves a t t h e same f r e q u e n c i e s as t h e b a r o t r o p i c t i d e s but with much s h o r t e r wave l e n g t h s . These i n t e r n a l t i d e s may i n t e r f e r e l i n e a r l y with t h e b a r o t r o p i c t i d a l c u r r e n t s , causing phase and amp1 i t u d e v a r i a t i o n s , depending upon t h e l o c a t i o n i n t h e Estuary and t h e d e n s i t y f i e l d i n the Estuary. By means of admittance c a l c u l a t i o n s on very s h o r t p o r t i o n s of t h e observed c u r r e n t s r e c o r d s , i t i s p o s s i b l e t o show t h a t t h e amp1 i t u d e s and phases of the " t i d a l c o n s t i t u e n t s " obtained from t h e harmonic analyses vary i n a way t h a t would be expected i f both b a r o t r o p i c and i n t e r n a l t i d e s were present i n t h e Estuary.

1.

INTROnUCTION The b a r o t r o p i c , o r s u r f a c e , t i d e s of t h e S t . Lawrence River have

been t h e s u b j e c t of an i n t e n s i v e amount of research over t h e past 15 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

190 years. As a r e s u l t , the physics of the surface t i d e s are f a i r l y well understood, i n the sense t h a t numerical models adequately represent the observed water l e v e l s and analytical models adequately predict the observed d i s t o r t i o n of the t i d a l wave as i t progresses upstream. Godin (1979) and LeBl ond (1978) give extended descriptions of the propagat ion of the surface t i d e in the whole system from two d i f f e r e n t viewpoints. Godin (1979) shows cotidal charts f o r two constituents K, and M,, b u t a brief description of the behaviour of the

M2

constituent i n t h e

lower p a r t of the system will give an indication of the t i d a l propagation. There i s an amphidromic point about half-way between Prince Edward Island and Anticosti Island in the Gulf of the S t . Lawrence. The M2 t i d e then progresses upstream i n the Lower Estuary, becoming a Kelvin wave, and continues as a Kelvin wave, past the Saguenay River and t h r o u g h the Middle Estuary, (Fig. 1) although i t loses most of the cross-channel height difference as the Estuary narrows.

By the time the wave has reached the I l e d'Orleans, the

rotational e f f e c t s have disappeared and the wave c r e s t i s v i r t u a l l y horizontal. The phase of the $'f surface t i d e a t the Saguenay River i s 220" and

a t Baie-St-Paul, near I l e aux Coudres, the phase i s 255.5", and so the wave takes 35.5x2.07 = 73.5 minutes t o make a passage o f 95 km. The Since the mean water depth i n mean c e l e r i t y i s therefore 77.6 k m / h r . the North Channel i s approximately loom, the phase speed should be a b o u t 110 k m / h r , and so the M2 surface t i d e in the North Channel could be considered t o be p a r t way between a standing and a progressive wave. The s i l l stretching from Pte au Pic t o the Morin Shoal, which shall b e called the Morin B a n k , provides a n adequate r e f l e c t o r as does the a b r u p t change i n direction of the channel as i t goes a r o u n d I l e aux Coudres.

On t h e s o u t h s i d e t h e p h a s e a t R i v i e r e du Loup i s 224.8",

while a t Pte aux Orignaux the phase i s i151.1", so t h a t the time of passage i s 54.4 minutes, which gives an average c e l e r i t y o f a b o u t 57.4 km/hr.

The mean depth i n the S o u t h Channel i s a b o u t 25 m, which gives

a wave speed of a b o u t 55 k m / h r and would indicate t h a t in the S o u t h Channel the M2 surface t i d e i s a simple progressive wave.

191

Figure 1 Location diagram. The Middle Estuary s t r e t c h e s from t h e Saguenay River t o I l e aux Coudres

192 Predictions of the tidal currents in the Middle Estuary have not

I t has been realized for a number of years t h a t the 1939 Tidal Current Atlas i s very inaccurate. The v e r i f i c a t i o n of the two-dimensional numerical models of Prandle and Crookshank (1974) , Ouellet and Cheylus (1971) , Aubin et a1 (1979) , and El Sabh et a1 (1979) consisted of matching observed with computed water l e v e l s , and l i t t l e attempt was made t o match the observed and computed t i d a l currents. This i s primarily because, until the S t . 1-awrence Tidal been very successful.

.

.

Current Surveys of 1974, 1975, and 1977, there has been an almost t o t a l

For some of the models, computed t r a j e c t o r i e s of the currents were shown, b u t these computed currents bear no more t h a n a passing r e l a t i o n t o observed currents and would be completely useless in any operational sense, such as computing the dispersion of oil s p i l l s , or f o r navigational t i d a l current a t 1 ases. lack of d a t a f o r v e r i f i c a t i o n of the tidal currents.

2.

HARMONIC ANALYSIS OF THE TIDAL CURRENTS.

A t o t a l of 44 current meter records have been obtained in the Middle Estuary in the course of the surveys conducted by the a u t h o r in

1974, 1975 and 1977.

All o f the plots and mooring d e t a i l s are t o be

found in the S t . Lawrence Data Reports (Budge11 and Muir, 1975; Muir, 1978; and Muir, 1979a) and these d e t a i l s will not be l i s t e d here. The preliminary analysis of the tidal current records was carried

o u t using the standard Canadian Hydrographic Service Tidal Streams Analysis program, which i s described by Godin ( 1 9 7 2 ) . The d a t a were f i l t e r e d and reduced t o hourly values which were then analysed f o r the appropriate tidal constituents, with no related constituents used. From the s t a t i s t i c s of the analyses, the harmonic analyses of the current records were very good as shown in Table 1. The residual currents were quite small, the coefficient matrices were wellconditioned, and the amount of explained variance was approximately 95%. The constituent l i s t s and standard plots may be found in the appropriate Data Reports.

I t was expected t h a t the t i d a l currents would progress up the

river as a simple Kelvin wave in the same fashion as the surface t i d e s , b u t t h i s i s not the case, as can be seen from Figure 2. This Figure gives the phase of the M2 constituent f o r the surface t i d e and f o r t h e t i d a l currents a t the location of the measurements.

The number in parentheses a f t e r the phase gives the depth of the current meter below c h a r t d a t u m in metres, and the arrow shows the d i r e c t i o n of rotation of the M2 el 1 ipse. If the assumption i s made t h a t the t i d a l currents are driven d i r e c t l y by the barotropic surface t i d e , then there are many anomalous features shown in Figure 2. F i r s t , the phases a t any given depth c a n n o t be contoured as a simple Kelvin wave, a n d , in f a c t , cannot be contoured in any meaningful way a t a l l .

Second, the phase speeds t h a t

can be calculated f o r a wave moving from mooring t o mooring vary considerably and do not match with the wave speeds t h a t can be calculated from the water level records.

For example, i t takes t h e

surface t i d e a b o u t 30 minutes t o travel from S t . Simeon t o Goose Cape, while the t i d a l current a t 5 m depth apparently takes 92 minutes t o cover the same distance a n d , in addition, i t arrives on the south shore a t Pte. Aux Orignaux a b o u t 47 minutes before i t a r r i v e s a t Goose Cape. Third, one would expect t h a t f o r a barotropic t i d e , the phase a t any given location should be constant t h r o u g h o u t the v e r t i c a l with, perhaps, a s l i g h t phase lead f o r the bottom currents due t o the e f f e c t s of f r i c t i o n (Soulsby, 1978), and t h i s does not happen. The presence of these anomalous features throws considerable d o u b t on the assumption t h a t the t i d a l currents are due simply t o a barotropic surface t i d e . 3. ADMITTANCE ANALYSIS OF THE TIDAL CURRENTS.

I n order t o check t h a t there was no e r r o r i n the harmonic analysis method and t o gain more information from the t i d a l current records, a n admittance analysis was carried o u t on a l l of the 1974 current meter records. Since t h i s method i s not widely known, a brief description of i t will be given here. The only complete description t h a t i s available

194

I

I 3 3 2 . 2(10)1

A 1977 CURRENT METER

d e p t h s in brackets

70'00

Figure 2

69*'40

Phase of t h e M2 t i d e .

195

i s t o be found in Godin (1976), where i t was used with success in the analysis of tidal currents in Robeson Channel. The basic idea i s the same as the tidal spectroscopy methods of Munk and Cartwright ( 1 9 6 6 ) , b u t very much shorter periods of record are used, and instead of using the equilibrium t i d e as an i n p u t function, a well-resolved water level record i s used. I t i s assumed t h a t there i s a linear relation between the tidal constituents derived from the water level record and the tidal constituents which make up the tidal current record for each frequency band. If there are two time s e r i e s which may be assumed t o be l i n e a r l y related, then the complex admittance ( o r the gain) between them, a t frequency 0 , i s given by

and the complex coherence i s given by

where Cxy i s the complex cross-spectrum between the two s e r i e s , x and y , and X and Y are the a u t o spectra of the two time s e r i e s . Given the amplitude and phase of the admittance, a component of the series y may be predicted from the corresponding component of the series x simply by multiplying the avplitude of the component of x by the modulus o f the admittance, and by adding the phase of the admittance t o the phase of x . Godin ( 1 9 7 6 ) gives the confidence l i m i t s f o r the estimates of the admittance and the phase. I n the case of the analysis of tidal current records, the input function i s the water level record f o r a nearby t i d e gauging s t a t i o n , and the u ( e a s t ) and v ( n o r t h ) components of the tidal currents are the independent o u t p u t records. So long as the i n p u t record can supply a well-resolved s e t of tidal constituents, i t i s then possible t o calculate G U ( a ) , y u ( a ) , G v ( o ) and y v ( a ) f o r a given bandwidth A , and from these t o calculate the component constituents f o r the tidal

196

On the S t . Lawrence, in the semidiurnal bands, f o r a bandwidth of 0.005 c y c l e s l h r ,

currents and thence the constituent e l l i p s e parameters.

the modulus of the coherence between the input t i d a l elevation and the t i d a l current components i s seldom l e s s t h a n 0.98. 4.

APPLICATION TO THE ST. LAWRENCE.

The reference water l e v e l s used f o r the application of the admittance method t o the 1974 S t . Lawrence d a t a , were the observed water levels a t S t . Jean Port Jol-i.

This s t a t i o n i s somewhat upstream

of the Middle Estuary, b u t has a very l o n g period of t i d a l record whose t i d a l constituents are very well resolvgd. The hourly water l e v e l s were used as the input function, while the f i l t e r e d , hourly current observations in u and v components were used as the o u t p u t functions. The admittances and coherences were calculated using a power-of-two Fast Fourier Transform.

The mean and trend were removed from the d a t a

using a l i n e a r , least-squares f i t and a cosine taper p u t on the f i r s t The raw spectral estimates were frequency smoothed t o produce estimates f o r b o t h 12 and 36 bands. The estimates from the 36 band analyses were used t o ensure and l a s t tenths o f the d a t a t o remove end e f f e c t s .

t h a t the admittance was n o t changing s i g n i f i c a n t l y within the t i d a l frequency bands, and t h a t the coherences were also constant within the t i d a l frequency bands. After the spectral analysis had been done f o r each of the records, the e l l i p s e parameters (Godin, 1972) were calculated f o r each of the d a t a s e t s using the admittances and the 55 t i d a l constituents from a one-year harmonic analysis of the St.-Jean-Port-Jol i water level record. Using these e l l i p s e components, the t i d a l currents were then predicted f o r each of the periods of record f o r the observed c u r r e n t s , and then the residual currents were calculated by subtracting the

predicted currents from the observed currents.

197 OBSERVED VARIANCE

HARMONIC RESIDUALS

ADMITTANCE RESIDUALS

012015

2319.23

7963.10

55.00

98.00

70.41

158.00

042005

4906.00

5852.45

47.00

49.00

52.10

54.22

5345.28

155.001

58.00

62.111

69.89

I I 7 8 . 9 3 1263.831

190.38

81.371

74.91

194.55

I

]042010

TABLE 1

I 2761.411

I

I

I

71.661

1203.1RI

I

I

Residual S t a t i s t i c s f o r 1 9 7 4 S t . Lawrence Moorings. Observed Variance is the variance of the observed t i d e , f o r e a s t and north components. R-u and R-v g i v e the range i n the r e s i d u a l s f o r the u and v components. V-u and V-v g i v e the variance in the r e s i d u a l s f o r the u and v components. Admittance r e s i d u a l s c a l c u l a t e d u s i n g 5 5 tidal constituents.

Table 1 shows a comparison between t h e r e s i d u a l s derived from t h e harmonic method and t h e r e s i d u a l s derived from t h e admittance method. The r e s i d u a l s from the harmonic method a r e s m a l l e r t h a n t h e r e s i d u a l s from t h e admittance method, and t h i s would i n d i c a t e t h a t t h e harmonic a n a l y s i s provides a b e t t e r a n a y l s i s procedure than t h e admittance method. The reason f o r t h i s i s not d i f f i c u l t t o f i n d . Table 2 provides the coherences computed i n t h r e e of t h e frequency bands f o r each record. The coherences a r e very high i n t h e semi-diurnal band, poorer i n the d i u r n a l band, and poorer s t i l l i n t h e q u a r t e r - d i u r n a l b a n d . Since most of the energy i n the t i d a l signal i s in t h e semidiurnal band, t h e signal t o n o i s e r a t i o i n t h e o t h e r bands i s too rnuct f o r the admittance method t o cope w i t h , and t h i s i s why t h e harmonic method provides a b e t t e r f i t , although both methods reduce t h e residuals t o acceptable l e v e l s .

198 Mooring Number

1

1

Diurnal Band

Semi-diurnal Band

1 .

U

1

Ouarter-diurnal Band

v

012005 012015

.911 .910

.431 .947

.988 .992

.810 .990

042005 04ZOlO

.961 .892

.915 .912

.988 .984

.997 .995

072015 072040

.557 .805

.947 .952

-995 .995

.997 .997

.832 .974

llZ0ll 112021 112056

.796 .919 .961

.979 .971 .934

.9R2 .990 .995

.998 .996 .995

.917

TABLE 2

.726

Coherence5 for three frequency bands, u and v components, between the 1974 St. Lawrence moorings and St.-Jean-Port-Joli.

Mooring Number

M

nizoo5 012015

97.2 136.1

042005 042010

140.3 121.4

072015 072040 llZ0ll 112021 112056

TABLE 3

.5R2 .802 .757

Harmonic Components m e g 5.1 1.6

Admittance Components M m 9 g 35.2 61.8

48.R 34.8

-8.2 -8.1

47.9 55.2

27.1 16.6

75.6 50.7

-6.0 1R.5

63.3 73.6

29.6 21.7

54.5 30.7 62.7

3.0 5.7 -1.9

74.6 59.8 54.2

10.3 19.0 -1.7

37.6 62.0

45.7 31.4

90.4 131.4

-

-9.6 -8.7

47.3 54.0

22.9 12.1

133.7 115.2

77.4 52.6

-5.9 19.4

62.0 73.4

25.1 18.8

56.0 31.8 64.6

3.0 6.1 -2.0

74.5 60.9 54.2

4.30 13.2 -5.4

1.9 .1

Comparison of ellipse components for the M2 derived from the harmonic analysis and the admittance analysis.

Table 3 gives a comparison o f the

M2

e l l i p s e parameters f o r each

o f the 1974 current meter records, and, since the coherences in the

semi-diurnal band are h i g h , these two s e t s o f constituents are q u i t e The admittance analysis shows, then, t h a t there i s no major f a u l t in the harmonic analysis o f the current meter records and t h a t

close.

a l l of the discrepancies mentioned previously require a physical

explanation.

199 The major advantage of the admittance method i s t h a t i t allows the analysis of r e l a t i v e l y short periods of record in which i t would be impossible t o separate the individual tidal constituents by means of the harmonic method. This i s possible since i t i s assumed t h a t the relationships between the constituents i n any given frequency band are known from the analysis of the water level record. Therefore, some reanalysis of the 1974 d a t a was done i n order t o examine the assumption t h a t the tidal currents are due t o a barotropic surface t i d e . The f i r s t of the reanalyses was performed on the two sixty-day records 74-12C-072015 and 74-12C-072040 which were 1 ocated in the middle of the Estuary, j u s t off Pointe Au Pic. These were analysed i n two, thirty-day sections and then in four, fifteen-day sections. Tables 4 and 5 give the r e s u l t s of the anaylses f o r each component along with the associated errors. The errors are i n degrees and associated w i t h the error in the estimate of the phase. I t should be noted from these two tables t h a t the amplitudes and phases of none of the bands are stable. T h i s i n s t a b i l i t y does not seem t o be related t o the coherence estimates. One would expect t h a t low coherences should be associated w i t h extreme estimates of the phase and amplitude, b u t t h i s does not seem t o be the case. The immediate inference t h a t can be drawn i s t h a t the tidal currents of the Middle Estuary are not very predictable. Using the analysed constituents from any particular one of the 15-day analyses, one could n o t predict the currents i n another of the 15-day periods with any degree of accuracy. The main c a r r i e r signal, which i s the semi-diurnal b a n d , would not be very f a r out in terms of amplitude, b u t i t s phase would c e r t a i n l y be incorrect and the modulations due t o other frequency bands would be considerably wrong. A second observation a b o u t the analyses i s t h a t there does not seem t o be any consistency in the variations in the t r a n s f e r function from one record t o the other. These two records were obtained from the same mooring, over the same time period. One of the meters was a t 15 metres depth and the other was a t 40 metres depth. If the vertical

200

velocity p r o f i l e due t o a barotropic surface t i d e i s considered t o be 1 i near o r semi -1 o g a r i thmic, then the r a t i 0s of the amp1 i tudes of the admittance should remain more o r l e s s constant and the phase differences should a1 so remain constant t h r o u g h o u t the v e r t i c a l , b u t neither of these things happen. The t h i r d observation t h a t may be made a b o u t the analyses concerns t h e coherences. If the t i d a l currents a t the two meters are caused by the same barotropic t i d e , then the signals should be related t o each other l i n e a r l y and the coherence between them should be exactly 1 in the absence of noise.

Since the same water level record i s used as the

input signal f o r b o t h t a b l e s , the coherence over any given period f o r the u-component, say, should be identical in each t a b l e . This does not happen, except possibly in the semi - d i urnal b a n d , where the coherences are so high t h a t i t i s d i f f i c u l t t o distinguish between them. One of the most puzzling features of t h i s f i r s t reanalysis of the d a t a i s the i n s t a b i l i t y in the amplitudes and phases of the admittances, which implies t h a t the amplitude and phase of the t i d a l constituents are variable with time. The major assumption of t i d a l theory i s t h a t the amplitude and phase of a t i d a l constituent must remain constant with time; otherwise, there could be no hope o f predicting the t i d e s .

A second reanalysis of the 1974 S t . Lawrence

d a t a was carried o u t t o determine whether o r not there was any pattern in the i n s t a b i l i t y of the admittance. This second reanalysis consisted of analysing three contiguous, non-overlapping segments of 236 hours each, f o r each of the 1974 current records. The length of 236 hours was chosen because i t i s exactly 19 M2 t i d a l cycles; i t contains p a r t of a spring t i d e and a neap t i d e cycle, a n d , w i t h 12 frequency bands f o r the spectral analysis, retains 19.5 degrees of freedom i n the estimates so t h a t they have a reasonable amount of significance.

For each of the 1974 records, the f i r s t three segments were analysed and t h e phases of the M2 constituent calculated. The r e s u l t s are given i n Table 6 where the difference in phase between the phase of the segment and the phase of

201

t h e whole r e c o r d i s g i v e n , a l o n g w i t h t h e e r r o r i n t h e c a l c u l a t i o n o f t h e phase. Mooring Number

I 1 PhPh;;

Record

Phase Error of segments

Phase Difference of segments

1

2

3

3

012005 012015

45.70 31.40

1.02 3.50

16.46 9.62

0.58 1.39

5.94 3.33

9.70 5.60

19.24 9.02

042005 042010

22.90 12.10

2.26 0.94

12.68 12.00

1.37 3.91

5.12 4.08

6.44 5.86

5.63 8.57

072015 072040

25.10 18.80

4.35 2.45

8.10 10.09

4.81 1.56

5.13 6.15 3.28

9.31 6.39 10.72

5.00 6.36 -.lo

5.00 3.47 7.00

7.13 4.92 4.49

4.65 8.06 2.68

112011 112021 112056 TABLE 6

-

I n s t a b i l i t i e s of the M2 phase from the 1974 moorings. Phase d i f f e r e n c e Phase Whole Record Phase segment.

-

The c a l c u l a t e d e r r o r s f o r t h e phase a r e q u i t e l a r g e f o r each case, which r e f l e c t s t h e s h o r t p e r i o d o f r e c o r d used f o r each o f t h e analyses.

However, t h e r e i s a p a t t e r n w h i c h emerges f r o m t h e phase

differences.

F o r a l l o f t h e r e c o r d s , t h e r e i s a d i s t i n c t phase s h i f t ,

which i s always i n t h e same d i r e c t i o n , f r o m t h e f i r s t t o t h e second t e n - d a y segment and t h e n a n o t h e r phase s h i f t , back a g a i n , f r o m t h e second t o t h e t h i r d t e n - d a y segment.

The m a g n i t u d e o f t h e s e phase

s h i f t s i s d i f f e r e n t f o r each o f t h e r e c o r d s , and most o f them, on a p u r e l y s t a t i s t i c a l b a s i s , a r e n o t s i g n i f i c a n t phase s h i f t s .

However,

t h e f a c t t h a t a l l o f them a r e i n t h e same d i r e c t i o n i s s i g n i f i c a n t .

It

would argue t h a t t h i s phase s h i f t is r e l a t e d t o t h e d i f f e r i n g p o r t i o n s o f t h e s p r i n g - n e a p c y c l e i n each o f t h e t e n - d a y r e c o r d s . Comparing t h e s e phase s h i f t s t o t h e

y

phase s h i f t s shown i n

Tables 4 and 5 f o r t h e s u c c e s s i v e 15-day segments i s r e v e a l i n g .

In the

15-day segments, t h e r e i s one c o m p l e t e s p r i n g - n e a p c y c l e and t h e phase s h i f t s a r e always i n t h e same d i r e c t i o n .

T h a t i s , t h e phase o f t h e

c o n s i s t e n t l y l a g s f r o m one 15-day segment t o t h e n e x t .

M2

I n t h e 10-day

segments, t h e r e i s an unequal p o r t i o n o f t h e s p r i n g - n e a p c y c l e i n each o f t h e r e c o r d s and t h e phase o f t h e

y

l a g s from t h e f i r s t t o t h e

second segment and t h e n r e t u r n s a l m o s t t o i t s o r i g i n a l v a l u e b y t h e

202

third segment. Taking the two together, i t seems t h a t the variation in the phase of the M2 i s not l i n e a r and t h a t t h i s variation may have b o t h a fortnightly component i n i t as well as a longer period component. To o b t a i n reasonable estimates of these variations i n the phase of the M 2 or any other constituent would require a large number of very long period tidal current records, and these are not available a t the present time. The fact t h a t the harmonic analysis and the admittance analysis, on the whole, agree, would argue t h a t there i s no e r r o r i n e i t h e r method of analysis and t h a t the observed inconsistencies in the t i d a l constituents and the i n s t a b i l i t i e s of these constituents are a real , physical phenomena i n the Middle Estuary of the S t . Lawrence. I t i s necessary, therefore, t o find an explanation for them. 5. THE PROPAGATION OF THE TIDAL CURRENTS.

The use of the admittance method f o r the analysis of the 1974 tidal current records has shown t h a t neither the amplitude nor the phase o f the tidal constituents a t a given location i n the Middle Estuary i s a constant. Since b o t h of the methods of analysis are essentially l e a s t squares, curve f i t t i n g exercises, and since neither o f the methods makes use of any physical principles, other t h a n those involved in choosing the frequencies f o r the curve f i t s , i t i s not the f a u l t of the analysis methods t h a t they give results which are inconsistent with o u r understanding of t i d a l processes. However, t h i s understanding of t i d a l processes r e s t s on the assumption t h a t the tidal currents in the Middle Estuary r e s u l t from the p r o p a g a t i o n of a simple barotropic tide. I t i s possible t o explain the observed phenomena i f t h i s assumption i s dropped, and the tidal currents are considered t o be the result of not o n l y barotropic tides b u t also baroclinic t i d e s . There are two major reasons for expecting t o have internal t i d e s present in the Middle Estuary. I n Muir (1982) i t was shown t h a t there i s considerable vertical density structure i n the Estuary a n d , i f there were internal t i d e s generated, these would be allowed t o propagate

203

because of t h i s vertical density s t r u c t u r e . The second reason f o r expecting the presence of internal t i d e s i s t h a t Forrester (1974) found them present in the Lower Estuary, where they were shown t o be generated by the a b r u p t change in depth associated w i t h the end of t h e Laurentian Channel, which i s a t the downstream end of the Middle Estuary. LeBlond and Mysak (1978), reporting on the work of Rattray and his co-workers (1960 o n w a r d ) , point out t h a t internal waves may be

generated by a step-1 ike change in topography and a1 t h o u g h the work of Rattray i s concerned w i t h the seaward propagatjon of these waves, t h e r e is also a landward propagation. I t i s t h i s l a n d w a r d , or upstream, propagating s e t of internal waves which would be found in the Middle Estuary. The internal t i d e s a f f e c t only the currents and do not a f f e c t the water l e v e l s . This would explain why i t i s possible t o predict the water l e v e l s accurately, b u t not the c u r r e n t s , by considering only t h e barotropic constituents. If we assume a uniform channel with a constant surface density and appropriate vertical density v a r i a t i o n , and t h a t a surface t i d e of frequency, 5 , i s generated a t one end of the semi-infinite channel which generates internal modes of the same frequency, then, a t a distance x from the o r i g i n , the current will be given by m

where; ai i s the amplitude of the i t h vertical mode a t depth z ; t i s time; ki i s the horizontal wave number associated with the i t h vertical mode; x i s the distance from the generation point and m i s the total number of vertical modes, with the zeroth mode being associated w i t h the barotropic t i d e .

Using standard trigonometric i d e n t i t i e s ,

equation ( 2 ) may be transformed i n t o

u ( x , z , t ) = AMP c o s ( o t + PHASE) where ;

(3)

204

and so the current a t any distance from the source could s t i l l be

described by a cosine of the appropriate frequency, b u t b o t h the amplitude and the phase of t h i s cosine would be functions of the amplitude and phase of the modes as well as a function of the distance from the source. To give an example of how t h i s process could affect an estuary such as the S t . Lawrence, assume t h a t a surface t i d e a t the M, frequency generates three internal modes a t x = 0. I f the wave properties are given in Table 7 , then equation ( 3 ) may be used t o calculate the amplitude and phase of a ' t i d a l constituent' a t various depths and distances from the source of the internal waves. The values are given i n Table 8. I t will be seen t h a t even a very simple example can produce the type of features t h a t are f o u n d i n the Middle Estuary. The phase speeds of the "tidal constituent" vary considerably and are n o t consistent with a simple progressive wave, there i s considerable v a r i a t i o n i n the amplitude, and the depth dependence of b o t h the amplitude and the phase are similar t o those observed. If a r e a l i s t i c topography and a cross-channel variation, such as due t o Poincare waves were introduced, the variations could very well look even more l i k e those observed in the Middle Estuary. If the S t . Lawrence were a uniform channel with a constant density s t r u c t u r e , then the amplitudes and phases of the internal waves could be calculated from a number of simultaneous current meter records i n a f a i r l y restricted area. This information could then be used t o predict the tidal currents a t any point in the Estuary from the knowledge of

205 Mode Number

Amplitude (m/s)

0 1

2 3

TABLE 7

1Om

0.50 0.21 0.15 0.08

0.50 0.17 0.03 0.04

Wave-number 4

0

7

0.50 -0.12 -0.02 0.03

0.00406 0.1093 0.23R9 0.3570

Wave parameters for the example i n Table 8

Distance

Surf.

(km)

TARLE 8

Surf.

Am l i t u d e Flflm 40m

I

Phase

Surf.

In m

40m

Phase Barotropic

A m p l i t u d e a r d p h a s e of a " t i d a l c o n s t i t u e n t " a t v a r i o u s d e p t h s and d i s t a n c e s from t h e s o u r c e , c a n p a r e d t o t h e p h a s e of t h e b a r o t r o p i c t i d e . Wave p a r a m e t e r s a r e a s i n T a h l e 7.

the amp1 i t u d e s , phases and wave number a t the source. However, quite a p a r t from the f a c t t h a t the S t . Lawrence i s not a simple channel and has large and i m p o r t a n t topographic f e a t u r e s , i t i s shown in Muir (1982) t h a t the horizontal and vertical density s t r u c t u r e i t s e l f i s quite complicated.

Since the density s t r u c t u r e i s a cornpl icated

function of b o t h space and time, then the wave numbers and the modal s t r u c t u r e a t the generation point would be a function of time and t h e propagation properties of the whole water mass would also he a function of b o t h space and time. Hence the ' t i d a l c o n s t i t u e n t s ' , as defined by Equation ( 3 ) , a t any given location will he a function o f time as well as a function of the density s t r u c t u r e which l i e s between the measuring p o i n t and the generation point.

206

6.

DISCUSSION

The purpose of t h i s paper was t o draw out some of the information a v a i l a b l e from t h e t i d a l c u r r e n t records t h a t have been c o l l e c t e d on

the Middle Estuary. The admittance method c u r r e n t s has been very useful in r e v e a l i n g records. However, the i n c o n s i s t e n c i e s a r e t h e t i d a l c u r r e n t s in the S t . Lawrence a r e

o f a n a l y s i s of t i d a l

the i n c o n s i s t e n c i e s in t h e s e due t o the assumption t h a t due t o s u r f a c e , o r b a r o t r o p i c , t i d e s alone. I t has been shown t h a t the i n c o n s i s t e n c i e s may be explained i n a q u a l i t a t i v e manner i f i t i s assumed t h a t t h e r e a r e b a r o c l i n i c , o r i n t e r n a l , t i d e s present which a l s o c o n t r i b u t e t o the observed t i d a l c u r r e n t s . Unless the i n t e r n a l t i d e s a r e taken i n t o account, i t will be impossible t o p r e d i c t t h e c u r r e n t s i n the Middle Estuary, even though t h e s t a t i s t i c s of t h e t i d a l analyses i n d i c a t e t h a t very good l e a s t - s q u a r e s f i t s a r e obtained. REFERENCES A U B I N , F . , T.S. MURTY, and M.I. EL-SABH, 1979. Numerical Simulation of t h e Movement and Dispersion of Oil S l i c k s i n the Upper S t . Lawrence Estuary: Preliminary R e s u l t s . Le Natural i s t e Canadien, 106: 37-44.

B U D G E L L , W.P. and L.R. MUIR, 1975. S t . Lawrence River Current Survey 1974 Data Report. Ocean & Aquatic S c i e n c e s , Cent. Region, Dept. Fish. E n v i r . , 335 pp. EL-SABH, M.I., T.S. M U R T Y , and L. LEVESQUE, 1979. Mouvement des Eauxs I n d u i t s Par La Mar6e e t l e Vent dans 1 ' E s t u a i r e d u Saint-Laurent. Le Natural i s t e Canadien, 106: 89-104. FORRESTER, W.D., 1974. I n t e r n a l Tides in S t . Lawrence Estuary. Marine Research, 32: 55-66.

GODIN, G., 1972. The Analysis of Tides. Toronto, 264 pp.

J.

U n i v e r s i t y of Toronto,

GODIN, G., 1976. T h e Reduction of Current Observations with t h e Help of t h e Admittance Function. Technical Note 14 - Mar. E n v i r . Data. Serv. Dept., Fish. E n v i r . , 13 pp. G O D I N , G . , 1979. La Mar6e Dans l e Golfe et 1 ' E s t u a r i r e du S a i n t Laurent. Le Natural i s t e Canadien, 106: 105-121.

207

L E B L O N D , P.H., 1978. On Tidal Propagation I n Shallow Rivers. Geophysical Res. 8 3 : 4717-4721. LEBLOND, P.H. and L.A. MYSAK, 1978. Waves in the Ocean. S c i e n t i f i c Pub1 ishing Co. , Amsterdam, 602 pp.

J.

Elsevier

R U I R , L . R . , 1978. S t . Lawrence Current Survey: 1975 Data Report. Ocean and Aquatic Sciences, Cent. Region, Dept. Fisheries and t h e Environment. 315 p p .

M U I R , L.R., 1979. S t . Lawrence River Oceanographic Survey: 1977 Data Report. Ocean and Aquatic Sciences, Cent. Region, llept. Fisheries and the Environment. Tidal , Met. and Current Meter D a t a , Vol 1, 199 p . , P r o f i l e D a t a , Vol. 2, 278 pp.

.

MUIR, L.R., 1982. V a r i a b i l i t y of Temperature, S a l i n i t y and Tidal YAveraged Density i n the Middle Estuary o f the S t . Lawrence, Atmosphere- Ocean, ( i n the p r e s s ) . M U N K , W. and n. CARTWRIGHT, 1966. Tidal Spectroscopy and Predict on. Phil. Trans. Royal SOC. London, Ser. A. , 259: 533-581.

O U E L L E T , M.Y. and M.J.F. CHEYLUS, 1971. Etude d u ModGle Mathgmatique du La Propagation Des Marees dans l e Fleuve S t . Laurent. Report CRE-71-05, Lava1 University. 37 pp. P R A N D L E , D. and N. CROOKSHANK, 1974. Numerical Model of the S t . Lawrence Estuary. J . Hyd. D i v . ASCE, 100: 517-529.

SOULSBY, R . L . , 1978. The Use of Depth Averaged Current t o Estimate Bed Shear Stress. Internal Document 26 , IOS, Crossway, Taunton, Somerset

.

208

SIMULATION OF THE LOW FREQUENCY PORTION OF THE SEA LEVEL SIGNAL AT YARMOUTH, NOVA SCOTIA D.L.

DEWOLFE AND R.H.

LOUCKS

N.S., and R.H. Loucks

Bedford I n s t i t u t e of Oceanography,Dartmouth, Oceanology L t d . , H a l i f a x , N.S. , Canada

ABSTRACT

I n t h e c o u r s e of necessary t o gap-fill

i t was

f i s h e r i e s oceanography r e s e a r c h ,

an e x i s t i n g t i d a l r e c o r d a t Yarmouth, N.S.

t o hindcast the record f o r a five-year ment of t h e t i d e gauge,

found and

period p r i o r t o the establish-

s p e c i f i c a l l y f o r low f r e q u e n c y ( p e r i o d s >12

hours) o s c i l l a t i o n s .

