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A comparative study of the simulation of daily streamflow sequences Thambirajah, Percy Anandarajah 1973

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A COMPARATIVE STUDY OF THE SIMULATION OF DAILY STREAMFLOW SEQUENCES by PERCY ANANDARAJAH THAMBIRAJAH B.Sc. (Hons.), U n i v e r s i t y o f S t r a t h c l y d e , S c o t l a n d , 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m bia Vancouver 8, Canada Date A p r i l 12, 1973 A B S T R A C T Using three years of da i l y streamflow and meteorologi-cal data from the Similkameen watershed at Princeton, B.C., the model parameters of the ex i s t i n g deterministic UBC Budget Model are evaluated. With these model parameters and the a v a i l -able meteorological data, the synthetic streamflow sequences are generated for the other seven years for the Similkameen watershed. These are subsequently compared with the actual flows. A separate s t a t i s t i c a l stochastic model i s developed by using the spectral analysis, and the three years of the same dail y flows are decomposed into 30 sub-harmonics or Fourier c o e f f i c i e n t s . By i n t e r p o l a t i n g the Fourier c o e f f i c i e n t s and by estimating the anticipated mean annual flows from the snow-pack data at Blackwall Peak, the synthetic traces of the d a i l y streamflow sequences are simulated for the other seven years. A f i r s t order Markovian model i s used to explain the random com-ponent. The comparative study i s then c a r r i e d out between the actual d a i l y streamflow sequences and those generated by the deterministic UBC Budget Model and the stochastic spectral model. In comparison with the stochastic spectral model, good f i t s are obtained with the fixed model parameters of the UBC Budget Model for the sequence of peaks for the simulated hydro-graphs of the intervening years. Since the winter melt factor i i i i i i n t he UBC Budget Model was assumed t o be a c o n s t a n t f o r t h i s a n a l y s i s , some e r r o r s o c c u r between the a c t u a l and the gener-a t e d c u m u l a t i v e volumes. With the d e t e r m i n i s t i c p e r i o d i c component o f the s p e c t r a l model, the r e c o n c i l i a t i o n between the c u m u l a t i v e volumes i s f a i r l y w e l l m a i n t a i n e d . S i n c e t h e r o l e o f o p e r a t i o n a l h y d r o l o g y i s n o t concerned w i t h the p r e -d i c t i o n o f a c t u a l f l o w s , the s t o c h a s t i c s p e c t r a l model s h o u l d be j u d g e d on i t s a b i l i t y i n p r e s e n t i n g the d e s i g n e r w i t h a s e r i e s of s y n t h e t i c t r a c e s t h a t are l i k e l y t o o c c u r d u r i n g the l i f e t i m e o f a p a r t i c u l a r p r o j e c t . TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES . . v i CHAPTER I . INTRODUCTION. . . . 1 1.1 OBJECTIVES OF RESEARCH 1 1.2 INTERACTIONS OF WEATHER SYSTEMS AND HYDROLOGIC PROCESSES 5 1.3 METHODS OF INVESTIGATING HYDROLOGIC PROCESSES. 10 I I . UBC BUDGET MODEL. 16 I I . 1 APPLICATION OF THE UBC BUDGET MODEL 16 11.2 EVALUATION OF MODEL PARAMETERS FOR UBC BUDGET MODEL 18 11.3 SIMULATION OF STREAMFLOW SEQUENCES USING THE UBC BUDGET MODEL 21 I I I . STOCHASTIC SPECTRAL MODEL . . . . . . . . . 23 111.1 APPLICATION OF THE STATISTICAL SPECTRAL MODEL. 23 111.2 EVALUATION OF THE SUB-HARMONICS FOR THE SIMILKAMEEN WATERSHED USING THE SPECTRAL MODEL 26 111.3 SIMULATION OF STREAMFLOW SEQUENCES USING THE SPECTRAL MODEL 2 7 111.4 SPECTRAL MODEL FOR THE SIMILKAMEEN WATERSHED AND THE ASSOCIATED SUB-ROUTINES 42 IV. RESULTS AND CONCLUSIONS . 46 IV.1 COMPARISON OF THE UBC BUDGET MODEL AND THE SPECTRAL MODEL 46 IV. 2 LIMITATIONS OF THE SPECTRAL MODEL 52 IV. 3 CONCLUSIONS 52 REFERENCES 56 APPENDIX I LIST'OF VARIABLES FOR THE SPECTRAL MODEL AND THE CALCOMP PLOT 59 LIST OF TABLES TABLE Page II.1 Areas and Elevations of Bands For UBC Budget Model 19 111.1 A. Values of Fourier C o e f f i c i e n t s Ftir the Similkameen Watershed . . . . . . . . . . 29 111.2 B. Values of Fourier C o e f f i c i e n t s For the Similkameen Watershed 30 IV.1 Measures of Central Location and Dispersion From 2nd March to 30th August, Similkameen Watershed 48 v LIST OF FIGURES FIGURE Page 1.1 Similkameen Watershed 3 1.2 Cyclonic Development According to the Bergen School. 8 1.3 General C l a s s i f i c a t i o n of Sequences i n Hydrology . . 11 2.1 Snow Elevation Relationship 20 3.1 Simulated Hydrographs for Similkameen Watershed, 1961-31 3.2 Simulated Hydrographs for Similkameen Watershed, 1962-32 3.3 Simulated Hydrographs for Similkameen Watershed, 1963-33 3.4 Simulated Hydrographs for Similkameen Watershed, 1964-34 3.5 Simulated Hydrographs for Similkameen Watershed, 1965-35 3.6 Simulated Hydrographs for Similkameen Watershed, 1966-36 3.7 Simulated Hydrographs for Similkameen Watershed, 1967-37 3.8 Simulated Hydrographs for Similkameen Watershed, 1968-38 3.9 Simulated Hydrographs for Similkameen Watershed, 1969-39 3.10 Simulated Hydrographs f o r Similkameen Watershed, 1970-40 3.11 Graph of Snowpack Water Equivalent Against Mean Annual Flows 41 4.1 Graph of Mean Flows Against Year 50 4.2 Graph of Standard Deviation Against Year 51 A C K N O W L E D G E M E N T Gratitude i s expressed to Dr. Michael C. Quick, B.Sc. , Ph.D., M.ASCE., P. Eng., Associate Professor, Depart-ment of C i v i l Engineering, for his constructive c r i t i c i s m s , encouragement and advice during the preparation of this research thesis. v i i CHAPTER I INTRODUCTION 1.1 OBJECTIVES OF RESEARCH The r e s e a r c h i s con c e r n e d w i t h the s h o r t a g e o f st r e a m -f l o w r e c o r d s f o r some waters h e d s and i t i s i n v o l v e d w i t h the p o s s i -b i l i t y o f g e n e r a t i n g d a i l y s y n t h e t i c s t r e a m f l o w sequences from t h e a v a i l a b l e r e c o r d s . Two s e p a r a t e approaches a r e t a k e n i n a n a l y s i n g and s o l v i n g t h i s problem. The f i r s t approach i s co n c e r n e d w i t h t h e use o f t h e l o n g e r m e t e o r o l o g i c a l r e c o r d s and the e x i s t i n g UBC Bud-get M o d e l . The second approach i s con c e r n e d w i t h the s p e c t r a l a n a l y s i s o f t h e a v a i l a b l e s t r e a m f l o w r e c o r d s and the use o f t h e e s t i m a t e d mean annual f l o w s from the m e t e o r o l o g i c a l d a t a . In e s t i m a t i n g the s y n t h e t i c sequences o f d a i l y s t r e a m -flows based on the i n t e r p o l a t e d v a l u e s o f the sub-harmonics o f the a v a i l a b l e s t r e a m f l o w r e c o r d s and the e s t i m a t e d mean annual f l o w s , i t s h o u l d be n o t e d t h a t t h e g e n e r a t e d sequences o f the s p e c t r a l model a r e n o t t h e d u p l i c a t i o n o f the a c t u a l f l o w sequen-ces o f the p a s t y e a r s . I f i t i s p o s s i b l e t o e v a l u a t e a c c u r a t e p a r a m e t r i c v a l u e s f o r the UBC Budget M o d e l , t h e n the g e n e r a t e d d a i l y s t r e a m f l o w sequences s h o u l d resemble the a c t u a l s t r e a m f l o w s e r i e s o f the p a s t y e a r s . T h i s r a i s e s the q u e s t i o n on the e v a l -u a t i o n o f the t e c h n i q u e s o r methods used f o r the g e n e r a t i o n o f the d a i l y s t r e a m f l o w sequences. But t h i s w ould e v i d e n t l y r e f l e c t 1 2 on the p a r t i c u l a r o b j e c t i v e s o f t h e h y d r o l o g i s t and on the a v a i l -a b i l i t y o f adequate m e t e o r o l o g i c a l r e c o r d s . From the few y e a r s of a v a i l a b l e s t r e a m f l o w d a t a and from the m e t e o r o l o g i c a l d a t a , the r e a l i s t i c model pa r a m e t e r s f o r the UBC Budget Model are f i x e d and improved on a t r i a l and e r r o r b a s i s u n t i l a good f i t i s o b t a i n e d f o r the a c t u a l and the r e c o n -s t i t u t e d h y d r o g r a p h s . The r e c o n c i l i a t i o n o f the c u m u l a t i v e f l o w s f o r the a c t u a l and r e c o n s t i t u t e d hydrographs i s the o t h e r r e q u i r e -ment . F o r t h e S i m i l k a m e e n R i v e r a t P r i n c e t o n , w i t h a n a t u r a l w a t e r s h e d a r e a o f 730 square m i l e s , t h r e e y e a r s o f s t r e a m f l o w d a t a f o r 1963, 1964 and 1970 are c o n c u r r e n t l y used w i t h t h e meteor-o l o g i c a l and snowpack d a t a f o r f i x i n g the model p a r a m e t e r s . Having a c c o m p l i s h e d t h i s , t h e s t r e a m f l o w sequences f o r the i n t e r v e n i n g y e a r s from 1961 t o 1970 are g e n e r a t e d by u s i n g the r e l e v a n t meteor-o l o g i c a l and snowpack d a t a . The g e n e r a t e d s t r e a m f l o w sequences a r e s u b s e q u e n t l y compared w i t h the a c t u a l s t r e a m f l o w s e r i e s . U s i n g th e a l t e r n a t i v e s t a t i s t i c a l s p e c t r a l model, the d a i l y s t r e a m f l o w s e r i e s f o r the y e a r s 1963, 1964 and 1970 are de-composed r e s p e c t i v e l y t o i t s sub-harmonics o r F o u r i e r c o e f f i c i e n t s . The a v a i l a b l e snowpack d a t a a t B l a c k w a l l Peak (6350 f e e t ) i s used t o e s t i m a t e the mean annual f l o w s f o r the i n t e r v e n i n g y e a r s , and by i n t e r p o l a t i n g t h e v a l u e s o f the F o u r i e r c o e f f i c i e n t s t o e s t i -mate the spectrum f o r the r e l e v a n t y e a r s , hydrographs a r e gener-a t e d and compared w i t h t h e a c t u a l h ydrographs and w i t h t h e hydro-Br i t i sh Columbia \j> i SCALE". 