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Hydraulic geometry of Green and Birkenhead rivers: Southwestern Coast Mountains, British Columbia Ponton, John Robert 1972

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HYDRAULIC GEOMETRY OF GREEN AND BIRKENHEAD RIVERS: SOUTHWESTERN COAST MOUNTAINS, BRITISH COLUMBIA by John R. P o n t o n , J r . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. John R. Ponton, J r . Department of Geography The University of Brit ish Columbia Vancouver 8, Canada Date 23 A p r i l 1972 i ABSTRACT Green and Birkenhead R i v e r s a r e l o c a t e d i n the s o u t h w e s t e r n C o a s t Mountains of British Columbia, and the d r a i n a g e i n b o t h b a s i n s i s s t i l l s t r o n g l y c o n t r o l l e d by glacial f e a t u r e s l e f t a f t e r t h e r e t r e a t o f the Vashon i c e s h e e t . R i v e r s l o p e s a r e imposed on the u p l a n d streams w h i l e the s l o p e o f the main v a l l e y streams i s a t l e a s t p a r t l y imposed by the g l a c i a l t o p o g r a p h y . D i s c h a r g e i n the streams i s dominated by snowmelt d u r i n g t he summer though peak d a i l y d i s c h a r g e s f r e q u e n t l y o c c u r i n a u t -umn d u r i n g autumn s t o r m s . A t - a - s t a t i o n h y d r a u l i c geometry c u r v e s were d e t e r m i n e d by l e a s t s q uare r e g r e s s i o n a n a l y s i s f o r f i v e s e c t i o n s from Water Survey o f Canada g a u g i n g r e c o r d s . V e l o c i t y shows a more r a p i d t h a n u s u a l r a t e o f a d j u s t -ment, and r e s i s t a n c e d e c r e a s e s more r a p i d l y t h a n the average as d i s c h a r g e i n c r e a s e s . R e s i d u a l v a l u e s appear t o be d i s t r i b u t e d about t h e r e g r e s s i o n l i n e s i n a s y s t e m a t i c manner s u g g e s t i n g t h a t t h e c h a n n e l form f l u c t u a t e s s y s t e m a t i c a l l y o v e r t i m e . S i m i l a r r e s u l t s were found f o r t e n o t h e r s e c t i o n s i n the s o u t h w e s t e r n C o a s t M o u n t a i n s . Downstream h y d r a u l i c g e o m e t r i e s were d e t e r m i n e d f o r Green R i v e r and B i r k e n h e a d R i v e r . B a n k f u l l d i s c h a r g e was assumed t o have a c o n s t a n t r e c u r r e n c e i n t e r v a l o f 2.33 y e a r s f o r b o t h b a s i n s . Channel w i d t h shows a g r e a t e r t h an u s u a l i n c r e a s e i n the downstream d i r e c t i o n w h i l e v e l o c i t y appears t o remain c o n s t a n t o r d e c r e a s e . i i TABLE OF CONTENTS Page C h a p t e r 1 R i v e r B a s i n C h a r a c t e r i s t i c s 1-1 I n t r o d u c t i o n . . . . x 1-2 Study A r e a 2 1-3 Geology 4 1-4 Morphometry 1 ° 1-5 P r e c i p i t a t i o n ^ 1- 6 D i s c h a r g e 2 0 C h a p t e r 2 Data C o l l e c t i o n and F i e l d T e c h n i q u e s 2- 1 I n t r o d u c t i o n 2 4 2-2 S e c t i o n a l Data C o l l e c t i o n 2 5 2-3 Downstream Data C o l l e c t i o n 2 7 2-4 Data A n a l y s i s 3 2 2- 5 R e g r e s s i o n A n a l y s i s 3 3 C h a p t e r 3 A t - A - S t a t i o n H y d r a u l i c Geometry 3- 1 H y d r a u l i c Geometry E q u a t i o n s 38 3-2 R e s i s t a n c e P r o p e r t i e s 4 3 3-3 Data V a r i a n c e 4 6 3-4 A d d i t i o n a l C o a s t Mountain Streams. 50 C h a p t e r 4 Downstream H y d r a u l i c Geometry 4-1 Flow Frequency and B a n k f u l l D i s c h a r g e 5 6 4-2 H y d r a u l i c Geometry E q u a t i o n s 58 4-3 Data V a r i a n c e 6 4 i i i C h a p t e r 5 Q u a s i - e q u i l i b r i u m i n C o a s t Mountain Streams 5-1 I n t r o d u c t i o n . 67 5-2 Q u a s i - e q u i l i b r i u m and Coast M o u n t a i n Streams 67 Photographs 70-74 B i b l i o g r a p h y 75-77 i v LIST OF TABLES Page Table I Summary of Morphometric Data H Table II Mean Monthly P r e c i p i t a t i o n (mm) and Temperature (°C) at Three Selected Stations near the Study Area 17 Table I I I Mean Monthly Discharge per Square Kilometer 18 Table IV A Comparison of Hydraulic Geometry Equations Derived from Lines F i t t e d by Hand and by Least Square Regression. 3 4 Table V At-A-Station Hydraulic Geometry 39 Table VI At-A-Station Hydraulic Geometry of A d d i t i o n a l Coast Mountain Streams 40 Table VII Comparison of the Exponents of At-A-Station Hydraulic Geometries 41 Table VIII Downstream Hydraulic Geometry 5 9 Table IX Comparison of the Exponents of Downstream Hydraulic Geometries 60 V LIST OF FIGURES Page Figure 1 Location of Study Area 3 Figure 2 Green River Basin 5 Figure 3 Birkenhead River Basin 6 Figure 4 Geology of the Green and Birkenhead Basins 7 Figure 5 Longitudinal P r o f i l e s 9 Figure 6 Number of Streams to Stream Order 13 Figure 7 Mean Stream Length to Stream Order v 14 Figure 8 Morphometry 16 Figure 9 Mean D a i l y Discharge a. Green and Birkenhead 19 b. Soo, Rutherford, and Green at Green Lake 21 Figure 10 Flood Frequency 22 Figure 11 Q 2 3 3 to Drainage Area 31 Figure 12 At-A-Station Hydraulic Geometry 42 Figure 13 Residuals of Sectional Area and Width, Green River at Green Lake 48 Figure 14 Residuals of Sectional Width, Birkenhead River 49 Figure 15 At-A-Station Hydraulic Geometry of Ten Streams i n Southwestern B r i t i s h Columbia a. S e c t i o n a l Width 51 b. Mean Sectional Depth 52 c. Mean Sectional V e l o c i t y > 53 d. Sectional Area 54 Figure 16 Downstream Hydraulic Geometry . 61 v i L I S T OF PHOTOGRAPHS Page Photograph 1 Soo R i v e r , l o o k i n g upstream n e a r g a u g i n g s e c t i o n 70 Photograph 2 R u t h e r f o r d C r e e k , l o o k i n g upstream n e a r g a u g i n g s e c t i o n "70 Photograph 3 B i r k e n h e a d R i v e r , l o o k i n g upstream a t ga u g i n g s e c t i o n and downstream s e c t i o n d u r i n g n e a r b a n k f u l l d i s c h a r g e 71 Photograph 4 B i r k e n h e a d Lake R i v e r between B i r k e n h e a d Lake and c o n f l u e n c e w i t h B i r k e n h e a d R i v e r , l o o k i n g upstream a t downstream s e c t i o n 71 Photograph 5 B i r k e n h e a d R i v e r above c o n f l u e n c e w i t h Tenas C r e e k , l o o k i n g downstream a t downstream s e c t i o n 72 Photograph 6 Owl C r e e k , l o o k i n g upstream a t downstream s e c t i o n 72 Photograph 7 Tenas C r e e k , l o o k i n g upstream a t downstream s e c t i o n . . 