A " n e i g h b o u r i n g " t i m e series, a t S a i n t J o h n , N.B., by

cross-spectral

analysis

d e s i r e d low f r e q u e n c i e s .

to

be

coherent

with

was determined Yarmouth

at

the

A f t e r s u i t a b l e low p a s s f i l t e r i n g and d e c i -

mation of t h e two t i m e s e r i e s f o r t h e l o n g e s t p e r i o d f o r which b o t h had

complete

lated.

records,

the

"admittance"

The r e s u l t i n g g a i n and phase

between t h e

information was

Transformed i n t o 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 , input. response

two was c a l c u then Fourier

w i t h S a i n t John as

The S a i n t John r e c o r d was t h e n f i l t e r e d w i t h t h e impulsef u n c t i o n w e i g h t s which produced

a s y n t h e t i c low f r e q u e n c y

s i g n a l f o r Yarmouth which d i f f e r e d from t h e a c t u a l by a b o u t 5%. The paper d e s c r i b e s i n d e t a i l t h e t h e o r y and methods used i n t h i s novel a p p l i c a t i o n , t o g e t h e r w i t h t h e r e s u l t s .

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

0

209 INTRODUCTION

I n t h e c o u r s e of f i s h e r i e s oceanography r e s e a r c h i n t o t h e p o s s i b l e i n f l u e n c e of e n v i r o n m e n t a l f a c t o r s on h e r r i n g y e a r - c l a s s

s t r e n g t h off

1978), it was noticed t h a t

Southwest Nova S c o t i a ( M e t u z a l s , e t a l . ,

t h e r e w a s a r e l a t i v e l y s t r o n g c o r r e l a t i o n between the. sea l e v e l s i g n a l a t H a l i f a x and t h e h e r r i n g y e a r - c l a s s correlation further,

strength.

To e x p l o r e t h i s

i t was c o n s i d e r e d d e s i r a b l e t o i n v e s t i g a t e t h e

u s e of t h e s e a l e v e l s i g n a l a t Yarmouth as a p o s s i b l e i n d i c a t o r of t h e oceanographic s i t u a t i o n y e a r t o year.

However,

t h e p e r i o d of

r e c o r d a v a i l a b l e a t Yarmouth was r e l a t i v e l y s h o r t and t h e r e c o r d its e l f c o n t a i n e d gaps.

It w a s d e c i d e d t o d e v e l o p a means of s i m u l a t i n g

sea l e v e l s a t Yarmouth, b o t h t o f i l l g a p s i n t h e r e c o r d of o b s e r v a t i o n s and t o e x t e n d t h e s i g n a l beyond t h e p e r i o d of r e c o r d (from 1960 t o 1 9 6 6 ) , t h u s g i v i n g a c o n t i n u o u s low f r e q u e n c y s i g n a l from 1960 t o The d a t a a t

1978.

S a i n t John,

continuous except f o r s m a l l gaps,

would be used as a " l e a d i n g i n d i c a t o r " ,

due t o t h e s i m i l a r i t y and

p r o x i m i t y of t h e two t i d a l s t a t i o n s .

DATA

The d a t a , o b t a i n e d from Marine E n v i r o n m e n t a l Data S e r v i c e s (MEDS) i n Ottawa,

c o n s i s t e d of

(1960-1976)

and Yarmouth, N.S.

hourly t i d a l heights

f o r S a i n t John,

N.B.

(1966-1978).

PROCEDURE

The p r o c e d u r e ,

i n capsule,

is t o take the cross-spectra

Yarmouth and S a i n t John a s d e f i n i n g t h e the

two s i g n a l s .

transfer

The impulse

function with

Saint

between

transfer function r e l a t i n g

r e s p o n s e ( t i m e domain) form of John

as

input

yields

the

the

simulated

Yarmouth s i g n a l as o u t p u t . Now we c a n d e s c r i b e t h e p r o c e d u r e i n more d e t a i l . record

complete

selected. t o be period

i n both

Yarmouth

and

A p e r i o d of

S a i n t John o b s e r v a t i o n s w a s

A c t u a l l y , t h e l o n g e s t such p e r i o d was s e l e c t e d .

t h e b a s i s on which selected

was

the cross-spectra

somewhat

more

than

a

This w a s

were c a l c u l a t e d .

The

year

and

in

duration

210 i n c l u d e d two f r e s h e t s e a s o n s .

T h i s t u r n e d o u t t o be a d i s a d v a n t a g e

because t h e f r e s h e t of t h e S a i n t John R i v e r i s t h e one f e a t u r e which S a i n t John s i g n a l s w i l l

t h e Yarmouth and

f u t u r e t h i s w i l l be a v o i d e d .

In

n o t have i n common.

For e a c h s i g n a l and f o r t h e s e l e c t e d

p e r i o d , a s t r o n o m i c a l l y p r e d i c t e d v a l u e s of sea l e v e l s were s u b t r a c t These r e s i d u a l s were low-pass

ed.

intervals

from

t i d a l signal. gains

and

one-hour

f i l t e r e d and decimated t o s i x - h o u r

intervals,

which

effectively

removes

the

C r o s s - s p e c t r a were t h e n c a l c u l a t e d and t h e r e s u l t i n g

phase

differences

were

obtained.

These

are

listed

in

John

to

T a b l e 1.

TABLE

1.

Coherence, Yarmouth.

gain

and

phase

difference,

Saint

( N e g a t i v e phase means Yarmouth Leads.)

Freq. Band

Coh.

Gain

Phase

0

0.69

0.73

-11.6"

1

0.94

1.12

2

0.95

1.09

-

3

0.93

0.96

4

0.93

0.95

5

0.83

6

Freq. Coh.

Gain

Phase

9

0.58

0.44

- 5

4

10

0.63

0.49

-10

8

11

0.74

0.60

1

12

0.47

0.45

-32

8

13

0.51

0.47

-1 5

0.75

1

14

0.20

0.17

-33

0.89

0.82

0

15

0.36

0.21

-1 9

7

0.86

0.88

1

16

0.30

0.19

17

8

0.72

0.59

-

Rand

3

-16

F o l l o w i n g Holloway ( 1 9 5 9 ) ,

we can proceed from t h e d e f i n i t i o n o f

t h e impulse r e s p o n s e ( t i m e domain)

function t o the a c t u a l algorithm

f o r i t s c o m p u t a t i o n u s i n g a s i n p u t t h e v a l u e s of g a i n and phase d i f f e r e n c e s ( T a b l e 1) i n t h e f r e q u e n c y domain, as f o l l o w s :

211 w(t)

=-,I"

G ( f ) exp [ i ( Q ( f )

where G ( f ) = G(-f)

... w ( t )

=

- I-" G(f)

-

271 f t ) ] df

(1)

and @ ( f ) = - a ( - f ) exp [ i ( @ ( f )

-

271 f t ) ] d f

0

+

("

- 271 f t ) ] df

G ( f ) exp [ i ( Q ( f )

0

Now s u b s t i t u t i n g f ' = -f w(t) =

i n (2) gives

(-G(f')

exp [ - i ( @ ( f ' )

I" G ( f )

exp [ i ( G ( f )

-

0

+ 0

("

= 2

-

271 f ' t ) ] d f '

271 f t ) ] d f

- 271 f t ) df

G ( f ) cos ( Q ( f )

0

= -1

m [G(o) c o s CYo)

+ 2

2m

+

1

G(q) c o s {@P(q>-

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

=

,<- =

expected value s t a n d a r d d e v i a t i o n o f o b s e r v e d d a t a and i t s residues

N

=

R

=

number o f d a t a p o i n t s

I

A exp(iG) -

'A

exp(iZ)

I

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 c e n t e r o f t h e e a r t h . Therefore,

t h e f o r m e r i s c a l l e d t h e s u r f a c e t i d e and t h e

l a t t e r i s called the geocentric t i d e .

According t o t h e r e s u l t

o f P a r k e ( 1 9 7 8 ) , t h e d i f f e r e n c e between t h e s e two t i d e s i s

n e q l i q i b l e f o r t h e semi-diurnal

tide.

The phase l a q o f t h e

d i u r n a l s u r f a c e t i d e i s about 15" l a r g e r t h a n t h a t o f t h e q e o c e n t r i c t i d e , and i t s a m p l i t u d e i s 15% l a r q e r .

222

T a b l e 2 shows t h e r e s u l t s o f t h e h a r m o n i c a n a l y s i s . The s t a n d a r d e r r o r o f t h e a n a l y s i s i s l a r q e l y due t o t h e l a r q e and t h e ill c o n d i t i o n o f

standard d e v i a t i o n i n t h e residues, the covariance m a t r i x .

The e s t i m a t e d v a l u e s o f t h e t i d a l

component d i f f e r s i q n i f i c a n t l y f r o m t h e e x p e c t e d v a l u e s .

The

d i s c r e p a n c y i s about 90 cm i n t h e d i u r n a l f r e q u e n c y band, w h i c h i s about t w i c e t h e a m p l i t u d e o f t h e s t r o n q e s t d i u r n a l component, K 1 .

The d i s c r e p a n c y f o r t h e s e m i - d i u r n a l

tide i s

about 30 cm w h i c h i s o n l y o n e - t h i r d o f t h a t o f t h e d i u r n a l tide,

and i s about h a l f o f t h e a m p l i t u t e o f t h e s t r o n q e s t

semi-diurnal

t i d e , M2.

I n qeneral, the discrepancy i s well

w i t h i n the estimated e r r o r .

V II

CONCLUSION This study concludes t h a t t h e major lobe o f t h e

s p e c t r u m of t h e d a t a s a m p l e r w i t h an i r r e q u l a r s a m p l i n q i n t e r v a l r e p e a t s a t an i n t e r v a l 1/MT where MT i s d e f i n e d as 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 s .

A t any f r e q u e n c y

o u t s i d e t h e major lobe, t h e spectrum m a i n t a i n s a l e v e l o f about 5% o f t h e m a j o r l o b e . The 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 shows t h e e f f e c t o f a l i a s i n q and, t h e r e f o r e , t h e c o v a r i a n c e v e c t o r and t h e r e s u l t o f t h e h a r m o n i c a n a l y s i s c o u l d be b i a s e d .

The

a l i a s i n q i s caused b y t h e f a c t t h a t t h e d a t a c a n ' t be smoothed p r i o r t o t h e a n a l y s i s t o r e d u c e t h e u n d e s i r e d components i n t h e s p e c t r u m .

The m a j o r s o u r c e s o f e n e r q y

c o u l d be t h e l a r q e r e s i d u e s o f t h e g e o i d a l h e i g h t i n t h e sea s u r f ace he i9 h t d a t a. . The r e s u l t o f t h e a n a l y s i s i s u n s a t i s f a c t o r y due t o

t h e l a r g e noise l e v e l i n t h e data.

The s t a n d a r d e r r o r of t h e

223

e s t i m a t e i s about 5 0 cm, which i s about equal t o t h e a m p l i t u d e o f K 1 and about o n e - h a l f t h e a m p l i t u d e o f M2 i n t h e area. REFERENCES Brown, R.D., and M.K. H u t c h i n s o n , 1980: Ocean t i d e determination from sate1 l i t e a l t i m e t r y . Presented a t COSPAR/SCOR/IUCRM Symposium on Oceanoqraphy f r o m Space, Venice, I t a l y , May 26-30. Ku, L.F., 1982: The c o m p u t a t i o n o f t i d e s f r o m GEOS-3 a l t i m e t e r data. Maul, G.A. and A . Yanaway, 1977: Deep sea t i d e d e t e r m i n a t i o n f r o m GEOS-3. NASA CR-141435. 1978: G l o b a l n u m e r i c a l models o f t h e open ocean Parke, M.E., t i d e s M2, S z r K 1 on an e l a s t i c e a r t h . Ph.D. t h e s i s , U n i v . o f C a l i f o r n i a , San D i e q o . 1980: T i d e s on t h e P a t a g o n i a n s h e l f f r o m t h e Parke, M.E., SEASAT r a d a r a1 t i m e t e r . P r e s e n t e d a t C O S P A R / S C O R / I U C R M Symposium on Oceanoqraphy f r o m Space, Venice, I t a l y , May 26-30. Rapatz, W.J. and W.S. H u q g e t t , 1975: P a c i f i c Ocean o f f s h o r e t i d a l program. P r e s e n t e d a t I C G U m e e t i n q i n Grenoble, France. 1979: G l o b a l Ocean t i d e s , P a r t 11: The S c h w i d e r s k i , E.W., s e m i - d i u r n a l p r i n c i p a l l u n a r t i d e (Mz), A t l a s o f t i d a l c h a r t s and maps. Naval S u r f a c e Weapon C e n t e r , NSWC TR 79-414. Won, I . J . and L.S. M i l l e r , 1978: Oceanic q e o i d and t i d e s o b t a i n e d f r o m GEOS-3 s a t e l l i t e d a t a i n t h e N o r t h w e s t e r n A t l a n t i c Ocean. NASA CR-156845.

224

T . E . 1JNNY l l n i v e r s i t y of Waterloo

INTRODUCTION l i y d r o log i c d a t a pr i iiiar i

and t o o t h e r \ r a r i a l i l e s i n c

l i ~ . d i - 1oo g i c

II>~Llrolog \‘;l

1

c t I\c c n

11

t irnc s c r i i ‘ s . d i ffc\rcnt.

Reprinted from Time Series M e t h o d s in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

@

225

45

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<

TIM

(11) NIGER R I V E R ( Y e a r l y d a t a ) a t \rIj2i.ltY: (Time F c r i o d 1940-1966)

I h Yl..Al?S

F i g . 1 . Group f o r i n a t i o n i n ) . e a r l y mid nionthly t i m e s e r i e s o f f l o ~ I n c o n t r a s t , tlic monthly time s e r i e s a r e marAcd 11)- s e a s o n a l it), ‘i’herc i s a l s o t h e prcx-

a c c o r d i n g t o t h e g e o p h y s i c a l year ( F i g . l a ) .

s e n c e o f c h a r a c t e r i s t i c g r o u p s o f h i g h arid lori v a l u e s aillong t h e !‘cars and a l s o w i t h i n t h e y e a r . The d a i l y time series a r e d i f f e r e n t froin n i o n t h l > ~xiid ).early t inre

series i n t h e s e n s e t h a t t h e s e t i m e s e r i e s a r c c h a r a c t c r i s c d 11). tlic o c c u r r e n c e o f sharp peaks and e x p o n c n t i : i l deca!-.

The c : i u s c - c f f e c t

r e l a t i o n s h i p i n r a i n f a l l - r u n o f f process i s s t r o n g e r i n t h e s h o r t

iii-

t e r v a l time s e r i e s . The p e r i o d i c i t i e s a r c i n d u c e d i n t h e s t r e a n i f l o w time s e r i c s liy

t h e geophysical cycle.

This is reflected i n thc o e c ~ i r ~ c ~ ~ ofi ehigl: c

p r e c i p i t a t i o n and h i g h r u n o f f d u r i n g tlic

nionths, tind lo\c pre-

smiiiicr

c i p i t a t i o n and low r u n o f f d u r i n g t h e w i n t e r iiioriths i n N o r t h c r r i c 1 i mates.

Means a n d v a r i a n c e s o f h y d r o l o g i c

larger i n

sLiinnier

tiiiic

a n d snial ler i n w i n t e r riioriths

scric’s

.

c i t y i n time a l s o o c c u r s i n t h e forin o f trc>nd i n the

:ire

foiiiicl

I ~ ’ u r t ! i c ~ rlion-hciiiiogcri, rl;it:i

;is \ i c ’ 1 1

;is

iii

forin o f g r a d u a l and suddcri v a r i a t i o n s i n t h e stoc1i;ist ic- n:itiiw

226 o f t h e JLita. 'l'he i i i i p l i c a t i o n a r i s i n g from t h e p r e s c n c c o f p e r s i s t e n c c , e s p c c i -

t h a t c o r r e s p o n d i n g t o prolonged wet and d r y p e r i o d s , i s s i g n i -

all!.

f i c a n t from t h e p o i n t o f \.icw of Liater resources . d e s i g n and o p e r a t i o i l s .

Both s h o r t - t e r m p e r s i s t c n

and 1 ong- t crm

p e r s i s t e n c e a r e i i i t c g r a l p a r t s o f h y d r o l o g i c t i m e s e r i e s which t l l e r c b y bccoiiie d i f f e r e n t from those tiine s c r i e s found i n o t h e r d i s c i p 1 i n e s swh

stock-niarket anal>.sis.

;IS

An 1 i i s t o r i c : r l 1). r c c o r d e d s t r c a n i f l o i i tiinc s c r i c s c m lie considcred

t o lie d c r i \ . e i i frorii

;I

s t o c l i a s t i c g e n e r a t i n g nicchanisiri t h a t e v o l v e s

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 , \<:is f i r s t h r o i i g l ~ t t o the, : i t t c i i t i o n of c n g i n c c ~ - sI)?

t h e pionceriiig s t u d i e s of (Ilurst, 1951, 1 9 S h ) on l o n g

tcrni s t o r a g e rcqiiireiiicnts i n the> N i l c I h s i n .

232 Through a n e x t e n s i v e a n a l y s i s of g e o p h y s i c a l time s e r i e s (cons i s t i n g o f a n n u a l v a l u e s ) , i n c l u d i n g t h e e x t r e m e l y long time s e r i e s o f t h e a v e r a g e a n n u a l f l o w on t h e r i v e r k i l e , and also o f normal i n d e p e n d e n t s e r i e s g e n e r a t e d by v a r i o u s e x p e r i m e n t s , t l u r s t d e r i v e d the

re 1a t i o n s h i p RK/SN

01

H N

where N i s t h e l e n g t h o f d a t a .

I n the above, R

N

i s t h e ad u s t e d

r a n g e and i t c a n h e e x p r e s s e d f o r an a n n u a l time s e r i e s {x , i = l , 2 , . . . ,

N) o f

and s t a n d a r d d e v i a t i o n N

l e n g t h K y e a r s , w i t h mean x

The term R ./S

i \ , N

[=i N) i s

the adjusted rcscalcd range.

h i

S"

as

c s t iriiatc of

H, d e n o t e d by K l l , was d e f i n e d a s

Hurst observed t h a t t h e value of the exponent I1 i n r e l a t i o n s h i p (0) h a s on the a v e r a g e a v a l u e o f 0.7.3 f o r t h e g e o p h y s i c a l t i m e s e r i e s , and 0 . 5 f o r t h e normal i n d e p e n d e n t s e r i e s .

In hydrologic litcratur-c,

t h e d i s c r e p a n c y i n t h c v a l u e s o f t h e e x p o n e n t i n h y d r o I O g i c a 1 time

s e r i e s and t h a t i n a l l i n d e p e n d e n t s e r i e s has lice11 called t h e llurs-i phenomenon and t h e e x p o n e n t i n r e l a t i o n s h i p Hurst c o e f f i c i e n t .

((I)

i s noii I\rlo\in as t h e

A v a l u e o f t h e ilurst c o e f f i c i c n t g r e a t e r t h a n

0.5 i s c o n s i d e r e d t o i n d i c a t e l o n g - t e r m p c r s i s t e n c c . 1)arnmcter v a l u e s f o r t h e l l u r s t c o e f f i c i e n t d e r i v e d frorii h i s t o r i

c a l r e c o r d i s found s e n s i t i v c t o non-hoiiiogerieities

i n d a t a inc1udi:ig

c h a n g e s i n t h e p r o b a l ~ il i s t i c laws d e f i n i n g t h e g c n e r ; i t iiig of t h e d a t a .

c i e n t g r e a t e r than O.S. to

iiic~c~iiatiisiii

Klemes [ 1 9 7 4 ) h;is sho\\,ri t h a t i n d c ~ p c n d c n t s c r i e s

s i s t e r i c e o f o r d e r zero) w i t h f l u c t w t i n g A r c f e r c ~ n c ciiia!. ~

iiic;ins

-

c s h i I 1 i tcil

a l s o be ~ti:iclc

;it

;in

(~CI.-

Il-coct

this point

Wing (19Sl) who, w h i l e coirimcnting on I l u r s t ' s o r i g i n a l p : i p c > r cx-

pre.;sed doulit a b o u t I l u r s t ' s f i n d i n g s a n d i 1 n p 1 i c d t h a t p e r h a p s

233 d i s c o n t i n u i t i e s i n t h e r e c o r d could have caused an H - c o e f f i c i e n t greater than 0.5. E x t e n s i v e a n a l y s i s of l a r g e assemblage of r e c o r d e d d a t a by ( H u r s t , 1951) encompassing many g e o p h y s i c a l phenomena, e . g . , r a i n f a l l , runo f f , l a k e l e v e l s , t r e e r i n g s and mud v a r v e s , showed t h a t groups of high and low v a l u e s t e n d e d t o o c c u r more f r e q u e n t l y i n n a t u r a l e v e n t s than i n p u r e l y random e v e n t s .

I n a d d i t i o n , H u r s t observed w i t h p a r -

t i c u l a r r e f e r e n c e t o annual streamflow time s e r i e s t h a t groups a s s o c i a t e d w i t h s t r e t c h e s of f l o o d s and d r o u g h t s o c c u r r e d w i t h o u t any r e g u l a r i t y e i t h e r i n t h e i r d u r a t i o n or i n t h e time of o c c u r r e n c e ( F i g . 1 ) . T h i s , t h e n , i s t h e fundamental d i f f e r e n c e between n a t u r a l s t r e a m flow time s e r i e s and o t h e r p u r e l y 'man-made'

s e r i e s such a s those

derived from random p r o c e s s e s , a u t o r e g r e s s i v e p r o c e s s e s and f r a c t i o n a l Gaussian n o i s e s e q u e n c e s . On Some of t h e Commonly Used Models f o r Hydrologic Time S e r i e s C e r t a i n s t o c h a s t i c p r o c e s s e s have been s u g g e s t e d by h y d r o l o g i s t s as models f o r time s e r i e s .

S.pecified s t a t i s t i c s of t h e chosen p r o -

cess a r e a d j u s t e d t o have n u m e r i c a l v a l u e s e q u a l t o t h a t of e q u i v a l e n t p a r a m e t e r s e v a l u a t e d from t h e observed s e r i e s .

The term used

i n h y d r o l o g i c l i t e r a t u r e i s p r e s e r v a t i o n . Thereby i t i s meant t h a t a l l sample f u n c t i o n s o b t a i n e d from t h e p r o c e s s d e l i v e r t h e same p a r a -

meter v a l u e s .

Considered s i g n i f i c a n t i n t h i s c o n n e c t i o n are one o r

more of t h e f o l l o w i n g :

Mean o f t h e s e r i e s , v a r i a n c e , c o r r e l a t i o n

c o e f f i c i e n t s a t l a g 1 and a t h i g h e r l a g s , and Hurst c o e f f i c i e n t . Two commonly used models a r e d i s c u s s e d below. Mandelbrot and van Ness (1968a) d e f i n e f r a c t i o n a l Brownian motion p r o c e s s (fBm) a s : 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.

.

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 <

....,

266

km - km-l

-

-

dk, wm -

Wm-l

= dw and c 2

mn

=

2 S ( w n , k )dkdw where m

S ( w , k ) i s t h e wave number-frequency s p e c t r a l d e n s i t y f u n c t i o n of t h e random f i e l d 5 d e f i n e d on t h e r a n g e k 2 0 and - - < w < w . The p h a s e a n g l e s

4

a r e mutually independent s t o c h a s t i c v a r i a mn Eq. (1) is a b l e s , homogeneously d i s t r i b u t e d o v e r [ O , ~ I T ] . d i s c r e t i z a t i o n of a s t o c h a s t i c i n t e g r a l giving t h e s p e c t r a l r e p r e s e n t a t i o n of a s t a t i o n a r y random f i e l d ,

(see e.g.

Wong,

1 9 7 1 ) . I n o u r c a s e t h e random f i e l d i s r e a l and n o r m a l l y d i s t r i b u t e d which e n a b l e s us t o s t a r t from a l e s s g e n e r a l s h a p e of t h e s p e c t r a l r e p r e s e n t a t i o n . The e n v e l o p e f u n c t i o n R ( x , t ) and phase f u n c t i o n O ( x , t ) a r e d e f i n e d by c ( x , t ) = R ( x , t ) cos O ( x , t ) where

S(x,t)

i s t h e H i l b e r t transform of m

r;

=

R(x,t) s i n O(x,t)

(2)

g i v e n by

co

which h a s s i m u l a r p r o p e r t i e s a s 5 . For p o i n t s c l o s e t o a f i x e d p o i n t ( x , t ) we may w r i t e a s a f i r s t a p p r o x i m a t i o n < ( x , t ) = 0

0

R ( x , t ) C O S { ~ ' ( X -)X + 0

6 ( t - t 0) + 0) where

0

=

a O ( x o , . t o ) / a t , 0' =

a 0 ( x , t ) / a x and 0 = O(x , t o ) .So, a p p r o x i m a t e l y , t h e i r r e g u l a r 0

0

0

waves behave l o c a l l y and i n s t a n t a n e o u s l y a s a r e g u l a r wave propagating i n t h e p o s i t i v e x-direction with amplitude R , phase

0 , wave number 0 ' and f r e q u e n c y - 0.We f u r t h e r assume t h a t t h e i r r e g u l a r waves g e n e r a t e l o c a l l y and i n s t a n t a n e o u s l y a Lagrangian v e l o c i t y a s i f t h e r e i s a r e g u l a r p r o g r e s s i v e wave on t h e s u r f a c e w i t h above-mentioned v a l u e s f o r t h e a m p l i t u d e , p h a s e , f r e q u e n c y and wave number.

I t can b e r e a d i l y d e r i v e d by u s i n g

methods a s i n d i c a t e d by e . g . P h i l l i p s t h a t a harmonic wave

5

= R cos

(kx cos

x

+

ky s i n

x

- w t

+ 0) which

h a s an a n g l e

x

w i t h t h e p o s i t i v e x - a x i s g e n e r a t e s t h e f o l l o w i n g Langrangian v e l o c i t y components i n t h e x-,

y- and z - d i r e c t i o n r e s p e c t i v e l y

267 k wR c o s h k ( z + d ) X

u ( ~ , t , w , R , k , k ,O) = X

2

wR 2k

k R w cosh2 k(z+d) X

s i n J)

+

COSJ)

k s i n h kd

Y

sinh2 k(z+d) X

+

s i n h 2 kd

c o s J)

s i n h 2 kd

k wR c o s h k ( z + d ) v(z,tIw,Rrkx,k Y

,O)

=

k s i n h kd

X

+

s i n h L kd

s i n h L kd

w (x , t ,w,R,k

COSJ)

k

Y

,o)

=

i n which $ = k x - w t

wR s i n h k ( z + d ) sin+ s i n h kd

+ 0

and where x = ( x , y , z ) and k

X

=

k cos

x

x

= k s n a r e t h e wave number components i n x- a n d yY d i r e c t i o n , C o n s e q u e n t l y , t h e L a g r a n g i a n v e l o c i t y components

and k

u1 and w

g e n e r a t e d l o c a l l y a n d i n s t a n t a n e o u s l y by t h e one1 d 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 a r e d e r i v e d from ( 4 ) by s e t ting w = -

6, k

= 0'

and k

X

Y

=

0 . The mean v a l u e o f t h e h o r i -

z o n t a l c u r r e n t i s g i v e n by 03

E{ul) =

1 dR

0

m

-03

2Tl

m

d6

1

-03

dO'

1

d O u(x,t, - b,R,O',O)

(5)

p(R,@,O,O')

0

i n which p ( R , O , O , O ' ) i s t h e

j o i n t d e n s i t y of R , O , O a n d 0 ' .

d e n o t e s a m a t h e m a t i c a l e x p e c t a t i o n . The v a r i a n c e of t h e 2 2 2 h o r i z o n t a l c u r r e n t i s d e t e r m i n e d from 0 = E{ul 3 - ( E { u l ) ) u1 The mean v a l u e and t h e v a r i a n c e of t h e v e r t i c a l v e l o c i t y com-

E(.)

.

ponent w

a r e c a l c u l a t e d i n a s i m i l a r way. The e v a l u a t i o n of 1 t h e j o i n t p r o b a b i l i t y d e n s i t y p(R,O,O,O') i s done i n t h e u s u a l

way a s i n d i c a t e d e . g . -

b y R i c e . The j o i n t d e n s i t y

I

p ( < , t , < , < , < ' , i ' ) of 5 ,

and t h e i r p a r t i a l d e r i v a t i v e s

i,i,<'

I

and

<

w i t h r e s p e c t t o t i m e and s p a c e i s d e t e r m i n e d f i r s t . T h i s

d e n s i t y i s j o i n t l y normal. For n o t a t i o n a l c o n c e n i e n c e w e s e t

268

-

I

z1

=

c,z2 =

<,

z3

-

z4 =

=

c1

z5 = 5 and z

(6)

6 = 5'

Then we f i n d

i n which z i s a column v e c t o r w i t h components z l ,

...I z

and 6 its t r a n s p o s e . M r e p r e s e n t s t h e c o v a r i a n c e m a t r i x g i v e n by

T .

z

-

M = -

E(z z

2 1

)

2

Substituting eqs. ions f o r z l I

.....

E(Z2)

...,

( 1 ) and ( 3 ) i n ( 6 ) and t h e r e s u l t i n g e x p r e s s z

6

i n ( 8 ) yields

where

d e f i n e s t h e s p e c t r a l moments o f t h e s p e c t r a l d e n s i y f u n c t i o n S ( w , k ) o f t h e random f i e l d

<(x,t). S u b s t i t u t i n g ( 9

i n (7) yields

a f t e r some a l g e b r a 1 P(Zl'Z2'

m a - ,

")

( 2 ~ )B

exp

c

-

1 2 + 7 I B ( z 2+z 0 1 4

2B

269

( 2 ) i n (6) and t h e r e s u l t i n g e x p r e s s i o n s f o r

S u b s t i t u t i n g eq. zl,

...,

z6 i n e q . ( 1 1 ) y i e l d s a f t e r m u l t i p l y i n g w i t h t h e

Jacobian a ( z

1'

z )/a(R,R,R',@,@,@= ') R * . * I

3

and a f t e r p e r f o r -

6

ming a c o n s i d e r a b l e amount o f a l g e b r a an e x p r e s s i o n f o r t h e following j o i n t d e n s i t y 3 R p ( R , R , R ' , O , O , O ' ) = ___ exp[ 3 (2n) B

+

B22 (R2+R2o2)

-

- 2B3 ( & + R 2 0 0 ' )

1 2 2B

__ {BOR2+2B1R20-2B2R20'+

+

2

2

Bq ( R ' +R20' )

)I

The p r o b a b i l i t y d e n s i t y p ( R , O , @ , O ' ) i s found by i n t e g r a t i n g t h i s e x p r e s s i o n w i t h r e s p e c t t o R and R ' , b o t h o v e r

(-m,-).

I t i s found t h e n from t h e r e s u l t i n g e x p r e s s i o n t h a t 0 i s i n d e -

pendent o f t h e o t h e r v a r i a b l e s and h a s a homogeneous d i s t r i b u t i o n over t h e r a n g e [ 0 , 2 1 ~ 1 .So we may w r i t e 1

p(R,@,O',O) = p ( O ) p(R,O,O') =

-

271

p(R,@,O')

(13)

where p(R,O,O')

R

=

3 exp

27l(booB)

c-

R ~

2 2

CBg+2B10-2B20'+B2202-2B3001+

2B

+B40 2 } 1

(14)

where a s may be s e e n from ( 1 2 ) t h e c o e f f i c i e n t s B " ,

Bl, e t c . are

which a r e d e f i n e d by ( 1 0 ) . i j However f o r s e a waves an e x p r e s s i o n f o r t h e s p e c t r a l d e n s i t y d e f i n e d by t h e s p e c t r a l moments b

S ( w , k ) i s i n g e n e r a l n o t a v a i l a b l e and w e o n l y have t h e f r e q u e n c y

spectrum S ( o ) . For t h a t r e a s o n an approximate e x p r e s s i o n f o r S ( w , k ) i s d e r i v e d by assuming t h a t t h e f r e q u e n c y and t h e wave

number a r e r e l a t e d by t h e d i s p e r s i o n r e l a t i o n

o2

=

gk(1

+ -1Tk

P9

t a n h kd

(15)

where g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , T t h e s u r f a c e t e n s i o n ,

270 d t h e d e p t h and p t h e d e n s i t y of t h e f l u i d . By d e n o t i n g t h e positive solution

for k of

( 1 5 ) by k

D(w) t h e a p p r o x i m a t i o n

=

f o r S ( w , k ) can be w r i t t e n a s S ( w , k ) = S ( W ). 6 ( k - D ( u ) )

(16)

where 6(.) i s t h e D i r a c f u n c t i o n . A s e r i o u s r e s t r i c t i o n of t h e v a l i d i t y of t h i s approximation i s t h a t

( 1 5 ) i s only v a l i d f o r

c o n s t a n t a t m o s p h e r i c p r e s s u r e above t h e waves which i s i n g e n e r a l not t h e case i n p r a c t i c a l circumstances. After a considerable amount o f a l g e b r a it may be shown t h a t f o r S ( w , k ) w i t h a s h a p e

(16) expression (14) reduces t o

a s i n eq.

p ( R , @ , O ' ) = p(R,O) 6(0' - D ( - O ) )

(17)

where

Expression (17) i s c o n s i s t e n t with t h e d i s p e r s i o n r e l a t i o n k = D(w) B2/B4

.

In d e r i v i n g ( 1 7 ) we s h o u l d make u s e o f t h e p r o p e r t i e s

-+ 0 , B3/B4

-+ 0 and B/B4'

+

0 a s S(w,k)

-f

S(w). 6(k-D(w)).

The j o i n t d e n s i t y p(R,O,O,O') may now be d e t e r m i n e d from e q s . ( 1 3 1 , ( 1 7 ) and ( 1 8 ) . S u b s t i t u t i n g t h i s e x p r e s s i o n i n ( 5 ) and p e r f o r m i n g some i n t e g r a t i o n s w e f i n d f o r t h e mean v e l o c i t y of t h e h o r i z o n t a l component o f t h e r e s i d u a l c u r r e n t m

E{u

} 1

=-

-

6k c o s h 2 k ( z + d ) k d ( b 2 0 + 2b100

+

5/2 d'

(19)

The v a r i a n c e o f t h e h o r i z o n t a l c u r r e n t may b e d e t e r m i n e d from 2 2 2 U = E{ul 1 - (E{ul}) where E{ul 1 may be d e t e r m i n e d i n a n u 1 a n a l o g o u s way a s E{u }. W e f i n d f i n a l l y 1

271 I n a s i m i l a r way w e may d e t e r m i n e t h e mean v a l u e of t h e v e r t i c a l v e l o c i t y component. W e f i n d , a s e x p e c t e d : E(wl) = 0

(21)

The v a r i a n c e o f t h e v e r t i c a l v e l o c i t y becomes

When, a s i s i n g e n e r a l t h e c a s e f o r s e a waves, t h e spectrum h a s o n l y non-zero v a l u e s f o r

w > 0

f o r waves p r o p a g a t i n g i n t h e

p o s i t i v e x - d i r e c t i o n t h e s p e c t r a l moments a p p e a r i n g i n ( 19) , ( 2 0 ) and ( 2 2 ) a r e d e f i n e d by

b.