11N.- 5 MILES / AREA 730 SQ. M l . / '*•<*. \ o k a n a g a n \ range CANADA 49 aN UNITED \ STATES PASS W a s h i n g t o n 3MILKAMEEN WATERSHED FfG.1.1. 4 graphs o b t a i n e d from the UBC Budget Model. The s t a t i s t i c a l t e s t s t h a t c o u l d be a p p l i e d t o b o t h the UBC Budget Model and the s t o c h a s -2 t i c s p e c t r a l model a r e the X and t h e F t e s t s . I n i t i a l l y the r e s e a r c h was i n t e n d e d t o be a p p l i e d t o a p r a c t i c a l s i t u a t i o n l i k e the T r o u t Creek w a t e r s h e d , w h i c h i s a d j a -c e n t t o t h e S i m i l k a m e e n w a t e r s h e d . T h i s i s a r e g u l a t e d w a t e r s h e d and d i f f i c u l t y was e x p e r i e n c e d i n f i t t i n g t h e UBC Budget Model t o t h i s w a t e r s h e d . Hence the S i m i l k a m e e n w a t e r s h e d i s s e l e c t e d f o r i n v e s t i g a t i n g the p o s s i b i l i t y o f g e n e r a t i n g s t r e a m f l o w sequences and f o r c a r r y i n g out a c o m p a r a t i v e s t u d y on the advantages- and l i m i t a t i o n s o f t h e s p e c t r a l model. The UBC Budget Model i s a d e t e r m i n i s t i c model, and s i n c e i t t a k e s i n t o c o n s i d e r a t i o n the mean d a i l y t e m p e r a t u r e , p r e c i p i t a t i o n , pan e v a p o r a t i o n , a r e a l d i s t r i b u t i o n o f the snow-pack w a t e r e q u i v a l e n t and the band s w i t c h t i m e s , more r e a l i s t i c h ydrographs are o b t a i n e d . However, i t i s q u i t e d i f f i c u l t t o e s t i -mate the a r e a l d i s t r i b u t i o n o f the snowpack s i n c e the number o f gauging s t a t i o n s are l i m i t e d . The r e c o n s t i t u t e d h y drograph c o v e r s a p e r i o d o f 182 days, from the 2nd o f March t o t h e 3 0 t h o f A u g u s t . The s t a t i s t i c a l s p e c t r a l model o r s t o c h a s t i c model i s used f o r g e n e r a t i n g s y n t h e t i c t r a c e s o f d a i l y s t r e a m f l o w sequences and i t s h o u l d n o t be c o n s i d e r e d as an a n a l y t i c a l s o l u t i o n . I t i s a s t o c h a s t i c model s i n c e i t i s the c o m b i n a t i o n o f a d e t e r m i n i s t i c p e r i o d i c component and a random component. The random component e x p l a i n s t h e r e s i d u a l s i n the form 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 o r M a r k o v i a n model. 5 1.2 INTERACTION OF WEATHER SYSTEMS AND HYDROLOGIC PROCESSES D u r i n g t h e p r e l i m i n a r y a n a l y s i s , f e a s i b i l i t y s t u d i e s on the p o s s i b i l i t y o f i n c o r p o r a t i n g a. random component i n the f l o w model f o r the m i g r a t o r y low and h i g h p r e s s u r e systems, w h i c h c o n t r o l the weather and p r e c i p i t a t i o n , were c a r r i e d o u t . A t t e m p t s were a l s o made t o f i n d a r e l a t i o n s h i p between p r e c i p i t a t i o n and the weather p a t t e r n s . B e t t e r d e f i n i t i o n s o f the i n t e r a c t i o n o f the v a r i o u s g e o p h y s i c a l phenomena s h o u l d e n a b l e th e h y d r o l o g i s t o f t h e f u t u r e t o e v a l u a t e more r e a l i s t i c p a r a m e t e r s f o r 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. W i t h th e i n c r e a s e i n the t ime s c a l e , i t has been found t o be d i f f i c u l t t o prove the e x i s t e n c e o f d e t e r m i n i s -t i c and p e r i o d i c t r e n d s i n the g e o p h y s i c a l phenomena. N e v e r t h e -l e s s , h y d r o l o g i s t s are a b l e t o e v a l u a t e r e a l i s t i c model p a r a -meters r e l a t e d t o t i m e s e r i e s i n the o r d e r o f h o u r s , days and months. The c h a r a c t e r i s t i c approach i n s y n o p t i c m e t e o r o l o g y i s i n v o l v e d w i t h t h e use o f weather maps showing th e g e o g r a p h i c a l d i s t r i b u t i o n o f t e m p e r a t u r e , p r e s s u r e , w i n d , h u m i d i t y and weather p a t t e r n s . These a r e i s s u e d by t h e weather s t a t i o n s a t s e v e r a l a g r e e d t i m e s d u r i n g the day. The subsequent t a s k i s t o a n a l y z e the e v e r - c h a n g i n g d i s t r i b u t i o n o f the weather system and t o p r e -d i c t f o r t h c o m i n g changes by e m p i r i c a l e x t r a p o l a t i o n . The most i m p o r t a n t f a c t o r s t h a t c o n t r o l t h e w ind and the weather a r e the t r a v e l l i n g d e p r e s s i o n s o r c y c l o n e s and t h e a n t i c y c l o n e systems. These d e v e l o p as e n t i t i e s w i t h i n a m a t t e r 6 of hours arid go t h r o u g h a t y p i c a l l i f e c y c l e o f m a t u r i t y , decay and d i s s o l u t i o n w i t h i n a few days. C y c l o n e s a r e a r e a s o f low b a r o m e t r i c p r e s s u r e and a n t i c y c l o n e s are a r e a s o f h i g h baromet-r i c p r e s s u r e . The number o f c y c l o n e s and a n t i c y c l o n e s v a r y from t i m e t o t i m e . The c o n t i n u o u s sequence of m i g r a t o r y h i g h s and lows move more o r l e s s eastwards i n t h e m i d d l e and h i g h e r l a t i -t udes o f t h e n o r t h e r n hemisphere. The i n t e n s i t y , d u r a t i o n and speed o f t h e m i g r a t o r y c e l l s v a r y from day t o day. They t e n d t o be more numerous, s t r o n g e r and move a t a g r e a t e r speed d u r i n g w i n t e r t h a n i n summer. The changes o f t h e g e o s t r o p i c winds w i t h e l e v a t i o n i s o f v i t a l i m p o r t a n c e i n u n d e r s t a n d i n g the s t r u c t u r e and mechanism o f t h e c y c l o n i c and a n t i c y c l o n i c d i s t u r b a n c e s . T h i s i s of p a r t i c u l a r s i g n i f i c a n c e i n w e a ther f o r e c a s t i n g . Abnormal d i s -t r i b u t i o n o f t e m p e r a t u r e a l s o r e s u l t s i n the changes o f t h e geo-t r o p h i c w i n d w i t h e l e v a t i o n . The s e a t o f t h e weather i s c o n s i d e r e d as the m i d d l e o r upper t r o p o s p h e r e and f o r l e v e l s where the p r e s s u r e i s 300 t o 500 m i l l i b a r s . T h i s r e g i o n i s i n the h e a r t o f t h e upper w i n d s . S t a g -n a t i o n o f t h e upper winds g e n e r a l l y causes s t a g n a t i o n i n t h e p r o -g r e s s i o n o f w e a t h e r . The f r o n t , w h i c h i s n o r m a l l y r e p r e s e n t e d by a s i n g l e boundary l i n e , i n d i c a t e s the narrow bands o f t r a n s i t i o n between the warm and c o o l a i r masses. The c o n t i n u a t i o n o f t h e boundary t h r o u g h the atmosphere r e s u l t s i n a f r o n t a l s u r f a c e , and t h i s causes the warmer a i r mass t o l i e above the c o o l e r a i r mass. 7 The u p s l i d i n g a t t h i s s u r f a c e r e s u l t s i n the f o r m a t i o n o f c l o u d s and r a i n . The d y n a m i c a l t h e o r y has been a p p l i e d t o e x p l a i n the c y c l e o f development, m a t u r i t y and decay, but so f a r no c o n v i n c -i n g e x p l a n a t i o n p e r t a i n i n g t o the t r u e c h a r a c t e r i s t i c b e h a v i o u r o f t h e f r o n t s has been made a v a i l a b l e . B j e r k n e s (17) e x p l a i n s t h e c y c l o n i c development a c c o r d -i n g t o t h e Bergen s c h o o l o f m e t e o r o l o g i s t s and t h i s i s shown i n F i g u r e 1.2. C o n s i d e r i n g t h e n o r t h e r n hemisphere, a t y p i c a l d e p r e s -s i o n i s d i a g n o s e d as a d i s t u r b a n c e g o i n g t h r o u g h a l i f e c y c l e o f f o r m a t i o n , m a t u r i t y and decay. I n the i n i t i a l s t a g e s the warm and c o o l a i r masses l i e a l o n g s i d e each o t h e r i n a q u i e s c e n t s t a t e and t h e r e i s a r e l a t i v e l y s l o w movement. The warm a i r mass moving from west t o e a s t e v e n t u a l l y s h e a r s p a s t the r e l a t i v e l y s low moving c o o l e r a i r mass. The c h a r a c t e r i s t i c g r a d u a l f a l l i n p r e s -s u r e o v e r an a r e a , g e n e r a l l y s e v e r a l hundred square m i l e s , w i t h t h e subsequent c i r c u l a t i o n o f winds and t h e w a v e - l i k e d i s t o r t i o n o f the f r o n t , i n d i c a t e s the f o r m a t i o n o f a d e p r e s s i o n . The f r o n t i s c a r r i e d a l o n g by t h e winds and t h e a r e a ahead o f the c e n t r e i s c a l l e d t h e warm f r o n t . T h i s warm f r o n t w i l l r e p l a c e the c o l d a i r by the warm a i r . To t h e west o f t h e c e n t r e , the e x i s t i n g c i r c u -l a t i o n r e p l a c e s the warm a i r by t h e c o l d a i r at t h e c o l d f r o n t . The warm s e c t o r i s t h e a r e a between the warm and t h e f o l l o w i n g c o l d f r o n t . The above p r o c e s s w i l l be m a i n t a i n e d by a c o n t i n u o u s f a l l i n p r e s s u r e o r t h e deepening o f the d e p r e s s i o n , a r e s u l t i n g i n c r e a s e i n t h e speed o f the w i n d , o f t e n r e a c h i n g to the s t r e n g t h 8 Initiation: shallow depression and wave like distortion of the polar front. STAGE 1 CYCLONIC D E V E L O P M E N T A C C O R D I N G TO B E R G E N S C H O O L (after Bjerknes) FIG. 1.2. 