73 Photograph 8 Upper P o o l e Creek s e c t i o n , l o o k i n g upstream a t downstream s e c t i o n 73 Photograph 9 Green R i v e r above Green L a k e , l o o k i n g downstream a t downstream s e c t i o n 74 Photograph 10 21 M i l e C r e e k , l o o k i n g downstream a t downstream s e c t i o n 74 ACKNOWLEDGEMENT Many people a s s i s t e d with the c o l l e c t i o n of the data and gave h e l p f u l advice during the work on t h i s t h e s i s . Though they are too numerous to thank i n d i v i d u a l l y , t h i s work would not have been f i n i s h e d without t h e i r help. Thanks are due e s p e c i a l l y to Miss Sharon Kennedy who a s s i s t e d i n the c o l l e c t i o n o f most of the data and Mr. Robert G i l b e r t who k i n d l y gave advice, h i s time, and some of h i s data. Mr. 0. Nagy of the Water Survey of Canada was most h e l p f u l i n g i v i n g the gauging s t a t i o n data. The comments and suggestions of my advisor, Dr. H.O. Slaymaker, and of Dr. M.A. Church were gr e a t l y appreciated. F i e l d work was funded through the U.B.C. Committee on Research Funds. v i i i SYMBOLS USED IN TEXT A Channel c r o s s s e c t i o n a l a r e a a b C o e f f i c i e n t i n r e l a t i o n w=aQ b b Exponent i n r e l a t i o n w=aQ c C o e f f i c i e n t i n r e l a t i o n d=cQ^ D 84 G r a i n d i a m e t e r e q u a l t o o r l a r g e r p a r t i c l e s t h a n 84% o f t h e bed d Mean d e p t h , A/w e E r r o r t erm o f r e g r e s s i o n e q u a t i o n f Exponent i n r e l a t i o n d=cQf f Darcy-Weisbach r e s i s t a n c e f a c t o r g C o e f f i c i e n t i n r e l a t i o n A=gQ h Exponent i n r e l a t i o n A=gQ k C o e f f i c i e n t i n r e l a t i o n v=-kQm M Rank o f each y e a r l y peak f l o o d m Exponent i n r e l a t i o n v=kQm < N Number o f y e a r s o f r e c o r d i n f l o o d f r e q u e n c y a n a l y s i s n Number o f o b s e r v a t i o n s Q Stream d i s c h a r g e 22.33 Stream d i s c h a r g e w i t h a 2.33 y e a r r e c u r r e n c e i n t e r v a l R H y d r a u l i c mean depth 2 R C o e f f i c i e n t o f d e t e r m i n a t i o n from r e g r e s s i o n a n a l y s i s s Water s u r f a c e s l o p e t 2 C o e f f i c i e n t i n r e l a t i o n s=tQ V Mean v e l o c i t y w Water s u r f a c e w i d t h Exponent i n r e l a t i o n f f * Q z Exponent i n r e l a t i o n s=tQ I n t e r c e p t o f r e g r e s s i o n e q u a t i o n S l o p e o f r e g r e s s i o n e q u a t i o n V a r i a n c e Chapter 1 RIVER BASIN CHARACTERISTICS 1-1 INTRODUCTION A l l natural streams are subject to a wide range of discharges. Since each channel cross section must transmit the water passed i n t o i t from upstream, the hydraulic c h a r a c t e r i s t i c s of the channel must adjust to accommodate each new discharge. The discharge i s essent-i a l l y independent of the stream s e c t i o n while the remaining h y d r a u l i c parameters, sediment s i z e and concentration, width, depth, v e l o c i t y , slope, and roughness are dependent, to varying degrees, on the d i s -charge . The mutual adjustment of these parameters has been termed the hydraulic geometry of stream channels (Leopold and Maddock, 1953). Though t h i s term was f i r s t used by Leopold and Maddock i n 1953, s i m i l a r studies had been c a r r i e d out f o r many years before by engineers i n v e s t -i g a t i n g i r r i g a t i o n canals (cf. f o r example Kennedy, 1895; Lacey, 1929; I n g l i s , 1949). The objective of these i n v e s t i g a t i o n s was to design a canal i n a l l u v i a l material which was s t a b l e ; one i n which there was no s i g n i f i c a n t scour or deposition f o r the l i m i t e d range of flows the canal c a r r i e d . Such a canal was s a i d to be " i n regime". Leopold and Maddock extended the canal i n v e s t i g a t i o n s of channel form to natural streams i n the midwestern United States. Since t h e i r o r i g i n a l paper, other h y d r a u l i c geometry studies have been c a r r i e d out i n a wide v a r i e t y of r i v e r environments. These studies have found that, 2 though streams carry a wider range of flows than canals, many streams are i n regime or i n a state of" "quasi-equilibrium". Hydraulic geometry studies concentrate on the mean response of the hydraulic parameters of a stream channel as r e f l e c t e d i n the channel form. The processes which lead to the varying channel forms embody com-plex e f f e c t s which are not adequately r e f l e c t e d i n the mean channel r e -sponse. The analysis of the stream's hydraulic geometry makes no attempt to enumerate these various processes. However, by looking at the mean response of the channel, some general statements about the channel form-ing processes can be made. The mean c h a r a c t e r i s t i c s remain of some s c i e n t i f i c and a great deal of p r a c t i c a l i n t e r e s t . This t h e s i s w i l l examine the hydraulic geometry of two streams i n the Coast Mountains of southwestern B r i t i s h Columbia. Except f o r the work by Day (1970), the hydraulic geometry of streams i n s i m i l a r high energy environments has been neglected. 1-2 STUDY AREA The two r i v e r basins chosen f o r t h i s i n v e s t i g a t i o n were Green River basin and Birkenhead River basin. Both basins are located near Pember-ton, B r i t i s h Columbia, approximately 145 km north of Vancouver, B r i t i s h Columbia ( f i g . 1). Birkenhead River drains the region to the north of Pemberton and flows south i n t o L i l l o o e t Lake, 15 km east of Pemberton. Green River drains the area to the southwest of Pemberton and flows northward r e c e i v i n g two major t r i b u t a r i e s from the west, Soo River and Rutherford Creek, before i t joins L i l l o o e t River 5 km east of Pemberton. These two basins were chosen because of t h e i r a c c e s s i b i l i t y by comparison with the majority of basins i n the Coast Mountains. They also c o n s t i t u t e part of the l a r g e r L i l l o o e t River basin where work on Figure 1, Location of Study Area Scale r. 1,000,000 GAUGING STATIONS USED FOR AT-A-STATION HYDRAULIC GEOMETRIES 1 BIRKENHEAD RIVER 6 CHEAKAMUS RIVER 2 GREEN RIVER 7 SENTINEL CREEK 3 RUTHERFORD CREEK 8 MASH ITER CREEK 4 SOO RIVER 9 MAMQUAM RIVER 5 GREEN RIVER AT GREEN LAKE 10 CAPILANO RIVER 11 MOSQUITO CREEK 12 SEYMOUR RIVER 13 NORTH ALOUETTE RIVER 14 JACOBS CREEK 15 KANAKA CREEK ISO-DISCHARGE CONTOURS: MEAN ANNUAL DISCHARGE xlO 2(m3/s)/km2 SourceTopographic. NTS. 1:1.000.000 map sheet NM -9/ lO Iso-discharge Contours after Slaymaker 4 sediment s o u r c e s and f l u v i a l s e d i m e n t a t i o n i s b e i n g c a r r i e d o u t ( G i l b e r t , 1972; Slaymaker and G i l b e r t , 1972). D i s c h a r g e d a t a were a l s o a v a i l a b l e f o r b o t h s t r e a m s . Four g a u g i n g s t a t i o n s have been l o c a t e d i n the Green b a s i n a t v a r i o u s t i m e s ( f i g . 