=

10

(-1)

m

i

wi

S ( o ) dw

(23)

-m

THE MEAN AND VARIANCE O F THE RESIDUAL CURRENT GENERATED BY TWO-

DIMENSIONAL IRREGULAR SURFACE WAVES. A three-dimensional

normally d i s t r i b u t e d random f i e l d

r ; ( x , y , t ) r e p r e s e n t i n g two-dimensional i r r e g u l a r 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-y p l a n e can be g i v e n by m

m

00

Which i s a g a i n a d i s c r e t i z a t i o n o f a s t o c h a s t i c i n t e g r a l g i v i n g t h e s p e c t r a l r e p r e s e n t a t i o n of a s t a t i o n a r y random f i e l d . I n t h i s e x p r e s s i o n w e have

dk

x

=k

x

-k n+l

X

and c n

1mn

=

,

2 S(wn, k X~

k

)

Ym

dk dk dw. The p h a s e X

Y

a n g l e s a r e m u t u a l l y i n d e p e n d e n t s t o c h a s t i c v a r i a b l e s , homogeneously d i s t r i b u t e d o v e r CO,2nJ. The envelope f u n c t i o n R ( x , y , t ) and t h e

272 phase f u n c t i o n O ( x , y , t ) a r e d e f i n e d by S ( x l y , t ) = R ( x , y r t ) COSO(xryrt) where

<

S ( x , y , t ) = R ( x , y , t ) s i n O ( x , y , t ) (25)

i s t h e H i l b e r t t r a n s f o r m o f 5 which i s o b t a i n e d from

( 2 4 ) by r e p l a c i n g t h e c o s i n e by s i n e . The two-dimensional i r r e g u l a r wave f i e l d i s l o c a l l y and i n s t a n t a n e o u s l y approximated by t h e r e g u l a r wave R c o s ( 0 ' x

+

0' y

Y

X

and wave numbers 0 '

+ 6t +

0) with frequency -

6

= a0/ax and 0 '

= W/ay i n t h e x- and y. Y d i r e c t i o n . T h i s wave g e n e r a t e s l o c a l l y and i n s t a n t a n e o u s l y a X

r e s i d u a l c u r r e n t ( u2 , v 2 , w 2 ) w i t h u2

=

~ (-x , t -, O,R,O' , 0 ' X

v2 = v ( ... ) and w2

=

w(.

.. )

,O), Y

where ( u , v , w ) i s d e f i n e d i n (4). The

f i r s t and second moments o f t h e s e v e l o c i t y components may be

,O'

determined when t h e j o i n t d e n s i t y p(R,O,O,O' X

)

i s given.

Y

T h i s d e n s i t y may be d e t e r m i n e d i n a s i m i l a r way a s t h e d e n s i t y p(R,O,O,O') f o r t h e one-dimensional wave f i e l d , by s t a r t i n g from t h e j o i n t normal d i s t r i b u t i o n p ( c , t , i , ; , < ' X

ri'x,<' Y

,i'Y ) .

The

moments of t h e s p e c t r a l d e n s i t y S(w,k ,k ) a p p e a r a s p a r a m e t e r s X

Y

i n t h i s d e n s i t y . However, s i n c e f o r s e a waves i n g e n e r a l o n l y t h e d i r e c t i o n a l spectrum i s given a s u i t a b l e approximation S ( w , k ,k ) h a s t o be d e v i s e d which g i v e s a l s o an a p p r o x i m a t e exX

Y

p r e s s i o n f o r p ( R , @ , O , O ' ,O' ) . The d i r e c t i o n a l spectrum i s someX

Y

t i m e s given i n t h e form

where

x

i s t h e a n g l e w i t h t h e main d i r e c t i o n o f t h e wave f i e l d

which i s i n o u r c a s e t h e x - a x i s .

I t i s noted t h a t S ( w , x )

gives

o n l y t h e l o c a l t i m e b e h a v i o u r o f t h e two-dimensional s u r f a c e waves i n t h e v a r i o u s d i r e c t i o n s and i s o n l y a f u n c t i o n o f t h e f r e q u e n c y w and t h e r a t i o k /k X

of t h e wave number components of Y

t h e harmonic components o f t h e s u r f a c e waves. N o i n f o r m a t i o n i s given w i t h r e s p e c t t o t h e wave numbers k o f t h e harmonic compon e n t s . To meet t h i s l a c k of i n f o r m a t i o n we r e l a t e t h e wave number t o t h e f r e q u e n c y 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 . C o n s e q u e n t l y ,

273 we s e t

(27)

S ( U r k x r k ) = S ( W , X ) - 6 ( k - D(w)) Y Where w e assumed a change of v a r i a b l e s ( k

X

of t h e r e l a t i o n s t g

x

k /k

=

Y

X

and k = (k

X

,

k ) + ( X , k ) by means

+

Y

k

2 4 Y

)

.

In t h e s a m e

way a s f o r t h e o n e - d i m e n s i o n a l waves i t c a n be shown t h a t t h e

,O'

d e n s i t y p(r,O,O,O' X

)

c a n be w r i t t e n i n a s h a p e which i s

Y

consistent with (27) :

I t remains t o determine t h e j o i n t d e n s i t y p ( R , O , O , O '

X

) which

r e f e r s only t o t h e s p a t i a l behaviour of t ( x , y , t ) i n t h e x-direction. C o n s e q u e n t l y , t h i s p r o p a b i l i t y d e n s i t y i s r e l a t e d t o t h e random f i e l d C ( x , t ) which i s o b t a i n e d from (24) dy d e l e t i n g t h e y-dependent t e r m : W

W

i n which w e s u b s t i t u t e d k given by c 2 mn

=

2s (u,,x,)

=

k cos

x.

The c o e f f i c i e n t s c

dwdx where dx =

xm + l

-

Xm

.

mn

The d e n s i t y

p ( R , @ , O , O ' ) i s d e t e r m i n e d i n a siffiilar way as t h e d e n s i t y X

p(R,O,O,O') f o r t h e o n e - d i m e n s i o n a l wave f i e l d . ' W e f i n d

with

2

exp

C-

R 7

2BL i n which t h e c o e f f i c i e n t s a r e as d e f i n e d i n e q . However, f o r t h e s p e c t r a l moments b

i j

(12)

we s h o u l d t a k e

are

274

By s u b s t i t u t i n g

(26) w e o b t a i n

where w e assumed a g a i n t h a t p o s i t i v e f r e q u e n c i e s c o r r e s p o n d w i t h waves i n t h e p o s i t i v e x - d i r e c t i o n .

The mean v e l o c i t y component

i n t h e x - d i r e c t i o n i s d e t e r m i n e d from m

2.rr E(u2} =

d0

0

m

0

--m

i n which we t a k e k

m

a3

1 d 6 1 dR

=

do'

D(-O).

1 do'Y

--m

-m

u(x,t,-;,R,@' -

X

,o'

,@). Y

S u b s t i t u t i n g ( 4 ) and (30) y i e l d s

a f t e r some a l g e b r a

The v a r i a n c e i s c a l c u l a t e d from a2

U

=

2

E{u2 } - ( ~ { u , } )where ~ 2

2 E{u 2 } may be d e t e r m i n e d i n a s i m i l a r way a s E(u

2

1.

W e find

finally 2

E{u 2

+

1 } = boo3

m

--I

-2 2 0 cosh k(z+d)

~ db 4k s i n h kd

+ b36)

2

12 B (boob2 + 2 b 1 0 6 + b 2 0 ) 5/2

+

4 4 2 2 3B4i2{3cosh k ( z + d )+ 3 s i n h k ( z + d )+2cosh k ( z + d )s i n h k ( z + d )1

- 2

4 1 6 s i n h kd

5 (B2fB30) *{

3 (B2

-2 7/2 B (boo0 +2b100+b20) 3

+

1 ( b o o '02 +2b100+b20)5/2"

(36)

275 I t i s e a s y t o v e r i f y t h a t E{v } = 0 and E(w } = 0 . I t i s f u r t h e r 2 2 e a s i l y s e e n from eqs. ( 4 ) t h a t between t h e h o r i z o n t a l v e l o c i t y

u1 o f t h e one-dimensional wave f i e l d and t h e h o r i z o n t a l v e l o c i t y components u2 and v relation u

1

=

u

verse velocity v

2

2

2

+

o f t h e two-dimensional v

2

wave f i e l d t h e

e x i s t s . The v a r i a n c e a

i s t h e r e f o r e g i v e n by

u

=

V

of t h e trans-

2

E{uI2}

-

E(u2

2

1.

v2 F i n a l l y , w e may d e r i v e t h a t t h e v a r i a n c e s o f t h e v e r t i c a l v e l o c i t y components a r e i d e n t i c a l f o r t h e one- and two-dimension a l wave f i e l d .

NUMERICAL RESULTS AND CONCLUSIONS.

I n f i g u r e s l n u m e r i c a l r e s u l t s a r e g i v e n f o r t h e mean and standard d e v i a t i o n s of t h e h o r i z o n t a l t r a n s p o r t v e l o c i t y f o r a one-dimensional wave f i e l d f o r s e v e r a l w a t e r d e p t h s and wind s p e e d s u s i n g t h e Pierson-Moskowitz wave spectrum: 4 S ( w ) = 0.0081 g / w e-'.74(g/V~) w > 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 <

0

above t h e s e a

s u r f a c e . I n f i g u r e s 2 +he s t a n d a r d d e v i a t i o n of t h e v e r t i c a l t r a n s p o r t v e l o c i t y i s g i v e n f o r v a r i o u s wind s p e e d s and w a t e r depths. I n performing t h e s e c a l c u l a t i o n s w e d i d n o t bother a b o u t t h e f a c t whether o r n o t s u c h a spectrum c a n be r e a l i z e d on a s e a o f f i n i t e d e p t h , t h e more a s o n l y t h e f i r s t two moments of t h e spectrum a r e needed i n t h e s e c a l c u l a t i o n s .

I t i s noted t h a t

t h e v a r i a t i o n s o f t h e t r a n s p o r t v e l o c i t y a b o u t i t s means v a l u e a r e i n t h e r a n g e of 5 t o 15 p e r c e n t o f t h e wind v e l o c i t y . T h i s i n d i c a t e s t h a t i r r e g u l a r waves have a h i g h d i s p e r s i v e e f f e c t . I t i s s e e n from e q s .

( 1 9 ) , ( 2 0 ) and ( 2 2 ) t h a t t h e wave spectrum i n -

f l u e n c e s t h e mean v a l u e and t h e v a r i a n c e s o f t h e t r a n p o r t v e l o c i t i e s by t h e v a l u e s o f t h e s p e c t r a l moments b

b l o and b

20' The i n t e g r a n d s i n t h e i n t e g r a l e x p r e s s i o n s f o r t h e s e moments converge t o z e r o a t l e a s t a s w

-3

as w

-f

a.

00

For t h e f r e q u e n c y range

which g i v e s t h e major c o n t r i b u t i o n t o t h e s e i n t e g r a l s t h e s u r f a c e

276

I

-.z 2-

I

.

I - i - A - r I -

. - - / /

-

/

I

/

I

I

1

J

I

W I N D S P E E D 10 M/SEC D E P T H 10 AND 40 M MEAN _ _ _ _ STANDARD DEV.

v

=

I

I

1

,

Ot

I

t - - N

/

I I

I

,'D=40 M

I

I

I 1

J

1

I

2.5

.O

7.5

5.0

P E R C E N T A G E OF W I N D S P E E D

0

M/SEC .o

-

1

I

I

.o

3.0

2.0

I

I

I

/

I I h

_

I

I

5

3

I

I

10

15

P E R C E N T A G E OF W l N D S P E E D

I

I,/

I

,--. E

v

I

I

I

I

I

I

0-

I

I I I

To-

I

t---cu LL

W l N D S P E E D 30 M/SEC

I I

W

a

1

I

I I!?-

D=40 M I

'Fig.

!

I

I

1

I

1 . Mean a n d s t a n d a r d d e v i a t i o n of t h e h o r i z o n t a l r e s i d u a l c u r r e n t for a o n e - d i m e n s i o n a l i r r e g u l a r f i e l d .

277

STANDARD D E V .

(M/SEC)

WIND SPEED i 0 M/SEC D E P T H 10 AND 4 0 M

STANDARD D E V .

(M/SEC)

20 M / S E C

STANDARD D E V .

(M/SEC)

F i g . 2 . S t a n d a r d d e v i a t i o n of v e r t i c a l t r a n s p o r t v e l o c i t y .

278 tension term in the dispersion relation is negligible as compared to the gravity term. Consequently, for the one-dimensional irregular waves only the gravity waves give a significant contribution to the mean and variances of the transport velocities. It is noted that the approximation (16) for the spectrum s ( o , k ) is only valid for constant atmospheric pressure above the waves. More accurate expressions for S ( w , k ) should give better results for the mean and variance of the transport velocity. With respect to the twodimensional irregular wave field it is seen from eqs. (12), (35) and (36) that the mean and variance of the transport velocity in the x-direction are in addition to boo, b10 and b20 also determined by the spectral moments bol, bll and b02 which are given by

Since according to the dispersion relation we have for gravity waves k

- w2 as

w

+ w

the integral expression b02 diverges. Using

the dispersion realtion (15) which includes the surfaces tension we have k

-

w as w

-f

-.

Then bO2 has a finite value. Our conclu-

sion may be that the approximation (27) for the two-dimensional waves has as a consequence that the ripples which arewaves with a lenght less than around 0.1 meter and which cover less than one percent of the total energy play a decisive part. This does not agree with our physical intuition. To eliminate the influence of the higher frequencieswe collected in table 1 some results which are obtained by restricting the interval of integration for the spectral moments to the range [0,1.51. For a velocity of 20m/s this range covers 99.2% of the wave energy. More reliable results may be obtained by prescribing a more accurate expression for the frequence-wave number spectrum S(w,kx r k y ) . TABLE 1.

Mean and variance of transport velocity at the surface for a Pierson-Moskowitz wave spectrum. The integration in the spectral moment expression has been restricted to the frequency range

279 0

5 w 5

1.5. Wind speed in 20m/s.

Depth in meters

10 20 40

1-dim. wave field

2-dim. wave field

E{ul}

2 ou

E{u2}

.684 .343 .217

1 5.576 2.937 1.876

.594 .300 .195

2 oU 2 4.242 2.356 1.649

REFERENCES Alfrink, B.J. and Vreugdenhil, C.B.., 1981. Residual Currents, Analysis of Mechanisms and Model Types. Delft Hydraulic Laboratory, report R1469-11. Cramer, H. and Leadbetter, M.R., 1967. Stztionary and Related Stochastic Processes. John Wiley & Sons, New York. Phillips, O . M . , 1977. The dynamics of the Upper Ocean. Cambridge University Press, Cambridge. Rice, S.O. Mathematical Analysis of Random Noise. In: Wax, N. (Editor), 1954. Selected Papers on Noise and Stochastic Processes. Dover, New York. Wong, E., 1971. Stochastic Processes in Information and Dynamic Systems. Mc Graw-Hill, New York.

280

GENERATION OF WEEKLY STREAMFLOW DATA FOR THE R I V E R DANUBE-RIVER MA IN- SY STEM EXPERIENCES WITH AN AUTOREGRESSIVE M U L T I V A R I A T E MULTILAG MODEL L.A.

SIEGERSTETTER AND W. WAHLIB

Technical U n i v e r s i t y of Munich

ABSTRACT In search for an optimal operating strategy for a complex hydrological system simulation runs required hundreds of years of synthetic weekly streamflow data. A multivariate multilag model was used for the generation of both time and space correlated hydrological series for eight gauging stations. As the model assumes the input to be normally distributed standardized variables extensive data transformation had to be performed. Stepwise regression is applied to k e e p the number of parameters defining the multiple autoregressive process as low as possible. 1.

INTRODUCTION

Statistical simulation of river flows is considered a powerful tool in the design and operation of water resources projects. Quite often historical records are too short to serve as a secure basis for the analysis of the behaviour of the system in qucstion. ‘I’hus, long synthetic flow series have to b e gcneratccl which truly reproduce the characteristics of the original d a t a . I urtherm o r e , for complex systems cross-correlated serics f o r 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

281

various locations may have to be considered simultaneously. As a consequence a multivariate model is required describing not only the internal structures o f the respective series but also a l l spacial interactions. A well known multivariate approach was introduced by G . K . Young and W . C . Pisano when they extended a first order autoregressive process (AK(l)-process) suggested by N . C . klatalas. llowever, flow series usually prove to possess significant autocorrelation coeffients for time lags higher than lag-I and should not be treated as A K ( 1 ) processes to fully use all available information. In the following a higher order autoregressive model and its application as a planning tool is discussed. 2.

TlIE MOl)llI,

A multivariate autoregressive model that theoretically allows f o r any p time lags is given by the matrix equation

,(i)

=

/I1 >; ( i - 1 )

where Y i i ) , x(i-l)

A1, 112, 1: (i).

*2

+

3

... ,

x (i-2)

---

7

Ap, H:

+

...

x Ci-P) :

+ Ap

x(i-P)

+

H.1': (i) (1)

(nl-vectors containing flow measurements for time intervals i , ..., i-p for all n gauges

(n n)-matrices with model parameters (n)-vector representing white noise order of the autorcgressive process

Persistence of the runoff is rnodc1lcd by assuming that any flow realization is influenced by p predecessors observed at the same and a11 other gauges. A n indepcndcnt stochastic component 13. 1 , l i ) a d c ~ c t to ~ the autoregressive part o r eq. ( I )

282

accounts for all such variance that cannot be derived from the history of the runoff process. 2.1

Estimation of Model Parameters ____ In comparison with an univariate model of the same order the number of model parameters in eq. (1) is increased by n 2 . Practical application, however, demands that the runoff structure be reproduced in the most economical way, i.e. with the lowest possible number of coefficients. Stepwise linear regression was therefore used in the estimation of the matrices Al, . . . , Ap. Matrix equation (1) is split into separate row requations each being interpreted as a linear regression of the form

+ a2

-

(i-2)+ a 2 . J32

j , l x1

X

(i-2) + . . . + a 2 . x (i-2) + J,n n

2

... (2)

=

n

al. (i-l)+ k=1 J,k Xk

where

k = 1 a2j,k k'

... ,

i=p+l,

n:

... ,

m: k=l,

1,

2 K=

Ci-p)

1 aPj,k k'

regression coefficient for the pth predecessor at gauge k

aPj ,k:

j=1,

(i-2) + . . . +

m:

row number in eq. (1) index total number of observations

... ,

n:

gauge number

Stepwise regression searches eq. (2) for the optimal conbination of independent variables in the description of

283

the response variable. Finally, a dependence is found that ( i ) contains significant regression coefficients only (ii) minimizes the variance of deviations from observed to cornputr; values for the response variable. Remaining (significant) regression coefficients are inserted in matrices A l , ... , Ap of e q . ( 1 ) while a11 ot\er matrix elements a r e assumed to be zero. In estimating the stochastic matrix H the difference between the observed values for the response variable and the respective comDuted values is calculated f o r each time interval as

Provided that the vectors E ( i ) consist of (0,l)-normally distributed random numbers the expected value of B - H T can be derived from D a s

with

and

I: T :1'. 13 , L .

identity matrix transpose matrices of H and Ii

1:

=

[I:"',

~('1,

. .. ,

I)

=

[I)"),

D"),

... ,

(1)

,

... ,

E (m-p)]

1) (m-p) ]

B can then easily be found using a n iterative procedure described by G.K. Young and W.C. Pisano. 2.2

Data Transformation In general the observed flow series cannot be used in the calibration of parameters without statistical manipulation as the suggested model requires stationary and normally distributed input data. A double data transformation approximately fulfils these requirements:

284

( i ) To achieve normality the following simple mathe-

matical functions were used nx

=

x1/ 2

(5)

nx

=

x1/3

(6)

nx

=

log(x)

(7)

A more complex function transforms the distributions Pearson-Type I 1 1 into a normal distribution nx

6 cs cs { [ - 2 K + 1 1

= -

where

cs

cs:

coefficient of skewness

K:

standardized parameter of Pearson-Type I 1 1 distribution

(ii) The second transformation elimates cyclical components by a seasonal standardization

nx :

normalized runoff values

rx : cyclically standardized runoff values (residuals) nx Z ,sz: mean and standard deviation of nx in time interval z Z: cyclical index (z = 1 , 2 , . . . , 5 2 for weeks)

-

The residuals rx can now be used as input data for the model. Consequently, the algorithm will produce synthetic residuals which obviously have to undergo the inverse transformations to (i) and (ii) to represent the desired artificial runoff series.

APPLICATION OF TIIE MODIJI, The model was tested for a complex hydrological and water resources system (Fig. 1 ) . Water .from the Danube watershed will b e transfered to the Rivcr Main watershed 3.

285

MAIN

1

N URN BERG

CLEINE

son-

I

RESERVOIR

...

BROMEACH-

OONAU

I:ig.

1.

0REGENSBURG

Schema o f t h c R i v e r D a n u b e - R i v e r Main System

286

to improve its water balance. The system includes the Main-Danube-Canal, three reservoirs and various natural or artificial connecting ducts. To obtain optimal strategies for the long term operation of the system a computer routine has been developed o n the basis of a weekly simulation. As input data numerous 100-year synthctic series for 8 water gauges in the respective areas were required. Following a n extensive data analysis and pilot runs an autoregressive third order process was considered suitable for the generation of the synthetic series. Stepwise regression reduced the number of significant elements in the matrices 121, A2 and A? to appr. 40 percent of the maximum possible value with a minimum of 14 percent for A3. It should be noted that A3 was basically a diagonal matrix which indicates that spacial dependencics were neglectable at lag-3 and higher in this specific case. 4. RESU1,lS 4.1 Transformations The gencrating algorithm choses the most suitablc transformation function out of (5) - (8) with optimality being defined as

the lowest possible skewness is observed in the transformed data (ii) thc null hypothesis of a normal distribution is not rejected at the 5 percent level. (i)

When using just onc transformation function for a complete runoff record it was found that while some of the weekly series fulfilled the a b o v c criteria othcrs wcrc significantly rejected. Ilcncc, a s p c c in1 transformation was s c lectcd for each wcck at cach station with extrcmcly satisfying results as only 4 out of a total of 410 wcckly distributions did not meet rcquiremcnts ( i l and ( i i ) s i n i u l tancoulsy. In F u c h cascs condition [ i ) w a s consirlcrctl to

287

to be sufficient. Whereas normality of the residuals could be achieved in good approximation for most weeks unrealistically high synthetic runoff values were in some few cases observed with log and Pearson-Type 111. The result was an overestimation of the skewness of the artificial data. The explanation may be purely mathematical as the inverse 3 transformations contain ex and x . In general, a more sophisticated estimation (maximum likelihood) of the population parameters should be used if extreme values are of m a j o r importance. In addition, the skewness criterion may be responsible for the production of too high runoff values as it is very sensitive to extreme values in the observed data. After an effort to cope with a single large observation the inverse transformation can lead to a relatively high proportion of large synthetic values. In the final version of the algorithm the log-transformation (7) was used in 64 percent of all cases. Second came Pearson-Type I 1 1 (8) with approximately 30 percent whereas functions (5) and ( 6 ) were selected in about 2 p e r cent and 4 percent of the weeks, respectively. 4.2

Statistical Parameters _____ .___-____-

The statistical parameters mean, standard deviation and skewness were computed f o r both the generated residuals and the synthetic runoff. Constant comparison of these parameters clearly showed theeffects of the chosen transformation and the quality of the model. Figures 2 and 3 present the results of a typical 100-year production run for the gauge 'Jluttendorf' with respect to means arid standard deviations. Figure 4 compares the rcsul ts for five 100-years series with one 500-year series. It should be noted that 100-year series still show fairly large sanpling deviations which have to h e consitlercd in a n y application.

288 _____

STRFAIIFLOW

G E N E R A T I O N OF W E E K L Y GAUGt ~-~-

Fig.

2.

DONAU-MAIN

. I i U E T T E N D O R F /' R E G N I T Z M E A N VALUES _ _ - 100 SYFITHETIC HISTOR[CAL 1977 ~ _ _RECORD 1 9 3 0-__________-

-

YFARS

~

Weekly means a t g a u g e f l u t t e n d o r f si mu 1a t i o n )

G E N E R A T I O N O F WEEKLY STREAMFLOW GAUGt

sYsTtri

HUETTtNnOKF

/

DONAU-MA I N S Y S T E M STANDARD D E V I A T I O N S

RFGNITZ

HISTORTC4L RECORD 1920-1977 -~ ~ - _~

(100-year

-

-

_

100 S Y N T H E T I C Y E A R S

1

Fig.

3 . Weckly s t a n d a r d d c v i a t i o n s a t g a u g c I l i i t t c n d o r f

( 1 9 9 - y e a r simulation)

289

GENERATION OF WEEKLY STREAMFLOW DONAU-MAIN SYSTEM GAUGE : SCHWEINFURT / MAIN MEAN VALUES _ _ _ 500 SYNTHETIC YEARS HISTORICAL RECORD 1930-1977 -. . . . . MINIMUM AND MAXIMUM VALUES FROM FIVE 100-YEAR SERIES

"i i

Fig. 4. Weekly means at gauge Schweinfurt (500-year simulation and extreme values out of five 100-year simulations __

~

_____

GtNERATION OF WEEKLY STREAMFLOW GAUGE : HUETTENDORF / REGNITZ HISTORICAL RECORD 1 9 3 0 - I 9 7 7

rK E

BE

EI

TR

l

NE

HU

PE

DUNAU-MAIN SYSTEM CORRELOGRAMS _ _ _ 1 0 0 SYNTHETIC YEARS -__

L

SF

c

KE

LAG

-K E

-

BE

€1

TR

NE

2

HU

PE

SF

LAG - 3

* BE

€1

TR

NE

HU

PE

SF

KE

BE

El

TR

NE

HU

PE

SF

Fig. 5. Auto- and crosscotrelation coefficients at gauge i-!uttendorf for time lags 0 to 3 (100-year simulation)

290

Correlation Structure In Figure 5 the reproduction of the original correlation structure of the hydrological system is presented for time lags 0 to 3. There is some indication that correlation is overestimated for higher time lags. IIowever, this can only be observed for the synthetic runoff, i.e. after inverse transformation, whereas residuals reflect the correlation structure precisely and without systematic deviations. Iiow transformations do affect correlation is still to be investigated. 4.3

4.3 Neeative Runoff When sampling from a symmetrical distribution function with range -.a < x < +co negative values are likely t o occur for long series with a probability depending on mean and standard deviation. The proportion of negative runoff produced for the Danube-Main-System was around 0,05 percent. Without significantly affecting the statistical properties of the series the values were equalled nought. In multivariate simulation negative values of inter-

mediate runoff can be observed, especially if runoff sequences possess high coefficients of variation together with almost identical means as in the case of the gauges along the River Altmuhl. ‘The number of negative values generated totalled approximately 4 percent. It should be mentioned that negative intermediate runoff was found in the Altmuhl observations, too. At all other gauges no such irregularities occurred. 5.

CONCLUSION

A multivariate regression model for the generation of both time and space correlated synthetic runoff sequences is described. Time dependence can lie reproduced for any

291

number of time lags required. In an application weekly runoff series were generated. For the necessary transformation of the observed data into normal distribution various functions were tested with log (x) and Pearson-Type I 1 1 proving to be most suitable. The results with respect to the reproduction of statistical parameters as means, standard deviations, auto- and crosscorrelation coefficients are shown for some typical sample sequences. It should be mentioned that the model has also been used for the generation of monthly series as well as sequences of decades with satisfying results. REFERENCES Fiering, b 1 . B . : Multivariate Technique for Synthetic IIydrology. Journal of the Hydraulics Division ASCE 9 0 (1964) NO. IIY 5, p p . 43 - 60. Kindler, J., W. Zuberek: On some Multi-Site Multi-Season Streamflow Generation Models. International Institute for Applied System Analysis, RM-76-76, Laxenburg 1967. Matalas, N-C. : Mathematical Assessment of Synthetic IIydrology. Water Resources Research 3 (1967) No. 4, pp. 937 - 945. O'Connell, P.E.: Multivariate Synthetic Hydrology: A Correction. Journal of the Hydraulics Division ASCE 99 (1973) NO. HY 12, p p . 2393 - 2396. I'egram, G.G.S., W. James: MiJltilag Multivariate Autoregressive Model for the Generation of Operational Hydrology. Water Resources Research 8 (1972) No. 4 , p p . 1074 -1076. Schneider, K., R. Ilarboe: Anwezdung des Young-Pisano-Modells zur Erzeugung gleichzeitiger kiinstlicher Abflufireihen fur verschiedene Stellen in einem Einzugsgebiet. Wasserwirtschaft 69 (1979) No. 7/8, pp. 219 - 225. Schramm, M.: Zur mathematischen Darstellung und Simulation des naturlichen Durchflufiprozesses. Acta Hydrophysica 19 (1975) NO. 2/3, p p . 77 - 191. Young, G . K . : Discussion of "Mathematical Assessment of Synthetic Hydrology" by N.C. Matalas. Water Resources Research 4 (1968) No. 3, p p . 681 - 682. Young, G . K . , W . C . Pisano: Operational IIydrology lJsing Residuals. Journal of the Hydraulics Division ASCE 94 (1968) NO. fIY 4 , p p . 909 - 923.

292

PROBABILISTIC CHARACTERIZATION OF POINT AND MEAN AREAL RAINFALLS VAN-THANH-VAN NGUYEN AND JEAN ROUSSELLE U n i v e r s i t e du Quebec a C h i c o u t i m i , C h i c o u t i m i , Quebec and E c o l e P o l y t e c h n i q u e de M o n t r e a l , M o n t r e a l , Quebec, Canada

INTRODUCTION I n f o r m a t i o n on r a i n f a l l d i s t r i b u t i o n s o v e r t i m e and space a r e both important i n various types o f hydrologic studies concerning the determination o f runoff characteristics.

The o b j e c t i v e o f t h e

s t u d y t o be r e p o r t e d was t o c o n s i d e r , f r o m a p r o b a b i l i s t i c p e r p e s t i v e , two o f t h e r a i n f a l

characteristics essential f o r the plan-

n i n g and d e s i g n o f u r b a n d r a i n a g e systems: and t h e a r e a l c o r r e c t i o n o f s t o r m s

t h e temporal p a t t e r n

.

F i r s t , t h e p r o b a b i l i s i c c h a r a c t e r i z a t i o n o f temporal storm p a t t e r n s was i n v e s t i g a t e d w h e r e i n a s t o r m was d e f i n e d as an u n i n t e r r u p t e d sequence o f c o n s e c u t i v e h o u r l y r a i n f a l l s .

A stochastic

model was d e v e l o p e d t o d e t e r m i n e 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 s o f r a i n f a l l a c c u m u l a t e d a t t h e prld o f each t i m e u n i t w i t h i n a t o t a l storm duration.

S e c o n d l y , a t h e o r e t i c a l methodo ogy i s p r o p o s e d

t o e s t a b l i s h a r e l a t i o n s h i p between t h e r a i n f a l l a t a f xed p o i n t and t h e a s s o c i a t e d mean r a i n f a l l o v e r a geograph c a l l y

i x e d area.

TEIIPORAL STORM PATTERN C o n s i d e r an i n t e r v a l o f t i m e w h i c h c o n s i s t s o f n h o u r s .

Mith

the d e f i n i t i o n o f a storm stated i n the previous section, the prob 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 a t e m p o r a l s t o r m p a t t e r n can be achieved by f i n d i n g 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 f u n c t i o n o f accum u l a t e d r a i n f a l l amounts a t t h e end o f each h o u r w i t h i n t h e n - h o u r storm duration. 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

0

293

Let Mn be defined as the number of consecutive rainy hours starting from the first hour of the n-hour period. 3y this definition the random variable ?In can assume the values 0, 1 , 2,...n. Let E~ denote the hourly rainfall depth in the v-th rainy hour. The accumulated amount of rainfall S(n) during Mn consecutive rainy hours can be defined as Mn s(n) = 1 cV v=o Then the distribution function Fn(x) of Sln) can be written as fol 1 ows (Nguyen and Roussel 1 e, 1981a) : n Fn(x)

=

P{S(n)<x 1

=

1

k=O

P{XKlx, PIn

=

k}

(2)

k in which P{-l denotes the probability, Xo = 0 and XK = 1 E~ for v=o k = 0, 1 , Z,...,n. In the following, for hourly rainfall process, we assume that: i ) the hourly rainfall depths E ~ ,E ~ , . . . E are independent, n identically distributed random variables with PIE S x 1 = 1 - ewXx V for every v = 1 , 2,...n; i i ) c l , E~,...En are indeDendent of PIn; and i i i ) the sequences of rainy hours can be represented by first and second-order Markov chains. Under these assumptions, we can write (Yguyen and Rousselle, 19Sla): n

where r(k) = (k-l)!, An illustrative application o f the theoretical model to an actual record of hourly rainfall has been made, in which the Dorval Airport hourly rainfall data for 1943-1974 have been used (Nguyen and Rousselle, 1981a). For purposes o f illustration, Figure 1

294 shows c o m p a r i s o n s between t h e e m p i r i c a l and t h e o r e t i c a l d i s t r i b u t i o n s f o r t h e p e r i o d s s t a r t i n g f r o m t h e second h o u r o f t h e day and c o n s i s t i n g o f n h o u r s , record f o r July.

r! =

2, 10 b y u s i n g t h e h o u r l y r a i n f a l l

I t was f o u n d t h a t t h e b i g g e r t h e v a l u e o f n t h e

larger the discrepancies.

A more d e t a i l e d d i s c u s s i o n o f t h e a -

greement between o b s e r v e d and t h e o r e t i c a l r e s u l t s can be f o u n d i n Nguyen and Roussel 1 e ( 1 981 a ) . I n summary, t h e s t o c h a s t i c model p r o p o s e d i s more g e n e r a l and more f l e x i b l e t h a n t h o s e u s e d i n p r e v i o u s i n v e s t i g a t i o n s .