9 o f g a l e f o r c e s and by a c o n t i n u o u s d i s t o r t i o n o f the f r o n t s . T h i s p r o c e s s i s t e r m i n a t e d w i t h the f o r m a t i o n o f an ' o c c l u s i o n * when the c o l d f r o n t g r a d u a l l y o v e r t a k e s t h e warm f r o n t . C i r r u s c l o u d s appear ahead o f the moving d e p r e s s i o n and t h i s i s f o l l o w e d by a l t o s t r a t u s and n i m b o s t r a t u s c l o u d s , accompanied w i t h f a l l i n g r a i n . The whole system may c o v e r s e v e r a l hundred square m i l e s and e x t e n d outwards i n a r a d i a l d i r e c t i o n from the c e n t r e o f t h e d e p r e s s i o n . In h y d r o m e t e o r o l o g y , l o n g range s t u d i e s w i l l r emain e m p i r i c a l u n t i l some w o r k i n g s o l u t i o n i s d i s c o v e r e d . To v e r i f y the e x i s t e n c e o f some p e r i o d i c v a r i a t i o n i n the weather, a c h a r a c -t e r i s t i c rhythm must be r e c o g n i z e d . S e e k i n g c o r r e l a t i o n s w i t h o u t a good p h y s i c a l r e a s o n i n g i s a hazardous o p e r a t i o n . S u b s e q u e n t l y i t may be p r o v e d t o be an a c c i d e n t a l c o i n c i d e n c e w i t h o u t any p r e -d i c t i v e v a l u e . The drawbacks i n s y n o p t i c m e t e o r o l o g y a r e r e l a t e d to t he l i m i t e d scope f o r t h e p r e d i c t i o n o f the mean d a i l y temper-a t u r e and the t o t a l p r e c i p i t a t i o n f o r t h e coming month. However t h e t h e o r e t i c a l e x p l a n a t i o n o f t h e i n t e r a c t i o n o f the v a r i o u s g e o p h y s i c a l phenomena w i l l s t i l l r e main a f o r m i d -a b l e t a s k . S t u d i e s have been made w i t h some r e a s o n a b l e s u c c e s s i n c o r r e l a t i n g p r e c i p i t a t i o n w i t h the phase o f the moon, and the mean annual d i s c h a r g e w i t h the sunspot c y c l e . The sunspot c y c l e o c c u r s once i n I l k y e a r s . A l l g e o p h y s i c a l phenomena, such as the p r e c i p i t a t i o n , r u n o f f , e v a p o r a t i o n , groundwater l e v e l s , l a k e l e v e l s , sediment 10 t r a n s p o r t and r i v e r w a t e r q u a l i t y , have d e t e r m i n i s t i c components, w h i c h a r e b a s i c a l l y p e r i o d i c s e r i e s . The main c y c l i c f r e q u e n c y f o r monthly v a l u e s i s 1/12 and t h e c y c l i c f r e q u e n c y f o r d a i l y v a l u e s i s 1/365. A complex p e r i o d i c component i s p r e s e n t i n t h e d a i l y d i s c h a r g e s a f f e c t e d by the m e l t i n g o f i c e and snow, and i n the d a i l y c y c l e o f r a d i a t i o n , t e m p e r a t u r e and e v a p o r a t i o n . These complex p e r i o d i c components a r e a l s o a s s o c i a t e d and combined w i t h t h e s t o c h a s t i c components o f the g r e a t e r and s m a l l e r e f f e c t s on the s e r i e s . 1.3 METHODS OF INVESTIGATING HYDROLOGIC PROCESSES D e t e r m i n i s t i c models a r e g e n e r a l l y u sed f o r s h o r t - r a n g e a n a l y s i s and t h e s e are c o n c e r n e d w i t h the i n p u t - o u t p u t t y p e o f m o d e l l i n g . S t o c h a s t i c models a r e used f o r l o n g - r a n g e s t u d i e s , and t h e main o b j e c t i v e i s t o produce a s e t o f t r a c e s s u f f i c i e n t l y l o n g and s t a t i s t i c a l l y i d e n t i c a l t o the h i s t o r i c a l r e c o r d . However, most models a r e i n f l u e n c e d by t h e p a r t i c u l a r aims and i n t e r e s t s and the background o f t h e h y d r o l o g i s t . The r o l e o f o p e r a t i o n a l h y d r o l o g y i s not c o n c e r n e d w i t h the p r e d i c t i o n o f f l o w s , but the emphasis i s r a t h e r i n p r e s e n t i n g the d e s i g n e r w i t h a spectrum o f s y n t h e t i c t r a c e s , each of w h i c h i s e q u a l l y l i k e l y t o be the a c t u a l sequence d u r i n g the l i f e o f a p a r t i c u l a r p r o j e c t . S t o c h a s t i c models a r e f r e q u e n t l y used i n r e s e r v o i r y i e l d s t u d i e s , and the s t a t i s t i c s o f i n t e r e s t s are r e l a t e d t o runs and r u n sums. These models a r e g r a d u a l l y becoming more a c c e p t e d as t o o l s f o r h y d r o l o -g i c p l a n n i n g and d e s i g n . 11 DETERMINISTIC S E R E S PERIODIC NON PERIODIC SINUSOIDAL COMPLEX PERIODIC ALMOST PERIODIC TRANSIENT STOCHASTIC SERIES STATIONARY NON STATIONARY. ERGODIC SPECIAL CLASSIFICATION NON ERGODIC G E N E R A L C L A S S I F I C A T I O N O F S E Q U E N C E S IN H Y D R O L O G Y (a f ter Yevjevich ) F I G . 1.3. 12 1.3.1 D e t e r m i n i s t i c T e c h n i q u e s . The w a t e r s h e d i s t r e a t e d as a h y d r o l o g i c system w i t h an i n p u t c o n s i s t i n g o f p r e c i p i t a t i o n and a c c u m u l a t e d w i n t e r snowpack, and an o u t p u t c o n s i s t i n g o f r u n -o f f . The f o r m u l a t i o n o f the h y d r o l o g i c model i s e v o l v e d on t h e b a s i s o f t h e p r i n c i p l e o f the c o n s e r v a t i o n o f mass, and comprises t h e component p r o c e s s e s i n v o l v i n g s o i l m o i s t u r e d e f i c i t , s u b - s u r -f a c e r u n o f f , f a s t r u n o f f , s l o w r u n o f f , pan e v a p o r a t i o n and s t o r -age i n t h e w a t e r s h e d . 1.3.2 S t o c h a s t i c T e c h n i q u e s . The p a r t i c u l a r mathemati-c a l t e c h n i q u e s d e v e l o p e d f o r i n v e s t i g a t i n g s t o c h a s t i c h y d r o l o g i c p r o c e s s e s a r e : (a) Moving Average M o d e l . T h i s i s e x p r e s s e d as: v t = a l £ t + a2et_1 + . . . + a m e t _ m + 1 ( 1 . 1 ) where al»a2»' • *am a r e t h e w e i g h t s " » m i s t h e number o f s i g n i f i c a n t moving a v e r a g e s ; E i s t h e random v a r i a b l e . F o r example, a r e l a t i o n s h i p c o u l d be d e v e l o p e d between the a n n u a l r u n o f f , u and the a n n u a l e f f e c t i v e p r e c i p i t a t i o n e. The w e i g h t s a^, a 2 , . . •> a m must be p o s i t i v e and e q u a l t o u n i t y , and m i s t h e e x t e n t o f t h e c a r r y - o v e r r e l a t e d t o t h e w a t e r r e t a r d -a t i o n c h a r a c t e r i s t i c s o f t h e p a r t i c u l a r w a t e r s h e d . (b) Method o f A u t o c o r r e l a t i o n C b e f f i c i e n t s . The use o f a u t o r e g r e s s i v e t e c h n i q u e s i n t h e m o d e l l i n g o f t i m e s e r i e s 13 is a well-developed s t a t i s t i c a l procedure. This model may be expressed as: vt = f ( u t - i > y t - 2 ' * ' y t - k ^ + e t C 1- 2) where f ( ) i s a mathematical function; k i s an integer; e t i s a random variable. The t y p i c a l form of the l i n e a r autoregression model of th the m order i s expressed as: H = a l y t - l + a 2 y t - 2 + • • • + Vt-m + e t C I ' 3 ) where a,, a 7 , . . .a are the regression c o e f f i c i e n t s ; is a random va r i a b l e . For m = 1, the l i n e a r autoregression model i s trans-formed into the f i r s t order Markovian process, and this i s ex-pressed as: y t = a y t - l + e t ( I ' 4 ) where a i s the Markovian process c o e f f i c i e n t ; e t i s a random var i a b l e . (c) Method of Spectral Densities. For a discrete sample se r i e s , the model may be expressed as: 14 3 = J where y i s the mean o f t h e s e r i e s ; A j and B.. are the a m p l i t u d e s o r F o u r i e r c o e f f i c i e n t s ; ^ i s the p e r i o d of c y c l i c i t y w i t h j = 1, 2, . . ., m; e t i s a random v a r i a b l e . The above model i s c h a r a c t e r i z e d by an o s c i l l a t o r y o r r e g u l a r form o f v a r i a t i o n , w h i c h i s p r e s e n t i n t h e d i u r n a l , season-a l and s e c u l a r changes o f the h y d r o l o g i c phenomena. (d) Method o f Mean Ranges. T h i s i s d e f i n e d as t h e maxi-mum d i f f e r e n c e o f the sum o f d e v i a t i o n s from the mean f o r t h e v a l u e s o f a s u b - s e r i e s 1, 2, . . ., n. The e x p e c t e d v a l u e i s de-pendent on t h e s i z e o f t h e s u b - s e r i e s , ER R = f ( n ) . T h i s t e c h n i q u e i s used f o r i n v e s t i g a t i n g h y d r o l o g i c p r o c e s s e s and i s r e l a t e d t o w a t e r s t o r a g e p r o b l e m s , and w a t e r s u r p l u s and d e f i c i t p r o b l e m s . (e) Method o f Mean Runs. The run i s d e f i n e d as the sum of d e v i a t i o n s above o r below a g i v e n v a l u e o f t h e s e r i e s , known as the c r o s s i n g l e v e l . The e x p e c t e d v a l u e o f t h e r u n - l e n g t h o r t h e e x p e c t e d v a l u e o f the run-sum may be r e l a t e d t o t h e c r o s s i n g l e v e l x Q , o r t o t h e p r o b a b i l i t y o f o c c u r r e n c e q o f v a l u e s s m a l l e r t h a n o r e q u a l t o x Q . T h i s t e c h n i q u e i s used f o r i n v e s t i g a t i n g the de-pendence o f h y d r o l o g i c sequences, and t h i s i s c l o s e l y r e l a t e d t o t h e v a r i o u s problems o f extremes such as d r o u g h t s , f l o o d s , w a t e r s u r p l u s and d e f i c i t s . 