2 ) . These were e s t a b l i s h e d i n the mid 1910's and abandoned i n t h e l a t e 1940's. The s t a t i o n on B i r k e n h e a d R i v e r was e s t a b l i s h e d i n 1945 and was i n o p e r a t i o n u n t i l 1971 ( f i g . 3 ) . In a d d i t i o n t o t h e d a t a c o l l e c t e d i n t h e main s t u d y a r e a , d a t a from t e n o t h e r g a u g i n g s t a t i o n s i n the s o u t h w e s t e r n C o a s t Mountains were c o l l e c t e d i n o r d e r t o examine r e g i o n a l s i m i l a r i t i e s i n s e c t i o n a l geometry ( f i g . 1 ) . 1-3 GEOLOGY L i t t l e d e t a i l e d g e o l o g i c work has been done i n e i t h e r b a s i n . From an a e r i a l r e c o n n a i s s a n c e o f t h e a r e a w i t h o c c a s i o n a l s a m p l i n g o f s u r f a c e e x p o s u r e s , Mathews ( p e r s o n a l communication) mapped t he b e d r o c k o f the r e g i o n : f i g . 4 p r e s e n t s a s i m p l i f i c a t i o n o f h i s map. A s m a l l a r e a o f T e r t i a r y and Q u a t e r n a r y v o l c a n i c s i s found i n t h e B i r k e n h e a d b a s i n w i t h the r e s t o f the bed r o c k i n the b a s i n b e i n g d i v i d e d about e q u a l l y between the two major g r o u p s . Only about one t h i r d o f the Green b a s i n i s under-l a i n by t he m e t a v o l c a n i c and metasedimentary r o c k s , and none o f t h e more r e c e n t v o l c a n i c s a r e f o u n d . The major g l a c i a l v a l l e y s i n b o t h b a s i n s a r e c o v e r e d w i t h Q u a t e r n a r y u n c o n s o l i d a t e d m a t e r i a l s . The bedrock g e o l o g y would appear t o have o n l y a minor i n f l u e n c e on the streams w i t h i n t h e r e g i o n . Some r e a c h e s o f the streams appear t o flo w a l o n g c o n t a c t s between t h e two major rock t y p e s , and a s h o r t s e c t i o n o f the B i r k e n h e a d b a s i n d i v i d e f o l l o w s a c o n t a c t ; however, most streams appear u n a f f e c t e d by changes i n b e d r o c k . There a r e no marked changes i n c h a n n e l p a t t e r n as streams c r o s s c o n t a c t s ; t h e d r a i n a g e d e n s i t y does n o t appear t o change as the r o c k type changes; and s t r e a m Figure 2 Green River Basin Scale 1:250,000 Contour Interval 500 m Gauging Sections Downstream Sections Permanent Ice and Snowfields SourcerN.T. S. 1:250,000 map sheet 92J 6 Figure 3 Birkenhead River Basin Scale 1:250,000 Contour Interval 500m Figure 4 Geology of the Green and Birkenhead Basins Scale 1:400,000 Source:Slaymaker after Mathews 8 patterns are not obviously related to the structure of the area. Glaciation has left the most marked imprint on the character of the drainage within each basin* Mathews (1958) , working in the Mount Garibaldi area 15 km to the southwest of the Green basin, found that the maximum height of the Vashon glaciation was approximately 2100 m. The Vashon Stade began 25,000 B.P. and the Coast Mountains appear to have been continuously glaciated until the final retreat 10,000 B.P. (Ryder, 1 9 7 2 ) . Only traces of this last regional glaciation are found, though well developed cirques found below the maximum height of glaci-ation may pre- or post-date the Vashon maximum. In the lower reaches of the four major streams in the Green and Birkenhead basins, Green, Soo, Rutherford, and Birkenhead, water falls are found as well as canyon like sections. These features are an in-direct result of glaciation and are also found on many of the smaller tributaries. Hanging valleys are another feature common to the region. The effect of glaciation on the longitudinal profiles of the major streams is seen in fig. 5. Soo, Green, and Birkenhead a l l have low slopes at approximately 650 and 850 m with a sharp increase in slope below this level. Lower slopes along Green and Birkenhead Rivers also appear at approximately 500 m. Mathews (1958) noted a similar stepped profile for Cheakamus and Monmouth valleys. The sharp breaks between the relatively smooth sections of the profiles do not appear to be re-lated to changes in bedrock. The sharp decrease in slope at the mouth of both Green and Birkenhead Rivers is a result of the post-glacial valley f i l l of Lillooet River. The other breaks in slope may be a result of base level control on the depth of erosion of the tributary glaciers by the main valley glaciers, but this is not clear. Whatever Figure 5 10 the p r e c i s e o r i g i n , i t i s c l e a r that the r i v e r s do not c o n t r o l the gradient of t h e i r long p r o f i l e s . A s i m i l a r stepped pattern can also be seen i n the hypsometric i n t e g r a l s ( f i g . 8). Drainage d i v i d e s at the heads of the major g l a c i a l v a l l e y s i n both basins are poorly defined. A swamp at the southwestern end of A l t a Lake acts as the divide at the head of the main g l a c i a l v a l l e y i n the Green basin. Place Creek, which enters the main g l a c i a l v a l l e y of the Birkenhead basin very near the d i v i d e , appears to have flowed i n t o the adjoining basin at one time: abandoned channels flowing towards Gates Lake i n the adjoining basin are s t i l l apparent. Though f l u v i a l modifications have taken place, the g l a c i a l topography s t i l l dominates the streams wi t h i n both basins. 1-4 MORPHOMETRY Following the techniques o u t l i n e d by St r a h l e r (1952, 1956, and 1957), stream order and lengths i n each basin were recorded (Table I ) . Basin area, area covered by permanent i c e and snowfields, and basin perimeter were measured. A random sample of 110 points w i t h i n the Birkenhead basin and 150 points within the Green was taken, and at each point e l e v a t i o n and aspect were recorded. The 150 points within the Green basin were di v i d e d proportionately among the three subbasins f o r which there were also discharge records. In the Soo bas i n , 50 random points were sampled; 38 points were sampled i n the Rutherford and Green at Green Lake basins and 24 i n the remaining area. A l l o f the data were c o l l e c t e d from the N.T.S. 1:250,000 topo-graphic map with a contour i n t e r v a l of 500 f t . This i s the only map which covers the e n t i r e area; however, drainage on t h i s map i s shown i n considerably more d e t a i l than on the 1:50,000 p r o v i s i o n a l N.T.S. TABLE I SUMMARY OF MORPHOMETRIC DATA Birk e n h e a d Green Soo R u t h e r f o r d Green a t Green Lake B a s i n a r e a , km 2 2 656 841 268 180 171 G l a c i a l a r e a , km 4.8 70.8 31.7 28.0 5.4 P e r c e n t g l a c i e r i z e d 0.7 8.4 11.8 15.5 3.2 P e r i m e t e r , km 115 155 91 73 68 Number o f o r d e r streams 1 380 245 69 83 35 2 85 58 16 20 10 3 24 8 3 2 2 4 4 3 1 1 1 5 1 1 Mean st r e a m o r d e r l e n g t h , km 1 0.92 1.21 1.45 0.96 1.43 2 1.65 2.13 1.72 1.81 2.11 3 3.02 6.56 5.83 5.00 11.25 4 8.28 19.38 28.13 16.25 5 31.87 8.34 Geometric mean b i f u r c a t i o n r a t i o 4.42 3.96 4.10 4.36 3.27 Drainage d e n s i t y , km/km * 0.96 0.70 0.73 0.94 0.57 C i r c u l a r i t y r a t i o 0.623 0.440 0.407 0.424 0.465 R e l i e f , m 2305 2680 2255 2286 1990 Mean e l e v a t i o n , m 1390 1385 1387 1442 1420 Hypsometric i n t e g r a l 0.522 0.466 0.477 0.538 0.391 T o t a l l e n g t h o f s t r e a m s / u n g l a c i e r i z e d a r e a . 