Using

t h e m e t h o d o l o g y r e p o r t e d h e r e , a t e m p o r a l s t o r m p a t t e r n can be c h a r a c t e r i z e d i n terms o f t h e t i m e o f s t o r m occurence, t h e t o t a l s t o r m d u r a t i o n , t h e t o t a l s t o r m d e p t h , and t h e p r o b a b i l i t y o f o c curence o f c u m u l a t i v e r a i n f a l l s w i t h i n t h e storm. AREAL CORRECTION FACTOR

I n t h e p l a n n i n g and d e s i g n o f u r b a n d r a i n a g e systems, e n g i n e e r s have been t h w a r t e d i n a t t e m p t i n g t o r e l a t e an e x p e c t e d mean r a i n f a l l over a j u r i s d i c t i o n t o a h i s t o r i c a l record a t a f i r s t - o r d e r weather s t a t i o n .

T h i s i s because r e s e a r c h has been c o n f i n e d t o

s p a t i a l p r o p e r t i e s o f m o v i n g s t o r m s r a t h e r t h a n t o c a t c h m e n t s and r e f e r e n c e r a i n g a g e s f i x e d i n space.

T h a t i s , a t t e n t i o n has been

f o c u s e d on e i t h e r a m o v i n g a r e a c o u p l e d w i t h a m o v i n g r e f e r e n c e p o i n t o r a f i x e d a r e a and a m o v i n g r e f e r e n c e p o i n t .

! J i t h t h e mo-

v i n g r e f e r e n c e p o i n t b e i n g t h e peak p o r t i o n o f t h e s t o r m r a i n f a l l a t each p a r t i c u l a r t i m e i n t e r v s l , i t f o l l o w s t h a t a r e a l mean r a i n f a l l s o e v a l u a t e d w i l l a l w a y s be an a t t e n u a t i o n o f t h e r e f e r e n c e point rainfall. R e p o r t e d i n t h i s p a p e r a r e t h e f i n d i n g s f r o m t h e works b y Nguyen e t a l . ( 1 9 8 1 b ) where mean r a i n f a l l s f o r a f i x e d a r e a and t h e a s s o c i a t e d r a i n f a l l f o r a f i x e d p o i n t i n t h a t a r e a have been analyzed from a p r o b a b i l i s t i c perspective. The a p p r o a c h a d o p t e d c o n s i s t s o f assuming an e x p o n e n t i a l d i s t r i b u t i o n f o r p o i n t r a i n f a l l d e p t h a t each r a i n g a g e , and f r o m t h i s d e r i v i n g t h e o r e t i c a l l y t h e d i s t r i b u t i o n f u n c t i o n f o r mean a r e a l

295

I 1.0

THEORETICAL

n=to HOURS

n = 2 HOURS

0.0

0.2

0.4

0.6

0.8

I0

CUMULATIVE R A I N F A L L (iN.)

Fig. 1

0.0

0.2

0.4 0.6

CUMULATIVE

R E T U R N PERIOD ( y e a r s )

t-I

2.0

1.0 0.5

0.1 0.05

0.2 I 1 I 1 0.0001 0.001 0.01 EXCEEDANCE PROBABILITY

1.0

R A I N F A L L (iN.1

Comparison between theoretical and empirical distributions of accumulated rainfalls in the n-hour period. ( A ) n = 2 hours, (B) n = 10 hours.

5.0

Fig. 2

0.8

0.01

11

0.05

Theoretical and empirical probability distributions of areal correction factor.

296 r a i n f a l l values.

The a r e a l c o r r e c t i o n f a c t o r s a r e t h e n computed

w i t h t h e same l e v e l o f exceedance p r o b a b i l i t y f o r p o i n t and a r e a l means r a i n f a l l s . An i l l u s t r a t i v e n u m e r i c a l a p p l i c a t i o n has been made f o r M o n t r e a l I s l a n d , w i t h an a r e a o f 466 km2 (Plguyen e t a l . ,

1981b).

The gage

a t D o r v a l A i r p o r t was chosen a s t h e r e f e r e n c e gage because i t i s t h e o n l y f i r s t - o r d e r weather s t a t i o n i n t h e immediate r e g i o n o f Yontreal.

The T h i e s s e n p o l y g o n method was used t o compute t h e

mean r a i n f a l l o v e r M o n t r e a l I s l a n d b y u s i n g t h e h o u r l y r a i n f a l l d a t a f r o m t h e 5 r e c o r d i n g r a i n gages a p p r o p r i a t e t o t h e T h i e s s e n method.

The l o n g e s t c o n c u r r e n t r e c o r d f o r t h e s e 5 gages was f o r

t h e p e r i o d September 1, 1969 t h r o u g h O c t o b e r 31, 1974; b u t , f o r t h e moment, o n l y h o u r l y r a i n f a l l s i n t h e summer season ( J u n e t h r o u g h September) have been c o n s i d e r e d . F i g u r e 2 shows t h e c o m p a r i s o n between t h e e m p i r i c a l and t h e theoretical d i s t r i b u t i o n s f o r areal correction factor.

A more

c o m p l e t e d i s c u s s i o n o f t h e r e s u l t s has been d e t a i l e d b y Nguyen e t a1

.

(1981b).

From t h e s e r e s u l t s , i t was f o u n d t h a t t h e a r e a l c o r -

r e c t i o n f a c t o r i s n o t c o n s t a n t f o r a l l r e t u r n p e r i o d s and i t s V a l ues may n o t be l e s s t h a n one as was t h e n e c e s s a r y outcome u n d e r t h e premises o f p r e v i o u s i n v e s t i g a t i o n s . REFERENCES Nguyen, V.T.V. and R o u s s e l l e , J . , 1981a. A S t o c h a s t i c Model f o r t h e Time D i s t r i b u t i o n o f H o u r l y R a i n f a l l Depth. \ l a t e r Resour. __ Res., V o l . 17, No. 2, p p . 399-409. Nguyen, V.T.V., R o u s s e l l e , J. and McPherson, H.B., 1981b. EvaCan. l u a t i o n o f A r e a l Versus P o i n t R a i n f a l l w i t h Sparse Data, ___ J o u r . C i v . Eng., V o l . 8, No. 2, pp. 173-178.

297

A RAINFALL-RUNOFF MODEL FOR DAILY FLOW SYNTHESIS

M. Mimikou, C i v i l E n g i n e e r i n g Department, Athens T e c h n i c a l U n i v e r s i t y and P u b l i c Power C o r p o r a t i o n , A t h e n s , Greece A. Ramachandra Rao, S c h o o l 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 , West L a f a y e t t e , I N 47907, U.S.A.

ABSTRACT A dynamic s t o c h a s t i c model f o r s y n t h e s i s o f d a i l y f l o w s and f l o o d

hydrographs a t a c e r t a i n s i t e i n a r i v e r r ea c h i s p re se n te d i n t h i s paper.

The i n p u t s t o t h e model are t h e i n f l o w s a t an u p s t r e a m p o i n t o f

t h e r e a c h and t h e r a i n f a l l on t h e b a s i n between t h e u p s t r e a m and down-

stream p o i n t s .

The model c o n s i s t s of t h r e e p a r t s :

an inflow-outflow

t r a n s f e r f u n c t i o n model, a Kalman f i l t e r f o r 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 o f t h e d r a i n a g e b a s i n and a second o r d e r a u t o r e g r e s s i v e model f o r t h e n o i s e component.

The model f o r 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

t o t h e o u t f l o w w a s n e c e s s a r y t o c o r r e c t t h e o u t f l o w s g i v e n by t h e i n flow-outflow t r a n s f e r f u n c t i o n o f t h e s y s t e m and i t e n a b l e s p r o p e r a c c o u n t i n g of t h e l a t e r a l f l o w i n t o t h e r e a c h . using observed d a t a .

The model i s t e s t e d by

D a i l y f l o w s and f l o o d h y d r o g r a p h s a t t h e downstream

l o c a t i o n s i m u l a t e d by u s i n g t h e model h a v e been shown t o a g r e e w i t h t h e h i s t o r i c d a t a w i t h remarkable accuracy.

Besides, t h e synthesized d a t a

p r e s e r v e t h e h i s t o r i c a l mean, skewness, k u r t o s i s , lag-one a u t o c o r r e l a t i o n and i n f l o w - o u t f l o w 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% con-

f i d e n c e l e v e l and t h e v a r i a n c e a t t h e 95% c o n f i d e n c e l e v e l .

The model

i s found t o b e v e r y e f f i c i e n t i n e x t e n d i n g d a i l y f l o w r e c o r d s , estimati n g m i s s i n g d a i l y d a t a and f l o w h y d r o g r a p h s , and f o r s i m u l a t i n g d a i l y r e s e r v o i r i n f l o w s i n real t i m e , e t c .

The model i s e s p e c i a l l y u s e f u l

f o r s y n t h e s i z i n g f l o w s from b a s i n s of i r r e g u l a r h y d r o l o g i c a l character-

i s t i c s and f o r which o n l y l i m i t e d d a t a are a v a i l a b l e .

INTRODUCTION E s t i m a t i o n of d a i l y f l o w s i s i m p o r t a n t f o r e f f i c i e n t management o f

water r e s o u r c e s y s t e m s .

E s t i m a t i o n of d a i l y f l o w s a t a l o c a t i o n where

Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editor6 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

298 a s t a b l e r a t i n g curve can be e s t a b l i s h e d i s a r o u t i n e t a s k (Nemec (1972)).

However, b e t t e r methods a r e needed t o e s t i m a t e d a i l y flows a t

s i t e s where flows are n o t gaged o r a t gaged s i t e s where f i e l d measure-

ments a r e not a v a i l a b l e because of v a r i o u s r e a s o n s such a s equipment The p r e s e n t paper d e a l s w i t h t h e development and t e s t i n g a

failure.

s t o c h a s t i c model f o r e s t i m a t i o n of d a i l y flows and f l o o d hydrographs. The model was developed t o g e n e r a t e d a t a which a r e used f o r d e s i g n i n g a dam i n Greece and f o r u s e i n o p e r a t i n g t h e r e s e r v o i r once i t i s b u i l t . The model considered h e r e i n h a s t h r e e components. i s an input-output t r a n s f e r f u n c t i o n model.

The f i r s t component

The r e s i d u a l s from t h i s

model a r e shown t o b e c o r r e l a t e d w i t h p r e c i p i t a t i o n thereby i n d i c a t i n g t h e presence of l a t e r a l i n f l o w 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 sequence. The l a t e r a l i n f l o w i s e s t i m a t e d from t h e r e s i d u a l sequence by u s i n g Kalman f i l t e r techniques and t h i s c o n s t i t u t e s t h e second component of t h e model.

The r e s i d u a l sequence from t h e second component of t h e model

is a l s o usually correlated.

T h i s r e s i d u a l sequence can b e e a s i l y modeled

by a low o r d e r Autoregressive Model which c o n s t i t u t e s t h e t h i r d component of t h e model.

The r e s i d u a l s from t h e AR model can be shown t o be

white and without p e r i o d i c i t i e s . The model i s t e s t e d by u s i n g t h e d a i l y p r e c i p i t a t i o n and flow d a t a from t h e Aoos r i v e r between Vovoussa ( i n l e t ) and Konitsa ( o u t l e t ) measuring s t a t i o n s i n Northern Greece (Fig. 1 ) .

The model i s c a l i b r a t e d

by u s i n g d a i l y inflows and outflows of t h e r e a c h and p r e c i p i t a t i o n measured d u r i n g 1971-75.

The model performance i s v e r i f i e d by u s i n g t h e

d a t a measured d u r i n g 1975-77 which are n o t used f o r c a l i b r a t i n g t h e model.

The accuracy of t h e model t o e s t i m a t e d a i l y flows and t o estimate

f l o o d hydrographs are t e s t e d and t h e performance of t h e model i s shown t o be s a t i s f a c t o r y . The importance of e s t i m a t i o n of l a t e r a l i n f l o w s , t h e e x t r a c t i o n of h y d r o l o g i c a l l y meaningful i n f o r m a t i o n from r e s i d u a l sequences and t h e l i m i t a t i o n s of simple t r a n s f e r f u n c t i o n models are d i s c u s s e d i n t h e paper.

Models of t h e t y p e developed i n t h i s paper a r e compared t o con-

c e p t u a l models (Crawford and L i n s l e y (1966)) w i t h s p e c i a l r e f e r e n c e t o watersheds i n which 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 v a r y c o n s i d e r a b l y and

299

LEGEND

i

," /

'..

0

G A G E .

R A I N

/ WATER FLOW

LEVEL

AND

M U S U R E M E N 1.

.'--,

\-/--

0 Iliochori

\ L--. \

Aoos

Fig. 1

and

Y

basin

river K

./

o

n

i

Vovoussa ( i n l e t i 1

'

between t

s

a

Vovoussa

(

i 1

a

where l i m i t e d h y d r o l o g i c a l d a t a are a v a i l a b l e . The p a p e r i s o r g a n i z e d as f o l l o w s . d i s c u s s e d i n t h e second s e c t i o n . discussed i n t h e t h i r d s e c t i o n . given i n t h e f o u r t h s e c t i o n . v e r i f i c a t i o n are g i v e n .

The s t r u c t u r e o f t h e model i s

The d a t a used i n t h e s t u d y are b r i e f l y The d e t a i l s o f model c a l i b r a t i o n are

I n t h e f i f t h s e c t i o n r e s u l t s o f model

A set o f c o n c l u s i o n s a r e g i v e n i n t h e l a s t

section.

MODEL DESCRIPTION

The 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. The w a t e r s h e d between i n l e t ( i ) and o u t l e t ( j ) i s r e p r e s e n t e d a s a The i n p u t s t o t h e s y s t e m are t h e i n f l o w s q i , t and the basin r a i n f a l l P The o u t f l o w s q are t h e o u t p u t s from t h e s y s t j ,t tern. Assuming a l i n e a r r e l a t i o n s h i p between t h e o u t p u t a t t i m e t and

b l a c k box s y s t e m .

.

one of t h e i n p u t s q

i,t

a t t i m e s t-b,

t-1-b,

..., t - r n - b , . , . ,

their relation

can be w r i t t e n as: 'j,t

=

'o'i,t-b

+

'lqi,t-1-b

+...+ umq i,t-m-b +...+

St

(1)

300 where, Uo, U1,

..., Urn,...,

are the weights of the impulsive response

function of the system, b is the delay time between input and output, and St is a residual series. (1970), equ.(l)can ‘j ,t

=

(U

+

UIB

Using the notation of Box and Jenkins

be rewritten as:

+. . .+

U Bm m

+. ..)

qi,t-b

+

St = U(B)B

where B is the backwards shift operator and U(B)

b qi,t

+

St

(2)

is the transfer function

of the system. This transfer function may be written as a product of two polynomials of B of orders r and s by the following expressions:

..

U(B)=6-1(B)~(B).

( 3 ) 6(B)=1-?i1B-.

. .-6rBr.. . ( 4 ) u(B)=w

where, 61,62,...,6r, and w ,wl,...,ws,

.

-w B-. .-wsBs

0

1

(5)

are two sets of parameters.

Therefore, the final form of the inflow-outflow transfer function model is given by:

‘j,t

=

(1-6 B1

...-GrBr) -1

(W

-W B0 1

...-wsBs)B bqi,t+St.

(6)

For constant inflow values, when the system progressively reaches an equilibrium, the outflow-inflow ratio, the gain g of the system, is given by equ. (6a) as: g = (1 - 61

-

.. . -

6r)

-1

(Wo

-

w1

-

. .. - us)

(6a)

The parameters of the model in equ. (6) is estimated in the present study by using the cross-correlation coefficients between input and output sequences. The nonstationarity of the daily flows series is removed in order to estimate the parameters by differencing the time series.

It should be noted that by differencing the inflow and outflow

series the parameters of the model of equ. (6) do not change.

The

transfer function or the impulse response function was estimated by using equ. (7) :

301

........

yii (m)

1

.............. X

.............. ........ 'ii ( 0 ) where, yij(m)

j]

(7)

-

i s t h e i n f l o w - o u t f l o w c r o s s - c o v a r i a n c e f u n c t i o n , yii(m)

is

t h e i n f l o w a u t o c o v a r i a n c e f u n c t i o n and l a g m i s s e l e c t e d s o t h a t Um approaches z e r o .

The p a r a m e t e r s 6i and w

j

of t h e model i n equ. (6) are

e s t i m a t e d by e q u a t i n g t h e c o e f f i c i e n t s o f B i n equ.

u.

J

,

= 0

U. = 6 U +...+ GrUj-,+wo J 1 j-1

u.J

=

u.J

= 6

+...+ 6r u.J-r

6 u 1 j-1

u +...+6ruj-r 1 j-1

W '

j-b

.

J < b

'

J = b+l, b+2

,

J > 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 <'

2tJ

F i g u r e 4 . Low f r e q u e n c y band p r i n c i p a l components a t S t a t i o n 7A. The second p r i n c i p a l component ( t o p ) i s d i s p l a c e d 50 cm/s above t h e second ( b o t t o m ) . t i d a l c u r r e n t s a r e d o m i n a t e d b y t h e Golden Gate and have s i m i l a r directions.

Further discussion o f the e f f e c t o f d i f f u s e r orienta-

t i m and s i m u l a t i o n o f t h e e f f e c t o f c u r r e n t s on d i l u t i o n f o r t h i s o u t f a l l a r e g i v e n i n R o b e r t s , 1980, and t h e m e c h a n i c s o f w a s t e w a t e r d i s p e r s i o n a r e d i s c u s s e d i n Koh and B r o o k s , 1975. F o r t h e f i r s t few h o u r s f o l l o w i n g i t s e s t a b l i s h m e n t , t h e w a s t e f i e l d i s a d v e c t e d b y ocean c u r r e n t s and d i f f u s e d b y o c e a n i c turbulence.

T h i s turbulence i s contained i n t h e h i g h frequency

band, and so t h i s band a f f e c t s t h e w a s t e w a t e r f o r t h e s e t i m e s c a l e s . Although 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 a x i s o f t h e h i g h f r e q u e n c y band i s s i m i l a r t o t h a t o f t h e t i d e , i . e . p o i n t i n g

3 57

towards t h e Golden Gate, t h e d i r e c t i o n i s n o t p a r t i c u l a r l y s i g n i f i c a n t , as t h i s band i s a l m o s t i s o t r o p i c .

The t i d a l band a l s o

a f f e c t s t h e w a s t e f i e l d on t h i s t i m e s c a l e i n two ways.

F i r s t , the

c u r r e n t s on t h i s t i m e s c a l e r e s p o n s i b l e f o r a d v e c t i n g t h e wastef i e l d a r e most p r o b a b l y t i d a l .

Second, t h e e x c u r s i o n o f t h e

wastefield i s also t i d a l . F o r v e r y l o n g p e r i o d s o f days t o months o r even y e a r s , t h e b u i l d u p o f p o l l u t a n t s i n t h e v i c i n i t y o f t h e d i s c h a r g e i s governed by t h e l o w frequency c u r r e n t s . by t h e s i t e and f l u s h t h e a r e a .

These a r e t h e c u r r e n t s which sweep The d i r e c t i o n o f t h e l o w frequency

band i s towards t h e Northwest, and i s u n r e l a t e d t o t h e t i d a l and h i g h e r frequency c u r r e n t s . than 10 cm/s.

Typical v e l o c i t i e s (Figure 4) are less

Considerable v a r i a b i l i t y e x i s t s i n t h e c u r r e n t s ,

and i t i s apparent f r o m F i g u r e 4 t h a t two d i s t i n c t p e r i o d s e x i s t . Up u n t i l about 400 hours a c o n s i s t e n t , w e l l - d e f i n e d f l o w t o t h e Northwest e x i s t s .

From about 400 hours t o 620 hours, however, t h e

c u r r e n t becomes v e r y slow, w i t h an i l l - d e f i n e d d i r e c t i o n . Because o f t h e g r e a t importance o f these l o w frequency c u r r e n t s , t h e y were i n v e s t i g a t e d i n more d e t a i l .

To do t h i s a lowpass f i l t e r

w i t h a h a l f - p o w e r c u t o f f o f 0.60 cpd was a p p l i e d t o t h e N o r t h e r l y and E a s t e r l y c u r r e n t components o f a l l meters o p e r a t i n g d u r i n g t h i s period.

12.

These meters were numbers 5, 6, 7A, 78, 8; 11, and

A m u l t i p l e p r i n c i p a l component a n a l y s i s was t h e n a p p l i e d t o

t h e r e s u l t i n g 14 t i m e s e r i e s .

(The m u l t i p l e p r i n c i p a l component

a n a l y s i s i s d e s c r i b e d i n d e t a i l by Koh, 1977, and a l s o i n Kundu, e t a l , 1976, where i t i s known as an E m p i r i c a l Orghogonal F u n c t i o n , o r EOF a n a l y s i s . )

B r i e f l y , t h e p r i n c i p a l components a r e formed

by t h e p r o d u c t o f t h e e i g e n v e c t o r s o f t h e c o v a r i a n c e m a t r i x o f the o r i g i n a l time series.

The p r i n c i p a l components a r e arranged

from one t o f o u r t e e n i n o r d e r o f d e c r e a s i n g v a r i a n c e . The r e s u l t s o f t h e m u l t i p l e p r i n c i p a l component a n a l y s i s a r e shown i n F i g u r e s 5 and 6.

F i g u r e 5 shows t h e f i r s t f i v e p r i n c i p a l

components, and F i g u r e 6 r e p r e s e n t s t h e d i r e c t i o n and magnitude o f t h e c o n t r i b u t i o n o f t h e f i r s t p r i n c i p a l component t o t h e t o t a l

358

90 80

70 60

-

50

cn

40

E

u

-

30

0 W

w 20

a

v)

10

0 -10

-

- 20 -30 0

I

5

15

10

20

25

TIME ( D a y s )

F i g u r e 5. F i r s t f i v e p r i n c i p a l components o f l o w f r e q u e n c y c u r r e n t s . Arranged i n d e c r e a s i n g o r d e r o f v a r i a n c e f r o m bot t om t o t o p .

l o w f requency c u r r e n t a t each s t a t i o n .

(The a c t u a l c o n t r i b u t i o n

o f t h e f i r s t p r i n c i p a l component i s fo u n d b y m u l t i p l y i n g t h e v e c t o r o f F i g u r e 6 by t h e f i r s t p r i n c i p a l component o f F i g u r e 5 ) . Note t h a t because t h e f i r s t p r i n c i p a l component i s p r e d o m i n a n t l y n e g a t i v e , t h e l o w fre q u e n c y c u r r e n t s f l o w i n a d i r e c t i o n o p p o s i t e t o t h e v e c t o r s shown i n F i g u r e 6.

The f i r s t p r i n c i p a l component

i n t h i s case accounted f o r 76% o f t h e t o t a l v a r i a n c e , and t h e f i r s t f o u r p r i n c i p a l components accounted f o r 91% o f t h e t o t a l variance.

3 59

GOLDEN

GATE

.2 5\

"\

p. I' 2

SCALE : f t 0 so00 10000

F i g u r e 6. Magnitude and d i r e c t i o n o f c o n t r i b u t i o n s o f f i r s t p r i n c i p a l component o f low frequency c u r r e n t s t o t o t a l c u r r e n t . The f i r s t p r i n c i p a l component shows t h e c h a r a c t e r i s t i c s o f t h e c u r r e n t s a t S t a t i o n 7 A ( F i g u r e 4 ) t o be common t o a l l s i t e s . That i s , t h e r e i s a p e r i o d o f r a p i d f l u s h i n g towards t h e Northwest, which ends a t about 400 hours ( a p p r o x i m a t e l y day 1 7 ) , f o l l o w e d by a p e r i o d o f s l u g g i s h c i r c u l a t i o n f r o m day 17 t o day 25.

Investiga-

t i o n o f s i x months o f r e c o r d showed t h e predominant c u r r e n t p a t t e r n over t h i s p e r i o d t o be towards t h e Northwest.

w i i l t h e r e f o r e be termed t h e "normal" p a t t e r n .

This current pattern Flany p e r i o d s

360

e x i s t e d , i n some cases l a s t i n g f o r s e v e r a l weeks, however, when t h e c u r r e n t was v e r y s m a l l . "abnormal

.

These p e r i o d s w i l l be termed

'I

The m u l t i p l e p r i n c i p a l component a n a l y s i s has a l s o been a p p l i e d t o t h e normal and abnormal p e r i o d s s e p a r a t e l y .

I t was

fo und t h a t f o r t h e normal p e r i o d , t h e d i r e c t i o n s and magnitudes o f t h e f i r s t p r i n c i p a l component were v e r y s i m i l a r t o t h o s e shown i n F i g u r e 6 w i t h a g e n e r a l l y d e c r e a s i n g magnitude t o t h e S o u t h e a s t . F o r t h e abnormal p e r i o d , however, t h e d i r e c t i o n s and magnitudes o f t h e f i r s t p r i n c i p a l component v a r i e d c o n s i d e r a b l y f r o m s i t e t o s i t e , w i t h no w e l l - d e f i n e d f l o w f i e l d .

The p i c t u r e t h a t emerges

i s t h a t t h e c u r r e n t s can be d i v i d e d i n t o two p e r i o d s .

First, a

w e l l - d e f i n e d f l o w t o t h e N o rth w e s t e x i s t s i n w h i c h t h e p r i n c i p a l component a n a l y s i s would be v e r y u s e f u l i n d e s c r i b i n g t h e f l o w field.

The c u r r e n t s a r e h i g h l y c o r r e l a t e d i n space and t i m e d u r i n g

t h i s period.

Second, a p e r i o d i n which t h e c u r r e n t s a r e slow and

no w e l l - d e f i n e d c i r c u l a t i o n p a t t e r n e x i s t s .

The c u r r e n t s a r e n o t

w e l l c o r r e l a t e d f o r t h i s p e r i o d , and t h e m u l t i p l e p r i n c i p a l compon e n t approxima ti o n s would be p o o r. These abnormal p e r i o d s a r e o f c o n s i d e r a b l e i m p o r t a n c e t o t h e e nvironment al i mp a c t o f t h e o u t f a l l .

During these periods f l u s h -

i n 3 o f p o l l u t a n t s f r o m t h e a r e a i s v e r y slow, and as t h e y can e xt end f o r s e v e r a l weeks a s i g n i f i c a n t b u i l d - u p o f c o n t a m i n a n t s c o u l d occur.

Simulations o f w a s t e f i e l d t r a n s p o r t f o r these periods

showed t h a t d u r i n g t h e normal p e r i o d , t h e w a s t e f i e l d moved s t e a d i l y t o t h e Northwest, w h i l e moving back and f o r t h w i t h t h e t i d e a l o n g an a x i s p o i n t i n g a p p r o x i m a t e l y t o t h e Golden Gate.

During the

abnormal p e r i o d t h e w a s t e f i e l d s t i l l moved back and f o r t h w i t h t h e t i d e , b u t was n o t t r a n s p o r t e d o u t o f t h e immediate d i s c h a r g e s i t e .

SUllPlARY AND CONCLUSIONS An a n a l y s i s o f c u r r e n t s measured n e a r a proposed ocean o u t f a l l

i s present ed i n o r d e r t o examine t h e e f f e c t s o f t h e c u r r e n t s on t h e d e s i g n and t r a n s p o r t o f t h e d i s c h a r g e d sewage.

The c u r r e n t s

361

were discussed i n terms o f t h e i r low, t i d a l , and h i g h frequency content.

I t was shown t h a t t h e t i d a l c u r r e n t s a f f e c t t h e waste-

water on t i m e s c a l e s o f m i n u t e s t o hours a f t e r r e l e a s e f r o m t h e diffuser.

The h i g h frequency c o n t e n t a f f e c t s t h e wastewater on

t i m e s c a l e s o f hours, p r i m a r i l y t h r o u g h t u r b u l e n t d i f f u s i o n .

The

low frequency c u r r e n t s a f f e c t t h e l o n g - t e r m b u i l d u p o f p o l l u t a n t s . Two d i s t i n c t p e r i o d s o f l o w f r e q u e n c y c u r r e n t b e h a v i o r e x i s t e d One had f a i r l y r a p i d c u r r e n t s t o t h e l o r t h w e s t , d u r i n g which r a p i d f l u s h i n g o f the discharge s i t e existed.

These c u r r e n t s would be

w e l l approximated by t h e f i r s t component o f a m u l t i p l e p r i n c i p a l component a n a l y s i s .

D u r i n g t h e o t h e r p e r i o d s , which can l a s t f o r

s e v e r a l weeks, t h e c u r r e n t s were slow and p o o r l y d e f i n e d .

Poor

f l u s h i n g would be expected f o r these p e r i o d s , and t h e m u l t i p l e p r i n c i p a l component a n a l y s i s may n o t be useful f o r these t i m e s . REF ERE illC ES

Koli, R . C . Y . , 1977, " A n a l y s i s o f m u l t i p l e t i m e s e r i e s by p r i n c i p a l components w i t h a p p l i c a t i o n t o ocean c u r r e n t s o f f San F r a n c i s c o , " Tech. Nemo. 77-4, LJ.Pl. Keck L a b o r a t o r y o f Hydraul cs and ' l a t e r Resources, C a l i f o r n i a I n s t i t u t e o f Technology. Koh, R . C . Y . , and Brooks, N.H., 1975, " F l u i d Flechanics o f !.lastewate:D i s p o s a l i n t h e Ocean," Ann. Rev. o f F l u i d Mechan cs, V o l . 7: 187-211. Kundu, P.K. e t a l . , 1976, Ylodal decomposition o f t h e v e l o c i t y f i e l d near t h e Oregon Coast," J . P h y s i c a l Oceanograohy, 5: 683- 7 04. Roberts, P.J.W., 1979, " L i n e Plume and Ocean O u t f a l l D i s p e r s i o n , " J . H y d r a u l i c s D i v i s i o n , ASCE, 105 (HY4): 313-331. Roberts, P.J.W., 1980, "Ocean O u t f n D i l u t i o n : E f f e c t s of C u r r e n t s , " J . H y d r a u l i c s D i v i s i o n , ASCE, 106 (HY5): 769-782.

362

SHOULD WE SEARCH FOR PERIODICITIES I N ANNUAL RUNOFF AGAIN?

ANDERS W I LLEN V a s s d r a g s r e g u l a n t e n e s F o r e n i n g , P.O.

Box 1 4 5 , N-1371 A s k e r , N orw a y

ABSTRACT T h i s paper d e a l s w i t h a n n u a l s e r i e s o f r u n o f f a n d p r e c i p i t a t i o n . T h e s e a r c h for h i d d e n p e r i o d i c i t i e s i n a n n u a l h y d r o c l i m a t o l o g i c a l

s e r i e s t h a t t o o k place e a r l i e r , h a s b e e n r e p l a c e d b y a " s t o c h a s t i c modeling" approach d u r i n g t h e l a s t 15 y e a r s . According to t h i s s c h o o l , t h e c o s i n e c o m p o n e n t s o n e may p o s s i b l y f i n d i n a n n u a l s e r i e s

a r e a r e s u l t (almost) e n t i r e l y o f c h a n c e , a n d h e n c e s h o u l d n o t he u s e d i n a n y m o d e l i n g or f o r e c a s t i n g p r o c e d u r e s . W h i l e r e c o g n i s i n g t h e i m p o r t a n c e o f s t o c h a s t i c e l e m e n t s i n h y d r o c l i m a t o l o g i c a l phenome n a , t h e c o n c l u s i o n of t h i s paper i s t h a t t h e u s e o f simple h a r m o n i c c o m p o n e n t s c o u l d b e j u s t i f i e d i n sane cases, i f u s e d b y c a r e . T h i s c o n c l u s i o n is s u p p o r t e d b y a n a l y s i s o f some s e r i e s f r o m a n d i n t h e n e i g h b o u r h o o d o f Norway a n d a d i c c u s s i o n o f a r g u m e n t s f o r a n d a g a i n s t t h e u s e of c o s i n e c o m p o n e n t s i n s t o c h a s t i c m o d e l i n g of h y d r o c l i m a t o l o g i c a l p h en o men a.

I n t h e a n a l y s i s t h e split-sample

t e c h n i q u e was a p p l i e d , w h e r e e a c h s e r i e s was d i v i d e d i n t o t w o or t h r e e p a r t s . F o r most periods t h e p h a s e s a n d a m p l i t u d e s were d i f f e r e n t , a r e s u l t t h a t w a s expected. However, a p e r i o d o f a b o u t 1 9 y e a r s

was f o u n d i n b o t h p a r t s of a N o r w e g i a n r u n o f f r e c o r d o f 1 0 8 y e a r s , a n d b o t h t h e a m p l i t u d e s a n d p h a s e s were i n a c c o r d . M o r e o v e r , s i m i l a r r e s u l t s were f o u n d f o r sane o t h e r s e r i e s .

INTRODUCTION T h e s e a r c h f o r p e r i o d i c i t i e s i n a n n u a l r u n o f f was more or l e s s 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

363 abandonned some 1 5 y e a r s ago. The new i d e a , i n t r o d u c e d by s c i e n t i s t s i n t e r e s t e d i n s t o c h a s t i c processes, w a s t h a t t h e p e r i o d i c i t i e s t h a t may e x i s t i n h i s t o r i c a l s e r i e s a r e a r e s u l t (almost) e n t i r e l y o f c h a n c e . I n s t e a d o f l o o k i n g f o r p e r i o d i c i t i e s , o n e s h o u l d r e g a r d runo f f a s a s t o c h a s t i c process o f o n e k i n d or a n o t h e r . D i f f e r e n t models h a v e b e e n s u g g e s t e d , from t h e v e r y simple s t a t i o n a r y w h i t e n o i s e model t o c o m p l i c a t e d models l i k e f o r example F r a c t i o n a l G a u s s i a n

Noise. The u s e o f s i m p l e c o s i n e - c o m p o n e n t s a s p a r t o f a s t o c h a s t i c model was (almost) n e v e r a p p l i e d , however. T h e r e a r e s e v e r a l a r g u m e n t s a g a i n s t t h e u s e o f s u c h c o m p o n e n t s , f o r example: a ) The e x t r a c t e d h a r m o n i c components a r e not s t a t i s t i c a l l y s i g n i f icant. b ) I f t h e split-sample

t e c h n i q u e is a p p l i e d , t h e p e r i o d l e n g t h s ,

p h a s e s and a m p l i t u d e s w i l l b e q u i t e d i f f e r e n t i n d i f f e r e n t sub-

samples of a r e c o r d . T h i s a r g u m e n t t h e n would m a k e c o s i n e components q u i t e useless i n forecasting procedures.

c ) Even i f a c o s i n e component r e a l l y would e x i s t i n t h e " r e a l " p r o c e s s t h a t p r o d u c e s r u n o f f a t a s i t e , t h i s component o n l y explains very l i t t l e of t h e total v a r i a t i o n i n the annual values. The l a s t a r g u m e n t i s n o t v e r y s t r o n g . I t is c e r t a i n l y t r u e , t h a t

it is u s u a l l y n o t p o s s i b l e t o make a n y d e t e r m i n i s t i c p r e d i c t i o n s o f n e x t y e a r r u n o f f by u s i n g c o s i n e c o m p o n e n t s , a t l e a s t no p r e d i c t i o n s w i t h c o n f i d e n c e . However

,

similar arguments c o u l d be r a i s e d a g a i n s t

e s s e n t i a l l y a i l models i n use today. T u r n i n g t o a r g u m e n t a ) , t h e l a c k of s t a t i s t i c a l s i g n i f i c a n c e o n e u s u a l l y o b t a i n s may v e r y w e l l b e a r e s u l t o f t h e s h o r t r e c o r d ( s ) a t hand. Owing t o t h i s it i s u s u a l l y d i f f i c u l t t o o b t a i n s i g n i f i c a n c e f o r any k i n d o f model o r c o e f f i c i e n t , and o f t e n e v e n t h e s i m p l e s t a t i o n a r y w h i t e n o i s e model seenis r e a s o n a b l e a c c o r d i n g t o s t a n d a r d

t e s t s o f s i g n i f i c a n c e . However

,

there are several f a c t s t h a t indi-

c a t e s t h a t t h e s i m p l e s t a t i o n a r y w h i t e n o i s e model i s too simple. And i f t h e t a s k is t o e s t i m a t e f o r example t h e p r o b a b i l i t i e s of c o m b i n a t i o n s o f d r y y e a r s , t h e computed r i s k s o f t e n w i l l b e too low,

364 i f t h e small ( b u t p r o b a b l y r e a l ) d e p a r t u r e s from t h e s t a t i o n a r y w h i t e n o i s e model a r e n o t t a k e n i n t o a c c o u n t . The a u t h o r c o n s i d e r s p o i n t b) t o be t h e s t r o n g e s t argument a g a i n s t t h e u s e o f c o s i n e components, and t h e a n l y s i s below is devoted p a r t l y t o t h i s argument.