15 D e t e r m i n i s t i c and p r o b a b i l i s t i c models do n o t conform c l o s e l y w i t h the n a t u r a l phenomena, and s u b s e q u e n t l y l e a d t o the o v e r d e s i g n i n g o r u n d e r d e s i g n i n g o f h y d r o l o g i c p r o j e c t s . However the a c t u a l c h o i c e o f a m a t h e m a t i c a l model would i n e v i t a b l y r e -f l e c t on the p a r t i c u l a r aims of t h e h y d r o l o g i s t . CHAPTER II UBC BUDGET MODEL II.1 APPLICATION OF THE UBC BUDGET MODEL The UBC Budget Model consists of a main program and eight separate sub-routines. Using three years of streamflow and meteorological data from the Similkameen watershed for the years 1963, 1964 and 1970, the Model parameters are evaluated on the basis of the re c o n s t i t u t i o n of the actual hydrograph. The winter melt i s a factor which varies from year to year and i t affects the f i n a l volume of the estimated runoff. For this p a r t i c u l a r research project the winter melt factor i s assumed to be a constant, so that the f e a s i b i l i t y studies for generating and comparing the synthetic d a i l y streamflow sequences for the other years using only meteorological and snowpack data could be carri e d out. The UBC Budget Model i s expressed i n terms of inputs, outputs and transformations. The term 'black box' refers to the system that transforms the inputs into outputs, and thi s form of simulation could be considered as an i n d i r e c t i n v e s t i g a t i o n of the response or behaviour of the watershed. The input consists of data for temperature, p r e c i p i t a t i o n , snowpack water equiva-lent and pan evaporation. This data i s available from the Monthly Record of Meteorological Observations., Canada and from the B r i t i s h 16 17 Columbia Snow Survey B u l l e t i n . D a i l y s t r e a m f l o w v a l u e s f o r t h e Simi l k a m e e n w a t e r s h e d at P r i n c e t o n a re a v a i l a b l e from the Surface Water Data, B r i t i s h Columbia and from the Water Resources Paper -Surface Water Data for P a c i f i c Drainage. P r i n c e t o n i s s e l e c t e d f o r 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 d a t a s i n c e no o t h e r meteoro-l o g i c a l s t a t i o n i s l o c a t e d i n the i n t e r i o r o f the w a t e r s h e d . B l a c k w e l l Peak (6,350 f e e t ) p r o v i d e s 13 y e a r s o f snowpack d a t a from 1960 t o 1972, and t h i s s t a t i o n i s s e l e c t e d s i n c e the d a t a from Copper M o u n t a i n (4,300 f e e t ) d i d not produce r e p r e s e n t a t i v e h y d r o g r a p h s . Pan e v a p o r a t i o n d a t a i s t a k e n from Summerland, and t h i s i s t h e o n l y a v a i l a b l e s t a t i o n i n t h e i n t e r i o r o f B r i t i s h C o l umbia. The a r e a l d i s t r i b u t i o n o f snowpack w a t e r e q u i v a l e n t on the d i f f e r e n t e l e v a t i o n bands i s a p p r o x i m a t e d by u s i n g a semi-l o g a r i t h m i c p l o t s i m i l a r t o t h a t used i n t h e C a r r s L a n d i n g Water-shed ( 9 ) . The o t h e r r e l e v a n t i n p u t d a t a a r e t h e a r b i t r a r y s o i l m o i s t u r e d e f i c i t s , l a n d a r e a s , e l e v a t i o n bands, e l e v a t i o n s o f the m e t e o r o l o g i c a l s t a t i o n s , and the c o n t r o l p arameters f o r the g r a -p h i c a l o u t p u t . The te m p e r a t u r e i s l a p s e d o v e r the d i f f e r e n t e l e v a t i o n bands w i t h t h e I n t e r n a t i o n a l S t a n d a r d Atmosphere v a l u e o f 3.5°F/1,000 f e e t . The o t h e r s u b - r o u t i n e s a r e c o n c e r n e d w i t h the d i v i s i o n o f r u n o f f i n t o f a s t o r s u r f a c e f l o w , medium o r i n t e r f l o w and slow o r groundwater r e c h a r g e . The d a i l y s t r e a m f l o w i s e v e n t u a l l y r o u t e d t o a c h a n n e l system. The amount and form of t h e d a i l y p r e c i p i t a t i o n i s e v a l u a t e d from t h e p r e c i p i t a t i o n e l e v a t i o n r e -l a t i o n s h i p . The p o t e n t i a l e v a p o t r a n s p i r a t i o n a t each mid-band 18 e l e v a t i o n i s e s t i m a t e d from t h e monthly pan e v a p o r a t i o n d a t a and the maximum t e m p e r a t u r e d a t a . The c o m p u t a t i o n o f the a r e a l r e -c e s s i o n o f the snowpack and the d e p l e t i o n o f the snowmelt r u n o f f p o t e n t i a l a re c a r r i e d o u t . The d a i l y w a t e r budget a t each e l e -v a t i o n band i s e v a l u a t e d from t h e d a i l y r u n o f f e x c e s s e s from the snowmelt and t h e r a i n f a l l i n p u t s a f t e r h a v i n g g i v e n c o n s i d e r a t i o n to t h e s o i l m o i s t u r e d e f i c i e n c i e s , l o s s e s t o t h e groundwater r e -charge and t o t h e d a i l y e v a p o t r a n s p i r a t i o n . H a v i n g g i v e n a l l o w -ance f o r the i m p e r m e a b i l i t y f a c t o r , t h e e f f e c t i v e d a i l y b a s i n i n p u t i s computed from t h e c u m u l a t i v e band e x c e s s e s . F i n a l l y , the d a i l y r u n o f f i s g e n e r a t e d on t h e b a s i s o f t h e u n i t h y drograph c o n v o l u t i o n . I I . 2 EVALUATION OF MODEL PARAMETERS FOR THE UBC BUDGET MODEL, The c o n s i s t e n t model par a m e t e r s f o r t h e Sim i l k a m e e n w a t e r s h e d f o r the y e a r s 1963, 1964 and 1970 are e v a l u a t e d from t h e Monthly Record of Meteorological Observations, Canada, t h e B r i t i s h Columbia Snow Survey B u l l e t i n , Surface Water Data, Bri-t i s h Columbia and from t h e Surface Water Data for P a c i f i c Drain-age, The l a p s e r a t e i s t a k e n as 3.5°F/1,000 f e e t , t h e v a l u e o f the w i n t e r m e l t i s assumed t o be a c o n s t a n t , and t h e e l e v a t i o n con-s t a n t s f o r b o t h t h e s n o w f a l l e l e v a t i o n r e l a t i o n s h i p and the r a i n -f a l l e l e v a t i o n r e l a t i o n s h i p are assumed t o be 30,000. The c o n s t a n t f o r t h e b a s e f l o w i s t a k e n as 100 c f s or 200 a c r e f e e t , t h e u p l a n d a r e a i s not s u b j e c t e d t o t h e u p l a n d s t o r a g e r e g u l a t i o n , and t h e 19 d i s t r i b u t i o n o f the snowpack w a t e r e q u i v a l e n t on t h e d i f f e r e n t e l e v a t i o n bands a r e e v a l u a t e d a c c o r d i n g t o the s e m i - l o g a r i t h m i c p l o t as shown i n F i g u r e 2.1. TABLE I I . 1 AREAS AND ELEVATIONS OF BANDS FOR UBC BUDGET MODEL BAND NO FROM Cfeet) TO Cfeet) ELEVATION OF MID-BAND Cfeet) AREA (sq. miles) 1 2,000 3,000 2,500 12 2 3,000 4,000 3,500 44 3 4,000 5,000 4,500 251 4 5,000 6,000 5,500 278 5 6,000 7,000 6,500 124 6 7,000 8,000 7,500 20 7 8,000 9,000 8,500 3 The emphasis on the v i t a l s i g n i f i c a n c e o f the d i s t r i b u t i o n o f the i n i t i a l snowpack w a t e r e q u i v a l e n t on each of the band areas s h o u l d n o t be o v e r l o o k e d . T h i s i s the c r u c i a l a s p e c t , w h i c h i s f u r t h e r a c c e n t u a t e d by the s c a r c i t y o f r e p r e s e n t a t i v e snowpack gauging s t a t i o n s i n t h e Similkameen w a t e r s h e d . Hence the e v a l u -a t e d d i s t r i b u t i o n o f the snowpack w a t e r e q u i v a l e n t may not be 20 I I 1000 2000 3000 U000 5000 6000 ELEVATION (Fee t ) . S N O W E L E V A T I O N R E L A T I O N S H I P (after Pipes) FIG. 2 . 1 . 21 r e p r e s e n t a t i v e o f the a c t u a l a r e a l d i s t r i b u t i o n . O ther f a c t o r s p e r t a i n i n g t o the non-homogeneity o f s o i l s and v e g e t a t i o n c o u l d a f f e c t the d i s t r i b u t i o n o f the snowpack. The f o l l o w i n g v a r i a b l e s o f the model par a m e t e r s a r e ad-j u s t e d t o f i t the r e c o n s t i t u t e d hydrographs o f the Sim i l k a m e e n w a t e r s h e d f o r the y e a r s 1963, 1964 and 1970: (a) the maximum f r a c t i o n o f i m p e r m e a b i l i t y ; (b) the u n i t h y d r o g r a p h shape p a r a m e t e r ; (c) the a d d i t i o n a l shape p a r a m e t e r f o r t h e i n t e r f l o w ; (d) the c o n s t a n t f o r the i n t e r f l o w s t o r a g e r e s e r v o i r . The v i s u a l i n s p e c t i o n o f the a c t u a l and s i m u l a t e d h y d r o -graphs and the r e c o n c i l i a t i o n o f the a c t u a l c u m u l a t i v e volumes and the s i m u l a t e d c u m u l a t i v e volumes a re the b a s i s o f e v a l u a t i n g the above v a r i a b l e s t h r o u g h a s e r i e s o f r e p e t i t i v e computer r u n s . For the Sim i l k a m e e n w a t e r s h e d the v a l u e s f o r t h e f o l l o w -i n g v a r i a b l e s are c o n s i d e r e d as most a p p r o p r i a t e : (a) the maximum f r a c t i o n o f i m p e r m e a b i l i t y - 0.60; (b) the u n i t h y d r o g r a p h shape parameter - 1.80; (c) the a d d i t i o n a l shape parameter f o r the i n t e r f l o w - 0.00; (d) the c o n s t a n t f o r the i n t e r f l o w s t o r a g e r e s e r v o i r - 0.10. I I . 