12 maps. G i l b e r t (personal communication) was able to determine from sampling a i r photos and some ground reconnaissance that f i r s t order streams on the map were probably no greater than second order. I f t h i s i s the case, the unusually low drainage density (Table I) suggests that much of the runoff occurs as i n t e r f l o w or i n unconcentrated flow over bedrock. The high cover of permanent i c e and snowfields also influences the drainage density, but the data quoted i n Table I applies only to areas of the basin not covered by permanent i c e and snowfields. The area of permanent i c e and snowfields i n the Birkenhead basin i s approx-imately 3.8 sq. km greater on the p r o v i s i o n a l maps than on the N.T.S. topographic map. Though t h i s i s a.large percentage increase, the a c t u a l increase i s small and i t l i k e l y i n d i c a t e s the order of magnitude by which the area of permanent i c e and snowfields i n the other basins would increase with l a r g e r scale maps. The Birkenhead system seems to follow the laws of stream order and stream length reasonably c l o s e l y ( f i g s . 6, 7); however, Green River and i t s sub-basins do not appear to follow these r e l a t i o n s as c l o s e l y . This i s l i k e l y a r e s u l t of the l a r g e r g l a c i a l area. A stream coming from a g l a c i e r can only be counted as a f i r s t order stream though i t may be d r a i n i n g an area which could support a second or t h i r d order stream. Though the geology of the Birkenhead basin i s more var i e d , there does not appear to be a change i n drainage density between the two major rock types (sec. 1-2). The geometric mean b i f u r c a t i o n r a t i o s are a l l close to the random value of four predicted by Shreve (1966) and also tend to i n d i c a t e a lack of strong s t r u c t u r a l c o n t r o l . Though the maximum r e l i e f i n the Birkenhead basin i s lower than i n the Green basin, i t s mean el e v a t i o n i s s l i g h t l y greater and the hypso-13 5 0 0 Figure 6 Number of Streams to Stream Order 100 h o XI E 10 1 + X 1 2 3 4 5 Stream Order • Birkenhead + Rutherford o Green at Green Lake *• Green x Soo 14 Figure 7 Mean Stream Length to Stream Order 100 + Rutherford o Green at Green Lake * Green x Soo 15 metric integral is larger. The large difference in maximum relief in the two basins is due to Wedge Mountain which, at 2900 m, is consider-ably higher than other peaks in the area. This high point also tends to make the hypsometric integral slightly atypical, but the lower in-tegral and lower mean elevation of the Green do reflect the larger area covered by lower glacial valleys. Slopes with a southern aspect are more frequent in the Birkenhead basin while northern slopes are dominant in the Green (fig. 8 ) . This leads to a more rapid snow melt in the Birkenhead basin (Table III, fig. 9 a ) . The more nearly circular shape of the Birkenhead may allow a more rapid concentration of storm and meltwater runoff. 1.5 PRECIPITATION Because of the mountainous terrain of the two basins, precipitation can be expected to vary with elevation and exposure throughout each basin. A general trend of decreasing precipitation from western to eastern borders of the basins should also be expected due to the domi-nant west to east storm path and the orographic effect of the mountains. Only one weather station, Alta Lake, is located in the study area, but two others have been in operation nearby (fig. 1 ) . There is a trend for a decrease in precipitation from west to east (Table II). However, the data are so limited that the whole area must be regarded as hydrologi-cally homogeneous for discharge purposes (c.f. ch. 4 ) . Estimated mean annual precipitation in the Green basin ranges from 750 mm to a high in the mountains which may exceed 3750 mm while the maximum precipitation in the Birkenhead basin probably does not exceed 2500 mm. The majority of the precipitation falls between October and February as snow (Table II). Relative Height 91 TABLE I I MEAN MONTHLY PRECIPITATION (mm) AND TEMPERATURE (°C) AT THREE SELECTED STATIONS NEAR THE STUDY AREA1 A l t a Lake Pemberton Meadows B r a l o r n e P r e c i p . % P r e c i p . Temp. P r e c i p . % P r e c i p . Temp. P r e c i p . % P r e c i p . Temp. Jan u a r y 223.0 15.4 -4.3 122.9 13.3 -4.8 65.0 9.9 -7.5 F e b r u a r y 162.5 11.3 -2.3 102.1 11.0 -1.7 54.6 8.3 -4.3 March 114.8 7.9 0.4 67.6 7.3 3.3 44.5 6.8 0.6 A p r i l 92.2 6.4 4.9 37.9 4.1 8.8 32.3 4.9 4.4 May 47.8 3.3 8.9 32.8 3.5 13.1 32.3 4.9 8.7 June 58.9 4.1 12.7 27.7 3.0 16.2 38.6 5.9 12.0 J u l y 30.5 2.1 14.9 22.9 2.5 18.1 29.7 4.5 14.4 August 52.6 3.6 14.5 27.9 3.0 17.3 30.0 4.6 13.8 September 89.9 6.2 11.9 59.7 6.4 13.4 40.4 6.2 10.7 O c t o b e r 169.9 11.8 6.3 125.7 13.6 7.9 79.5 12.1 4.7 November 185.2 12.8 1.2 126.8 13.7 2.1 95.5 14.6 -1.3 December 217.7 15.1 -1.7 173.0 18.7 -2.6 112.3 17.2 -6.1 T o t a l 1445.0 100.0 927.0 100.0 654.0 100.0 Mean 120.4 5.6 77.3 7.6 54.6 4.2 Normal p r e c i p i t a t i o n and te m p e r a t u r e f o r the p e r i o d 1931 - 1960 as computed by the Department o f T r a n s p o r t M e t e o r o l o g i c a l Branch i n the Monthly R e c o r d . TABLE I I I MEAN MONTHLY DISCHARGE PER SQUARE KILOMETER^ B i r k e n h e a d Green Soo R u t h e r f o r d Green a t Green Lake J a n u a r y 11.9 18.5 19.7 18.0 18.0 F e b r u a r y 13.5 16.2 19.0 18.9 17.1 March 11. 3 17.9 20.5 17.2 17.9 A p r i l 21.5 39.6 51.6 37.4 36.5 May 67.6 93.9 125.2 95.4 83.1 June 110.3 141.2 167.1 154.1 114.7 J u l y 86.9 135.3 157.8 154.5 102.9 August 40.9 94.7 116.4 94.4 54.0 September 25^3 61.1 81.8 64.9 32.9 Oc t o b e r 24.3 50.1 71.2 59.5 37.7 November 23.1 33.6 43.5 32.2 28.3 December 17.7 25.8 31.4 30.8 26.2 D i s c h a r g e i s i n m / s e c x 10 U n d e r l i n e d d i s c h a r g e s a r e not s i g n i f i c a n t l y d i f f e r e n t from the s i m i l a r l y u n d e r l i n e d v a l u e s o f t h a t month a t the 5% l e v e l when Duncan m u l t i p l e range t e s t s a r e a p p l i e d t o the d a t a . Figure 9a Mean Daily Discharge ol i 1 L_ i i I ' • ' ' » O 60 120 180 240 300 365 Days 20 1-6 DISCHARGE The d i s c h a r g e o f the streams i n the two b a s i n s f o l l o w s a y e a r l y regimen which would be e x p e c t e d from the n a t u r e o f the p r e c i p i t a t i o n i n p u t and the t e m p e r a t u r e regimen ( f i g s . 