ANALYSIS METHODS I n t h i s s t u d y o n l y s i m p l e methods a r e u s e d . A number o f p e r i o d l e n g t h s h a s been a p p l i e d t o t h e s e r i e s . For e a c h p e r i o d l e n g t h and s e r i e s a l e a s t s q u a r e p r o c e d u r e h a s been a p p l i e d a c c o r d i n g t o :

T

c r i t ( p ) = min

D =

(A

+

k=l

A ,A,B

2

c

A-sin 2 z k + B'cos P

P

Q )

(1)

k

0

r u n o f f y e a r number k

Qk k

y e a r number

T

l e n g t h of t h e s e r i e s (number o f y e a r s )

P

length of p e r i o d ("wavelength")

Ao,A,B

parameters

I t s h o u l d be n o t i c e d t h a t t h i s is n o t q u i t e t h e same a s c o n v e n t i o -

n a l F o u r i e r - a n a l y s i s , b e c a u s e i n t h a t method t h e s e r i e s is decomposed i n t o a number o f harmonic components, where a l l p e r i o d l e n g t h s c o u l d be w r i t t e n a s p=T/N

(N i n t e g e r , N = 1 , 2 ,

...,T/2.

Here w e have more

freedom i n t h e c h o i c e o f p, we c o u l d f o r example have T=50 and ~ 2 0 . For f i x e d v a l u e s o f p eq.

(1) is a l i n e a r l e a s t s q u a r e problem.

A number o f v a l u e s of p were t r i e d , and a s a r e s u l t t h e c r i t e r i o n

( c r i t ) was o b t a i n e d a s a f u n c t i o n of p f o r each s e r i e s . I n t h e o u t p u t of t h e computer programe w e a l s o o b t a i n e d t h e harmonic components i n t h e form

QPk = A o

+

C-COS

[y

(k-phg

365 v a l u e o f t h e c o s i n e component f o r y e a r k-1900

QPk C

a m p l i t u d e o f t h e c o s i n e component p h a s e . Here t h e r e f e r e n c e y e a r f o r p h a s e s i s 1 9 0 0 . T h i s

Ph

means t h a t t h e h a r m o n i c component o b t a i n s its maximum a t y e a r s 1900+m

ph, m i n t e g e r

One p o s s i b i l i t y i n t h e s e a r c h f o r p e r i o d i c i t i e s would b e t o s i m p l y choose t h e v a l u e o f p t h a t g i v e s t h e b e s t f i t , r e s t r i c t i n g p t o “ r e a s o n a b l e ” v a l u e s . Argument b ) a b o v e s t r o n g l y o p p o s e s t h i s method however. I n s t e a d t h e split-sample t e c h n i q u e was a p p l i e d , where t h e r e c o r d s were d i v i d e d i n t o t w o o r t h r e e p a r t s . Then t h e a n a l y s i s method was a p p l i e d t o t h e d i f f e r e n t p a r t s a s w e l l a s t o t h e whole r e c o r d . T h i s method r e q u i r e s l o n g r e c o r d s , and h e n c e s t r o n g l y r e s t r i c t s t h e choice o f s t a t i o n s to use.

RESULTS AND CHOICE O F STATIONS U n f o r t u n a t e l y o n l y v e r y few s t a t i o n s w i t h l o n g e n o u g h r e c o r d s o f

g o d q u a l i t y e x i s t s i n Norway. F o r Elverurn ( 1 4 7 km n o r t h o f Oslo) a t r i v e r G l o m m a w e h a v e a r u n o f f r e c o r d s t a r t i n g i n 1 8 7 2 a n d a t As n e a r

Oslo t h e p r e c i p i t a t i o n r e c o r d s t a r t s i n 1 8 7 4 , b o t h t h e s e s t a t i o n s a r e s t i l l i n o p e r a t i o n . T h e r e e x i s t s some p r e c i p i t a t i o n r e c o r d s s t a r t i n g

i n 1 8 9 0 : ~and ~ v e r y few s t a r t i n g e a r l i e r t h a n t h a t . The q u a l i t y of t h e l o n g e r p r e c i p i t a t i o n s e r i e s c a n be q u e s t i o n n e d however. A p r e c i p i t a t i o n s e r i e s is v e r y much i n f l u e n c e d by v e r y l o c a l phenomena

( a s f o r example c h a n g e i n v e g e t a t i o n i n t h e i m m e d i a t e

v i c i n i t y o f t h e g a u g e ) and by c h a n g e s i n s h i e l d i n g p r o c e d u r e s . A s t h e r i v e r p r o f i l e s i n Norway u s u a l l y a r e s t a b l e a n d a s r u n o f f i s a n i n t e g r a t o r o v e r t h e catchment area, r i v e r runoff i s p r o b a b l y a b e t t e r i n d i c a t o r o f long-term v a r i a t i o n s o f water a v a i l a b i l i t y t h a n p r e c i p i tation. S t a r t i n c j w i t h t h e s t r e a m f l o w r e c o r d f r o m E l v e r u m (1872-1979) was s p l i t u p i n t o t w o p a r t s , 1872-1922 a n d 1923-1979. d i f f e r e n t v a l u e s o f p i n eq.

it

Applying

(l), t h e r e s u l t was ( a s e x p e c t e d ) t h a t

no s i n g l e v a l u e o f p g a v e a h a r m o n i c component t h a t e x p l a i n e d v e r y much o f t h e t o t a l v a r i a n c e i n r u n o f f . Anyhow, c r i t ( p ) a s a f u n c t i o n

366 o f p was o b t a i n e d , and i n f i g . 1 i s shown w a s f u n c t i o n o f p 2 (R = ( t o t a l v a r i a n c e - e x p l a i n e d v a r i a n c e ) / t o t a l v a r i a n c e ) ) f o r t h e whole r e c o r d . I t s h o u l d be n o t i c e d t h a t t h e s t u d y and f i g u r e i s n o t v e r y d e t a i l e d f o r l o w v a l u e s o f p , where local maxima and minima may

0.30

0 25

0.20

0.1 5

0.10

0.05

0.01

+-I

5

F i g . 1. w

10

a

15

20

25

30

35

LO

45

50

P

s f u n c t i o n o f p e r i o d l e n g t h ( p ) f o r Elverum (1872-1979).

e x i s t n o t shown i n t h e f i g u r e . Anyhow, i f c o s i n e terms a r e t o be u s e d i n a s t o c h a s t i c m o d e l , t h e a u t h o r t h i n k s t h e y s h o u l d be used i n modeling long-term phenanena, i . e .

phenomena which may be t h e r e s u l t

o f l a r g e - s c a l e c l i m a t i c v a r i a t i o n s . S h o r t - t e r m phenomena ( w i t h " p e r i o d i c i t i e s " o f a b o u t a few y e a r s ) a r e e a s i l y c o n s i d e r e d a combined r e s u l t o f l o c a l e f f e c t s and c h a n c e , and hence a r e e a s i l y modeled by l o w o r d e r a u t o r e g r e s s i v e terms. I g n o r i n g p e r i o d l e n g t h s s h o r t e r t h a n 1 0 y e a r s , t h e f i g u r e shows p e a k s f o r p e r i o d s o f a b o u t 11, 1 3 , 20 and 30 y e a r s . These p e a k s c e r t a i n l y are not s i g n i f i c a n t according t o standard s t a t i s t k a l t e s t s , and i n d e e d do n o t e x p l a i n v e r y much o f t h e t o t a l v a r i a n c e .

367 Moreover, t h e a m p l i t u d e and/or p h a s e s for c o s i n e - t e r m s c o r r e s p o n d i n g t o t h e s e p e r i o d s were d i f f e r e n t i n t h e t w o p a r t s o f t h e r e c o r d , exc e p t f o r p a b o u t 20 y e a r s . C o n s i d e r i n g a p e r i o d l e n g t h of 3 1 y e a r s f o r example, t h e a m p l i t u d e is 8.86 i n t h e f i r s t p a r t and 21.32 second p a r t o f t h e r e c o r d ( u n i t : m 3/ s )

in the

.

The b e s t i n t e g e r v a l u e o f p n e a r 20 w a s 1 9 , and f o r t h i s v a l u e o f

p t h e following t a b l e w a s obtained.

TABLE 1

R e su 1t s f o r c o s i n e term w i t h p e r i o d l e n g t h 19 y e a r s f o r Elverum

(runoff

. Unit:

m3/s. ~~

~~

Series

1872-1922

1923-1979

1872-1979

Stand.dev.

43.47 243.20 13.75 9.79 0.0524

42.96 248.33 15.81 8.13 0.0689

43.08 246.40 14.58 8.81 0.0576

A0

C

Ph R2

Peaks n e a r a p e r i o d l e n g t h of 20 y e a r s a l s o were o b t a i n e d f o r t h e two p a r t s o f t h e s e r i e s , t h e b e s t i n t e g e r v a l u e s were 20 and 1 9 y e a r s f o r 1872-1922 and 1923-1979 r e s p e c t i v e l y . Looking a t t a b l e 1, one n o t i c e s t h a t f o r a c o s i n e - t e r m w i t h a p e r i o d l e n g t h of 19 y e a r s , t h e a m p l i t u d e s and p h a s e s a r e ( a p p r o x i m a t e l y ) c o n s i s t e n t i n t h e t w o p a r t s of t h e r e c o r d . T h i s c o n s i s t e n c e made t h e a u t h o r i n c l i n e d t o a c c e p t a c o s i n e term w i t h t h i s wavel e n g t h a s a " r e a l i t y " , a l t h o u g h indeed t h i s component e x p l a i n s v e r y l i t t l e o f t h e v a r i a n c e . I n f i g u r e 2 is shown t h e Elverum r e c o r d and

t h e c o s i n e term w i t h p=19 y e a r s f i t t e d t o t h e whole r e c o r d . A second-order

a u t o r e g r e s s i v e (AR-2) model was a l s o t r i e d . The

o r d e r of t h e A R - m o d e l w a s c h o s e n a f t e r a p p l y i n g t h e s p l i t sample t e c h n i q u e , t h e h i g h e s t o r d e r t h a t gave c o n s i s t e n c e i n t h e p a r a m e t e r s between t h e t w o p a r t s of t h e r e c o r d was a c c e p t e d . The AR-2 m o d e l

368 e x p l a i n e d s l i g h t l y more o f t h e v a r i a n c e t h a n t h e p e r i o d i c component 2 w i t h p=19 y e a r s ( R =0.0728 f o r t h e whole r e c o r d ) for y e a r s s t a r t i n g a t 1:st o f J a n u a r y . However, when c h a n g i n g t h e s t a r t i n g d a t e f o r t h e y e a r , t h e p a r a m e t e r s of t h e AR-2

m o d e l changed and sometimes a n

RL-value a s l o w a s 0.0360 w a s o b t a i n e d . For t h e c o s i n e - t e r m w i t h p=19 y e a r s , t h e p a r a m e t e r s remained e s s e n t i a l l y c o n s t a n t however. Turning t o t h e p r e c i p i t a t i o n s e r i e s from As (1874-1979) it was d i v i d e d i n t o two p a r t s , 1874-1924 and 1925-1979, and a n a l y z e d i n t h e same way a s Elverum.

0 40[

350

300

250

200

150 \:

1870

tea0

1890

tgoo

1910

1920

1930

I ~ L O 1950

F i g . 2 . The Elverum r u n o f f s e r i e s (1872-1979) term w i t h p=19 y e a r s . U n i t : m3/s.

1960

1970

1980

and t h e f i t t e d c o s i n e

A l s o t h i s r e c o r d had a peak i n R2 €or p n e a r 20 y e a r s for b o t h

p a r t s o f t h e record as w e l l a s for t h e whole r e c o r d . For p=19 y e a r s , t a b l e 2 was o b t a i n e d . O b v i o u s l y t h e a m p l i t u d e s were n o t q u i t e i n a c c o r d , b u t t h e p h a s e s

were n e a r l y c o i n c i d e n t and a l s o n e a r t h e p h a s e s o b t a i n e d for Elverum.

369 TABLE 2 .

R e s u l t s f o r c o s i n e term w i t h p e r i o d l e n g t h 1 9 y e a r s f o r As ( p r e c i p i t a t i o n ) . U n i t : mm.

Series

1874-1924

1 9 25-1979

1874-1979

141.32 754.48 32.85 9.42 0.0528

147.15 779.75 52.53 8.47 0.0668

144.37 768.06 44.25 8.86 0.0468

_______~

Stand. dev. A0 C

Ph R2

F o r t h i s p r e c i p i t a t i o n s e r i e s b e t t e r a g r e e m e n t were o b t a i n e d f o r p=14 years, with amplitudes 48.26,

36.69

and 41.20 and p h a s e s 9 . 7 3 ,

8.78

and 9 . 3 5 r e s p e c t i v e l y . One p r e c i p i t a t i o n s e r i e s f r o m Trerndelag i n Norway ( L i e n i n S e l b u 1 8 9 6 - 1 9 7 8 ) phases (5.90, 1896-1978)

4.90,

4.36)

a l s o showed i n t e r n a l a g r e e m e n t i n

f o r t h e series (1896-1936,

1937-1978 and

f o r a p e r i o d o f 19 y e a r s , b u t t h e a m p l i t u d e s d i f f e r e d e v e n

more t h a n f o r As. F o r t h e L i e n s e r i e s s h i e l d i n g p r o c e d u r e s h a s n o t been homogeneous h o w e v e r , a n d a l s o t h e s e r i e s i s somewhat s h o r t e r . A t l a s t t h e r u n o f f r e c o r d f r o m G o t a Alv r i v e r

(1807-1964) o b t a i n e d

from Unesco ( 1 9 7 1 ) was a n a l y z e d . The s t a t i o n is located i n t h e west o f Sweden sane 3 3 0 k m s o u t h o f Oslo. T a b l e 3 shows t h e r e s u l t s obtained f o r a cosine-term w i t h a wavelength o f 19 y e a r s .

TABLE 3 .

R e s u l t s f o r c o s i n e t e r m w i t h p e r i o d l e n g t h 1 9 y e a r s f o r G o t a Alv (runoff). Unit: m3/s. ~~~~~