3 SIMULATION OF STREAMFLOW SEQUENCES USING THE UBC BUDGET MODEL Hav i n g a s c e r t a i n e d the r e l e v a n t model par a m e t e r s f o r t h e Sim i l k a m e e n w a t e r s h e d f o r t h e y e a r s 1963, 1964 and 1970, hy d r o -22 graphs are s i m u l a t e d f o r the i n t e r v e n i n g seven y e a r s from 1961 to 1970 u s i n g m e r e l y d a t a from t h e Monthly Record of Meteoro-l o g i c a l Observations, Canada and the B r i t i s h Columbia Snow Sur-vey B u l l e t i n . These hydrographs are t h e n v i s u a l l y compared w i t h the sequence o f the a c t u a l h y d r o g r a p h s , a v a i l a b l e f r om the Sur-face Water Data, B r i t i s h Columbia, and from the Surface Water Data for P a c i f i c Drainage. The i n i t i a l runs g e n e r a t e d w i t h the snowpack w a t e r e q u i v a -l e n t d a t a from t h e Copper M o u n t a i n (4300 f e e t ) d i d n o t produce r e p r e s e n t a t i v e h y d r o g r a p h s . Hence B l a c k w a l l Peak (6350 f e e t ) i s s e l e c t e d f o r e v a l u a t i n g the d i s t r i b u t i o n o f the snowpack w a t e r e q u i v a l e n t . However t h i s snowpack d a t a i s o n l y a v a i l a b l e f o r t h e p e r i o d from 1960 t o 1972. The monthly pan e v a p o r a t i o n d a t a i s t a k e n from Summerland and t h i s g a u g i n g s t a t i o n i s n o t i n the v i c i n i t y o f the Similkameen w a t e r s h e d . F u r t h e r d a t a i s n o t a v a i l a b l e f o r some o f the i n t e r -v e n i n g months and t h e s e v a l u e s are a p p r o x i m a t e d from the gaug-i n g s t a t i o n a t the U n i v e r s i t y o f B r i t i s h C olumbia. The monthly l a k e e v a p o r a t i o n d a t a i s n o t a v a i l a b l e f o r 1961, and f o r the r e l e v a n t r u n the v a l u e s are assumed t o be s i m i l a r t o the v a l u e s o f 1962. CHAPTER I I I STOCHASTIC SPECTRAL MODEL I I I . l APPLICATION OF THE STATISTICAL SPECTRAL MODEL The s p e c t r a l a n a l y s i s and the a u t o c o r r e l a t i o n t e c h -n i q u e s a re the a l t e r n a t i v e approaches f o r the i n v e s t i g a t i o n o f h y d r o l o g i c p r o c e s s e s . These t e c h n i q u e s have n o t been w e l l e s t a b -l i s h e d i n w a t e r r e s o u r c e s economics and h y d r o l o g i c p l a n n i n g , and r e s e a r c h i s s t i l l b e i n g c a r r i e d out on the a p p l i c a t i o n i n t h e s e f i e l d s . The s p e c t r a l a n a l y s i s t e c h n i q u e s are w e l l e s t a b l i s h e d i n the s t u d y o f wave phenomena, v i b r a t i o n a n a l y s i s , e a r t h q u a k e s and s t r u c t u r a l r e s p o n s e . There i s some degree o f u n c e r t a i n t y i n i t s a p p l i c a t i o n t o the complex s e t t i n g o f t h e h y d r o l o g i c phe-nomena. S t o c h a s t i c models a r e used f o r the f o l l o w i n g c a s e s : 1. F o r the p r a c t i c a l and economic p l a n n i n g o f w a t e r s u p p l y and i r r i g a t i o n p r o j e c t s . 2. Most f r e q u e n t l y used i n r e s e r v o i r y i e l d s t u d i e s ; and t h e s t a t i s t i c a l a n a l y s i s i s r e l a t e d t o t he runs and r u n sums. 3. I n p r e s e r v i n g c e r t a i n e s t i m a t e s o f t h e s t a t i s t i c a l p r o p e r t i e s f r om a g i v e n r e c o r d , and t h e s e e s t i m a t e s a r e us e d f o r g e n e r a t i n g many equ-a l l y l i k e l y sequences. These sequences a r e us e d i n p l a c e o f the h i s t o r i c a l r e c o r d o r f o r t h e a n t i -c i p a t e d l i f e o f a p a r t i c u l a r p r o j e c t and f o r the development of o p t i m a l d e s i g n s . R o d r i g u e z (13) has emphasized t h a t s t o c h a s t i c models f o r s t r e a m f l o w g e n e r a t i o n a r e becoming more a c c e p t e d f o r p l a n n i n g 23 24 s t u d i e s s i n c e the concept o f d e t e r m i n i s t i c h y d r o l o g i c c y c l e s c o u l d be q u e s t i o n e d from the p h y s i c a l and s t a t i s t i c a l p o i n t o f view. The d a i l y and monthly h y d r o l o g i c d a t a t e n d t o i n d i c a t e a s t r o n g c y c l i c f e a t u r e . R o d r i g u e z (13) a l s o a s s e r t s t h a t s t a -t i s t i c a l models s h o u l d n o t be ju d g e d by the degree i n w h i c h the lower moments, namely the mean, v a r i a n c e and sometimes skew-ness a r e p r e s e r v e d , b u t r a t h e r by t h e i r a b i l i t y i n p r e s e r v i n g the c h a r a c t e r i s t i c s t h a t a r e o f s i g n i f i c a n c e , e.g., r e s e r v o i r y i e l d . S o l a r r a d i a t i o n i s the main c o n t r o l l i n g f a c t o r o f the t e r r e s t r i a l w e a t h e r . The e x i s t e n c e o f the annua l c y c l e i n b o t h t h e d a i l y and mo n t h l y h y d r o l o g i c d a t a i s a t t r i b u t e d t o the e f f e c t from the s o l a r r a d i a t i o n . S o l a r a c t i v i t y i s n o t a d e t e r m i n i s t i c p e r i o d i c p r o c e s s and random f i l t e r s such as t h e atmosphere and c l o u d s t h r o u g h w h i c h t h e s o l a r r a d i a t i o n a c t s , s h o u l d be c o n s i d -e r e d . The b a s i c p r i n c i p l e s f o r the s t a t i s t i c a l m o d e l l i n g o f the Similkameen w a t e r s h e d c o u l d be summarized as f o l l o w s . There a r e two d i f f e r e n t p r o c e s s e s , w h i c h when added up d e s c r i b e the o r i g i n a l h y d r o l o g i c s e r i e s . The f i r s t i s the d e t e r -m i n i s t i c component and t h i s i s due t o the average a n n u a l p e r i o d -i c i t y i n the d a i l y s t r e a m f l o w sequences. The d a i l y and monthly p e r i o d i c i t i e s i n the h y d r o l o g i c t i m e s e r i e s may not be d i s t i n c t o r remain c o n s t a n t from y e a r t o y e a r . The second p r o c e s s a c c o u n t s f o r t he random v a r i a t i o n s i n the atmosphere w i t h the p a s s i n g o f t i m e . The random component o r r e s i d u a l s can be a p p r o x i m a t e l y 25 e x p l a i n e d by 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 o r M a r k o v i a n model. The e x p l a n a t i o n o f the random component by t h i s t e c h n i q u e i s s t i l l s u b j e c t t o c o n t r o v e r s y . The s t o c h a s t i c model u s e d f o r s i m u l a t i n g the d a i l y sequences o f s t r e a m f l o w f o r t h e S i m i l k a m e e n w a t e r s h e d i s ex-p r e s s e d as X = y + a e ( I H . l ) T T X T where T """ 1 j ^ J • * •> ^ 9 w = b a s i c p e r i o d ; a = s t a n d a r d d e v i a t i o n o f x x T e x = s t o c h a s t i c component. The d e t e r m i n i s t i c component y T i s e x p r e s s e d mathemati-c a l l y as: m y = y + E (A.cos 2ir£.x + B . s i n 2 i r f . x ) ( I I I . 2 ) where y = mean of the s e r i e s ; .A. A- and B. a r e the F o u r i e r c o e f f i c i e n t s ; m = number o f s i g n i f i c a n t h a r m o n i c s . The F o u r i e r c o e f f i c i e n t s f o r the r e l e v a n t s e r i e s a r e e v a l u a t e d as: A- =4 £ (x - y ) cos i l L L ( I I I . 3 ) 3 w T _ i x x y w v J 2 w B. = - E (x - y ) s i n 2 T T J T ( I I I . 4 ) 26 where £. = i J w I t must be emphasized t h a t the s t a t i s t i c a l s p e c t r a l a n a l y s i s cannot be used e a s i l y o r d i r e c t l y i n water r e s o u r c e s economics and p l a n n i n g a t the p r e s e n t s t a t e o r l e v e l o f t h e s e t e c h n i q u e s . But i t p r o v i d e s a b e t t e r i n s i g h t i n t o the c r i t i c a l sequences o f f l o w , w h i c h h i s t o r i c a l d a t a a l o n e may not be a b l e to y i e l d . I I I . 2 EVALUATION OF THE SUB-HARMONICS FOR THE SIMILKAMEEN WATERSHED USING THE SPECTRAL MODEL U s i n g e q u a t i o n s I I I . 3 and I I I . 4 , t h e d a i l y s t r e a m f l o w s e r i e s f o r t h e S i m i l kameen w a t e r s h e d f o r the y e a r s 1963, 1964 and 1970 a r e r e s p e c t i v e l y decomposed t o the f i r s t 35 sub-harmonics o r F o u r i e r c o e f f i c i e n t s . Through a s e r i e s o f t r i a l and e r r o r t e c h -n i q u e s , the f i r s t 30 sub-harmonics appear t o g i v e a r e a s o n a b l y good v i s u a l f i t f o r t h e r e c o n s t i t u t e d h y d r o g r a p h u s i n g e q u a t i o n I I I . 2 . H a v i n g f i t t e d t h e p e r i o d i c or d e t e r m i n i s t i c component w i t h t h e sub-harmonics f o r the r e l e v a n t y e a r s , t h e u n e x p l a i n e d v a r i a n c e or r e s i d u a l s a r e a n a l y z e d by the m u l t i p l e r e g r e s s i o n t e c h n i q u e . The r e s i d u a l s are l a g g e d by one day, and 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 or M a r k o v i a n model i s f i t t e d f o r t h e r e s p e c -t i v e y e a r s . T h i s i s e x p r e s s e d as: e x = r l » e T - l + n t ( H I . 5 ) where 27 G = r e s i d u a l s ; r ^ = 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 ; n = independent random v a r i a b l e w i t h mean z e r o and a g i v e n v a r i a n c e . A random number g e n e r a t o r FRANDN i s used f o r g e n e r a t i n g the r e a l i s t i c p o s i t i v e v a l u e s f o r the s t o c h a s t i c component. T h i s i s e x p r e s s e d as n T = FRANDN (0.0) a (1-R 2) ( I I I . 6 ) where n T = independent random v a r i a b l e ; a = s t a n d a r d d e v i a t i o n ; 2 R = m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t . I I I . 3 SIMULATION OF STREAMFLOW SEQUENCES USING SPECTRAL MODEL. Fo r t h e i n t e r v e n i n g y e a r s from 1961 t o 1970, t h e mean annual f l o w s a t P r i n c e t o n a r e e s t i m a t e d l i n e a r l y from t h e snow-pack w a t e r e q u i v a l e n t d a t a a t B l a c k w a l l Peak (6350 f e e t ) , a v a i l -a b l e from t h e B r i t i s h C olumbia Snow Survey B u l l e t i n and from t h e known mean a n n u a l f l o w s f o r 1963, 1964 and 1970. T h i s i s shown i n F i g u r e 3.11. The 30 F o u r i e r c o e f f i c i e n t s f o r t h e r e l e v a n t i n t e r v e n -i n g y e a r s and f o r the e s t i m a t e d mean a n n u a l f l o w s o f t h e S i m i l k a -meen w a t e r s h e d a r e e v a l u a t e d by i n t e r p o l a t i n g t h e v a l u e s o f t h e F o u r i e r c o e f f i c i e n t s f o r the y e a r s 1963, 1964 and 1970. These i n t e r p o l a t e d v a l u e s are shown i n T a b l e s I I I . l and I I I . 