9 a , b ) . Minimum d i s c h a r g e i s r e c o r d e d d u r i n g the l a t e w i n t e r and maximum fl o w s a r e g e n e r a l l y a s s o c -i a t e d w i t h p e r i o d s o f maximum snow me l t i n t h e l a t e s p r i n g and e a r l y summer. P e r i o d s o f h i g h f l o w a l s o o c c u r d u r i n g the autumn though the g e n e r a l t r e n d i s f o r d e c r e a s i n g d i s c h a r g e . These autumn peaks a r e a s s o c i a t e d w i t h warm r a i n storms from the c o a s t which b r i n g heavy p r e -c i p i t a t i o n t o t h e a r e a and m e l t much o f t h e snow t h a t has a l r e a d y accum-u l a t e d . Storms o f t h i s t y p e a c c o u n t f o r t h e peak i n d i v i d u a l f l o w s r e -c o r d e d i n each b a s i n . I t would appear t h a t t h e s e storms a r e more i n t e n s e i n t h e w e s t e r n b a s i n s f o r o v e r h a l f o f t h e y e a r l y peak f l o w s i n the Soo o c c u r d u r i n g t h e autumn w h i l e l e s s t h a n one f i f t h o f the B i r k e n h e a d y e a r l y peak f l o w s have o c c u r r e d a t t h i s t i n e o f y e a r . E x c e p t f o r Green a t Green L a k e , the f r e q u e n c y d i s t r i b u t i o n c u r v e s o f t he peak d i s c h a r g e s f o r each b a s i n are s i m i l a r f o r t h e more f r e q u e n t e v e n t s ( f i g . 1 0 ) . The lower s l o p e o f t h e c u r v e f o r Green a t Green Lake r e s u l t s from t h e damping e f f e c t o f Green Lake on t h e magnitude o f peak f l o w s . The B i r k e n h e a d has a s t e e p e r s l o p e f o r extreme e v e n t s which may be r e l a t e d t o i t s more c i r c u l a r shape: i t does appear as i f t h e more c i r c u l a r b a s i n s have a s t e e p e r s l o p e f o r extreme e v e n t s . 2 The mean monthly d i s c h a r g e p e r km was computed f o r each b a s i n (Table I I I ) . Soo R i v e r has the l a r g e s t d i s c h a r g e i n t e n s i t y i n each month w h i l e the B i r k e n h e a d has the l o w e s t . T h i s may r e f l e c t t h e d i f -f e r e n t p e r i o d o f r e c o r d f o r each b a s i n ; b u t i f i t i s assumed t h a t no t r e n d e x i s t s , t h i s d e c r e a s e i n d i s c h a r g e from west t o e a s t r e f l e c t s 125 1 0 0 h CD E> CO -C o CO Figure 9b Mean Daily Discharge Years of Record Green at Green Lake 25 Rutherford 24 Soo 24 4 2 5 4 0 0 Figure 10 Flood Frequency Based on Daily Discharge 3 0 0 <D p> 5 2 0 0 | 100 ** * SoOx 6 x * X X ' B U t f c « T . + +T V o n a t G r e e n L a k e + + 4- + * G r e e n a t vai o o o o o o o o o o o o o o o 1 i _ L X + X 4-4-X to to 1.01 1.10 2 2.33 10 2 0 50 Recurrence Interval, Years the d e c r e a s e i n p r e c i p i t a t i o n . B i r k e n h e a d R i v e r has a s h a r p e r peak i n maximum d i s c h a r g e i n June which may r e f l e c t the more r a p i d snow m e l t , s m a l l e r a c c u m u l a t i o n o f snow, and s m a l l e r g l a c i a l a r e a . O n l y Ruther-f o r d Creek has an i n c r e a s e i n d i s c h a r g e i n t e n s i t y between June and J u l y r e f l e c t i n g i t s g r e a t e r g l a c i a l c o v e r . 24 Chapter 2 DATA COLLECTION AND FIELD TECHNIQUES 2-1 INTRODUCTION The mutual adjustment of the dependent hy d r a u l i c parameters to the varying discharges of a stream section may be considered i n two d i f f e r e n t ways. The discharge of a channel section w i l l vary through time; i t w i l l a lso vary with the l o c a t i o n of the channel s e c t i o n with-i n the channel system. The r e l a t i o n s h i p s between discharge and the dependent parameters as discharge changes through time at a given channel section are termed a t - a - s t a t i o n hydraulic geometry (Leopold and Maddock, 1953). The r e l a t i o n s h i p s which consider the adjustment of the dependent parameters with increasing discharge downstream are termed downstream hydraulic geometry. Since there are two d i s t i n c t types of hydraulic geometries for stream channels, two d i f f e r e n t groups of data need to be c o l l e c t e d f o r a complete analysis (cf. Leopold and Maddock, 1953). I d e a l l y the sections used f o r the downstream analysis would also be used for the sec t i o n a l analysis so that the s e c t i o n a l and downstream h y d r a u l i c geom-e t r i e s would be integrated i n t o a "three dimensional" p i c t u r e of channel adjustment. However, t h i s i d e a l approach i s often i m p r a c t i c a l . The measurements made at each se c t i o n can be approached i n two ways. The reach i n which the section i s located can be considered homo-geneous, and a number of measurements along the reach are made so that the average s e c t i o n a l form of that reach i s determined. Such an approach 25 minimizes the effect of chance occurrences such as a tree f a l l or a bank slump on the sectional geometry; however, only the average character of the reach will be recorded. The movement of a bar through the reach would not be noticed. The second approach considers the section as a two dimensional transect across the river. The movement of a bar through this section would be detected, but the section may not be representative of that portion of the river system due to some chance occurrence which had modified the two dimensional transect. The second approach was the one chosen for this thesis. Data col-lected at gauging stations with cableways is of the two dimensional type. Since the data for the at-a-station analysis came from gauging records, the second approach seemed the most suited. However, distinct problems do arise when cableways are not exclusively used (sec. 2 - 2 ) . By adopting the two dimensional approach, special care had to be taken in the choice of the location of the downstream sections; but less time was needed to measure each section. The pool and r i f f l e sequence is a distinct feature of a l l streams in the study area, and i t would be meaningless to compare the form of pools in one section of the system to the form of riffles in another section. An attempt was made to measure a l l downstream sections at a point which was transitional between a. pool and r i f f l e . An attempt was also made to choose sections which, from visual inspection, appeared to be representative of the channel system in that area. 2-2 SECTIONAL DATA COLLECTION No data for the determination of at-a-station hydraulic geometries were collected by the author. However, data were available for five sections within the basins; and on the basis of this data five sets of at-a-station hydraulic geometries were determined. These five sections 26 were g a u g i n g s t a t i o n s o f the Water Survey o f Canada ( f i g s . 