Series

1807-1871

1872-1964

1 8 7 2 - 1 9 22

1 9 23-1964

105.47 542.26 5.32 5.30 0.0013

97.32 529.10 28.68 11.66 0.0443

96.22 532.48 30.29 12.24 0.0527

99.51 523.61 28.39 10.81 0.0389

~~~~~

Stand-dev.

A0 C

Ph R2

370 O b v i o u s l y t h e r e s u l t s f o r t h e f i r s t p a r t o f t h e s e r i e s (1807-

,

1 8 7 1 ) d i f f e r e d from t h e r e s u l t s f o r t h e r e s t o f t h e r e c o r d . B u t f o r t h e t h r e e o t h e r a n a l y z e d p a r t s of t h e r e c o r d ( t h e f i r s t of w h i c h i s s i m p l y t h e c o m b i n a t i o n o f t h e l a s t t w o r e c o r d s ) t h e p h a s e s a n d ampl i t u d e s were a p p r o x i m a t e l y c o n s i s t e n t . And t h e s e t h r e e t i m e p e r i o d s c o v e r n e a r l y t h e same y e a r s a s t h e Norwegian s e r i e s d i s c u s s e d a b o v e . R e t u r n i n g t o t h e Elverum r e c o r d , eq.

( 2 ) w i t h p=19 y e a r s w a s

a p p l i e d , and t h e r e s i d u a l s were a n a l y z e d . T h e s e d i d n o t a p p e a r q u i t e l i k e w h i t e n o i s e , and s i m p l e l o w - o r d e r A r - m o d e l s

were a p p l i e d t o t h e

r e s i d u a l s . A g a i n , t h e c h o i c e o f o r d e r o f t h e AR-model was d e t e r m i n e d by t h e c r i t e r i u m , t h a t t h e h i g h e s t p o s s i b l e order t h a t g a v e c o n s i s t e n c e i n parameter estimates i n t h e t w o s u b s a m p l e s was a c c e p t e d . The r e s u l t was a s e c o n d - o r d e r m o d e l ,

i . e . t h e same o r d e r a s t h a t ob-

t a i n e d when a n a l y z i n g t h e s t r e a m f l o w d i r e c t l y . The r e s i d u a l s f r o m t h i s combined m o d e l a p p e a r e d r a t h e r w h i t e . T h e s u g g e s t e d s t o c h a s t i c

m o d e l f o r Elverum i s t h u s :

+

= QP

Qk

-

AA(l).(Q

k

k-1

QP

)+ AA(2)'(Qk-2

k-1

-

QP

)

k-2

+

EPS k

v a l u e o f h a r m o n i c component y e a r k

QPk

AA (l), AA ( 2 )

autoregressive coefficients

EPSk

random ,?umber

SEP S

s t a n d a r d d e v i a t i o n o f EPS

( w i t h z e r o mean) c o r r e s p o n d i n g t o y e a r k

I n t h i s p r e l i m i n a r y s t u d y , we s i m p l y u s e d t h e n o r m a l d i s t r i b u t i o n

for EPS. I n d e e d , t h e c h o i c e o f d i s t r i b u t i o n s h o u l d b e d e v o t e d a s t u d y , b u t t h i s is beyond t h e s c o p e of t h i s p a p e r . The p a r a m e t e r s a r e shown i n t a b l e 4 .

371 TABLE 4 .

Parameters o b t a i n e d f o r combined model for E l v e r u m . U n i t : m3/s.

Series

1872-1922

1923-1979

1872-1979

A0 C

243.20 13.75 9.79 0.2010 -0.1356 41.32 0.0966

248.33 15.81 8.13 0.2433 -0.1696 40.03 0.1320

246.33 14.58 8.81 0.2271 -0.1585 40 - 5 3 0.1150

Ph AA(1) AA(2) SEPS R2

DISCUSSION

Do not t h i s search f o r cosine-terms

i n a n n u a l s e r i e s means a b i g

r i s k , t h a t t h e c o s i n e - t e r m s which is applied h e r e a r e t h e r e s u l t s of

a q u i t e d i f f e r e n t p r o c e s s ? And d o n o t t h e m o d e l a p p l i e d mean t h a t w e f i x t h e period lengths ( i n generated sequences) to 19 y e a r s , while these l e n g t h s s h o u l d v a r y ? T h e r e a r e many c r i t i c a l q u e s t i o n s t o

r a i s e a g a i n s t t h e model a n d a n a l y s i s a p p l i e d , s a n e o f them were stated already i n the introduction. T u r n i n g t o t h e f i r s t q u e s t i o n ( t h e r i s k t h a t we c h o o s e a model t o f i t w h a t is i n r e a l i t y a r e s u l t o f a q u i t e d i f f e r e n t p r o c e s s )

,

this

r i s k e x i s t s w h a t e v e r model o n e may u s e .

C o n s i d e r i n g t h e l e n g t h o f " p e r i o d s " i n g e n e r a t e d s e q u e n c e s , it i s n o t f i x e d t o 1 9 y e a r s . Owing t o t h e r a t h e r l a r g e v a r i a n c e o f t h e r e s i d u a l s (EPS)

,

generated sequences w i l l contain "periods" with

d i f f e r e n t l e n g t h s . The most s e r i o u s problem c o n c e r n s t h e s t a b i l i t y of t h e model parameters when estimated from d i f f e r e n t p a r t s o f a

r e c o r d . How t o i n t e r p r e t e t h e r e s u l t s o f t h e split-sample t e c h n i q u e a p p l i e d t o t h e s e r i e s ? A s s t a t e d a b o v e , t h e a u t h o r h a s more c o n f i dence i n long records o f runoff than i n long records of p r e c i p i t a t i o n . Thus, t h e d i f f e r e n c e i n a m p l i t u d e s f o r even t h e l o n g prec i p i t a t i o n r e c o r d f r o m As s h o u l d be g i v e n lower w e i g h t t h a n t h e c o i n c i d e n c e for t h e E l v e r u m s e r i e s , a n d for t h e G o t a Alv record a s

372 r e g a r d s t h e y e a r s 1872-1922 a n d 1922-1964. And t h e s e two t i m e p e r i o d s c o v e r a b o u t t h e same y e a r s as d o t h e E l v e r u m s e r i e s . B u t w h a t a b o u t t h e s m a l l a m p l i t u d e f o r t h e 1807-1871 p a r t o f t h e G o t a Alv r e c o r d , i s t h i s n o t a c o n f i r m a t i o n t h a t w e s h o u l d n o t u s e

c o s i n e t e r m s ? I f w e r e a l l y a r e i n t e r e s t e d i n ( t h e o r e t i c a l ) l o n g term s t a t i o n a r y c o n d i t i o n s , t h e n c e r t a i n l y t h i s a r g u m e n t is a s t r o n g o n e . B u t i n p r a c t i s e , t h e main i n t e r e s t o f t e n is o n t h e n e x t 1 0 t o 20 y e a r s r a t h e r t h a n o n t h e n e x t 1 5 0 y e a r s . Then w e may h o p e , t h a t n o t o n l y t h e phases b u t also t h e amplitudes w i l l remain s t a b l e f o r t h e

t i n e s p a n of main i n t e r e s t . The f a c t t h a t t h e a m p l i t u d e s a n d p h a s e s e t c . , may b e t h e r e s u l t o f a p r o c e s s o f q u i t e d i f f e r e n t k i n d , d o e s n o t p r e v e n t u s from u s i n g them. C u r hope is, t h a t t h e r e a l p r o c e s s is i n a s t a t e o f " n e a r stationarity",

i.e.

i n a s t a t e where t h e r e a l s t a t e s o f t h e r e a l

process a r e c h a n g i n g v e r y s l o w l y . T h u s w e may h o p e , t h a t t h e para-

meters c h o s e n c a n b e c o n s i d e r e d s t a b l e f o r a t l e a s t sane y e a r s a h e a d . But o f course w e c a n n o t c o n s i d e r t h e parameters a s s t a b l e a s would be r - u i r e d

i f w e were t o u s e t h e m o d e l f o r d e t e r m i n i s t i c

p r e d i c t i o n s . A s stated above, s i m i l a r arguments a p p l y to m o s t o t h e r models.

A n o t h e r matter o f d i s c u s s i o n is how many parameters t o u s e i n a model. One s c h o o l f a v o u r s t h e u s e o f a s few parameters a n d a s simple models a s p o s s i b l e

,

o n l y t h e v e r y s i g n i f i c a n t p a r a m e t e r s s h o u l d be

accepted, t h e o t h e r p a r a m e t e r s m i g h t b e a r e s u l t of c h a n c e . T h i s

i d e a h a s i t s m e r i t s when d i s c u s s i n g s c i e n t i f i c r e s e a c h . S u p p o s e a s c i e n t i s t h a s f o u n d t h a t t h e 50-year

lag autocorrelation for

t e m p e r a t u r e u s u a l l y i s n e g a t i v e . Then, i f he wants t o claim t h i s a s t h e d i s c o v e r y o f a new c l i m a t i c t h e o r y , i n d e e d w e s h o u l d r e q u i r e s t r o n g e v i d e n c e b e f o r e a c c e p t i n g t h e t h e o r y . B u t i f t h e i n t e r e s t is

to estimate r i s k s o f combinations o f d r y y e a r s d u r i n g sane y e a r s a h e a d , t h e n t h e s e estimates may b e s e r i o u s l y b i a s e d ( p r o b a b l y too l o w ) i f r e j e c t i n g a l l parameters n o t p a s s i n g t h e c o n v e n t i o n a l

s t a t i s t i c a l s i g n i f i c a n c e tests. A t l e a s t p a r a m e t e r s , whose e s t i m a t e s appear s t a b l e u n d e r t h e split-sample t e c h n i q u e s h o u l d b e t a k e n i n t o

373 a c c o u n t , u n l e s s t h i s l e a d s t o e x c e s s i v e computing costs. C o n s i d e r i n g t h e p e r i o d l e n g t h of 19 y e a r s , H i b l e r I11 and J o h n s e n (1979) a n a l y z e d v e r y l o n g r e c o r d s (1244-1971)

from G r e e n l a n d ice

cores, and t h e y claimed t o have found a 20-yr c y c l e .

REFERENCES

H i b l e r 111, W.D. and J o h n s e n , S.J., 1979. The 20-yr c y c l e i n G r e e n l a n d ice core r e c o r d s . N a t u r e , 280 : 481-483. UNESCO, 1971. D i s c h a r g e o f S e l e c t e d R i v e r s of t h e Wold. Vol 11. Unesco, P a r i s .

374 STEP AHEAD STREAMFLOW FORECASTING U S I N G PATTERN ANALYSIS M.G.

GOEBEL and T . E .

UNNY

Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y o f Waterloo, Waterloo, O n t a r i o , Canada

S YN O P S I S

Persistence i n streamflow time s e r i e s i s u s u a l l y s p e c i f i e d through c o r r e l a t i o n c o e f f i c i e n t s o f o r d e r one and l a r g e r . a r e c a l c u l a t e d u s i n g t h e whole t i m e s e r i e s r e c o r d .

These c o e f f i c i e n t s However,

p e r s i s t e n c e e f f e c t s a r e bound t o be v a r i a b l e i n d i f f e r e n t p a r t s o f t h e record.

C o n s i d e r a t i o n o f s ma l l c o l l e c t i o n s o f s u c c e s s i v e d a t a as

d i s t i n c t e n t i t i e s e f f e c t i v e l y incorporates persistence i n analysis o f data.

T h i s i s t h e b a s i c method i n v o l v e d i n p a t t e r n a n a l y s i s .

The h i s t o r i c t i m e s e r i e s i s segmented i n t o a s e r i e s o f r e p r e s e n t a t i o n a l o b j e c t s which d e f i n e l o c a l shapes w i t h i n t h e t i m e s e r i e s . o b j e c t s a r e i n t u r n d e f i n e d by a s e t o f p a t t e r n v e c t o r s .

These

Cluster

a n a l y s i s i s used t o s p e c i f y t i m e i n d e x e d r e g i o n s o r c l a s s e s i n n-dimensional p a t t e r n space a c c o r d i n g t o which f u t u r e p a t t e r n s may be classified. The c u r r e n t s t r e a m f l o w o b s e r v a t i o n s a r e used f o r s t e p ahead f o r e c a s t i n g based on t h e c o n c e p t t h a t a c o m p l i m e n t a r y datum i s r e q u i r e d t o complete a p a t t e r n . INTRODUCTION St reamf low d a t a r e c o r d e d a t m o n t h l y i n t e r v a l s i s c h a r a c t e r i z e d b y t h e presence o f s e a s o n a l i t y caused by r e o c c u r r i n g g e o p h y s i c a l e v e n t s w i t h i n a year.Thus

one observes groups o f h i g h o r l o w f l o w s a c c o r d i n g

t o wet o r d r y p e r i o d s .

Interdependence o f s u c c e s s i v e f l o w d a t a i n

m o nt hly s t r e a m f l o w t i m e s e r i e s i s termed p e r s i s t e n c e and can be e x p l a i n e d by watershed s t o r a g e s as w e l l as b y l o n g t e r m m e t e o r o l o g i c a l Reprinted from T i m e Series Methods in Hydrosciences, by A.H. El-Shaarawiand S.R. Esterby (Editors) 0 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

375

events. Persistence in data is usually specified through correlation coefficients of order one and larger. However, because historical data is only a single realization of a stochastic sequence, correlation coefficients are bound to be variable with increases in the length of record with time. Furthermore, correlation coefficients are derived by scanning the data from beginning to end; thus if at all they represent the persistence effect, it could be nothing more than an averaged effect across the whole time series. On the other hand, in physical situations dealing with real world data, the persistence effect is bound to vary change in different par o f the record. An effective way of incorporating persistence in data analysis is through consideration of small collections of data as distinct entities in the analysis. Such an approach is taken through the use of pattern analysis. Pattern analysis is a well developed discipline by itself and it has been applied in diverse fields such as satellite imagery, medical diagnosis and cybernetics; however, it's application in hydrology is recent. Panu (1978) has developed a model to synthesize streamflow records using pattern analysis. Within the context of analysis of time dependent hydrologic data Panu, Unny, MacInnes and Wong (1981) give a well documented review. In simple terms, a pattern is a shape representation of a physical object. In this application a rising limb, the peak or the falling limb of a streamflow time series (a streamflow time waveform or STW) are a1 1 considered separate physical objects. Measurements on these objects define the shape or pattern of the objects. In practice the actual streamflow data is simply arranged into a set of n-dimensional pattern vectors with each pattern element being numerically equal to the field measurement. Selecting an appropriate dimensionality for the pattern vectors is based on the degree of interdependence between successive streamflow observations. For monthly data it would be difficult to justify any significant dependence beyond a two month period; hence,

376

t h r e e dimensional p a t t e r n v e c t o r s a r e adequate.

Overlapping the

o b j e c t s such t h a t each p a t t e r n v e c t o r c o n t a i n s o n l y one new datum a l l o w s a l l p o s s i b l e shapes t o be r e p r e s e n t e d .

Pattern analysis i s

c a r r i e d o u t on these p a t t e r n v e c t o r s and g i v e s i n t e r p r e t a b l e i n f o r m a t i o n regarding the nature o f the streamflow time s e r i e s . v e c t o r formation i s i l l u s t r a t e d i n F i g u r e 1. process t h a t maps a p a t t e r n

&

Pattern

Feature e x t r a c t i o n , a

onto a feature vector

L, i s

used t o reduce d i m e n s i o n a l i t y and enhance s e p a r a b i l i t y . was n o t r e q u i r e d , h e r e t h e r e f o r e f e a t u r e v e c t o r s ,Y+,

commonly

T h i s process

a r e synonomous w i t h

pattern vectors. Step ahead f o r e c a s t i n g i s c a r r i e d o u t by c o n s i d e r i n g t h e p a t t e r n found f r o m c u r r e n t and most r e c e n t o b s e r v a t i o n s as b e i n g i n c o m p l e t e i n t h e sense t h a t t h e f u t u r e o b s e r v a t i o n i s r e q u i r e d t o p r o v i d e t h e data t o complete t h e p a t t e r n v e c t o r .

The i n f o r m a t i o n g a t h e r e d t h r o u g h

p a t t e r n a n a l y s i s o f t h e h i s t o r i c a l r e c o r d i s used t o p r e d i c t t h e m i s s i n g datum.

Dixon (1979) used p a t t e r n a n a l y s i s t o e s t i m a t e d a t a

i n data sets w i t h missing observations.

The methodology p r e s e n t e d

h e r e i n d i f f e r s f r o m D i x o n ' s approach i n t h a t t h e s t e p ahead f o r e c a s t i s made by f i n d i n g t h e expected v a l u e o f t h e f u t u r e s t r e a m f l o w g i v e n thi possible c l a s s i f i c a t i o n o f t h e incomplete p a t t e r n . Data from t h e B l a c k R i v e r watershed (gauging s t a t i o n 02EC002, n e a r Washago, O n t a r i o ) was used t o s i m u l a t e s t e p ahead f o r e c a s t i n g .

For

t h i s r i v e r 30 y e a r s o f h i s t o r i c a l r e c o r d was used t o p e r f o r m c l u s t e r analysis.

An a d d i t i o n a l 28 y e a r s o f h i s t o r i c a l d a t a was t h e n used t o

make s t e p ahead f o r e c a s t s which were t h e n compared t o a c t u a l observations.

F u r t h e r f o r e c a s t s were always based upon t h e a c t u a l

observations r a t h e r than forecasted values. CLUSTER ANALYSIS A p a t t e r n c l a s s can be c o n s i d e r e d t o be a r e g i o n i n n-dimensional p a t t e r n space c o n t a i n i n g t h e s e t o f a l l p o s s i b l e p a t t e r n s which a r e s u f f i c i e n t l y a l i k e a c c o r d i n g t o some s p e c i f i e d measure.

Any mathematical d e s c r i p t i o n o f a c l a s s i s a s t a t i s t i c a l summary o f

377

STW 1916 for

Black River

....

1

T

I

I

I

I

I

I

1

I

I

I

F

M

A

M

J

J

A

S

O

N

D

.I

a)

100

(No n t hs )

H i s t o r i c a l Data

--

75 -50 --

2 5 --

0

A

L .I

M

F

5

F

M

A

52

M

A

53

c)

1

A

M

54

J

M

J

J

55

Representational Objects

b)

Figure

M

Pattern Vectors

Pattern Vector Formation From Streamflow Data For Black River

J

J

‘6

A

378

i t s member patterns. I n many instances of pattern analysis, including the one i n t h i s application the t r u e classes a r e u n k n o w n . All information concerning c l a s s s t r u c t u r e must be estimated from a training s e t of unlabelled sample patterns t h r o u g h a procedure referred t o a s clustering or unsupervised learning. Various clustering algorithms have been developed, the simplest and b e t t e r known of which are the K-means algorithm (Tou and Gonzales, 1974) and the Isodata algorithm (Ball and Hall, 1965). I n the clustering algorithm developed f o r monthly streamflow patterns i t was assumed t h a t there a r e K = 1 2 d i s t i n c t pattern classes. All the patterns from one time position are i n i t i a l l y assumed t o belong t o one g r o u p , G k ; t h a t i s they form an i n i t i a l c l u s t e r . The c l u s t e r center k i s a reference vector, m , calculated a s follows: Nk

m k = -1 Z Y N k j = 1 -j where N k i s the number of patterns ( f e a t u r e vectors Y . ) i n group -J k G . These i n i t i a l c l u s t e r s are characterized by a rather large variation. This variation i s described by a covariance matrix C k as fol 1ows : Nk

Another way o f examining the variation within a g r o u p i s t o calculate the distance from each pattern t o i t s reference vector. Using a Euclidean distance measure:

379

ntra group distance i s :

The mean

Nk

1

(4)

Another useful distance i s the i n t e r g r o u p distance or the distance k j between reference vector 3 and any other reference vector E :

n k

j

dE(" , m )

=

C Z

(mr

-

j 2m i ) 1'

(5)

i=l A boundary between two groups can be visualized as a distance half-

way between any two c l u s t e r centres. The hyperplane describing such a boundary i s a decision surface according t o which a p a r t i c u l a r pattern i s said t o belong t o one group o r the other. For example:

C1 ustering proceeds, as fol1 ows :

Determine i n i t i a l c l u s t e r centres and boundaries as noted above . 2 . Using 'Equation ( 3 ) , determine f o r every pattern the distance t o i t s own c l u s t e r centre and the distance t o the centre of the two c l u s t e r s t h a t are situated time contiguous on b o t h sides (i . e . , neighbouring c l u s t e r s ) . 3 . Reassign a l l the patterns t o any of the three time contiguous c l u s t e r s according t o the c r i t e r i o n of Equation ( 6 ) . T h e centres of these c l u s t e r s can be considered t o be the centres of the pattern classes of which the c l u s t e r s form a s u b g r o u p . The procedure ensures t h a t a proper t r a j e c t o r y progressing in time i s formed by connecting the centres o f the classes. The c l u s t e r s so formed are time-dependent c l u s t e r s a n d n o t global c l u s t e r s . 4. For the c l u s t e r s obtained in Step 3 above, new c l u s t e r centres and boundaries are calculated a n d the procedure repeated from Step 2 t o Step 3 w i t h the objective of reducing the variance of the groups, and thus the i n t r a c l u s t e r distances are a l s o reduced. 1.

380

Further, the i n t e r c l a s s distances are enhanced providing f o r increased s e p a r a b i l i t y of the classes. This clustering algorithm stops when the l a s t complete cycle of i t e r a t i o n was carried o u t and i t was found t h a t there was no f u r t h e r change i n the membership of any of the c l u s t e r s . Typically, about fourteen i t e r a t i o n s were required t o c l u s t e r the patterns. The final k c l u s t e r s represent sample estimates of the pattern classes ( W ) . These classes are characterized by a narrow multivariate probability distribution having a mean and some variance about the mean. The regions i n m-dimensional feature space representing each of the twelve classes a l s o have d e f i n i t e boundaries according t o which any new unlabelled pattern can be properly c l a s s i f i e d . Two typical c l u s t e r s f o r Black River are shown i n Figure 2 . The ordered progression from one class centre t o the next i s called a mean annual cycle (See Figure 3 ) . The outer segment of the curve h a v i n g large values of y l , y2 o r y3 o r combinations of a l l three, represent the regions in pattern space t h a t are equivalent t o peak flows. The two loops represent streamflows t h a t have a large annual peak flow (as occurs in the spring r u n o f f ) and a smaller peak occurring annually i n the f a l l . A larger distance between two adjacent reference vectors on the annual cycle s i g n i f i e s a greater v a r i a b i l i t y of the member patterns within the c l a s s . Cluster boundaries bisect the distance between adjacent reference vectors. Clusters may be r e l a t i v e l y close together i n terms of distance i n the 3-dimensional feature space. However, i f the reference vectors a r e on d i f f e r e n t loops, they are f a r apart i n time ( i . e . , they are n o t time contiguous). STEP AHEAD FORECASTING

Forecasting i s based on a l l of the information gathered t h r o u g h the use o f c l u s t e r analysis. The most recent pattern i s considered t o be incomplete in t h a t i t has two known features, y1 and y 2 , b u t the t h i r d feature, y3, representing the step ahead flow, i s u n k n o w n .

381

Figure 2

Typical Clusters Represented by Objects from the STW f o r Black River. (Not a l l objects are shown)

125 ,3

100

75 A

--. m (I:

E

v

3

50

4 LA

25 P a t t e r n Vector

----0

1

y3

50

-w

m

E

v

25

8

--

3

0

0

I

I

1

Reference Vector

382

1

Figure 3 .

Mean Annual C y c l e For t h e B l a c k R i v e r enpresented i n thr--i dirli?psional_ F'carur? sr)' ? & - . ~ ' - 0

3 83

A step ahead forecast i s made by estimating the value o f y3.

The i n i t i a l step i s t o " c l a s s i f y " the incomplete pattern. Because the t h i r d element of the pattern vector i s missing, proper classification i s n o t always possible. However, classes t o which the-pattern i s l i k e l y t o belong can be determined with a f i n i t e probability. The classes t o which an unlabelled pattern may belong are called candidate classes. These candidate classes are determined u s i n g the f i r s t two elements of the incomplete pattern. Forecasting i s carried o u t based on the probability of the occurrence of y3 w i t h i n a candidate c l a s s as well a s the probability o f occurrence of the candidate class within the pattern space. Thus the forecast i s the expected value o f y3. I t was observed in the c l u s t e r analysis t h a t patterns tended t o cluster i n groups whose member patterns were derived from times of origin t h a t are the same o r neighbouring months of the year. The time of origin of a pattern i s therefore an a p r i o r i information for determination of candidate c l a s s e s . Two c r i t e r i a must be satisfied f o r a c l a s s t o be a candidate c l a s s :

For the features y1 and y2 from an incomplete pattern, a f i n i t e value f o r y3 must e x i s t w i t h i n the c l a s s boundaries. 2. The probability of the class must be non-zero given the months of origin o f the incomplete pattern class t o be a candidate c l a s s . 1.

Pattern c l a s s i f i c a t i o n i s based on the a posteriori probability of the class which can be found using Bayes Theorem. Written i n terms o f the application:

384

k where p(y, ,y21u ) i s the c l a s s - c o n d i t i o n a l p r o b a b i l i t y d e n s i t y and i s e s t i m a t e d from the marginal p r o b a b i l i t y a s f o l l o w s :

ull

“12

i s the p a r t i a l covariance m a t r i x of c l a s s w k which i s

The a p o s t e r i o r i p r o b a b i l i t i e s f o r a l l k G 12 c a n d i d a t e c l a s s e s a r e such t h a t

zK

P(,

k

IY1 ? Y 2 )

=

1

(10)

k I t must be noted here t h a t a l l p r o b a b i l i t i e s a r e based on c a l c u l a t i o n s using d a t a from the c l u s t e r a n a l y s i s .

Should a s l i g h t l y

d i f f e r e n t set o f p a t t e r n s be used f o r c l u s t e r i n g then i t i s quite l i k e l y t h a t the a p o s t e r i o r i p r o b a b i l i t i e s w i l l change given t h e same incomplete p a t t e r n . Such a case would occur a s new streamflow measurements c o n t i n u o u s l y update t h e h i s t o r i c a l r e c o r d . This s i t u a t i o n i s common t o any model t h a t uses a f i n i t e and, in most c a s e s , 1 imi t e d amount of samples.

385 The p a t t e r n c l a s s e s a r e assumed t o have a m u l t i v a r i a t e normal The m i s s i n g f e a t u r e y3 o f an

d i s t r i b u t i o n o f sample p a t t e r n s .

i n complet e p a t t e r n i s fo u n d b y c a l c u l a t i n g t h e e x p e c t e d v a l u e o f F o r any y1 and y2 f r o m a p a r t i c u l a r c a n d i d a t e c l a s s t h e r e i s y3. a p r o b a b i l i t y a s s o c i a t e d w i t h any y3. T h i s c o n d i t i o n a l p r o b a b i l i t y

is: P(Y1 'Y2 ,Y3) P(Y31Yl ,Y,Ik

=

P(Y, ,Y$ k

where

P(Yl ,YZyY3)

k

1

--

Zn3" and

P(Y, YY2)

k

=

e

-+(Y-! -

k T k-1 ) cc 1

(Y-m _ - k)

detCCkl'

k

P(Y, YY2IW )

as i n Equat ion (8) Not a l l value s o f y3 a r e f e a s i b l e .

F e a s i b l e v a l u e s o f y3 a r e

d e t ermined by f i n d i n g p o i n t s on t h e boundary o f t h e c l a s s i n m-dimensional f e a t u r e space a l o n g t h e l i n e g i v e n b y y1 and y2. These values f o r y3 range from, say, A t o B where A i s on t h e boundary between c l a s s k- 1 w k and w .

wk

and wkt1

and B i s on t h e boundary between

The expect ed v a l u e o f y 3 g i v e n t h e c a n d i d a t e c l a s s i s : B

386

The variance of y3 can a l s o be calculated by simply including an additional term in the expression above. I n a l l cases the estimate of the most probable value o f y3 i s weighted according t o the class a posteriori probabilities. The expected value of y3 i s :

k= 1

STEP AHEAD FORECASTING RESULTS

The methodology described above was applied t o an additional 28 years o f historical data t o produce a s e t of step ahead forecasts based on actual observations. This made i t possible t o compare the step ahead forecasts w i t h the streamflows t h a t actually occurred. I t i s c l e a r t h a t there a r e always some forecast e r r o r s . These e r r o r s being the difference between predicted and observed streamflows are assumed t o be independent random variables. The mean overall e r r o r for the e n t i r e s e r i e s of forecasts i s g i v e n a s follows: €

=

-

n

1 n

* E

i=l

i

(14)

where n i s the number of step ahead forecasts. The standard deviation i s calculated as: s

=

c n

A - -E 2) / (n-1)1'

(Ei

i=l

The two sided Student's t Test can be used t o t e s t whether the overall e r r o r 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 zero. The t t e s t i s a s follows:

IL.5 I S

>

tn-1;0,025

387

The f o r e c a s t e r r o r s were a l s o e v a l u a t e d on a monthly b a s i s u s i n g t h e method g i v e n above. A 5% s i g n i f i c a n c e l e v e l was used f o r the c r i t i c a l value o f t . R e s u l t s f o r the Black River a r e given in the following t a b l e : Summary of Forecast E r r o r s by Months f o r Black River Month

S

a l1

tn-l ; O . 025 2.052 2.052 2.052 2.052 2.052 2.052 2.052 2.052 2.052 2.052 2.052 2.052

S

January February March April May June July August September October November December Overall

-0.07 -1.93 -0.33 3.54 8.03 3.05 -0.36 2.17 0.89 -1.77 -3.81 -3.37 0.50

11.27 6.44 20.46 26.76 13.49 7.88 6.46 3.03 3.17 11.47 10.83 12.95

0.032 1.583 0.086 0.699 3.150 2.048 0.292 3.790 1.493 0.816 1.860 1.379

13.2

0.70

I

1.960

Figure 4 shows the f i r s t two y e a r s of h i s t o r i c a l d a t a used f o r the simulation of f o r e c a s t i n g .

Superimposed on t h e time s e r i e s a r e the

s t e p ahead f o r e c a s t s i n c l u d i n g the 2 s t a n d a r d d e v i a t on lim t s a s c a l c u l a t e d by the f o r e c a s t i n g methodology. CONCLUDING REMARKS

A methodology i s given f o r s t e p ahead streamflow f o r e c a s t i n g

using p a t t e r n a n a l y s i s which accounts f o r varying p e r s i s t e n c e e f f e c t ! within the time s e r i e s . C l u s t e r i n g the hi t o r i c d a t a provides the information t h a t c h a r a c t e r i z e s the streamf ows. The a c t u a l f o r e c a s t i n g procedure uses t h i s information t o complete a p a t t e r n v e c t o r on a p r o b a b i l i s t i c b a s i s . This approach g ves s a t i s f a c t o r y r e s u l t s f o r the case s t u d y presented h e r e .

388

\-.

Observed Streamflow

i

S t e p ahead f o r e c a s t s .

*

Mean flow and 2 s t a n d a r d d e v i a t i o n s on e i t h e r s i d e .

130 120

110 100

90

.. .-. . .. ..*

I

80

,--.70 (11

60

2

l

1

-

?

3

0

:.

1 ;

--m E v

. .. ...

:.

50

i 40

-

.. ! ..

-

30 T

20

10

0

r

J

,

l

l

,

,

l

1946

Figure

4.

,

,

,

l

,

,

l

,

,

1

D J F

F

,

,

,

,

(

D

1947

S t e p Ahead F o r e c a s t e d Streamflows Superimposed on t h e Observed T i m e S e r i e s For the Black River

389

Further developments in t h i s use of pattern analysis will be t o extend the methodology t o weekly and daily streamflow time s e r i e s . Pattern vectors of higher dimensionality would then be needed. In a d d i t i o n pattern vectors can be used t o incorporate other pertinent

information i n the analysis. This would include such data as snow accumulation, temperatures, the amount o f water stored in the watershed, o r weather forecasts from independent sources. Data from t r i b u t a r i e s o r other rivers i n the region could be included as we1 1 . ACKNOWLEDGEMENTS

The senior author gratefully acknowledges the support provided during the research period by grants from the National Science and Engineering Research Counci 1 of Canada. REFERENCES

Ball, G . H . , and Hall, D.J., "Isodata, An I t e r a t i v e Method o f Multivariate Analysis and Pattern Classification", Proceedings of the IFIPS Congress, 1965. Dixon, J.K., "Pattern Recognition with Partly Missing Data", I E E E Trans. on Systems, Man and Cybernetics, Vol . SMC-9, No. 10, October 1979. Panu, U.S., "Stochastic Synthesis of Monthly Streamflows Based on Pattern Recognition", Doctoral Dissertation, Department Of Civil Engineering, University of Waterloo, Canada, May, 1978. Tou, J.T., and Gonzalez, R . C . , "Pattern Recognition Principles", Addison-Wesley Publishing Company, Massachusetts, 1974, rev. 1977. Unny, T . E . , Panu, U.S., MacInnes, C . D . , and Wong, A . K . C . , "Pattern Analysis and Synthesis o f Time-Dependent Hydrologic Data", Advances i n Hydroscience, Academic Press, New York, Vol. 1 2 , P . P . 195-297, 1981.

3 90

WALSH SOLUTIONS I N HYDROSCIENCE

Z E K A I SEN C i v i l E n g i n e e r i n g F a c u l t y , Technical U n i v e r s i t y o f I s t a n b u l , Turkey

ABSTRACT

O r t h o g o n a l Walsh f u n c t i o n s h a v e p o t e n t i a l a p p l i c a t i o n p o s s i b i l i t i e s i n hydrosystem problems such as t h e s t a t i s t i c a l d e s c r i p t i o n , s i m u l a t i o n , real time p r e d i c t i o n , p o r o u s m e d i a d e s c r i p t i o n , 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 etc. T h i s paper p r e s e n t s , F i r s t o f a l l , t h e f u n d a m e n t a l s o f these square-wave n d -1 f u n c t i o n s w h i c h a s s u m e a l t e r n a t i v e l y o n l y +aI v a l u e s o v e r a s p e c i f i e d time i n t e r v a l d u r i n g t h e o p e r a t i o n o f h y d r o s y s t e m c o n c e r n e d . The most a t t r a c t i v e p r o p e r t i e s o f them are t h a t t h e y a r e p i e c e w i s e l i n e a l i n e a r , orthonormal and t h e i r m u l t i p l i c a t i o n s r e q u i r e s i m p l e mathematical o p e r a t i o n s . F o u r d i f f e r e n t w a y s o f Walsh f u n c t i o n a p p l i c a t i o n s These are; i n t h e hydroscience are described herein. ( 1 ) t h e a n a l y s i s o f s t o c h a s t i c - p e r i o d i c time s e r i e s w i t h a minimum a f f o r d o f c o m p u t a t i o n s a n d g r e a t a c c u r a c y ; (2) s t a t i s t i c a l d e s c r i p t i o n o f p o r o u s media geometry and i t s comparison w i t h t h e F o u r i e r f u n c t i o n s ; ( 3 ) a n a d a p t i v e W a l s h p r e d i c t o r c o u p l e d w i t h t h e Kalman t e c h n i q u e c a p a b l e o f m a k i n g r e a l time p r e d i c t i o n s V i s i t i n g P r o f e s s o r , King A b d u l a z i z U n i v e r s i t y , F a c u l t y o f E a r t h S c i e n c e s , P.0. B o x 1744 J e d d a h , Kingdom o f S a u d i Arabia.

-

Reprinted from Time Series M e t h o d s in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors, o 1982 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

391

and its a p p l i c a t i o n f o r a m o n t h l y s e q u e n c e o b s e r v e d i n T u r k e y ; a n d (4) Walsh f u n c t i o n s a s a n a l t e r n a t i v e t o t h e f i n i t e element m e t h o d w h i c h a r e e x t e n s i v e l y e m p l o y ed i n t h e solutions of various d i f f e r e n t i a l equations r e l a t e d t o g r o u n d water f l o w .

INTRODUCTION The Wal8h f u n c t i o n 8 a r e a s e t o f two v a l u e d

o r t h o g o n a l f u n c t i o n s t h a t p r o m i s e p o t e n t i a l use i n v a r i o u s h y d r o s c i e n c e problems. The o r i g i n a l d e f i n i t i o n o f Walsh f u n c t i o n s h a s b e e n p r e s e n t e d b y W a l s h (1923) u s i n g a s e t o f r e c u r s i v e r e l a t i o n s . I n g e n e r a l , t h e Walsh f u n c t i o n s form a n o r d e r e d s e t o f s q u a r e w a v e s t a k i n g two s p e c i f i e d v a l u e s +aI n d -1 o n l y . Tkdo a r g u m e n t s , namely, time p e r i o d , t , a n d a n o r d e r i n g number, n , are r e q u i r e d f o r t h e i r d e f i n i t i o n , h e n c e n - t h o r d e r W a l s h f u n c t i o n i s d e n o t e d b y WAL(n,t). The d i s c r e t e version o f t h e s e funotions has been given i n a compact f o r m by Brown(1977) as, WAL(0,n) = I f o r n = 1,2, ,No I f o r n = 1,2, ,N/2 WAL(1,n) = (1) f o r n = (N/2)+1,(N/2)+2, ,N [I

.. ..

WAL(m,n) = WAL(rm/2],2n) .WAL(m

-

..

2[m/2]

,n>

where N d e p i c t s t h e t o t a l n u m b e r o f o r t h o g o n a l f u n c is the integer p a r t t i o n s i n a complete set; and o f t h e a r g u m e n t . An a l t e r n a t i v e d e f i n i t i o n o f t h e Walsh f u n c t i o n s h a s b e e n p r o p o s e d by P a l e y ( 1 9 3 2 ) a s a p r o d u c t o f a n o t h e r o r t h o g o n a l s e t c a l l e d Rademacher (1922) f u n c t i o n s . However, P a l e y ' s d e f i n i t i o n i s d i f f e r e n t from W a l s h l s o n l y i n t h e o r d e r i n g . Paley d e f i n i t i o n o f W a l s h f u n c t i o n s has b e e n s u m m a r i z e d by S e n ( 1 9 8 1 ) . On t h e b a s i s o f E q . ( l ) t h e r e s u l t i n g comp-

[.I

392

l e t e e e t o f W a l s h f u n c t i o n s f o r N=8 is s h o w n i n F i g u r e 1. WAL(0,B) WAL(1,B)

ct=f-t=p

I ,

WAL(2,8) WAL(3,8)

I

1

I

I

1

F i g u r e I.Walsh f u n c t i o n s f o r N = 8. A necessary requirement f o r a successful application

o f W a l s h f u n c t i o n s is t h a t N=Zq where q i s a n y c o n v e n i e n t p o e i t i v e i n t e g e r power. T h e o r t h o g o n a l i t y p r o p e r t y o f W a l s h f u n c t i o n s is g i v e n b y t h e f o l l o w i n g e q u a t i o n

N xWAL(i,n).WAL( j , n ) = n=O

c

0

for

( i # j)

1

for

( i = j)

(2)

As i s o b v i o u s f r o m F i g u r e 1 t h e W a l s h f u n c t i o n s a r e

s y m m e t r i c a b o u t t h e i r mid time p o i n t s , Hence, t h i s p r o p e r t y e n a b l e 8 a symmetry r e l a t i o n s h i p a s , WAL(i,n) = WAL(n,i)