2. The s t o c h a s t i c component i s s e l e c t e d on the b a s i s o f t h e assumption-28 t h a t the random component f o r the y e a r 1970 i s r e p r e s e n t a t i v e o f the r e s p e c t i v e e s t i m a t e d mean a n n u a l f l o w s over the p e r i o d under c o n s i d e r a t i o n . The s y n t h e t i c s t o c h a s t i c hydrographs f o r the i n t e r v e n -i n g seven y e a r s a re then g e n e r a t e d from t h e i n t e r p o l a t e d v a l u e s of t h e F o u r i e r c o e f f i c i e n t s and from t h e e s t i m a t e d mean annual f l o w s . E q u a t i o n I I I . l i s u s e d f o r g e n e r a t i n g t h e s y n t h e t i c s t r e a m f l o w sequences. The s t o c h a s t i c h ydrographs a r e e v e n t u a l l y compared w i t h t h e d e t e r m i n i s t i c UEC hydrographs and w i t h t h e a c t u a l h y d rographs f o r t h e r e s p e c t i v e y e a r s . The Calcomp p l o t i s used f o r the c o m p a r a t i v e s t u d y . TABLE III.1 EVALUATED AND INTERPOLATED A-j VALUES OF FOURIER COEFFICIENTS FOR THE SIMILKAMEEN WATERSHED 1966 $ 1970 557 5 528 cfs. 1969* 649 cfs. 1962 5 1965 711 § 759 cfs. 1963 794 cfs. 1961 5 1968* 898 § 901 cf s. 1967* 940 cfs 1964 1120 cfs. -723 -705 -693 -675 -877 .973 -1482 430 418 409 396 572 659 1097 -134 -118 -106 - 89 -234 ' -306 -'667 -125 -159 -181 -214 - 82 - 16 315 276 273 271 276 199 165 7 -289 -262 -244 -217 -204 -197 - 163 212 179 157 123 150 164 230 -113 - 60 - 24 29 - 34 - 65 - 222 18 - 43 - 85 -143 - 64 - 25 • 174 40 88 120 168 152 144 102 - 54 20 24 - 35 - 11 1 59 43 28 18 2 - 3 - 6 - 18 - 33 13 45 90 69 58 3 5 - 62 -106 -173 -120 - 94 39 20 73 108 160 112 88 - 33 - 23 - 45 - 39 - 80 - 51 - 37 34 16 15 15 13 5 1 - 18 - 14 8 24 46 29 21 - 20 25 - 25 - 57 -107 - 60 - 37 80 - 50 - 2 31 79 32 9 - 108 78 43 20 - 16 11 25 93 - 90 - 53 - 28 10 - 9 - 19 - 66 82 60 45 22 20 20 15 - 48 - 45 - 43 - 40 - 23 - 14 30 - 18 - 16 14 - 11 - 25 - 32 - 67 80 61 48 28 43 50 85 -108 - 78 - 58 - 28 - 38 - 43 - 68 104 73 52 20 22 23 27 - 76 - 63 - 54 - 41 - 24 - 15 28 42 16 - 2 • - 28 - 42 - 49 - 82 * Interpolated Values . Evaluated for 1963. 1964 and 1970. TABLE III.2 EVALUATED AND INTERPOLATED B i VALUES OF FOURIER COEFFICIENTS FOR THE SIMILKAMEEN WATERSHED 1966 $ 1970 557 5 528 c f s . 1969* 649 c f s . 1962 5 1965* 711 § 759 cf s. 1963 794 c f s . 1961 § 1968 898 5 901 c f s . 1967* 940 c f s . 1964 1120 c f s . 327 317 310 300 320 330 381 -553 -546 -540 -533 -562 -577 -649 585 533 499 447 537 582 806 -482 -433 -400 -350 -420 -454 -627 272 218 182 128 230 280 534 - 80 - 41 - 15 25 - 62 -106 -323 - 60 -129 -175 -244 -149 -101 138 117 153 177 213 178 160 73 -111 -114 -116 -119 -117 -115 -109 66 57 52 44 78 95 180 - 26 19 49 94 28 - 5 -170 - 8 - 76 -120 -188 - 86 - 35 222 33 61 79 107 40 7 -161 - 42 - 44 - 46 --49 1 26 151 37 - 1 - 26 - 62 - 70 - 74 - 93 - 29 34 70 137 118 109 60 20 61 -115 -196 -150 -127 - 10 - 20 20 48 88 67 56 2 31 - 1 - 22 - 53 - 40 - 34 - 2 - 38 - 27 - 19 - 8 1 5 26 21 39 50 67 37 22 - 54 18 - 16 - 38 - 71 - 28 - 7 102 - 73 - 49 - 32 - 6 - 33 - 47 -115 113 77 52 15 39 50 109 -124 - 91 - 68 - 33 - 37 - 38 - 48 96 72 55 30 23 19 1 - 43 - 36 - 32 - 24 - 2 9 63 - 12 - 23 30 - 41 - 54 60 - 91 49 42 37 31 49 58 101 - 69 - 54 - 44 - 28 - 37 - 41 - 63 Interpolated values. P u o l i i o r o / 1 - F ^ v I D A - * 1 a&A o n J 1 0 7 0 11.0 n 92.5 8 74.0 X LO LL-u o _ J L L 55.5 H 37.0 H 18.5 -J S I M U L A T E D H Y D R O G R A P H S F O R S f M I L K A M E E N W A T E R S H E D 1961 F I G . 3.1'. 0.0 Actual flows UBC budget model Stochastic model Mean flow 898 c.ts. • DAY 111.0 1 92.5 H S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1962 740 H FIG. 3 . 2 . 55.5H - Actual flows ••• UBC budget model -- Stochastic model Mean flow 711 c.ls. 0 —r 4 5 90 135 180 D A Y 225 270 315 360 CM 111.0-1 92.5-g 7 U H X LO U 55.5 H o L L 37.0 -18.5 -0.0 1A 'V V / , . X 0 45 ~T~ 90 \ 0 S I M U L A T E D H Y D R O G R A P H S FOR S I M I L K A M E E N W A T E R S H E D 1963 F I G . 3 . 3 . Actual flows UBC budget model Stochastic model Mean flow 79k c.f.s. ( Used for evaluat ing model parameters and Four ier coefficients ) 135 "l8o" 2F5 1^0 315 360 DAY 111.0-1 DAY S I M U L A T E D H Y D O G R A P H S F D R S I M I L K A M E E N W A T E R S H E D 1964 FIG. 3 . 4 . Actual flows UBC budget model Stochastic model Mean flow 1120 els. i ! ( Used for evaluating model parameters and Fourier coef f i c ien ts ) 7 2 ? 270 315 ilo 11101 92.5 H o f t . O H o x or ^55.5 o u_ 37.0-f 18.5 A 0J0 S I M U L A T E D H Y D O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1965 F I G . 3 , 5 . . Actual flows UBC budget model Spectral model Mean flow 759 c.fs. 0.0 4 5.0 90.0 135.0 180.0 DAY 2250 270.0 315.0 —r~ 360.0 CM cn 111.0 92.5" o o 74.0-LL 55.5 o LL. 37.0-18.5-0.0 S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1966 F I G . 3 . 6 . Actual f lows UBC budget model Stochastic model Mean f low 557 c.f.s. A W W 0 1^ 45 90 — I r -135 180 DAY — r ~ 225 270 315 360 111.0 o o LO L L ( J O _ J L L 37.0 18.5 S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1967 F I G . 3. 7. Actual f lows UBC budget model Stochastic model Mean flow 940 cf.s 270 315 360 111.0 92.5-74.0-55.5-37.0-18.5-S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1968 FIG. 3 . 8 . Actual f lows UBC budget model Stochastic model Mean f low 901 c.f.s. CO 360 DAY 111.0-92.5H S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1969 74.0 H FIG. 3 . 9 . 55.5H 37.0 18.5 H N \ \ ^ -\ f \ , \ •• 0.0 A? 90 Actual flows UBC budget model Stochastic model Mean flow 649 cf.s. 135 180 225 270 315 DAY vo 111,0-1 92.5 H S I M U L A T E D H Y D R O G R A P H S F O R S I M I L K A M E E N W A T E R S H E D 1970 o o LO L L O 74.0 55.5H 37.0 18.5 FIG. 3. 10. 0.0 Actual flows UBC budget model Stochastic model Mean f low 528 c.f.s. ( Used for evaluating model parameters and Fourier coeff icients ) o 0 45 i 9 0 135 180 225 270 315 360 DAY Data from Blackwall Peak 6350 ft. / / / / / / / / / O G R A P H O F S N O W P A C K W A T E R E Q U I V A L E N T A G A I N S T M E A N A N N U A L F L O W S FIG. 3. 11. 100 200 300 400 500 600 700 800 M E A N ANNUAL F L O W S C F S . 900 1000 1100 111.4 SPECTRAL MODEL FOR THE SIMILKAMEEN WATERSHED AND THE ASSOCIATED SUB-ROUTINES The computer program f o r t h e above model i s e x p r e s s e d as: $RUN * WATFIV 3 = -P $ COMPILE REAL Q l (30,500), QGS ( 5 0 0 ) , CUMQG, A(40), B ( 4 0 ) , QT ( 5 0 0 ) , RES(500), QGSR(5,100) READ, ( A ( I ) , B ( I ) , I = 1,30) Z = RANDN (7.7) R E S ( l ) = 7.6 DO 7 1 = 1 , 365 SI = I J = I + 1 N = I 8 RES(J) = 0.562750 * RES(N) + FRANDN ( 0 . 0 ) * 182.843 IF (RES (J) . LT . 0.0) N = J IF (RES (J) . LT . 0.0) GO TO 8 7 CONTINUE CUMQG = 0.0 DO 4 L = 1, 365 SL = L QT(L) = 0.0 DO 5 K = 1, 30 Ql (K, L) = A(K) * COS ((44.0 * K * L)/2555.0) + B(K) * SIN((44.0 * K * L)/2555.0) QT(L) = QT(L) + Q l (K,L) 5 CONTINUE QGS (L) = 528. 290 + QT(L) CUMQG = CUMQG + QGS(L) 4 CONTINUE DO 9 M = 1, 365 SM = M QGSR(M) = QGS(M) + RES(M) WRITE (3,103) SM, QGSR(M) 103 FORMAT (2F10.0) 9 CONTINUE RETURN END 42 43 $DATA $STOP $RUN * FORTRAN ( V a l u e s o f Aj and B_. o f the F o u r i e r c o e f f i c i e n t s ) DIMENSION Q(500) READ (4,100) ( Q ( I ) , 1 = 1 , 365) 100 FORMAT (F 10.0) DO 2 1 = 1 , 365 SI = I WRITE (2,105) S I , Q(I) 105 FORMAT (2F10.0) 2 CONTINUE RETURN END $ENDFILE $RUN - LOAD# 4 = SIMILKMEEN70 2 = -S $RUN * FORTRAN DIMENSION QG(200) READ (2,101) ( Q G ( I ) , 1= 1,182) 101 FORMAT (F 10.0) DO 3 I = 1,182 SI = I WRITE (1,106) S I , QG(I) 106 FORMAT (2F10.0) 3 CONTINUE RETURN END $ENDFILE $RUN - LOAD # 2 = SIMILKGEN70 1 = -T 44 $RUN * FORTRAN DIMENSION X ( 1 1 0 0 ) , Y(1100) 4 N = 0 DO 3 I = 1,365 READ ( 2 , 101, END = 200) X ( I ) , Y ( I ) 3 N = I 200 IF (N.EQ.O) GO TO 300 CALL SUMFUN (X, Y, N) GO TO 4 300 CONTINUE 101 FORMAT (2F 10.0) 7 M = 0 DO 6 J = 1,365 READ (3, 102, END = 400) X ( J ) , Y ( J ) 6 M = J 400 IF (M.EQ.O) GO TO 500 CALL GENFUN (X, Y, M) GO TO 7 500 CONTINUE 102 FORMAT (2F 10.0) 9 L = 0 DO 8 K = 1, 182 READ ( 1 , 103, END = 600) X ( K ) , Y(K) 8 L = K 600 IF (L.EQ.O) GO TO 700 CALL UBCFUN (X, Y, L) GO TO 9 700 CONTINUE 103 FORMAT (2F 10.0) CALL PLOTND STOP END SUBROUTINE SUMFUN (X, Y, N) DIMENSION X ( N ) , Y(N) DO 5 I = 1,N X ( I ) = X ( I ) / 4 5 Y ( I ) = Y ( I ) / 1 8 5 0 45 5 CONTINUE CALL AXIS ( 0 . , 0., 'DAY', -3, 8., 0., 0., 45.) CALL AXIS ( 0 . , 0., 'FLOW CFS', 8, 6., 90., 0., 1850.) CALL LINE (X, Y, N, 1) CALL SYMBOL (2.0, 6.0, 0.5, 'SIMULATED HYDROGRAPH FOR SIMILKAMEEN WATERSHED 1970', 0., 51) CALL PLOT ( 0 . , 0., 3) RETURN END SUBROUTINE GENFUN (X, Y, M) DIMENSION X(M), Y(M) DO 2 J = 1, M X ( J ) = X ( J ) / 4 5 Y ( J ) = Y ( J ) / 1 8 5 0 2 CONTINUE CALL DASHLN (0.06, 0.06, 0.06, 0.06) DO 10 I = 1, M CALL PLOT ( X ( I ) , Y ( I ) ,4) 10 CONTINUE CALL PLOT ( 0 . , 0., 3) RETURN END SUBROUTINE UBCFUN (X, Y, L) DIMENSION X ( L ) , Y(K) DO 11 K = 1, L X(K) = 1.311 + X(K)/45 Y(K) = Y(K)/1850 11 CONTINUE CALL DASHLN (0.5, 0.05, 0.05, 0.05) CALL PLOT ( X ( l ) , Y ( l ) , 3 ) DO 12 J = 2, L CALL PLOT ( X ( J ) , Y ( J ) , 4) 12 CONTINUE RETURN END $ENDFILE $RUN - LOAD # 2 = -S 3 = -P 1 = -T $ENDFILE $RUN PLOT: Q $ENDFILE PAR = LINED CHAPTER IV RESULTS AND CONCLUSIONS IV.1 COMPARISON OF THE UBC BUDGET MODEL AND THE SPECTRAL MODEL D u r i n g t h e i n i t i a l i n v e s t i g a t i o n s w i t h t h e UBC Budget Model f o r t h e a p p r o p r i a t e model p a r a m e t e r s o f the Similkameen w a t e r s h e d f o r t h e y e a r s 1963, 1964 and 1970, c o n s i s t e n t p a r a -m e t r i c v a l u e s are o b t a i n e d f o r t h e r e s p e c t i v e y e a r s . These v a l u e s can be c