2, 3 ) ; h e r e a l a r g e number o f measurements o f some o f t h e h y d r a u l i c p a r a m e t e r s a t v a r y -i n g d i s c h a r g e s e x i s t . S e v e r a l times each y e a r d u r i n g t h e o p e r a t i o n o f each s t a t i o n w i d t h , mean v e l o c i t y , and c r o s s s e c t i o n a l a r e a were measured, and t h e d i s c h a r g e was computed. The number o f o b s e r v a t i o n s range from 97 a t R u t h e r f o r d Creek t o 140 a t Green R i v e r (Table V ) . Measurements o f c h a n n e l s l o p e and bed m a t e r i a l s i z e were made by the a u t h o r . Data g a t h e r e d a t a ga u g i n g s t a t i o n may not be t h e b e s t s u i t e d f o r the c o n s t r u c t i o n o f s e c t i o n a l h y d r a u l i c geometryj however, the l e n g t h o f r e c o r d w i l l document s y s t e m a t i c changes i n the c h a n n e l form i f t h e y e x i s t . S i n c e t h e d a t a were c o l l e c t e d by the Water Survey i n o r d e r t o e s t a b l i s h the gauge h e i g h t - d i s c h a r g e r e l a t i o n s h i p , t h e e x a c t c r o s s s e c t i o n a t which t he measurements were made was n o t c r i t i c a l . T h i s may mean t h a t some measurements were made a t s l i g h t l y d i f f e r e n t p o i n t s a l o n g t he c h a n n e l from o t h e r s , and t h i s may a c c o u n t f o r some o f the v a r i a n c e o f t h e d a t a (Wolman, 1955). T h i s e r r o r w i l l be g r e a t e s t f o r streams which may be waded f o r s l i g h t l y d i f f e r e n t s e c t i o n s w i l l be more s u i t e d f o r measure-ment a t d i f f e r e n t d i s c h a r g e s . From t h e Water Survey s t a t i o n d e s c r i p t i o n s , t h i s e r r o r i s most l i k e l y t c be found i n t h e d a t a f o r Green R i v e r a t Green L a k e . Measure-ments a t t h i s s i t e were t a k e n by wading and from a cableway. The B i r k e n -head s e c t i o n may a l s o be a f f e c t e d by t h i s s i n c e some wading measurements were made; however, i t appears t h a t most measurements a t the s i t e were made from a cableway. Measurements a t Soo and Green were t a k e n from a cableway w h i l e those a t R u t h e r f o r d appear t o have been t a k e n from a b r i d g e . In a d d i t i o n t o t h e s e f i v e s e c t i o n s , d a t a f o r t e n o t h e r g a u g i n g s t a t i o n s i n the s o u t h w e s t e r n C o a s t Mountains were o b t a i n e d from the Water Survey of Canada ( f i g . 1, sec. 3-4). An attempt was made to s e l e c t s t a t i o n s which represented varying basin s i z e s and at which a reasonably large number of observations had been made. Unfortunately the s e l e c t i o n was very l i m i t e d . The number of observations range from 17 for Mamquam River to 123 f o r Seymour River (Table VI). The smaller streams (Sentinel and Jacobs Creeks) are waded while cableways are used to make measurements of the l a r g e r streams (Seymour, Capilano, Mamquam, and Cheakamus Rivers, and Mashiter Creek). The other sections are measured by wading at low flows and from cableways or bridges at high flows. No a d d i t i o n a l data on slope or material s i z e was c o l l e c t e d at these s i t e s . 2-3 DOWNSTREAM DATA COLLECTION The data used for the computation of the downstream hydraulic geom-e t r i e s was c o l l e c t e d by the author during the summer of 1970. Measure-ments wera made at sixteen sections, s i x on Green River and i t s t r i b u -t a r i e s ( f i g s . 2, 3). The sections measured were chosen i n an attempt to represent a v a r i e t y of basin s i z e s . However, a c c e s s i b i l i t y proved to be a major l i m i t i n g f a c t o r i n the choice of the actual f i e l d s i t e s . Except f o r the portion o f the r i v e r between the gauging s t a t i o n and the confluence with Green River, Soo River was completely i n a c c e s s i b l e . The discharge along t h i s a c cessible p o r t i o n generally makes wading impracti-c a l . Some of the upstream portions of Rutherford Creek are a c c e s s i b l e during the summer, but only when the discharge precludes wading. Scckeye Creek, which flows i n t o Birkenhead Lake, was accessible during the summer but the v e l o c i t y of the stream made wading impossible. Some other access i b l e s i t e s could not be used because of man made modifications of the channel. 28 At each measurement s i t e b a n k f u l l width, b a n k f u l l s e c t i o n a l area, and water surface slope were measured and a sample of the bed and bank material was taken. Bankfull flow was chosen as the discharge to be con-sidered at each section because of i t s reasonably constant recurrence i n t e r v a l (sec. 4-2) and because of the c l e a r evidence of t h i s flow that i s generally l e f t throughout the channel system (cf. Leopold and S k i b i t z k e , 1967). The value of b a n k f u l l discharges at each s i t e was derived from the r e l a t i o n s h i p between basin area and discharge. The b a n k f u l l mean v e l o c i t y was computed from the c o n t i n u i t y equation, v=Q/A. The cross section measured at each of the sixteen s i t e s was at a po i n t which was t r a n s i t i o n a l between a pool and a r i f f l e . This was done so that the v a r i a t i o n i n the dependent hydraulic parameters due to the pool and r i f f l e sequence would be removed from the data. The height of the b a n k f u l l flow along each bank was determined independently from e v i -dence such as debris and changes i n vegetation. In a l l cases the two points were within 15 cm of the same height. I t was assumed that the s t r a i g h t l i n e which connected these two points and which was perpendic-u l a r to the d i r e c t i o n of flow represented the b a n k f u l l water surface and the b a n k f u l l width. This distance was measured using a metric tape. The distance from t h i s l i n e to the bed of the stream was then measured with a P h i l a d e l p h i a rod at a number of points. The number of depth measurements and t h e i r spacing v a r i e d with the width of the s e c t i o n and the i n t r i c a c y of the bottom topography: the average number of measure-ments was approximately t h i r t y . Because of the generally large s i z e of the bed m a t e r i a l , i t was assumed that extensive scour d i d not take place at higher flows. These measurements and the b a n k f u l l width