(3)

The p r a c t i c a l i m p o r t a n c e of t h i s is t h a t t h e Walsh

t r a n e f o r m s a n d t h e i r i n v e r s e s r e p r e s e n t t h e same mathematical o p e r a t i o n ,

TIME SERIES REPRESENTATION An e f f e c t i v e u s e o f W a l s h f u n c t i o n s o c c u r s i n r e p r e s e n t i n g a time s e r i e s b y t h e s u p e r p o s i t i o n o f m e m b e r s o f a set of orthogonal simple functions, In order t o repres e n t a time s e r i e s X ( i = 1 , 2 , , , n > 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 << q l . X and X a r e 0 dimensional v e c t o r s r e p r e s e n t i n g a c t u a l and e s t i m a t e d c o n c u r r e n t events a t t h e q n locations, while X w i l l r e p r e s e n t e r r o r f r e e measurements made a t the

n

It i s important

l o c a t i o n s o f a m o n i t o r i n g network.

t h a t these

n

l o c a t i o n s be a subset o f t h e

q

location.

An i n f o r m a t i o n a l q u a n t i t y o f in t e r e s t i s t h e in f o r m a t i on t r a n s m i t t e d by t h e network measurements concern! ng Equation 2.4 I(X;Xn)

=

H ( X ) + H(Xn)

-

H(X,Xn)

X.

From

420

But as

Xn

H(X,Xn)

reduces t o

I(X;Xn)

i s a subvector o f H(X)

then t h e j o i n t entropy term

X

so t h a t

H(Xn)

=

Substituting

(4.1 f o r t h e measurement

Xn

Z

i n Equation 3.9 y i e l d s

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 e s t i m a t e about I(X;X)

I(x;xn)

=

-

-

[I(&;xn)

1

X

I(&;i)]

Under c o n d i t i o n s where t h e e s t i m a t e preserves t h e i n f o r m a t i o n n contained i n X n then I(&;Xi 1 = I ( E ; ~ ) ,or when t h e e r r o r i s independent o f b o t h t h e measurement and t h e estimate, then I(X;X) achieves an upper bound which i s given by

=

I(X;^x)

I(X;Xn)

=

H(Xn)

(4.2)

The e r r o r entropy i s given by Equation 3.8

H(E)

=

H(X)

-

I(X;Xn) + 1(c;Xn)

(4.3)

and achieves i t s minimum value when t h e e r r o r i s independent o f the measurement.

H(E)

=

H(X)

Equations 4.1,

-

Thus t h e lower bound f o r e r r o r entropy i s H(Xn)

4.2,

4.3,

(4.4) and 4.4 suggest t h e d e s i r a b i l i t y o f

maximizing 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 network measurements, and under c e r t a i n c o n d i t i o n s , t h e i r j o i n t entropy, when designing m o n i t o r i n g networks.

T h i s approach forms t h e b a s i s

o f t h e network design method suggested by Case1 t o n and Husain [1980].

The i m p l i c a t i o n s o f t h i s approach t o network design when

the (spatial) properties o f

X

a r e described by a m u l t i v a r i a t e

normal model a r e discussed by Caselton and Zidek L 1 9 8 l I .

5 MODELS INVOLVING SERIAL DEPENDENCY S e r i a l processes, where an event a t t i m e

t

i s influenced t o

some degree by p r i o r events, o f t e n a r i s e i n t h e hydrosciences. measure o f t h e e x t e n t o f t h i s i n f l u e n c e i s t h e i n f o r m a t i o n which i s t r a n s m i t t e d by p r i o r events concerning t h e n e x t event.

Xt

A

421

w i l l r e p r e s e n t a v e c t o r d e s c r i b i n g events a t t i m e t and Xt-l, e t c . r e p r e s e n t events a t e a r l ie r times. The in f o r m a t i onal Xt-2' ,Xt-2,. ,X ) f o r p r i o r events q u a n t i t y o f i n t e r e s t I(Xt;Xt-l

..

to time

i s g i v e n by Equation 2.4

t-r

,...,X t-r 1

I(Xt;Xt-l,Xt-2

H(Xt) + H(Xt-l,Xt-2

=

=

where

nHr

t-r

H(Xt,Xt-l H(Xt)

-

,... 't-r)

,xt-2."',x

t- r

(5.1 1

AHr

i s t h e i n c r e a s e i n j o i n t e n t r o p y when

the a r r a y

{Xt-l

,Xt-2

,...,X t-r I .

s t a t i o n a r y , an upper bound f o r

X t i s added t o

I n t h e case where t h e process i s AH r

can be e s t a b l i s h e d when

i s a " f r e e e x t e n s i o n " o f t h e process, i.e.

X

t

when

(5.2) and when t h e e x t e n s i o n i s n o t " f r e e " then

Thus, i f

for both

r

r

i s l a r g e enough t o ensure t h a t E q u a t i o n 5.2 h o l d s

and a l l l a r g e r values o f

r,

t h e n t h e maximum 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 by t h e p a s t process t o t h e n e x t event i s g i v e n by E q u a t i o n 5.1. be based upon t h e values o f

I f an e s t i m a t e o f t h e n e x t e v e n t i s t o

r

p a s t events t h e n t h e l o w e r bound

o f e r r o r e n t r o p y can be o b t a i n e d from E q u a t i o n 3.13.

s p e c i f y i n g t h e o p t i m a l e s t i m a t i o n model.

without

422

H(

E)

(5.4)

The lower bound of error entropy i s given, in this case, by the increase in the joint entropy caused by adding the next event. 6

INFORMATION TRANSMISSION FOR SOME SPECIFIC ESTIMATION MODELS

The previous sections have shown how certain bounds on information transmission and error entropy can be established w i t h o u t the estimation model or error characteristics being specified. I n this final section, the information transmission characteristics of some commonly specified types of estimation model are mentioned. A requirement of many estimation models i s t h a t the error be Gaussian and unbiased. The relationship between the joint entropy of an m dimensional error vector and the error covariance matrix R E f o r Gaussian errors i s given by H(E)

=

1 1 0 g ~ { ( 2 1 r det(RE)l e)~

(6.1 1

where det( 1 denotes the determinant. Equation 6.1 confirms t h a t minimizing the determinant of the error covariance matrix i s synonymous with minimizing error entropy. Thus, the general case of regression, where errors are required t o be unbiased and uncorrelated and are commonly assumed t o be Gaussian, also conforms t o the minimization of error j o i n t entropy and the maximization of information transmission I ( X ; X ) . The Kalman f i l t e r i s representative o f a more complex type of dynamic model which has received attention in the hydrosciences. A

423 A property o f t h e optimal l i n e a r f i l t e r i s t h a t t h e optimal

estimate and i t s e r r o r a r e u n c o r r e l a t e d (Gelb C19741).

Since t h e

h

process

{X,,?)

A

i s j o i n t l y Gaussian then t h e value o f

I(Xk;%)

i s zero and Equation 3.8 reduces t o

I(xk;xk 1 Tomita e t a l . [19761 show t h a t t h e optimal l i n e a r f i l t e r minimizes the determinant o f t h e e r r o r covariance m a t r i x and t h e r e f o r e , f o r Gaussian e r r o r , minimizes t h e e r r o r entropy.

Equation 6.2

confirms t h a t t h e i n f o r m a t i o n t r a n s m i s s i o n between t h e s t a t e and i t s e s t i m a t e i s a l s o maximized.

REFERENCES Case1 ton, W.F. and Husain, T., "Hydrologic Networks: I n f o r m a t i o n Transmission" , Journal o f t h e Water Resources P1 anni ng and Management D i v i s i o n , ASCE, Vol. 106, No. WR2, J u l y 1980, pp. 503-520. Caselton, W.F. and Zidek, J.V., "The Use o f a Proper Local U t i l i t y i n t h e Design o f M o n i t o r i n g Networks", Technical Report No. 81-11. I n s t i t u t e o f A p p l i e d Mathematics and S t a t i s t i c s , UBC, June 1981. Dyhr-Niel sen, M. "Loss o f I n f o r m a t i o n By D i s c r e t i z i n g H y d r o l o g i c Series", Hydrology Papers, Colorado S t a t e U n i v e r s i t y , Oct. 72. Gallager, R.G. " I n f o r m a t i o n Theory and Re1 i a b l e Communication", John Wiley 8 Sons, 1968. Press 1974. Gelb, A. " A p p l i e d Optimal Estimation", M.I.T. Hartmani s, J. "Appl i c a t i o n o f Some Basic I n e q u a l i t i e s f o r Entropy", I n f o r m a t i o n and C o n t r o l , Vol. 2, 1959, pp.199-213. Rai f f a , H. "Decision Analysis", Addison Wesley Pub1 is h i ng Company 1970. Shannon, C.E. "A Mathematical Theory o f Communications", B e l l Systems Technical Journal, Vol. 27, 1948, p.379-423, 623-656. Tomita, Y . , Ohmatsu, S., Soeda, T. "An A p p l i c a t i o n o f I n f o r m a t i o n Theory t o E s t i m a t i o n Problems", I n f o r m a t i o n and C o n t r o l , Vol. 32, 1976, pp.101-111. Weidemann, H.L., Stear, E.B. "Entropy a n a l y s i s o f parameter e s t i m a t i o n " , I n f o r m a t i o n and C o n t r o l , Vol. 14, 1969, pp.493506.

424

VALIDATION OF SYNTHETIC STREAMFLOW M O D E L S ~ DENNIS P. L E T T E N M A I E R ~A N D STEPHEN J. B U R G E S ~

INTRODUCTION I n r e v i e w i n g t h e l a s t two d e c a d e s o f work i n s t o c h a s t i c h y d r o l ogy, one f i n d s a n overwhelming predominance o f e f f o r t i n model dev e l o p m e n t , as opposed t o p a r a m e t e r e s t i m a t i o n ( c a l i ' r : a t i o n ) and verification.

Recent work, however, (Rlemes, e t a l . , 1 9 8 1 ; B u r g e s

and L e t t e n m a i e r , 1981) h a s d e m o n s t r a t e d t h a t t h e p r a c t i c a l i m p l i c a t i o n s of d i f f e r e n c e s i n model form may n o t b e n e a r l y as g r e a t as s u g g e s t e d by much of t h e work on model d e v e l o p m e n t .

Therefore,

i n c r e a s e d e m p h a s i s on model v a l i d a t i o n , which w e c o n s i d e r t o i n c l u d e c a l i b r a t i o n and v e r i f i c a t i o n , a p p e a r s i n o r d e r . The common a p p r o a c h t o i m p l e m e n t a t i o n of s t o c h a s t i c ( o r synt h e t i c ) h y d r o l o g i c models i s t o e s t i m a t e a p a r a m e t e r v e c t o r from a set of h i s t o r i c r e c o r d s , t h e n t o proceed as i f t h e e s t i m a t e d p a r a m e t e r s were t h e p o p u l a t i o n v a l u e s . F o r i n s t a n c e , a p a r a m e t e r * "T , em] i s e s t i m a t e d from a h i s t o r i c d a t a m a t r i x = [el, vector

s

T [Zi,

...

A

..., &], w i t h T

X , = (Xil,

...,

X . ) , w h e r e n i s t h e num1s b e r o f y e a r s o f h i s t o r i c r e c o r d and s i s t h e number of s i t e s . XT =

-1

Although t h e r e c o r d l e n g t h , n , i s i m p l i e d t o b e e q u a l f o r a l l s i t e s , t h i s n e e d n o t b e t h e case.

Further, t h e observations X

ij

a P r e s e n t e d a t I n t e r n a t i o n a l C o n f e r e n c e o n T i m e S e r i e s Methods i n H y d r o s c i e n c e s , Canadh C e n t r e f o r I n l a n d Waters, B u r l i n g t o n , O n t a r i o , O c t o b e r 6-8, 1981. b R e s e a r c h A s s o c i a t e P r o f e s s o r , Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y o f Washington, S e a t t l e , WA 98195. C

P r o f e s s o r , Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of Washi n g t o n , S e a t t l e , WA 95195.

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

425 may c o n s i s t of p s e a s o n a l f l o w volumes, i . e . , n with Xijk = x i j .

xij

..., xi j P 1

- {xijl,

kLl

h

The time-honored

a p p r o a c h t o model v a l i d a t i o n , once

S has

been

e s t i m a t e d , i s t o g e n e r a t e d m u l t i p l e s e q u e n c e s of s y n t h e t i c d a t a , Yi(R), i = 1, T ; where T i s t h e number of s e q u e n c e s g e n e r a t e d ,

...,

and R i s t h e l e n g t h of t h e s y n t h e t i c s e q u e n c e s ( n o t n e c e s s a r i l y equal t o n , t h e h i s t o r i c record length). From t h e m u l t i p l e s y n t h e t Mijk ( r ) a r e computed. For v a l i i c d a t a s e q u e n c e s , low o r d e r moments, d a t i o n of m u l t i s i t e s e q u e n c e s , c r o s s moments, u s u a l l y of t h e s e c m d o r d e r , may a l s o b e computed.

The a v e r a g e s of t h e s e e s t i m a t e d moments

o v e r t h e T s y n t h e t i c s e q u e n c e s a r e t h e n compared t o t h e ( s i n g l e ) moments e s t i m a t e d from t h e h i s t o r i c r e c o r d .

If the replication is, i n

some s e n s e , s a t i s f a c t o r y , t h e model i s c o n s i d e r e d t o b e v a l i d .

Al-

though q u a n t i t a t i v e measures of v a l i d a t i o n and model q u a l i t y a r e r a r e l y made, t h e e s t i m a t e d s t a n d a r d e r r o r of t h e s y n t h e t i c moments p r o v i d e s a p o s s i b l e measure f o r e v a l u a t i n g t h e q u a l i t y of r e p l i c a t i o n . A t l e a s t t h r e e d i f f i c u l t i e s a r e e n c o u n t e r e d i n s u c h moment-

oriented validations.

F i r s t , t h e h i s t o r i c record l e n g t h s are u s u a l l y

so s h o r t t h a t i t i s n o t p r a c t i c a l t o r e s e r v e any of t h e r e c o r d f o r A

validation, i.e.,

t h e h i s t o r i c moments and t h e p a r a m e t e r s e t

from

which t h e s y n t h e t i c s e q u e n c e s a r e d e r i v e d , (and s y n t h e t i c moments e s t i m a t e d ) a r e t h e same h i s t o r i c r e c o r d .

Thus t h e h i s t o r i c and syn-

t h e t i c moments a r e n o t i n d e p e n d e n t , and t h e d e g r e e of r e p l i c a t i o n which s h o u l d b e e x p e c t e d i s open t o q u e s t i o n .

Second, estimates of

moments beyond t h e f i r s t o r d e r a r e b i a s e d a s a r e s u l t of t h e p e r s i s t e n c e s t r u c t u r e i n h e r e n t i n h y d r o l o g i c t i m e s e r i e s a n d , i n some cases, s m a l l sample effects.

A d d i t i o n a l b i a s e s may b e i n t r o d u c e d

by t r a n s f o r m a t i o n s invoked t o a l l o w s y n t h e s i s of r a w s e q u e n c e s i n t h e normal ( g a u s s i a n ) domain, an i n t e r m e d i a t e s t e p i n many models. T h i r d , t h e m u l t i p l i c i t y of moments i n a m u l t i s i t e , m u l t i s e a s o n model, c o u p l e d w i t h s h o r t h i s t o r i c r e c o r d s from which model parame t e r s a r e e s t i m a t e d , make i t d o u b t f u l t h a t any p r o p e r l y p a r a m e t e r i z e d s t o c h a s t i c model c a n , o r s h o u l d , preserve a l l moments.

Clearly, i f

426 enough p a r a m e t e r s are i n c l u d e d i n t h e model ( l e a v i n g a s m a l l number of d e g r e e s of freedom) t h e s y n t h e t i c s e q u e n c e s become i n c r e a s i n g l y

similar t o t h e h i s t o r i c , u n t i l u n l t i m a t e l y t h e s y n t h e t i c s e q u e n c e s

are i d e n t i c a l t o t h e h i s t o r i c r e c o r d .

T h i s , of c o u r s e , s u b v e r t s

t h e p u r p o s e of s y n t h e t i c h y d r o l o g y , which i s t o p r o v i d e a r e p r e s e n t a t i o n of a l t e r n a t i v e , e q u a l l y l i k e l y s c e n a r i o s which a r e s t a t i s t i c a l l y r e p r e s e n t a t i v e of t h e p a s t .

When s u i t a b l e model i d e n t i f i c a t i o n

p r o c e d u r e s ( s u c h as t h e Akaike (1974) i n f o r m a t i o n c r i t e r i o n ) a r e i n voked t h e r e s u l t i n g model may b e u n a b l e t o p r e s e r v e some, o r e v e n many, moments.

T h i s i s p a r t i c u l a r l y t r u e when t h e model e s t i m a t i o n

p r o c e d u r e i s n o t moment-based

(e.g.,

maximum l i k e l i h o o d ) .

Although

t o some e x t e n t s u c h i n f i d e l i t y may r e p r e s e n t a s h o r t c o m i n g of t h e model u s e d , i n a more g e n e r a l s e n s e i t s i m p l y r e f l e c t s l i m i t a t i o n s imposed by s h o r t d a t a r e c o r d s .

ALTERNATIVE VALIDATION MEASURES

Sample moments, of c o u r s e , a r e n o t t h e o n l y p o s s i b l e v a l i d a t i o n measures.

O t h e r s , which w e t e r m s t a t i s t y c a l , i n c l u d e a u t o c o r r e l a t i o n s ,

as w e l l as o t h e r m e a s u r e s of model p e r s i s t e n c e s t r u c t u r e , s u c h a s

t h e Hurst c o e f f i c i e n t .

The b a s i c c o m p u t a t i o n a l a p p r o a c h i s t h e same

as t h a t o u t l i n e d a b o v e ; a v e r a g e s of t h e estimates o v e r many s y n t h e t i c t r a c e s a r e compared w i t h h i s t o r i c estimates. A l l s t a t i s t i c a l m e a s u r e s , moments and o t h e r s , h a v e a common

shortcoming:

t h e y do n o t r e f l e c t t h e u l t i m a t e u s L of t h e s y n t h e t i -

c a l l y generated sequences.

Although s p e c i f i c s v a r y , most a p p l i c a -

t i o n s of s y n t h e t i c h y d r o l o g y a d d r e s s p r o b l e m s e i t h e r o f e s t i m a t i n g t h e r e q u i r e d i n i t i a l o r i n c r e m e n t a l s i z e of a s t o r a g e f a c i l i t y , o r of e s t i m a t i n g p e r f o r m a n c e c h a r a c t e r i s t i c s

an e x i s t i n g ( o r p r o p o s e d ) s y s t e m .

(e.g.,

r e l i a b i l i t y ) of

Thus, v a l i d a t i o n m e a s u r e s which

r e f l e c t c o m p a t i b i l i t y of s y n t h e t i c and h i s t o r i c s e q u e n c e s w i t h r e s p e c t t o s y s t e m p e r f o r m a n c e o r s i z i n g may b e more r e l e v a n t t o acc e p t a n c e o r r e j e c t i o n of a model t h a n s t a t i s t i c a l m e a s u r e s . The two p e r f o r m a n c e - r e l a t e d v a l i d a t i o n m e a s u r e s u s e d h e r e a r e t h e z e r o f a i l u r e s t o r a g e c a p a c i t y , computed u s i n g t h e s e q u e n t p e a k

427 a l g o r i t h m , and t h e c r i t i c a l e x t r a c t i o n r a t e , o r maximum u n i n t e r r u p t e d w i t h d r a w a l rate f o r a f i x e d s t o r a g e s i z e . c u r v e t e c h n i q u e s are used.

I n b o t h cases, s i m p l e m a s s

Although s u c h t e c h n i q u e s a r e c r u d e i n

t h a t t h e y assume e a c h s t r e a m f l o w s e q u e n c e t o b e d e t e r m i n i s t i c , t h e estimates p r o v i d e d are h e l p f u l i n e s t i m a t i n g b a s e l i n e s t o r a g e r e q u i r e ments a n d / o r e x t r a c t i o n rates.

The s e q u e n t peak a l g o r i t h m , i n de-

t e r m i n i n g t h e s t o r a g e n e c e s s a r y t o m e e t a p r e s c r i b e d demand, i s app l i c a b l e t o s y s t e m e x p a n s i o n problems.

It h a s been found t o b e q u i t e

s e n s i t i v e t o l o n g t e r m p e r s i s t e n c e , o r low f r e q u e n c y e f f e c t s (Burges and L e t t e n m a i e r , 1 9 7 7 ) .

The c r i t i c a l e x t r a c t i o n r a t e i s more r e p r e -

s e n t a t i v e of performance of an e x i s t i n g system.

Other performance

measures, d i s c u s s e d by P a l m e r and L e t t e n m a i e r (1981) might a l s o b e used.

However, as d e m o n s t r a t e d by K l e m e s , e t a l . ( 1 9 8 1 ) , many

measures of s y s t e m p e r f o r m a n c e which a l l o w l i m i t e d s u p p l y s h o r t f a l l s , e.g.,

number, s e v e r i t y , o r l e n g t h of s h o r t f a l l s f o r a f i x e d demand

and r e s e r v o i r s i z e , are r e l a t i v e l y i n s e n s i t i v e t o model form.

Thus,

w h i l e t h e "no f a i l " a l g o r i t h m s used h e r e may n o t b e p h y s i c a l l y o r e c o n o m i c a l l y r e a l i s t i c f o r many s y s t e m s , t h e y are d e s i r a b l e as v a l i d a t i o n t o o l s i n t h a t t h e y are r e l a t i v e l y s e n s i t i v e t o model form and p a r a m e t e r magnitudes.

A f i n a l v a l i d a t i o n measure c o n s i d e r e d h e r e which i s , i n a s e n s e , s t a t i s t i c a l , i s t h e c r o s s i n g d i s t r i b u t i o n of t h e n o r m a l i z e d (deseas o naliz e d) flow sequences.

The c r o s s i n g d i s t r i b u t i o n i s t h e f r e q u e n c y

w i t h which a g i v e n l e v e l , e x p r e s s e d as s t a n d a r d d e v i a t i o n s from t h e mean, p a r t i t i o n s s e q u e n t i a l n o r m a l i z e d ( s e a s o n a l ) f l o w l e v e l s . Normalized f l o w s a r e computed by s u b t r a c t i n g t h e s e a s o n a l means from e a c h o b s e r v a t i o n and d i v i d i n g by t h e s e a s o n a l s t a n d a r d d e v i a t i o n . For example, a c r o s s i n g of n o r m a l i z e d l e v e l K i s s a i d t o o c c u r be-

+

*

*

;t

k;

< K and Ft+l > 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

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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

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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

.it :

EXCEEDAWCE P R O B A B I L I T Y 1 1 - P I

Season 2 ..................................................................

I. a 5 : .

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S K E l C O E F F I C I E W T I S U Y l E D FLOWS) CEDAR ( I N F S Y W ANYUAL SYWTHETIC F L O W

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ESTIMATE FROM H I S T O R I C RECORD

1.05..

1.05..

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C

C C C

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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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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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. . . . . . . . . . . . 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

cccc

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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

cc

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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

CEDAR

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ESTIMATE

SEASON

a

SYNTHETIC

FLOWS

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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4c.

cccc c c

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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).

.,

n

-

P

"

0

"

r

0

" "

"

PO PP

"0

P

D

0

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I

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,

. o .. . ". . . .. . .. . ... . .. . .. . . .. : . . : . : . L ! 0 1 0 . . . -0 -.- n0.

. 0. " .

0

n

0 P

I

I

.- 0. . . . . . . . . . " .......................... .. 0 " O 9 0 . 0 ". _ .0. . On.. ,n. n - - n r -

n

"

n

..

" P

u

P

P

0

441

0

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 O
F u r t h e r , t h i s r e l a t i o n may b e linear operators as: -1 m k m k k Z = C (l-C)C X = ( l - ~ C) c B X = ( l - c ) ( l - c B t t-k t xt k=O k=O The summation ( 4 ) c a n b e stopped a t t h e M-th s t e p , choosing M so t h a t ( l - c ) c k may b e c o n s i d e r e d z e r o f o r k g r e a t e r t h a n M . For t h e e v e n t s of the Sieve r i v e r , r e p o r t e d i n Mannocchi e t a l i i (1981) a n d i n Piccolo a n d Ubertini (19791, c w a s found t o b e e q u a l to 0.8 a n d therefore M w a s s e t e q u a l t o 15. The v a l u e of c w a s chosen by means of a t r i a l - a n d - e r r o r procedure i n o r d e r t o

-

458 minimize t h e sum of s q u a r e s of d i f f e r e n c e s b e t w e e n t h e r a i n f a l l ,0.9) a n d c u m u l a t e d e f f e c t f o r d i f f e r e n t v a l u e s of c ( c = 0 . 1 , 0 . 2 , . the observed discharges.

..

2 . 2 I n t e r v e n t i o n modeling defined by ( 3 ) allows a The i n t e r v e n t i o n v a r i a b l e 6;) b e t t e r f i t t i n g f o r some p a r t i c u l a r b e h a v i o u r s of t h e s y s t e m ( t h e d r a i n a g e b a s i n i n o u r c a s e ) by m e a n s of t h e v a r i a t i o n s p r o d u c e d on t h e o u t p u t l e v e l b y t h e p a r a m e t e r s of i t s t r a n s f e r f u n c t i o n . Some of t h e e f f e c t s w h i c h c a n b e r e p r o d u c e d a r e : d e l a y e d s h o c k s , new c o n s t a n t l e v e l s , t e m p o r a r y l e v e l v a r i a t i o n s , l i n e a r a n d combined l i n e a r e f f e c t s , e x p o n e n t i a l d e c r e a s e s . The n a t u r e of r a i n f a l l - r u n o f f phenomenon s u g g e s t e d a n i n t e r v e n t i o n model w h i c h g i v e s t o t h e s y s t e m a s u p p l e m e n t a r y step effect with exponential decreas e , o r a delayed impulse a t t h e b e g i n n i n g of t h e h y d r o g r a p h r i s i n g l i m b .

2.3 S t u d y c a s e s Some f e a t u r e s of t h e models f i t t e d , u p t o n o w , t o some events in four Italian r i v e r s a r e summarized i n t a b l e 1; figure 1 TABLE 1. T y p e s o f m o d e l s f i t t e d t o some e v e n t s i n f o u r I t a l i a n r i v e r s . RIVER (area)

TRANSFER

LAG ( m i n)

OF M O D E L P A R A M E T E R S

NUMBER

TIME

w (8) 6 (8)

3

0

3

1

T.MARCHIAZZA 2 ( K M 5.3)

3

0

3

1

6

0

3

2

-

6

0

-

-

2

-

-

-

5

-

-

-

-

-

-

REFERENCES

k-

N t

0

+

t~(8) e (6)

ANSELMO-UBERTINI

-

2

-

-

2

-

-

2

2

9

(ID.)

1

2

5

PICCOLO-UBERTINI

-

6 6

( 1978) (ID.)

0

2

-

-

-

2

1

1

1

7

(1979)

6

0

2

1

-

-

-

-

2

1

6

MANNOCCHI E T A L .

-

-

1

2

7

(1981)

-

1

2

7

(ID.)

1

2

6

ANSELMO ET A L .

(1981)

60

-

2

0

2

2 1

-

-

60

2

1

-

60

-

-

2

-

1

-

-

1

2

5

-

1

2

6

(ID.)

-

1

2

6

(ID.)

1

2

7

(ID.)

1

2

6

(ID.)

60

2

1

-

-

60

-

-

2

1

-

-

-

1

-

6

-

(8) 6 (8)

_J

6

6 F .TOLE 2 (KM 1535)

-

NOISE

d;)

Zt w (B) 6 (8)

X t

F.IMPICCAT1 2 (KM 7.6)

F.SIEVE 2 (KM 831)

FUNCTIONS

60

0

2

-

2

-

2

-

4 59

0-

t '

100CUMULATIVE 2

RAINFALL

t

(3/s)

c 500-

intervention

...'..... PERIOD OF

400

1

CALIBRAT

A

I

I

J

0

_____

OBSERVED

FLOWS

0TIME FIG. 1.

ORIGIN

Example o f

FORECASTING

............... F 0 R E C A ST 1 N G

F L O W S b y (2) OF

FLOWS b y (4)

FORECASTINGS

forecast

and U b e r t i n i , 1 9 7 9 ) .

u s i n g the i n t e r v e n t i o n v a r i a b l e < ( T ) t

(from:

Piccolo

shows a n example of r e a l time f o r e c a s t . I t may b e pointed o u t t h a t t h e s e t y p e s of models g a v e good r e s u l t s both f o r medium sized b a s i n s such a s Sieve a n d Toce r i v e r s a n d f o r s m a l l experimental catchments such a s Marchiazza a n d Fosso d e g l i Impiccati.

3. DISCUSS ION AND CONCLUSION A comparison between d i f f e r e n t models for t h e same e v e n t may g e n e r a l l y b e made by means of some u s u a l "goodness of fit" s t a t i s t i c s : S , sum of r e s i d u a l s q u a r e s , Q , Box-Pierce a n d Ljung-Box t e s t of r e s i d u a l s , R 2 , e x p l a i n e d v a r i a n c e by model. Yet, i n o u r c a s e , owing t o t h e o b s e r v e d p a t t e r n i n t h e r e s i d u a l s , p a r t i c u l a r a t t e n t i o n ought t o b e p a i d t o ( b e s i d e s r a n d o m n e s s ) , the s t a t i o n a r i t y of the r e s i d u a l s p r o c e s s , t h a t i s t o c o n t r o l l i n g i t s evolution with time. A s a n example of t h i s comparison, the

460 event of December 1 2 t h - l 8 t h , 1952 from t h e Sieve r i v e r w a s s i m u l a t e d u s i n g f i v e d i f f e r e n t models whose c h a r a c t e r i s t i c s a r e summarized i n t a b . 2 . T A B L E 2.

D i f f e r e n t m o d e l s f i t t e d t o t h e S i e v e e v e n t o f Dec. 1 2 t h - l 8 t h ,

--

1

(.015)

--

__

(.020)

(.025)

(.020)

.170

.074

.173

_

1

_

0

--

(TI

0

6

1 1

e 1

e

3

$3

Q R2 L

U

.017 1

.a53

(.046)

(.020)

-.375

-.030

--

--

--

__

--

__

_-

--

_-

--

(6.340)

----

(1.350)

-30.317

5.238

(.001)

_-

.980 (.005)

(.060)

(.047)

(.060)

(.049)

- .495

-.670

-.606

-.631

-.651

--

__

(.047)

-

_-

--

(.049)

.245

5 $1

.029

(.020)

__

e Nt

(.006)

-.306

1 1

-.051

.924

--

W

6

( .004) -.049 (.021)

(.005) -.025

W

W

--

0

6

zt

-

(.009)

_

__

5 (.004)

3

__

1 W

t

4

2

( .004) .013

MODEL

X

1952.

(.050)

--

.095

.224

(.040)

(.030)

(.023)

(.060)

(.027)

1.004 (.039)

.910

.916

.682

.g02

--

-

-

_-

11.8

19.8

16.1

-.145 17.5

16.1

.9a4

.9a4

.985

.986

.984

214.5

224.1

207.1

192.6

2 24.6

a

THE V A L U E S I N B R A C K E T S ARE THE C O N F I D E N C E L I M I T S .

461

xt

Z

t

y+

2 3 m

3 V

'Z 4 a, [L

+50

5 -50 1952 DEC.

FIG. 2 .

12

13

14

15

16

l!

1952. O b s e r v e d i n p u t ( X a n d Z ) a n d t t o u t p u t ( Y ) s e r i e s . P l o t o f t h e r e s i d u a l s from t h e f i v e d i f f e r e n t models t (1). ( 2 ) . (31, ( 4 ) , (5).

E v e n t o f S i e v e r i v e r , Dec. l Z t h - l E t h ,

I n f i g . 2 , b o t h i n p u t s e r i e s X, , Zt a n d o b s e r v e d o u t p u t Y, a r e p l o t t e d a g a i n s t t i m e , t o g e t h e r w i t h t h e r e s i d u a l s from e a c h model.

462 Numerical c o m p a r i s o n s between t h e f i v e models a r e shown i n t a b . 3. For e a c h of t h e t h r e e p e a k h y d r o g r a p h s c o n s t i t u t i n g t h e e v e n t , t h e maximum a b s o l u t e r e s i d u a l a n d t h e v a r i a n c e - f o r t h e r i s i n g a n d t h e f i r s t p a r t of t h e d e s c e n d i n g l i m b s e p a r a t e l y were c o n s i d e re d , as well a s t h e r e s i d u a l a t p e a k a n d th e to ta l variance d u r i n g the peak hydrograph. The s e p a r a t i o n of t h e h y d r o g r a p h i n t o r e c e s s i o n p h a s e ( " t a i l " ) a n d "peak" p h a s e , a n d , within t h e l a t t e r , between r i s i n g a n d d e s c e n d i n g l i m b , w a s m a d e t o t e s t w h e r e model improvements a r e more n e e d e d a n d w h a t t y p e of s u p p l e m e n t a r y i n f o r m a t i o n w o u l d b e more c o n v e n i e n t . T A B L E 3.

R a t i o s o f m o d e l p e r f o r m a n c e s r e f e r r e d t o m o d e l (1) FIRST PEAK

1.1 L . 1

3:l 4:1

5: 1

-

D

RES.

1.187 1.327 1.064 1.025 1.003 1.241 1.078

.778 1.179 2.556 .966 1.389 1.241 .006

.908 .781 .864 .921 .279 .175 .733

VAR.

.869

1.272

1.076

.504

VAR. RES. VAR. RES. VAR.

P

R

R

*(RES.) (R)

SECOND PEAK

.876 1.023 .871 .902 1.124 1.242 .747

RES.

*.

M A X I M U M A B S O L U T E R E S I D U A L AND ( V A R . )

A N D THE D E S C E N D I N G ( 0 ) L I M B .

D

.628 .570 .901 .926 .803 .644 .842 .780

THIRD PEAK

P

R

D

P

.642 .636 .498 .925 .490

1.987 4.156 1.177 1.431 2.127

.956 .586 .943 1.623 .933

2.386 3.178 1.405 1.483 1.171

.496 .588

3.266 2.033

1.002 .596

2.646 1.646

.693

4.320

.374

3.239

V A R I A N C E OF R E S I D U A L D U R I N G T H E R I S I N G

R E S I D U A L AT PEAK

AND

V A R I A N C E D U R I N G T H E WHOLE

P E A K HYDROGRAPH ( P I .

The p a t t e r n of r e s i d u a l s shown i n f i g . 2 s u g g e s t s t h a t t h e r e i s n o r e a l d i f f e r e n c e between t h e models. This is s u b s t a n t i a l l y c o n f i r m e d b y t h e r e s u l t s on t a b l e 3 ; a l t h o u g h a b e t t e r p e r f o r m a n c e of some models i n t h e s i n g l e p e a k s may b e seen ( p e r h a p s m o d e l s (5) a n d ( 3 ) i n t h e f i r s t , model ( 4 ) i n t h e second a n d model ( 1 ) i n t h e t h i r d p e a k ) no s u c h d i s t i n c t i o n c o u l d b e m a d e l o o k i n g a t t h e whole e v e n t . So t h e smoothing of e r r o r s a t p e a k w a s not a c h i e v e d ; i n t a b l e 4 t h e r e s i d u a l s t a n d a r d e r r o r s for "peaks" a n d "tails" a r e given: these values a r e a l m o s t c o n s t a n t from one model t o a n o t h e r . F o r m a l l y t h e d i f f e r e n c e between models ( 1 ) a n d ( 2 ) c o n s i s t s i n t h e f a c t t h a t t h e o u t p u t p a r a m e t e r i n model ( 2 ) i s m a s k e d i n

463 TABLE 4 .

3 Standard e r r o r s o f t h e r e s i d u a l s ( m / s ) . S.E.

MODEL

1

2

3

4

5

TOTAL

14.65

14.97

14.39

13.88

14.99

AT P E A K S

20.99

21.61

21.00

20.10

21.83

AT T A I L S

5.92

5.59

4.64

4.98

4.94

Z t itself:

i t becomes e v i d e n t w r i t i n g Z i n t e r m s of l i n e a r o p e r a t o r s a s shown b y ( 5 ) . So a c o n s t r a i n t ( 6 1 = c ) h a s b e e n i n t r o d u c e d i n t o t h e model w h i c h , f o r t h i s r e a s o n , d i m i n i s h e s i t s f l e x i b i l i t y . However, d u r i n g 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 , a l e s s s c a t t e r e d s e r i e s , l i k e Zt, p r o v i d e s b e t t e r r e s u l t s . The u s e of t h e i n t e r v e n t i o n v a r i a b l e c(T)led t o n o a p p r e c i t a b l e improvements i n t h e simulation s t e p , i f compared to t h e more s i m p l e a n d p a r s i m o n i o u s model ( 1 ) ( w i t h r a i n f a l l s o n l y i n i n p u t ) . N e v e r t h e l e s s a f a i r improvement i n f o r e c a s t i n g c o u l d b e observed (see fig. 1 ) . The Q t e s t v a l u e s i n t a b . 2 a r e w e l l below t h e c r i t i c a l t h r e s h o l d x 2 = 28.4, w i t h 20 d e g r e e s of freedom a t 90% p r o b a b i l i t y l e v e l , s o t h a t t h e h y p o t h e s i s of r a n d o m n e s s of t h e r e s i d u a l s would n o t b e r e f u s e d . However t h e r e s i d u a l s e r i e s c a n n o t b e t h o u g h t of a s s t a t i o n a r y i n f a c t t h e v a r i a n c e of r e s i d u a l s p r o v e d to be a p p r e c i a b l y different by s u b div id in g the sample i n "peak" a n d " t a i l " p e r i o d s ( t a b . 4). At p r e s e n t two main d e v e l o p m e n t s m a y b e f o r e s e e n i n f u t u r e research. By f i r s t new a u x i l i a r y i n p u t v a r i a b l e s m i g h t b e explored in order to reduce e r r o r s during the peaks (high-water p h a s e ) , such as a n antecedent precipitation index or a v a r i a b l e which s h o u l d b e c a p a b l e of " p e r c e i v i n g " t h e r i s i n g of d i s c h a r g e s ; i n t h i s l a s t s e n s e a u s e of " p e r l o g " f u n c t i o n ( A p p l e b y , 1965; 0 ' C o n n e l l , 1980) :

can be interesting. S e c o n d a r i l y i t w o u l d b e p r a c t i c a l t o i n t r o d u c e some h y d r o l o g i c a l i n s i g h t b y d i v i d i n g t h e model i n two p a r t s ( o r more a s i n Hino a n d H a s e b e , 1 9 8 0 ) , base-flow a n d r u n o f f f o r i n s t a n c e , a s t h e d i f f e r e n c e i n r e s i d u a l v a r i a n c e s may s u g g e s t two d i f f e r e n t b e h a v i o u r s i n t h e rainfall-flow r e l a t i o n s h i p . These improvements a r e r e l a t e d m a i n l y t o t h e s t r u c t u r e of t h e model. Y e t , a v e r y i m p o r t a n t p r o b l e m i s t o " c o n c e p t u a l i z e " t h e model ( K l e m e s , 19811,

464

t h a t i s t o f i n d r e l a t i o n s h i p s between t h e p a r a m e t e r s of t h e model a n d 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 of t h e d r a i n a g e b a s i n . Concept u a l i z a t i o n w o u l d a l l o w a w i d e r u s e of t h e s e models b y e x t r a p o l a t i o n a n d s i m i l i t u d e . By now, t h i s s t a g e h a s b e e n s c a n t i l y t a c k l e d s i n c e i t r e q u i r e s a good d e a l of i n s i g h t both i n t h e p h y s i c s of t h e phenomenon a n d i n t h e s t r u c t u r e of t h e model: t h i s would c o n s t i t u t e a long-term f i e l d f o r f u t u r e r e s e a r c h . REFERENCES Amorocho,

J.,

1973.

Nonlinear hydrologic analysis.

Advances i n H y d r o s c i e n c e , 9:

203-251. Anselmo,

Ubertini,

V.,

forecasting. Anselmo,

Melone,

V.,

L.,

1979.

Transfer

H y d r . Sc. B u l l . , Ubertini,

F.,

function-noise

model a p p l i e d t o

flow

24: 353-359. 1981.

L.,

Space-time

distribution of rainfall-

r u n o f f r e l a t i o n s 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 model.

I n t . Symp. o n R a i n -

f a l l - r u n o f f M o d e l i n g , M i s s i s s i p p i S t a t e U n i v e r s i t y , USA. ( I n p r i n t ) . Appleby,

F.V.,

1965.

U n p u b l i s h e d D.I.C.

Dissertation, Imperial College,

L o n d o n U-

niversity. B i d w e l l , V.J.,

1971.

R e g r e s s i o n a n a l y s i s o f n o n l i n e a r catchment system. Water Res.

Research, 7( 5): Box,

G.E.P.,

11 1 8 - 1 1 2 6 .

Jenkins,

1970.

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 .

H o l d e n Day, S. F r a n c i s c o , 553 p p . B o x , G.E.P.,

Tiao,

1975. I n t e r v e n t i o n a n a l y s i s w i t h a p p l i c a t i o n s t o economic

G.C.,

and e n v i r o n m e n t a l p r o b l e m s . J o u r n . Americ. S t a t i s t . Assoc.,

M.,

Greco,

Todini,

E.,

Gallati,

Maione,

F.,

U.,

Martelli,

S.,

Natale,

L.,

1977. M o d e l l o matematico d e l l e p i e n e d e l l ' A r n o .

70: 70-79. Panattoni, I.B.M.

L.,

Italia,

C e n t r o S c i e n t i f i c 0 d i P i s a , 1: Z l O p p , 2 : 4 9 6 p p , 3 : 7 9 p p . Hino,