o n s i d e r e d as r e a s o n a b l y good s i n c e the d a t a f o r the m onthly r e c o r d o f m e t e o r o l o g i c a l o b s e r v a t i o n s i s t a k e n from P r i n c e t o n , w h i c h i s a t the n o r t h e r n e x t r e m i t y o f t h e w a t e r s h e d . F u r t h e r the snowpack d a t a i s measured o n l y from one s t a t i o n a t B l a c k w a l l Peak (6350 f e e t ) , and the pan e v a p o r a t i o n d a t a i s t a k e n from Summerland and t h i s s t a t i o n i s not i n t h e v i c i n i t y o f the Sim i l k a m e e n w a t e r s h e d . W h i l e g e n e r a t i n g the s y n t h e t i c hydrographs w i t h the f i x e d model parameters o f the UBC Budget Model f o r the i n t e r -v e n i n g y e a r s , s u r p r i s i n g l y good f i t s f o r t h e sequence o f peaks are o b t a i n e d w i t h the a c t u a l h y d r o g r a p h s . Some e r r o r s o c c u r i n the c u m u l a t i v e volumes g e n e r a t e d by t h e UBC Budget Model. T h i s may be a t t r i b u t e d t o the snowpack d a t a a t B l a c k w a l l Peak, w h i c h may not be r e p r e s e n t a t i v e o f the a c t u a l snowpack d i s t r i b u t i o n over t h e e n t i r e w a t e r s h e d . The p r e c i p i t a t i o n d a t a f o r the UBC Budget Model i s t a k e n from P r i n c e t o n , and t h i s gauging s t a t i o n 47 i s not in the v i c i n i t y of Blackwall Peak. For the purpose of t h i s study the wintertnelt factor has been assumed to be a con-stant, and this l a r g e l y accounts for some errors in the cumula-ti v e volumes. In r e a l time operation the winter melt factor is evaluated by the feedback of the early season runoff response. More recent work has indicated that t h i s winter melt factor can be evaluated by running the model right through the previous year. The UBC Budget Model would function as a useful tool for the i n v e s t i g a t i o n of hydrologic processes i n research watersheds which are well instrumented for pan evaporation, snowpack and meteorological observations. The application of the s t a t i s t i c a l spectral model should not be judged from the basis of providing an analy-t i c a l s olution, but i n i t s a b i l i t y i n providing a sequence of synthetic traces, which a r e l i k e l y to occur during the l i f e of a p a r t i c u l a r project. There i s a f a i r degree of r e c o n c i l i a t i o n between the generated and the actual cumulative volumes for the synthetic hydrographs generated by the deterministic or periodic component of the spectral model. But the sequence of the recon-s t i t u t e d peaks is not comparable with that obtained from the UBC Budget Model. For the comparative study, the measures of central location and dispersion are evaluated from 2nd March to 30th August for the actual streamflow series and for the respective models. These parameters, shown in Table IV.1, provide a better perspective 48 TABLE IV.1 MEASURES OF CENTRAL LOCATION AND DISPERSION FROM 2nd MARCH t o 30th AUGUST SIMILKAMEEN WATERSHED (730 s q . m i l e s ) MEAN FLOW c f s STANDARD DEVIATION c f s YEAR FOR ACTUAL FLOWS UBC BUDGET MODEL STOCHASTIC SPECTRAL MODEL FOR ACTUAL FLOWS UBC BUDGET MODEL STOCHASTIC SPECTRAL MODEL 1961 1629 2121 1578 2319 2446 1391 1962 1128 1234 1300 1107 1210 1261 1963* 1195 1214 1352 1281 1221 1272 * 1964 1913 1916 2196 2332 2314 2256 1965 1315 1513 1348 1525 1615 1261 1966 944 1072 1152 890 1159 1370 1967 1645 2235 1667 2244 2582 1490 1968 1551 1822 1581 1595 1900 1391 1969 1157 1772 1244 1532 1949 1284 1970* 942 943 1123 1398 1290 1370 Used f o r e v a l u a t i n g F o u r i e r C o e f f i c i e n t s and UBC Model Parameters. 49 on the s i g n i f i c a n c e o f the f i t t e d models, i n a d d i t i o n t o t h e v i s u a l i n s p e c t i o n o f t h e s i m u l a t e d hydrographs as shown i n F i g u r e s 3.1 t o 3.10. The v a r i a n c e i s by f a r the most i m p o r t a n t measure o f s p r e a d . N o r m a l l y i t i s d e s i r a b l e t o supplement a statement o f the l o c a t i o n o f a d i s t r i b u t i o n on how c l o s e l y t h e d a t a i s c o n c e n t r a t e d about the mean. In F i g u r e s 4.1 and 4.2, the means and t h e s t a n d a r d d e v i a t i o n s are m o n o t o n i c a l l y p l o t t e d f o r t h e a c t u a l s t r e a m f l o w s e r i e s . The r e s p e c t i v e s t a t i s t i c a l p a r ameters f o r the UBC Budget Model and t h e s t o c h a s t i c s p e c t r a l model a r e s u b s e q u e n t l y p l o t t e d a g a i n s t t h e s e v a l u e s . The UBC Budget Model produces c o n s i s t e n t mea-s u r e s o f s p r e a d and p r e s e r v e s the v a l u e s o f the means f o r 1962, 1963, 1964, 1965, 1966, 1968 and 1970. There are some f l u c t u a t i o n s on the h i g h s i d e f o r the means o f 1961, 1967 and 1969. Fo r the g e n e r a t i o n o f t h e d a i l y s y n t h e t i c stream-f l o w sequences w i t h the s t o c h a s t i c s p e c t r a l model, u s i n g E q u a t i o n I I I . l , t he mean annual f l o w s f o r the i n t e r v e n i n g y e a r s are e s t i -mated from the gra p h i n F i g u r e 3.11 from the known v a l u e s o f the snowpack wa t e r e q u i v a l e n t o f B l a c k w a l l Peak. As shown i n F i g u r e 4.1 t h e r e a re some e r r o r s i n t h e v a l u e s o f t h e means from 2nd March t o 3 0 t h August f o r t h e s t o c h a s t i c s p e c t r a l model. I n F i g u r e 4.2 the s t a n d a r d d e v i a t i o n s t e n d t o f l u c t u a t e on the low s i d e f o r 1961 and 1967. T h i s i s e v i d e n c e d and c o n f i r m e d by t h e v i s u a l i n s p e c t i o n o f the s i m u l a t e d hydrographs i n F i g u r e s 3.1 and 3.7. F o r 1961 and 1967, the UBC Budget Model p r e s e r v e s the mea-sure o f d i s p e r s i o n , w h i l e the s t o c h a s t i c s p e c t r a l model f a i l s i n p r e s e r v i n g t h i s i m p o r t a n t s t a t i s t i c . 2800 1 • 2400 -2000 to <J 1600 H 3 1200 2 800-400-G R A P H OF M E A N F L O W S A G A I N S T Y E A R ( 2 nd. March - 30 th. August ) FIG. 4 . 1 . Actual mean flow UBC budget model Stochastic model Model parameters and Fourier coefficients are evaluated from 1963,1964, 1970 data. 0 1970 1966 — i 1 1 1— 1962 1969 1963 1965 YEAR 1968 1961 1967 1964 o 28001 2400 -co LL G R A P H O F S T A N D A R D D E V I A T I O N A G A I N S T Y E A R ( 2 nd. March - 30 th. August ) F I G . 4 . 2 . 2000-o 1600 -UJ a 1200 -a cr < a < i— LO 800 -400-Actual standard deviation UBC budget model Stochastic model Model parameters and Fourier coefficients are evaluated from .1963,1964,1970 data 0 1966 1962 1963 1970 1965 1969 YEAR 1968 196-7 1961 1964 52 IV.2 LIMITATIONS OF THE SPECTRAL MODEL Chow (3) a s s e r t s t h a t the b e h a v i o u r o f the h y d r o l o g i c phenomena changes w i t h t i m e a c c o r d i n g t o the law o f p r o b a b i l i t y as w e l l as w i t h the s e q u e n t i a l r e l a t i o n s h i p between the o c c u r r e n c e o f the phenomena. T h i s c o n f i r m s t h e s t o c h a s t i c n a t u r e o f t h e h y d r o l o g i c phenomena. But t h e r e i s some degree o f u n c e r t a i n t y i n s t o c h a s t i c models, and t h i s has r e s u l t e d i n the g r a d u a l accep-t a n c e o f t h e s e models by h y d r o l o g i s t s . S t o c h a s t i c models a r e employed i n p r e s e r v i n g c e r t a i n e s t i m a t e s o f the s t a t i s t i c a l p r o p e r t i e s o f a g i v e n r e c o r d , and t h e s e r e s u l t s are t h e n used f o r g e n e r a t i n g many e q u a l l y l i k e l y sequences. These s y n t h e t i c sequences c o u l d be used i n p l a c e o f the h i s t o r i c a l r e c o r d s , and a spectrum o f e q u a l l y l i k e l y o p t i m a l d e s i g n s c o u l d be d e v e l o p e d . With t h e i n t r o d u c t i o n o f computer t e c h n o l o g y , t e c h n i q u e s w h i c h have been l a r g e l y untapped due t o t h e voluminous c a l c u l a t i o n s a r e now b e i n g r a p i d l y e x p l o r e d . T h i s has e v i d e n t l y c o n t r i b u t e d t o the i n c r e a s i n g p o p u l a r i t y and use o f t h e s t a t i s t i c a l s t o c h a s t i c models. But a t the p r e s e n t l e v e l o f t h e s e s p e c t r a l a n a l y s i s t e c h n i q u e s , t h e s e models cannot be e a s i l y o r d i r e c t l y u s e d f o r s o l v i n g problems r e l a t e d t o water r e s o u r c e s economics and p l a n n i n g . IV.3 CONCLUSIONS The two approaches used f o r t h i s p a r t i c u l a r r e s e a r c h problem a r e q u i t e d i f f e r e n t and a r e n o t aimed a t p r o d u c i n g i d e n t i -c a l r e s u l t s . 