were used to compute the b a n k f u l l cross s e c t i o n a l area. Mean b a n k f u l l depth was then 29 defined as d=A/w, where A i s the channel cross s e c t i o n a l area and w i s the water surface width. Water surface slope was measured over a distance of eig h t to ten times the b a n k f u l l width. Though slope does change as discharge increases, t h i s change i s small (Wolman, 1955), and the mean water surface slope over t h i s distance at lower flows may be assumed to be equal to that of b a n k f u l l discharge. Slope changes i n t h i s area are l i k e l y to be even less s i g n i f i -cant f o r the channel slope i s at l e a s t p a r t i a l l y imposed by the g l a c i a l topography. At low flows, however, there i s a greater d i f f e r e n c e between the water surface slope of pools and r i f f l e s than at higher flow when the slope of the pools increases while that of the r i f f l e s decreases g i v i n g a more uniform slope to the whole reach. By measuring the average slope over the suggested distance of eight to ten times the b a n k f u l l width, at l e a s t one pool and r i f f l e sequence w i l l be included so that the average slope at b a n k f u l l w i l l be obtained (Leopold and Sk i b i t z k e , 1967). The slope was surveyed using a theodolite and the Ph i l a d e l p h i a rod. The sample of sediment s i z e at each s i t e was made by s t r e t c h i n g the metric tape diagonally from one bank to the other. Generally t h i s l i n e was over a distance of 100 meters and an a r b i t r a r y sample i n t e r v a l along the tape was chosen (usually one meter). The intermediate axis of the p a r t i c l e d i r e c t l y under each sample point on the tape was measured. One hundred measurements were made at each s i t e . No attempt was made to determine the e r r o r introduced ~into the measurements by the f i e l d methods used. Instrument errors should be very small, and the major sources of e r r o r are due to the assumptions and estimations which must be made. M i l l e r (1958) made successive measurements of b a n k f u l l width and mean b a n k f u l l depth along s t r a i g h t 30 sections of a stream with no tributaries at low flow. He found that the standard deviation of the measurements was as great as 15% of the mean width and 22% of the mean depth. These measurements, however, were apparently made without regard to their position within the pool and r i f f l e sequence. The pool and r i f f l e sequence will account for a portion of the variance found by Miller. By measuring the section at approximate-ly the same point within the sequence, the error should be decreased. In absence of better information, Miller's results can be viewed as a probable upper limit of the error. The bankfull discharge was not actually measured at any of the sites. The five gauging stations within the basins and a sixth on Lillooet River were used to establish a relationship between basin area and bank-ful l discharge. For this computation, the assumption that each basin was hydrologically similar had to be made. Flood recurrence intervals were calculated from the series of annual peak discharges using the formula recurrence interval = N+l/M where N equals the number of years of record and M is the rank of each annual peak within the series (fig. 1 0 ) . Least square regression was used to f i t an equation to the data which related bankfull discharge to basin area (fig. 1 1 ) . Though other workers (cf. Miller, 1958; Brush, 1961) have found that a power law relationship gives the best f i t to the data.- a linear relationship was found to give a better f i t for this 2 area. Though R for the regression was very high ( 0 . 9 8 ) , the standard error of estimate was 28.55 and the predicted values from the equation deviated by a maximum of 30% (Birkenhead River) from the observed values. This is due in part to the very small number of sample points and the assumption of hydrologic similarity. The deviations of the observed 6001 Q 2.33 Figure 11 to Basin Area Lillooet+ 400 co E CD D) i_ ca s: o co 200 Green-+• Birkenhead + S o o Rutherford+ +Green at Green Lake -L "2100 Basin Area, km 2 400 800 1200 1600 values from the regression equation reflect the hydrologic conditions of each basin (Tables II, III; figs. 9 a , b) and the deviation of the estimated discharges at each site from the actual bankfull discharges may be as great as the deviations of the observed values. In view of the variation in the frequency of bankfull flow, the exact recurrence interval of bankfull flow in this area is not known (sec. 4 - 2 ) . However, the similarity of the flood frequency curves (fig. 10) indicates that the recurrence interval is approximately the same for a l l stations. Birkenhead River at the gauging section was observed to be very near bankfull on one occasion. The discharge at that time equalled a flow with a recurrence interval of approximately 2.33 years. On the basis of this observation and the frequent use of Q2 as bank-full by other authors, a flow with this recurrence interval was chosen for the downstream hydraulic geometry analysis. Because of the lack of data, no rating curves could be constructed for the sites in order to check this value of bankfull recurrence nor could any other check be used. 2 - 4 DATA ANALYSIS In the past i t has been generally assumed that in self-formed alluvial channels the hydraulic geometry relations are power laws (e.g. fi~M9 ) (Leopold and Maddock, 1 9 5 3 ) . For this reason, logarithmic transformations of the data were made and least square regression was used to relate the dependent parameter to discharge. The significance of the correlation coefficient and the B coefficient (the exponent in the hydraulic geometry equation) was tested using the table of signifi-cant values of the correlation coefficient from Yamane (1967) and the and the F ratio respectively. That the relationships do not follow 33 power laws is a possibility, and the assumption that they do is a bias that is reflected in this thesis. A number of linear relationships were also computed using the regression model, and in well over 80% of the cases the power law yielded a higher degree of explanation. That yet another relationship exists is a possibility which was not investi-gated. It should also be noted that many of the channels in this area are not self-formed alluvial channels but have fixed boundaries. The mater-i a l which forms the channel boundaries would be placed in motion only during exceptionally high flows. This is likely the case for ten of the downstream sections and at least two of the gauging sections, Rutherford Creek and Green River. It may also hold for Birkenhead River and