M.,

H a s e b e , M.,

1980. F u r t h e r t e s t o f a p p l i c a b i l i t y o f t h e i n v e r s e d e t e c t i o n

m e t h o d a n d e x t e n s i o n t o h o u r l y h y d r o l o g i c d a t a . P r o c . o f t h e 3 r d I n t . Symp. on S t o c h a s t i c H y d r a u l i c s o f I A H R , McLeod, A . I . ,

K.W.,

Hipel,

(1)

model

L e n n o x , W.C.,

construction;

(2)

Tokyo, 1977.

129-140. Advances i n Box-Jenkins modeling:

applications.

Water

Resour.

Research,

13:

5 6 7-586.

V.,

Klemes,

1981.

S t o c h a s t i c m o d e l s o f r a i n f a l l - r u n o f f r e l a t i o n s h i p . I n t . Symp. o n

R a i n f a l l - r u n o f f M o d e l i n g , M i s s i s s i p p i S t a t e U n i v e r s i t y , USA. Mannocchi, F.,

M e l o n e , F.,

U b e r t i n i , L.,

(In print).

1981. R a i n f a l l - f l o w p r o c e s s e s b y t h e m u l -

t i p l e t r a n s f e r - n o i s e m o d e l s . X I X I n t . IAHR C o n g r e s s , New D e l h i , 4 : 2 9 - 3 6 . O'Connell,

P.E.

(Editor),

1980.

Real time hydrological

I n s t . o f H y d r o l o g y , W a l l i n g f o r d , U.K., Piccolo,

D.,

Ubertini,

L.,

1979.

f o r e c a s t i n g and c o n t r o l .

264 pp.

F l o o d f o r e c a s t i n g by i n t e r v e n t i o n t r a n s f e r s t o -

c h a s t i c m o d e l s . P r o c . X V I I I IAHR C o n g r e s s , C a g l i a r i , Sabatini,

P.,

Ubertini,

L.,

1978.

p i c c o l o b a c i n o . X V I Conv.

I t a l y , 5 : 319-326.

Ricostruzione e previsione d e i deflussi i n un

I d r a u l i c a e C o s t r . I d r a u l i c h e , T o r i n o , 8 2 7 : 1-12.

465

ANALYSIS OF WATER TEMPERATURE RECORDS USING A DETERMINISTIC STOCHASTIC MODEL]

-

BRUCE J . NEILSONAND BERNARD B . H S I E H ~ V i r g i n i a I n s t i t u t e o f Marine S c i e n c e and School of Marine S c i e n c e , The College of William and Mary, G l o u c e s t e r P o i n t , VA 23062

INTRODUCTION S i n c e 1954 d a i l y o b s e r v a t i o n s o f t h e w a t e r temperature i n t h e York River e s t u a r y have been made a t Gloucester P o i n t , V i r g i n i a by t h e s t a f f o f t h e V i r g i n i a I n s t i t u t e of Marine Science o f t h e College o f William and Mary. These o b s e r v a t i o n s were made a t t h e end o f a pier which extends 1 1 5 m . from t h e s h o r e l i n e and i s l o c a t e d approximately 200 m. downriver o f t h e G l o u c e s t e r P o i n t Yorktown c o n s t r i c t i o n .

(See Figure 1 ) Water depth a t t h i s s t a t i o n

i s 4 . 2 m. a t mean low water and measurements were made a t mid-depth. The time s e r i e s record v a r i e s o v e r t h e period o f o b s e r v a t i o n s . I n i t i a l l y d a i l y temperature extremes were monitored using a mercury maximum and minimum thermometer. In 1972 a c o n t i n u o u s l y r e c o r d i n g conductivity-salinity-temperature probe was i n s t a l l e d . In o r d e r t o have t h e maximum record l e n g t h p o s s i b l e and t o have compatible d a t a s e t s , o n l y t h e maximum and minimum temperatures from t h i s l a t t e r period were used. Thus, t h e primary d a t a were d a i l y extreme temperatures f o r t h e y e a r s 1954 t o 1977 i n c l u s i v e . Daily mean temperatures were c a l c u l a t e d by averaging t h e d a i l y extremes. For s h o r t d a t a gaps ( s a y a few days) d a t a from a back-up instrument were used. For more l e n g t h y p e r i o d s w i t h missing v a l u e s , t h e d a t a gaps were l e f t i n t h e r e c o r d . Data gaps were g r e a t e s t f o r t h e y e a r s 1964, 1968 and 1972. ' C o n t r i b u t i o r number 1038 from t h e V i r g i n i a I n s t i t u t e o f Plarine ScSence 'Present Address: Clemson U n i v e r s i t y , Clemson, S . C . 29632 Reprinted from r 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

466

J

I

I -

I

77OW

76OW Chesapeake Bay

FIGURE 1.

Chesapeake Bay, the York River and the l o c a t i o n o f t h e sampling s t a t i o n .

467

For s p e c i f i c days o f t h e y e a r , t h e d a i l y temperatures f o r the 24 years g e n e r a l l y were normally d i s t r i b u t e d . A few y e a r s were warmer (1959, 1975, 1 9 7 7 ) o r c o l d e r (1968, 1976) than normal and produced a l a r g e p o r t i o n o f t h e extreme values f o r the i n d i v i d u a l days o f t h e year.

The a u t h o r s a r e aware o f no anomalies i n t h e d a t a o r o f any

systematic e r r o r s i n t h e temperature measurements.

Therefore, lacking

evidence t o t h e c o n t r a r y , we assumed t h a t t h e d a t a s e t was s u i t a b l e f o r a n a l y s i s both i n c a l e n d a r y e a r segments and a s a s i n g l e , continuous 24 y e a r r e c o r d . PRELIM1 NARY ANALYSIS P r i o r s t u d i e s o f a s i n g l e y e a r o r a few y e a r s o f water temperature measurements have shown t h a t t h e annual c y c l e i s an extremely impor-

t a n t component o f water temperature v a r i a t i o n s i n r i v e r s (Kothandaraman , 1 9 7 1 ) and e s t u a r i e s (Thomann, 1967) a n d t h a t a harmonic with a period of one y e a r accounts f o r roughly 95% o f t h e t o t a l v a r i a n c e o f t h e r e c o r d . For t h a t r e a s o n , t h e f i r s t s t e p i n our s t u d y was t o examine t h e time s e r i e s using harmonic a n a l y s i s . When t h e d a t a were examined on a c a l e n d a r y e a r b a s i s , we found t h a t t h e mean temperatures and t h e amplitudes and phase a n g l e s o f t h e annual c y c l e were r e l a t i v e l y c o n s t a n t from y e a r t o y e a r , a s shown i n Table 1 . For higher o r d e r harmonics, t h e amplitude was an o r d e r o f magnitude s m a l l e r and both t h e amplitude and t h e phase a n g l e v a r i e d widely. The f i r s t harmonic o r annual component accounted f o r between 95% and 98.6% o f t h e v a r i a n c e , a s can be seen i n Table 2 . No higher o r d e r harmonic c o n s i s t e n t l y accounted f o r a s i g n i f i c a n t p o r t i o n o f t h e r e s i d u a l v a r i a n c e . When t h e 24-year record was analyzed as a s i n g l e time s e r i e s , t h e annual c y c l e was found t o account f o r 95.8% o f t h e v a r i a n c e . The o n l y harmonics which c o n t r i b u t e d more than 0 . 1 % o f t h e variance were t h e f i r s t ( 2 4 y r . p e r i o d ; 0 . 2 % ) , second ( 1 2 y r . p e r i o d ; 0 . 1 5 % ) , t e n t h (29 month p e r i o d ; O . l % ) , t w e n t y - f i r s t ( 1 4 month p e r i o d ; 0 . 1 3 % ) , a n d f o r t y - e i g h t h ( 6 month p e r i o d ; 0 . 2 % ) . Because t h e periods o f t h e harmonics a r e determined by t h e record l e n g t h , t h e power spectrum a l s o was examined t o s e e i f any cr>iponents 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

O
.

(2.1)

The object is to study the properties of the outflow sequence m

IY(t)

in terms of the assumed properties of the inflows {X(t)}.

The structural simplicity of the present model enables us to work in terms of the explicit formal solution of the stochastic difference equation (2.1), namely Y(t+l)

=

b

t+l Y(0)

t r + clOb X(t-r).

(2.2)

For a process that has been running for a long time we may replace

this by the asymptotic version Y(t+l)

=

r ccco b X(t-r). r=O

(2.3)

Our problem is then reduced to that of studying the weighted sums

lrbrX(t-r)

of the elements of the Markov Chain

{X(t)).

One method would be to utilise the fact that, for the particular Markov Chain chosen, the Laplace Transform of the weighted sums has a comparatively simple form (Phatarfod, 1976; Lloyd and Saleem, 1979), and tabulate that transform and invert it numerically, thus obtaining a table of the outflow probability distribution for each season. This was the method used (in a non-seasonal case) by Anis, Lloyd and Saleem (1979). Instead, however, in the present work we have concentrated on the skewness of the seasonal outflow distributions, and investigated the feasibility of obtaining these by direct calculation from (2.3).

3.

THE NON-SEASONAL VERSION OF THE INFLOW MARKOV CHAIN The non-seasonal version of the inflow process with which we

489

have worked is a homogeneous first-order Markov Chain in which the marginal distribution of each X(t)

is of the 2-parameter gamma form,

with scale parameter a and shape parameter p, the p.d.f. at x being Xp-le-x/cY

/aPr(p)

COO,

p>o,

(x>o)

(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

<

s) , a n d - s i m i l a r l y f o r

i

. Derive t h e non-zero s o l u t i o n s 0k M d e t . [ -E ( O ) A ( e ) - 0 I1 = 0 , such t h a t 10 I < k -

E(8)

-

of t h e equation 1.

I n t h e above

I

M = M

1

+ M

2' N = M (M +1)/2 1 1 2. For each

I t i s known t h a t i n general t h e r e a r e

+

Bk

M2(M2+1)/2

such s o l u t i o n s .

obtained i n 1, form t h e matrix

i t s eigen-values.

:(8)$(0)

and f i n d

Take t h a t eigenvalue which i s equal t o

and f i n d i t s corresponding (normalized) eigenvector. 3.

The 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 D i s given by N s m a . = P r { D =I} = 1 Z 01 j > 1, a . = P r . [ D =0] = 1 - c 13 (2) 3 s k k' S 1 j=l j

where t h e

Z's

a r e constants s a t i s f y i n g

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

Dw

N

l i n e a r equations. The

has a form similar t o (2); see

Phatarfod (1981a) f o r d e t a i l s .

Method D : Bottomless Dam Model (Mark 2) W e assume here t h a t t h e r e i s a dependence between t h e summer a n d

winter flows of t h e same y e a r , b u t t h a t t h e summer flows a r e independent of t h e flow of t h e winter of t h e previous year.

This

may n o t be a t e r r i b l y r e a l i s t i c model t o assume; however, i f we have s i x seasons, say, then it i s n o t u n r e a l i s t i c t o t a k e t h e s m a l l e s t c o r r e l a t i o n c o e f f i c i e n t t o be equal t o z e r o ; w e a r e t h u s assuming t h a t t h e i n p u t process s t a r t s a f r e s h each y e a r ; t h e i n p u t i n t h e f i r s t season has a c e r t a i n p r o b a b i l i t y d i s t r i b u t i o n and t h e i n p u t s i n t h e remaining f i v e seasons a r e governed by t h e f i v e t r a n s i t i o n p r o b a b i l i t i e s between t h e f i r s t and t h e l a s t ( s i x t h ) season.

506 The theory behind t h i s procedure i s given i n Phatarfod (1981b). The s t e p s f o r t h e case of two seasons a r e : 1. D i s c r e t i z e 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 annual i n p u t s . L e t i t be denoted by

i . e . P i = Pr(i-+

pO,pllp2,..-

2 . Derive t h e non-zero s o l u t i o n s 2 M p +p 0+p 8 +... = 8 , such t h a t 0 1 2 such s o l u t i o n s .

Bk

s +w

i&I.

of t h e equation

lekl

< 1.

In general we have

M

3. The e q u i l i b r i u m s o l u t i o n is given by

M

where t h e

m

are c o n s t a n t s s a t i s f y i n g

Y's

d i s t r i b u t i o n of

D W

M

l i n e a r equations.

The

h a s a s i m i l a r form: see Phatarfod (1981b) f o r

details.

NUMERICAL EXAMPLE

3.

We now use t h e above t h r e e methods f o r a s p e c i f i c i n p u t model. W e assume t h a t t h e inflows follow a Two-season Thomas F i e r i n g model. Denoting t h e mean, t h e variance and t h e c o e f f i c i e n t of skewness of t h e summer flows by winter flows by

u2,

c o e f f i c i e n t between n ' +I

by

p2,

L

ul,

O1

0;

and

y2

respectively, the correlation

Sn

and

Wn

by

and

y1

r e s p e c t i v e l y , those of t h e

p1

and t h a t between

Wn

and

t h e model i s given by t h e equations

For t h e model given i n ( 4 ) 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 s of given

and of

W

given

are gamma.

Sn

n ' Using t h e t a b l e s

'n-1 n of incomplete gamma function ( o r a l t e r n a t i v e l y t h e IMSL/MDGAi\l

a..

Subroutine) t h e p r o b a b i l i t i e s

17

e t c . can be c a l c u l a t e d .

(1979) This

I

e x e r c i s e i s n o t c a r r i e d o u t h e r e ; i n s t e a d , f o r t h e sake of consistency e t c . w e r e obtained by using t h e i j same sequence o f generated values as used for t h e simulation method

the transition probabilities

a

507 (Method A ) .

A sequence of 2030 values

obtained using t h e model ( 4 ) . and

W,

ditions.

(each) o f

Sn

and

Wn

were

The f i r s t 30 values (each) of

Sn were ignored t o e l i m i n a t e any e f f e c t of t h e i n i t i a l conThe remaining 2000 v a l u e s (each) of

Sn

and

Wn

were

used t o o b t a i n 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 s t o r a g e content by t h e Method A , with i n i t i a l c o n t e n t 2 and r e s e r v o i r s i z e s as well a s t o e s t i m a t e 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 For our example we have taken p2 = 2 . 0 ,

O2

=

etc.

pl - . 6 , U l = -3, Y1 = 1, P1 = 0 . 5 ,

1.333, y 2 = 1 . 0 , p 2 = 0.1.

so t h a t t h e d r a f t - r a t i o i s

a. i j

K = 3,4,

W e a l s o take

M1

= M2

= 1,

2/2.6 = 0 . 7 7 .

The Tables below compare t h e equilibium (cumulative) 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 s of t h e dam obtained by a l l t h e methods. shows, f o r example, t h a t f o r

K = 4 , t h e p r o b a b i l i t i e s o f emptiness

of t h e dam a t t h e beginning o f w i n t e r a r e methods A and B r e s p e c t i v e l y .

-029

and

On t h e o t h e r hand, i f

p r o b a b i l i t i e s o b t a i n e d by Methods C and D a r e respectively.

Table 2

.046, by K = 4.5,

.048 and

the

-052

This shows t h a t Methods C and D do overestimate t h e

s i z e of a dam f o r a r e q u i r e d p r o b a b i l i t y of emptiness; however, t h e d i f f e r e n c e i s n o t much.

I t i s f a i r l y obvious t h a t i f o u r u n i t of

water i s s o chosen t h a t w e would r e q u i r e a dam s i z e of about 1 0 units, then t h e four methods would give roughly e q u i v a l e n t answers.

508

TABLE 1

B

A

K = 3 0 0.5 1.0 1.5 2.0 2.5 3.0

.440 .569 -692 .799 .871 -925 .969

'

3.5 4.0 4.5

K = 4

K = 3

K = 4

-426 -556 .676 .778 .846 .go2 .938

.441

-423

-

-

-710

.867 .959

Prob.

Prob. Empty = -016

.681

-

-

.752

.727

.883

-865

.954

-948

.968

.959

-923

-

.

Prob Empty =.0406

B

C

K = 3

K = 4

.059 .3.5 .591 .712 .809 -881 .936

.056 .305 .5 74 -697 .791 -856 .go9

-058

-056

-

-

.594

.797

-

-

-

-

.706

.697

.954

.

-

- 5 70

.882

Prob Empty = .080

D

.2 84

-

.920

-

Prob. Empty =.041

-296

.765

-

.945 .971 Prob. Empty = .029

-

Prob. Prob. Empty Empty = - 0 2 4 =.032

K = 4

Prob. Empty = .064

-

-976

K = 3

3.5 4.0 4.5

.505 -

f

A

0 0.5 1.0 1.5 2.0 2.5 3.0

-

D

-521

.831

-971 -984

Empty = -031

C

Prob. Empty =.046

-

-

-

-852

.838

.923

-

.918 -

.952

.948

Prob. Empty =.048

Prob. Empty =.052

-

-

509

RE FE RENCES Dearlove, R.E. and H a r r i s , R.A., 1965. P r o b a b i l i t y of Emptiness, 111. Proc. Water Res. Assoc. Symp. on Reservoir Y i e l d , Oxford, Pap. 7. IMSL/GGAMS , 1976. I n t e r n a t i o n a l Mathematical and S t a t i s t i c a l L i b r a r i e s , Houston, Texas, Vol. 2 , 7 t h ed. Klemes, V., 1970. A two-step p r o b a b i l i s t i c model o f s t o r a g e r e s e r v o i r with c o r r e l a t e d i n p u t s . Water Resour. Res. , 6 ( 3 ) : 756-767. Kottegoda, N.T. , 1970. S t a t i s t i c a l methods o f River flow s y n t h e s i s I.C.E. Supplement 1 8 , f o r Water Resources Assessment. (paper 73395). K r i t s k i y , S . N . and Menkel, M . F . , 1940. Obobshchennye priemy r a s c h e t a r e g u l i r o v a n i y a s t o k a na usnove matematischeskoy s t a t i s t i k i . Gidrotekhnicheskoe s t r o i t e l s t v o . 2 : 19-24. Lloyd, E.H. , 1963. A p r o b a b i l i t y t h e o r y o f r e s e r v o i r s with s e r i a l l y J. Hydrol., 1: 99-128. correlated inputs. McMahon, T.A. and Mein, R.G., 1978. Reservoir c a p a c i t y and y i e l d . E l s e v i e r , Amsterdam, 2 1 3 pp. 1954. A p r o b a b i l i t y t h e o r y o f dams and s t o r a g e Moran, P.A.P., systems. Aust. J. Appl. S c i . 5 : 116-124. The i n f i n i t e l y deep dam with Seasonal P h a t a r f o d , R.M. , 1981a. SIAM. J. Appl. Maths., 4 0 : 400-408. Markovian inflows. The i n f i n i t e l y deep dam with Seasonal Phatarfod, R.M., 1981b. Markovian i n f l o w s 11. (under p u b l i c a t i o n ) . Phatarfod, R.M. and S r i k a n t h a n , R. , 1981. D i s c r e t i z a t i o n i n 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 with Markovian i n f l o w s . J. Hydrol., 52: 199-218. Prabhu, N . U . , 1958a. On t h e i n t e g r a l e q u a t i o n f o r t h e f i n i t e dam. Q. J. Math., Oxford, 9 ( 2 ) : 183-188. Savarenskiy, A. D. , 1940. Metod r a s c h e t a r e g u l i r o v a n i y a s t o k a . Gidrotekhnicheskoe S t r o i t e l s t v o . 2 : 24-28.

510

OPTIMAL ARMA MODELS FOR THE STATISTICAL ANALYSIS OF RESERVOIR OPERATING RULES J.W. D E L L E U R , M. GIN1 AND M . KARAMOUZ Civil Engineering, Purdue U n i v e r s i t y , West L a f a y e t t e , Indiana, USA ABSTRACT A time s e r i e s of mean annual flows with a h i g h Hurst c o e f f i c i e n t i s used. The Kitagawa search procedure f o r the optimal o r d e r p , q of ARMA models i s seen t o work w e l l . Simulation of t h e annual flow s e r i e s using a s l i g h t l y modified McLeod and Hipel ' s Waterloo Simulation

Frocedure 1 preserved the r e s c a l e d range and t h e Hurst c o e f f i c i e n t . Reservoir r e l e a s e s and s t o r a g e s were obtained f o r an o p e r a t i o n r u l e t h a t minimizes l o s s e s f o r a s p e c i f i e d p e n a l t y f u n c t i o n . d i s t r i b u t i o n s of individual e v e n t s and sequences 2 , 3 , . were obtained f o r flows , re1 e a s e s and s t o r a g e s .

Probability

. . ,20

events

Temporal di saggrega-

t i o n i s used f o r the generation of monthly s e r i e s .

These i n turn

a r e used f o r t h e development of optimal seasonal r e l e a s e r u l e s f o r t h e operation of t h e r e s e r v o i r s .

The r e s e r v o i r r e l i a b i l i t y i s then

estimated in two d i f f e r e n t ways: i n terms of t h e f r e q u e n c i e s of f a i l u r e y e a r s and of f a i l u r e months. DATA USED The annual s e r i e s examined i s f o r t h e Blacksmith Fork near Hyrum, Utah (1913-1957 from Yevjevich ( 1 9 6 3 ) , 1957-1970 from USGS Water Supply P a p e r s , 1970-1979 from Utah Power and Light Company). The p r o b a b i l i t y d i s t r i b u t i o n of annual flows was found t o be approximately normal. For n = 66 y e a r s o f record the mean was o n = 127.62 c f s , t h e standard d e v i a t i o n S = 4 2 . 7 2 c f s , the a d j u s t e d range aR, = 604.54 c f s , t h e r e s c a l e d a d j u s t e d ranges aR,/S = 14.15 and t h e Hurst c o e f f i cient

was estimated from K

=

log (aRn/S) / log ( n / 2 )

=

0.75.

The

a u t o c o r r e l a t i o n showed values a t t h e 95% s i g n i f i c a n c e l e v e l a t l a g s 1 Reprinted from Time Series Methods in Hydrosciences, by A.H. El-Shaarawi and S.R. Esterby (Editors) - Printed in The Netherlands

0 1982 Elsevier Scientific Publishing Company, Amsterdam

511

and 12 and t h e p a r t i a l a u t o c o r r e l a t i o n f u n c t i o n a t l a g s 7 and 17. FITTING ARMA MODELS

TO

THE ANNUAL S E R I E S

A u t o r e g r e s s i v e - m o v i n g a v e r a g e models were f i t t e d t o t h e s t a n d a r d -

- on)/S

i z e d s e r i e s Zt = ( Q t

where Qt i s t h e mean annual f l o w o f y e a r t.

The model i s

a h e r e t h e $ . ( j = 1 , 2 , . . . ,p) a r e 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 s and t h e 3 0 . ( j = 1 , Z , . . ,q) a r e t h e m o v i n g a v e r a o e c o e f f i c i e n t s and 8 = -1. J 0 The ARMA models were f i t t e d b y t h e method o f maximum l i k e l i h o o d ( m l e ) .

.

The s e a r c h f o r t h e o p t i m a l model f o l l o w s t h e method p r o p o s e d b y K i t a g a w a (1977) w h i c h i s based on t h e A k a i k e i n f o r m a t i o n c r i t e r i o n ( A I C ) and t h e de t e r m i na t i on coe f f ic ien t s R 2 g iven b y N C (p,q)

R2

=

where 3f

1 oE2

=

n I n (mle

uE2) t

2 (ptq)

(U& 2/~z2)

i s the variance of the residuals

E~

and o z 2 i s t h e v a r i a n c e

t h e s t a n d a r d i z e d annual s e r i e s , K i t a g a w a ' s ( 1 9 7 7 ) p r o c e d u r e was used t o s e a r c h f o r t h e o p t i m a l o r d e r

o f t h e ARMA m o d e l . I t s a d v a n t a g e i s t h a t t h e o p t i m a l v a l u e s o f p and q can, i n g e n e r a l , be d e t e c t e d w i t h o u t f i t t i n g t h e w h o l e s e t o f p o s s i b l e models w i t h a l l t h e c o m b i n a t i o n s o f t h e a u t o r e g r e s s i v e and m o v i n g average parameters. i f i e d where p, s e l e c t i o n o f p,

and q, and q,

A

r e g i o n defined by o

5 p 2 ,p,

0

5 q 5 9, i s Spec-

a r e t h e maximum v a l u e s o f p and q c o n s i d e r e d .

The

i s based on a s t u d y by H a s h i n o and D e l l e u r ( 1 9 8 1 )

a c c o r d i n g t o w h i c h pm = 11 t o 15, and q,

= 2 t o 3.

I n a d d i t i o n t h e models s h o u l d be checked f o r s t a t i o n a r j t y and i n v e r t ibility.

Those w h i c h a r e n o t s t a t i o n a r y o r i n v e r t i b l e o r f o r w h i c h t h e

model d i d n o t c o n v e r g e t o a s t a b l e s o l u t i o n a r e e l i m i n a t e d . ( 1 1 , O ) was found t o be t h e b e s t . l i s t e d i n Table 1.

The model

The b e t t e r a c c e p t a b l e models a r e

512

TABLE 1 .

BETTER ARMA MODELS FOR THE BLACKSMITH FORK ARMA OY2 4 ,O 3,2 8 ,O 8,2 11 ,o 13,O

A1 C -19.74 -20.36 -21.33 -26.02 -24.57 -27.47 -24.72

R2

0.302 0.349 0.378 0.471 0.491 0.527 0.536

q'* 2 20 52 55 100 131 143

*See following section

Before performing t h e s i m u l a t i o n s , d i a g n o s t i c checks were performed f o r t h e whiteness and n o r m a l i t y of the r e s i d u a l s .

P l o t s of t h e proba-

b i l i t y d i s t r i b u t i o n s o f t h e r e s i d u a l s on normal p r o b a b i l i t y paper showed t h a t t h e r e s i d u a l s a r e approximately normally d i s t r i b u t e d .

The p o r t -

manteau l a c k o f f i t t e s t passed a t t h e 10% l e v e l i n a l l c a s e s indicatinq t h e adequacy o f t h e models. SIMULATION O F ANNUAL SERIES The Waterloo Simulation Procedure 1 of McLeod and Hipel (1978) has been used because random r e a l i z a t i o n s of t h e underlying s t o c h a s t i c process a r e used as i n i t i a l v a l u e s , t h u s avoiding b i a s i n s i m u l a t i o n s .

For t h e g e n e r a t i o n of t h e f i r s t r terms ( r = max ( p , q ) ) , the procedure r e c u i r e s t h e approximation of t h e ARMA ( p , q ) model by a MA ( 9 ' ) model where t h e o r d e r q ' i s s e l e c t e d so t h a t the d i f f e r e n c e s between t h e t h e o r e t i c a l v a r i a n c e of t h e ARMA ( p , q ) model and t h e v a r i a n c e of t h e MA ( q ' ) model i s l e s s than a s p e c i f i e d e r r o r v a l u e . The o r d e r q ' i s shown I t i s seen t h a t q ' i n c r e a s e s i n Table 1 using an e r r o r l e v e l o f r a p i d l y a s p i n c r e a s e s . I n o r d e r t o keep t h e computational burden within r e a s o n , t h e ARMA (4,O) model was f i n a l l y s e l e c t e d a s i t r e q u i r e s only 20 moving average terms t o be c a l c u l a t e d . The model i s Z,

L

= 0.5125

Zt-l

- 0.0316 Zt-2

- 0.0388 Zt-3 + 0.1239 Zt-4

+

Et

(4)

One s l i g h t m o d i f i c a t i o n was introduced i n t h e procedure: when a random var a b l e produces a n e g a t i v e f l o w , t h e random v a r i a b l e i s d i s c a r d e d and t h e next random v a r i a b l e i s introduced i n t h e c a l c u l a t i o n s .

This i s

equ v a l e n t t o using a t r u n c a t e d d i s t r i b u t i o n o f t h e flows.

The number

513

of s i m u l a t e d s e r i e s was s e l e c t e d a s N

=

3,000 and t h e l e n g t h o f each

was n = 500 y e a r s . The p r e s e r v 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 s

i s checked b y u s i n g t h e

e m p i r i c a l c u m u l a t e d d i s t r i b u t i o n f u n c t i o n (ECDF) o f K f o r t h e 3000 simulations.

The mean H u r s t c o e f f i c i e n t i s 0.70,

i t s variance i s

0.0017 and t h e p r o b a b i l i t y t h a t t h e s i m u l a t e d K b e l a r g e r t h a n t h e historical

K i s 0.112, t h u s t h e H u r s t c o e f f i c i e n t i s p r e s e r v e d .

STATIST1 CAL CHARACTERISTICS

OF R E S E R V O I R OPERATION

The g e n e r a t e d i n f l o w s e r i e s a r e r o u t e d t h r o u g h a s i n g l e r e s e r v o i r o f known c a p a c i t y o p e r a t e d i n a c c o r d a n c e w i t h a r e l e a s e r u l e d e s i g n e d t o minimize t h e t o t a l losses from t h e operation.

The l o s s f u n c t i o n i s

d e f i n e d as a p i e c e w i s e e x p o n e n t i a l f u n c t i o n .

Within a specified safe

< RUP) t h e r e i s no l o s s as t h e r e l e a s e i s r e l e a s e r a n g e (RLOW 5 r e l e a s e -

l a r g e enough t o s a t i s f y t h e demand and y e t i s s m a l l enough t o p r e v e n t The loss f u n c t i o n i s t h u s d e f i n e d as

flooding. Loss(Rt)

=

A[exp(Rt/RUP)

- exp(l)]

Loss(Rt) = 0

i f RLOW

Loss(Rt) = B[exp(-Rt/RLOW)

-

exp(-l)]

i f Rt 3 RUP

(5a)

Rt 5 RUP

(5b!

-

if Rt 5 RLOW

(5d

where A and B a r e known c o n s t a n t s t h a t depend on t h e p r i c e o f t h e w a t e r and on how e x t e n s i v e t h e p r o p e r t y damage i s , and Rt i s t h e r e l e a s e during year t.

For annual f l o w s t h e values o f t h e c o n s t a n t s a r e taken

as f o l l o w s A = 3.88 x l o 5 , B

=

RLOW = 0 . 8 (mean annual f l o w ) .

1.58 x

lo6,

RUP = 1 . 2 (mean a n n u a l f l o w ) ,

The s a f e r a n g e i s t h u s w i t h i n 20% o f

t h e mean annual f l o w , and t h e v a l u e s o f A and B r e s u l t i n l o s s e s o f l o 6 u n i t s when t h e r e l e a s e i s z e r o o r t w i c e t h e mean a n n u a l f l o w . The o b j e c t i v e f u n c t i o n i s t o m i n i m i z e t h e t o t a l l o s s e s f o r t h e T y e a r s o f e x p e c t e d economic l i f e o f t h e r e s e r v o i r :

subject t o the following constraints: i)

t h e mass b a l a n c e o f t h e r e s e r v o i r ( c o n t i n u i t y )

514

where It=inflow during year t , St=storage a t the beginning of year t ,

where the superscripts min a n d max indicate the minimum o r maximum. Karamouz a n d Houck (1981 a ) solved t h i s problem as an i t e r a t i v e d i s c r e t e dynamic problem a n d regression analysis, using 20 d i s c r e t e storage volumes uniformly d i s t r i b u t e d between zero and f u l l reservoir capacity. They regressed the optimal storage , optimal release and concurrent inflow by means of the equation Rt = a I t + b S t + c

(8)

for d i f f e r e n t bounds on R T i n a n d R:ax BOUND) ( a I t + b S t

RTax

=

(1

R!ni

=

maximum [ O ;

f

f

as follows

c)

( 1 - BOUND) ( a I t + b St + c ) ]

I n ( 9 a , b ) the quantity ( a I t

f

(93)

(9b)

b St + c ) represents the release r u l e

obtained in the previous i t e r a t i o n . Combining equations ( 7 a ) and ( 8 ) one obtains Sttl

=

(1 - a ) I t

f

( 1 - b ) St - c

(10)

0, then Rt i s given by the release r u l e ( 8 ) , i f S t + l < O , then If Sttl S t t l i s s e t equal t o zero and R t = I t f St. The storage S t t l cannot exceed the reservoir capacity, CAP, and the excess i s released. Therefore i f Kt 5 I t + St - CAP then Rt = I t + St - CAP and St+, = CAP. Five hundred years of simulated annual flows were routed t h r o u g h reservoirs of se,yleral capacities a n d the annual releases a n d storages were obtained as explained. Figure 1 shows the empirical cumulative

probability d i s t r i b u t i o n s o f the flows and releases for 1 and 3 sequen-

515

t i a l years and of the storages f o r 1 , 2 , 3 , 4, 5, 6 , 10 and 20 sequent i a l years. These are shown f o r a storage coefficient (storage capacitylmean total annual runoff volume) of 1 . 4 and two d i f f e r e n t BOUND values. Similar d i s t r i b u t i o n s were obtained f o r storage coefficients o f 1.0; 0.5 and 0 . 2 , a n d f o r three BOUND values each time. The probability d i s t r i b u t i o n s of k sequential releases are seen t o l i e one above the other as k decreases and do n o t i n t e r s e c t each other. Thus, Prob ( k + 1 sequential releases < R i ) 5 P r o b ( k sequential Similar s t a t e releases < R i ) where R i i s a specified release value. ments can be written f o r the flows a n d the storages. Comparing the probability d i s t r i b u t i o n s f o r one year low flows, I i , Prob ( 1 year release 5 T i ) < Prob ( 1 year flow 5 I i ) , so the probabili t y of droughts i s decreased by the reservoir. Likewise, comparing the probabilities of exceedance f o r high flows, the probability o f floods i s decreased by the reservoir, as expected. Comparing the deviations between the p r o b a b i l i t i e s of inflows and of releases f o r the same flow values f o r several storage c o e f f i c i e n t s , the deviations are seen t o increase as the storage coefficient increases. Thus larqer reservoirs provide more control of the flows. Comparing the probability d i s t r i b u t i o n of storages a n d the values of the BOUND i t appears t h a t f o r higher values of BOUND the cumulative distribution increases more rapidly t h a n f o r the lower values of BOUND. For the i n f i n i t e BOUND values i t exhibits a larger percentage of f a i l ures (reservoir empty o r f u l l ) , f o r example, f o r the storage coeffic i e n t = 1 . 4 , the reservoir i s empty 14.4% and f u l l 1 6 . 2 % o f the time whereas there i s v i r t u a l l y no f a i l u r e w i t h the lower BOUND values. ANNUAL R E L I AB ILITY

The occurrence based annual r e l i a b i l i t y R a , i s defined a s the number of non-failure years expressed as a percentage of the t o t a l number of years i n the given period, i t i s thus equivalent t o the probability t h a t the reservoir will deliver the expected demand t h r o u g h o u t i t s l i f e t i n e without ipcurring a deficiency. The r e l i a b i l i t y c h a r a c t e r i s t i c s are computed f o r stationary conditions, t h a t i s f o r a lonu operation

516

I

q-

FIGURE 1.

P r o b a b i l i t y d i s t r i b u t i o n s of flows, r e l e a s e s and s t o r a p e s with s t o r a g e c o e f f i c i e n t of 1 .4 and BOUND o f i n f i n i t y ( t o p ) and 0.03 ( b o t t o m ) .

period n o t influenced b y i n i t i a l c o n d i t i o n s o f s t o r a g e . This i s done through t h e generation of 1000 r e p l i c a t e s e r i e s o f 500 y e a r s by means of t h e previously described ARPIA model. These inflow s e r i e s a r e routed through r e s e r v o i r s of various combinations o f s i z e s and d r a f t s making use of t h e previously developed r e l e a s e r u l e s and t h e r e l i a b i l i t y c h a r a c t e r i s t i c s a r e then c a l c u l a t e d .

The averages o f t h e s e

annual r e l i a b i l i t i e s a r e shown in Fig. 2 as a f u n c t i o n of the storacle r a t i o (storage/mean annual runoff volume) and d r a f t r a t i o (dr 't r a t e / mean annual f l o w ) f o r t h e s e v e r a l BOUND values used i n t h e r e l e a s e rules.

Nhen BOUND i s l a r g e , namely t h e maximum r e l e a s e i s not con-

517

s t r a i n e d , t h e r e a r e many r e s e r v o i r f a i l u r e s ( r e s e r v o i r empty o r f i l l e d ) , b u t t h i s r u l e c o n t r o l s t h e flows b e t t e r by e l i m i n a t i n g more e f f e c t i v e l y t h e extremes ( f l o o d s and d r o u g h t s ) . However, t h e amplitudes of t h e r e s e r v o i r f l u c t u a t i o n s a r e r e l a t i v e l y l a r g e . When t h e B O U N D i s s m a l l , namely t h e range of p e r m i s s i b l e r e l e a s e s i s small in the dynamic program, t h e number of f a i l u r e s i s very s m a l l , b u t t h e control of t h e extreme flows i s l e s s e f f e c t i v e . Usually t h e s e d i f f e r ences a r e most v i s i b l e when the s t o r a g e c o e f f i c i e n t i s 1 . 0 o r l a r u e r . The annual r e l i a b i l i t y of t h e r e l e a s e s i s very s e n s i t i v e t o t h e value of t h e d r a f t r a t i o and appears t o be e s s e n t i a l l y i n s e n s i t i v e t o t h e BOUND values. bIONTHLY RELIABILITY

The d i s a g g r e g a t i o n model of Mejia and Rousselle (1976) was used t o simulate t h e monthly flows. The procedures f o r t h e e s t i m a t i o n of t h e parameters and f o r generation a r e given in S a l a s e t a 1 .(1980) Chapter 8. The l o s s f u n c t i o n f o r monthly flow i s of t h e same form a s shown i n equ. ( 5 ) b u t with R U P = 1 . 2 (mean monthly f l o w ) , RLOW = 0 . 8 (mean monthly f l o w ) . The s a f e range i s thus within 20% of t h e mean monthly flow (averaged over the 1 2 months).

The values o f A and B a r e t h e

same a s before and r e s u l t in a l o s s o f l o 6 u n i t s when the r e l e a s e i s zero o r twice t h e mean monthly flow. The r e l e a s e r u l e s f o r monthly flows a r e o f t h e same form a s in equ. ( 8 ) where I t and R t r e p r e s e n t t h e inflow and t h e r e l e a s e during month

t and S t i s t h e s t o r a g e a t t h e beginning of month t . As b e f o r e , t h e r e l e a s e r u l e s were obtained by r e g r e s s i o n of t h e optimal r e l e a s e vs. the optimal s t o r a g e ( r e s u l t i n g from t h e d i s c r e t e dynamic program) and t h e c u r r e n t inflow (Karamouz and Houck, 1981 b ) . Four hundred y e a r s of monthly r e s e r v o i r o p e r a t i o n s have been computed by r o u t i n g t h e monthly flows through t h e r e s e r v o i r in accordance with the optimal r e l e a s e rules. The averages o f t h e 4800 months r e l i a b i l i t i e s a r e p l o t t e d in Fig. 3 as a function of t h e s t o r a g e r a t i o and of t h e d r a f t r a t i o f o r t h e s e v e r a l B O U N D values used in t h e d e f i n i t i o n of the release r u l e .

The B O U N D i s seen t o a f f e c t t h e r e l i a b i l i t y .

In

518 0.5

I .O

Storage Coeff.= I .4 100-

g-

0.2

80 -

-

-

BOUND

bp h 60c. ._ -

z 5 ._ m -

$

BOUND

000

0.03 a 0.01

-

0

0 4

0.09 0.01

O

0.06

a 0.01

40-

\

0.4

0.6

D r a f t Ratio

0.8

'

1.2

1.0 I

0.4

J

I

0.6

I

I

-

1.4 1

I

0.8

'

I

I .O

1

I .2

1

i

l

- f o r Storage Coeff. 112

I!O l

0.4

l

l

0.6

i

l

0.8

114

'-

l

l

1.0

I .O

0.5 l

l

1.2

l

l

1.4

0.2

Annual reliability.

FIGURE 2. Storaae Coeff.

F I G U R E 3.

I

1.4

Ok

016

0!4

I

= I .4

I .o

0.5

0.2

Monthly reliability.

> 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

<

p j G 0.4,

614

The s o l u t i o n i s g i v e n by d3; v a l u e ( d g ) = 1 . 3 The e x t r e m e - p o i n t m a t r i x i s c a l c u l a t e d as

M

0.4 0.2 =[ 0.2 0.2

0.2 0.2 0.4 0.2

0.2 0.2 0.2 0.4

;I;

0.2 0.2

]

F i n a l l y , t h e p r o d u c t o f t h e above m a t r i c e s Cuijl

A

=

0.8 [1.4 1.2

M i s g i v e n by

1.0

1.0 1.4 1.0

1.2 1.4

1.4

The s o l u t i o n i s g i v e n by d2 o r d3; Value ( d 2 ) = Value ( d 3 ) = 1 . 0 * * The s o l u t i o n w i t h r e s p e c t t o I 1 f o r d = ( 0 , 0 . 5 , 0 . 5 ) , value ( d ) = 1.2 Therefore,

[Value ( I 1 )

+

v a l u e ( I 2 ) ] = 1.2

-

1.0

=

0.2.

REFERENCES Behara, M., K o f l e r , E. and k n g e s , G. (1978) Entrophy and i n f o r r n a t i v i t y i n d e c i s i o n s i t u a t i o n under p a r t i a l i n f o r m a t i o n , S t a t i s t i s c h e H e f t e , 19, p. 124-130. K o f l e r , E. and Menges, G. ( 1 9 7 6 ) En t s che idun gen be i un vo 1 1 s t a n d i ge r I n f o r m a t ion. Sp r i n ge r-Ve r l a g , B e r l i n - H e i d e l b e r g , New York. K o f l e r , E. and k n g e s , G. (1979) The s t r u c t u r i n g o f u n c e r t a i n t y and t h e maxEmin-principle. Operations Research Verfahren , 34.