53 The UBC Budget Model i s c l a s s i f i e d as a d e t e r m i n i s t i c or p a r a m e t r i c model and the l a n d phase o f the h y d r o l o g i c c y c l e i s b r o k e n i n t o s e v e r a l components. The major components are i n -f i l t r a t i o n , e v a p o t r a n s p i r a t i o n , a q u i f e r r e s p o n s e and t h e r o u t i n g o f t h e s t r e a m f l o w . E m p i r i c a l a p p r o x i m a t i o n s are used f o r d e f i n -i n g t h e p r o c e s s e s t h a t c o n t r o l t h e s e components. The main empha-s i s i s towards a b e t t e r u n d e r s t a n d i n g of t h e p h y s i c a l laws govern-i n g t h e components and i n the at t e m p t t o make t h e e m p i r i c a l model approximate more c l o s e l y t o t h e u n d e r l y i n g p h y s i c a l s i t u a t i o n . The UBC Budget Model has been d e v e l o p e d t o h e l p i n the s o l u t i o n o f v e r y p r a c t i c a l o p e r a t i o n a l p roblems. I t s u s e f u l n e s s has been de m o n s t r a t e d i n many d i f f e r e n t t y p e s o f s i t u a t i o n s . Some o f t h e a p p l i c a t i o n s o f p a r a m e t e r i c models a r e concerned w i t h t h e d e s i g n c a p a c i t i e s o f s m a l l r e s e r v o i r s , t h e d e t e r m i n a t i o n o f t h e e f f e c t o f c h a n n e l improvements upon t h e f l o o d f r e q u e n c y c h a r a c t e r i s t i c s o f a catchment, t h e e f f e c t o f u r b a n i z a t i o n on f l o o d p e aks, t h e e x t e n s i o n o f s t r e a m f l o w r e c o r d s f o r s m a l l b a s i n s on t h e b a s i s o f r a i n f a l l r e c o r d s as w e l l as f o r t h e s o l u t i o n o f problems p e r -t a i n i n g t o a q u i f e r r e s p o n s e . S t o c h a s t i c m o d e l l i n g i s concerned w i t h t h e s t a t i s t i c a l s i m u l a t i o n o f a measured r e s p o n s e o f a system. H a v i n g d e r i v e d the p a r a m e t e r s , the s t a t i s t i c a l model i s used f o r g e n e r a t i n g many e q u a l l y l i k e l y time s e r i e s . These a r e used i n the d e s i g n o f w ater r e s o u r c e s systems. S t o c h a s t i c models are w i d e l y used f o r systems a n a l y s i s and s y n t h e s i s , p a r t i c u l a r l y f o r r e s e r v o i r p l a n n i n g s t u d i e s and f o r t h e s i m u l a t i o n o f i n p u t s t o complex 5 4 systems. I t has w i d e s p r e a d use i n the a n a l y s i s o f the re s p o n s e o f complex systems. F o r t h i s p a r t i c u l a r r e s e a r c h p r o b l e m , t h e UBC Budget Model appears t o be more v e r s a t i l e s i n c e adequate m e t e o r o l o g i c a l and snowpack d a t a a r e a v a i l a b l e . G e n e r a l l y , i t i s a b l e to p r e -s e r v e t h e means and t h e s t a n d a r d d e v i a t i o n s o f t h e s t r e a m f l o w s e r i e s o f t h e r e l e v a n t y e a r s . The s t o c h a s t i c s p e c t r a l model may prove t o be a p p l i c a b l e t o water s h e d s w h i c h do not have ade-quate m e t e o r o l o g i c a l d a t a . As mentioned e a r l i e r , s p e c t r a l a n a l y s i s has been s u c c e s s f u l l y a p p l i e d t o c e r t a i n problems i n dynamics o f systems such as ea r t h q u a k e a n a l y s i s , e l e c t r i c a l and sound t h e o r y , and many ty p e s o f wave s t u d y . In a l l t h e s e s i t u a t i o n s t h e r e a r e c l e a r - c u t cause and e f f e c t r e l a t i o n s h i p s . The s t o c h a s t i c spec-t r a l model u s e d f o r t h i s a n a l y s i s does not t a k e i n t o c o n s i d e r -a t i o n the cause and e f f e c t r e l a t i o n s h i p f o r the d a i l y s t r e a m f l o w sequences. I t t h e r e f o r e appears t h a t as c u r r e n t l y p r e s e n t e d the s p e c t r a l a n a l y s i s t e c h n i q u e s do not go v e r y f a r i n t h e s o l u -t i o n o f problems p e r t a i n i n g to w a t e r r e s o u r c e s economics and p l a n n i n g . Perhaps w i t h f u r t h e r r e s e a r c h and r e c o g n i z i n g t h e cause and e f f e c t r e l a t i o n s h i p , the t e c h n i q u e s may be c a p a b l e o f u s e f u l development, b u t t h i s i s u n c e r t a i n . M a t h e m a t i c a l models a r e p l a y i n g a v i t a l r o l e i n hydro-l o g y , and f u t u r e p r o g r e s s w o u l d i n e v i t a b l y depend on the d e v e l o p -ment and i n n o v a t i o n o f t h e s e t e c h n i q u e s . Dawdy ( 4 ) a s s e r t s t h a t the d i v i s i o n o f m a t h e m a t i c a l m o d e l l i n g i n t o s t o c h a s t i c , p a r a m e t r i c 55 and systems s t u d i e s i s q u i t e a r b i t r a r y , and i n r e a l i t y a l l approaches b l e n d t o g e t h e r . Many problems i n h y d r o l o g y r e q u i r e computers i n o r d e r t o handle l a r g e q u a n t i t i e s o f d a t a and t o r e a l i z e any l e v e l o f a c c u r a c y . The t e c h n i q u e used f o r a r r i v -i n g a t a s o l u t i o n o f a p a r t i c u l a r problem may prove t o be more i m p o r t a n t t h a n the type o f m a t h e m a t i c a l model used. T h i s em-p h a s i z e s the urgency and need o f b e i n g aware o f m a t h e m a t i c a l advances i n o r d e r t o keep a b r e a s t o f t h e l a t e s t t e c h n i q u e s u s e d by modern h y d r o l o g i s t s f o r t h e s o l u t i o n o f problems. R E F E R E N C E S B r i t i s h Columbia Snow Survey Bulletin, Water I n v e s t i -g a t i o n s B ranch, Water Resources S e r v i c e , Department o f Lands, F o r e s t s and Water R e s o u r c e s , V i c t o r i a , B.C. C a r l s o n , R. F., A. J . A. McCormack and D. G. Watt s . " A p p l i c a t i o n o f L i n e a r Random Models t o Four A n n u a l S t r e a m f l o w S e r i e s , " Water Resources Research, V o l . 6, No. 4 (1970), pp. 113-130. Chow, V. T., and S. J . K a r e l i o t i s . " A n a l y s i s o f S t o -c h a s t i c H y d r o l o g i c Systems," Water Resources Research, V o l . 6, No. 6 (1970), pp. 1569-1582. Dawdy, D. R., and G. I . K a l i n i n . " M a t h e m a t i c a l Model-l i n g i n H y d r o l o g y , " Bulletin o f the I n t e r n a t i o n a l Asso-c i a t i o n o f S c i e n t i f i c H y d r o l o g y (December, 1971), pp. 25-30. F i e r i n g , M. B. , and B. B. J a c k s o n . Synthetic Streamflows:, A m e r i c a n G e o p h y s i c a l U n i o n (1971). H o l l i s , G. E., and L. F. C u r t i s . "The Use o f A e r i a l Photography i n H y d r o l o g y , " Proceedings, The I n s t i t u t i o n of C i v i l E n g i n e e r s , P a r t 2, Re s e a r c h and Theory (December, 1972), pp. 679-680. K a r e l i o t i s , S. J . , and V. T. Chow. " A n a l y s i s o f R e s i d -u a l H y d r o l o g i c S t o c h a s t i c P r o c e s s e s , " Journal of Hydro-logy ( F e b r u a r y , 1972), pp. 113-130. Monthly Record of Meteorological Observations, Canada. P i p e s , A. An Analysis of the Carrs Landing Watershed, Department o f Lands, F o r e s t s and Water R e s o u r c e s , V i c t o r i a , B.C., 1971. 5 6 57 (10) Q u i c k , M.C, and A. P i p e s . daily and Seasonal Runoff Forecasting With A Water Budget Model. I n t e r n a t i o n a l Symposia on the R o l e o f Snow and Ice i n H y d r o l o g y , B a n f f , A l b e r t a , 1972, pp. 1-9. (11) Quimpo, R. G. " A u t o c o r r e l a t i o n and S p e c t r a l A n a l y s i s i n H y d r o l o g y , " Journal of the Hydraulics Division, ASCE (March, 1968), pp. 363-371. (12) Quimpo, R. G. " S t o c h a s t i c A n a l y s i s o f D a i l y R i v e r F l o w s , " Journal of the Hydraulics Division, ASCE ( J a n -u a r y , 1968), pp. 43-57. (13) R o d r i g u e z - I t u r b e , I . , D. R. Dawdy and L. E. G a r c i a . "Adequacy o f M a r k o v i a n Models w i t h C y c l i c Components f o r S t o c h a s t i c S t r e a m f l o w S i m u l a t i o n , " Water Resources Research, V o l . 7, No. 5 (1 9 7 1 ) , pp. 1127-1143. (14) Spillway Design Floods for the Fraser River Basin, E n g i n e e r i n g D i v i s i o n , Water P l a n n i n g and Management Br a n c h , Environment Canada, Ottawa, 1972, pp. 1-29. (15) Streamflow Forecasting, Water Survey o f Canada, Atmos-p h e r i c Environment S e r v i c e , C a l g a r y , A l b e r t a , March, 1972, pp. 1-202.. (16) Surface Water Data, British Columbia. Water Survey o f Canada, I n l a n d Waters B r a n c h , Department o f Energy, Mines and R e s o u r c e s , Canada. (17) S u t c l i f f e , R. C. Weather and Climate. B r i t a i n : Cox and Wyman L t d . , 1966, pp. 111-193. (18) Toebes, C , and V. Ouryaev. Representative and Ex-perimental Basins. UNESCO, 1970, pp. 1-348. (19) UNESCO. The Use of Analog and Digital Computers in Hydrology. IASH/AIHS ( 1 9 6 9 ) , pp. 349-754. (20) Water Resources Paper. Surface Water Data for Pacific Drainage. Department o f Energy, Mines and R e s o u r c e s , Water R e s o u r c e s Branch. 58 Y e v j e v i c h , V. Stochastic Processes in Hydrology. Water Resources P u b l i c a t i o n s , C o l o r a d o , 1972, pp. 1-125. Y e v j e v i c h , V. Probability and Statistics in Hydrology. Water Resources P u b l i c a t i o n s , C o l o r a d o , 1972, pp. 1-285. APPENDIX 1 LIST OF VARIABLES FOR THE SPECTRAL MODEL AND THE CALCOMP PLOT A(K) B(K) CUMQG FRANDN Ql (K,L) Q(I) QG(I) QGS(L) QGSR(M) QT(L) RES(J) A j v a l u e s o f t h e F o u r i e r c o e f f i c i e n t s B. v a l u e s o f t h e F o u r i e r c o e f f i c i e n t s C u m u l a t i v e volumes o f t h e g e n e r a t e d s t r e a m f l o w sequences c f s days Random number g e n e r a t o r V a l u e o f t h e p e r i o d i c component f o r each subharmonic o f t h e a r r a y c f s A c t u a l s t r e a m f l o w s e r i e s c f s D a i l y s t r e a m f l o w sequences g e n e r a t e d by the UBC Budget Model c f s P e r i o d i c o r d e t e r m i n i s t i c component f o r the d a i l y s t r e a m f l o w sequences g e n e r a t e d by t h e s p e c t r a l model c f s D a i l y s y n t h e t i c s t r e a m f l o w sequences g e n e r a t e d by the s t o c h a s t i c s p e c t r a l model c f s C u m u l a t i v e v a l u e s o f t h e subharmonics o f the p e r i o d i c component c f s S t o c h a s t i c component o f the d a i l y s t r e a m f l o w sequences g e n e r a t e d by t h e random number g e n e r a t o r FRANDN c f s 5 9 

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