Green River at Green Lake. At the other sections i t is quite likely that the slope is at least partly imposed. Channels with fixed boundaries may not adjust according to a power law though i t would appear from the anal-ysis of the data that they do. 2-5 REGRESSION ANALYSIS Least square regression is used throughout this thesis to determine the power equation which best fits the data. This technique can be viewed as simply a means of fitting an equation to data or as a statistical technique from which certain inferences about the data may be made. Until recently few workers have attempted to do more than determine the equation which best fits the data by using "visual regression": the line which appears to best f i t the data is simply drawn through the scatter of data points. As the data scatter increases, the chance of drawing by eye the line to give the equation which minimizes the variance decreases (c.f. Table IV, Miller's data). If the worker has a preconceived bias of TABLE IV A COMPARISON OF HYDRAULIC GEOMETRY EQUATIONS DERIVED FROM LINES FITTED BY HAND AND BY LEAST SQUARE REGRESSION1 Stream a b R2 c f R2 k m R2 2 M i l l e r , Pecos R i v e r , hand f i t t e d 1.20 0.59 0.38 0.30 2.10 0.13 r e g r e s s i o n e q u a t i o n 0.27 0.58 0.85 -0.78 0.23 0.50 6.61 0.25 0.01 3 Brush , L i t t l e J u n i a t a , hand f i t t e d 1.15 0.55 0.28 0.38 2.60 0.07 r e g r e s s i o n e q u a t i o n 1.29 0.53 0.98 0.25 0.39 0.99 3.17 0.07 0.30 B r u s h , S t a n d i n g S t o n e , hand f i t t e d 0.41 0.65 0.42 0.25 4.30 0.10 r e g r e s s i o n e q u a t i o n 0.50 0.63 0.98 0.39 0.27 0.79 4.52 0.12 0.22 B r u s h , S h a v e r , hand f i t t e d . 1.50 0.47 0.29 0.34 1.90 0.21 r e g r e s s i o n e q u a t i o n 1.49 0.47 0.95 0.31 0.33 0.81 2.24 0.20 0.55 w=aQ M i l l e r , 1958 f 3 d=cQ B r u s h , 1961 35 what the equation should be, i t may be reflected in the line he draws to describe the equation. By using least square regression to determine the equation, this possibility of worker bias is removed, and the equat-ion will minimize the squared deviations of the data from the regression line. Tne least square regression model is also a statistical technique from which inferences may be drawn, but as such i t is only as powerful as the degree to which the data used f i t its underlying assumptions. The general form of the model is Y.=«+BX.+e.. The X.s are observable 1 1 1 1 fixed independent variables which can be measured without error,ecis an unknown location parameter, and B is an unknown parameter reflecting the effect of X. Y. is a random dependent variable associated with X^ , and e. is an unobservable random variable with a mean of zero and a 1 variance of 6*". This variance remains constant for a l l X., and e. and l i e^ are uncorrelated i f i * j (cf. Krumbein and Graybill, 1965). The value of oc and of B may be found so that Y is related to X. This relationship depends not only on the values of Y and X but also on the conditions set upon the relationship. In the least square model the calcualted function-al relationship must pass through the means of X and Y, and the squared deviations from the regression line are minimized. The hydraulic geometry data seems to f i t the underlying assumptions of regression only poorly. Discharge (X) can not be measured without some degree of error. This error may be small at a section, but in the downstream direction i t may vary by as much as 30% from the "real" value. Discharges at a section may be considered to be reasonably independent of each other i f sufficient time is allowed between each measurement. For most of the collected discharges there were at least several weeks between readings; however, these discharges can not be considered com-36 pletely independent for discharge is a function of basin size, basin characteristics, and climatic factors. In the downstream direction the dependence of the discharges is even stronger. The discharge at P is clearly dependent on the discharge at P-AP, some point upstream of P. The deviations from the calculated regression lines are due to both measurement errors and variance within the parameters. These sources of variance may be assumed to be essentially random (Scheid-egger and Langbein, 1966). Therefore i t can be assumed that the mean value of the deviations about each X is zero. However, the variance of the deviations about each X will not be the same for a l l Xs. The pos-sible values of the dependent parameters will clearly cover a larger range for a discharge of 1,000 m3/s than for one of 10 m3/s though the logarithmic transforms will tend to reduce this. It is also likely that the deviations about X^  are partially correl-ated with those about X_,. If an unusual amount of scour occurs during one high flow, the values of the dependent parameters will reflect this event at the following low flows. Mean depth may show a consistent positive deviation from the regression equation until another high flow changes the channel section. The correlation among deviations will be greater at a section than in the downstream direction i f the downstream measurements are made a reasonable distance apart. The sectional devi-ations may reflect the last formative discharge. The downstream sections may also reflect the last formative discharge, but each section will have responded slightly differently to this flow due to differences in the character of the section. Since the data only poorly fits the assumptions of regression, inferences and p r o b a b i l i s t i c statements which can be made from the analysis can not be applied to the data i n a rigorous manner or with 2 a great deal of confidence. R can s t i l l be used as a measure of the degree to which the equation f i t s the data (Yamane, 1967), but c o n f i -dence i n t e r v a l s constructed about the c o e f f i c i e n t s or t e s t s that B i s d i f f e r e n t from zero, f o r example, can not be r i g o r o u s l y applied. I t would appear that the technique does a i d i n d i s c r i m i n a t i n g between two s i m i l a r , but d i f f e r e n t sets of data. The change i n the gauging se c t i o n f o r Green River at Green Lake i s c l e a r l y seen i n the r e s i d u a l s ( f i g . 13) even though equations f o r the two sets of data can not be considered s t a t i s t i c a l l y d i f f e r e n t . However, l e a s t square regression would appear to be most useful as a means of f i t t i n g an equation to the data i n as objec t i v e a fashion as p o s s i b l e . 38 Chapter 3 AT-A-STATION HYDRAULIC GEOMETRY 3-1 HYDRAULIC GEOMETRY EQUATIONS Width, cross s e c t i o n a l area, mean depth, and mean v e l o c i t y were r e l a t e d to discharge at the f i v e gauging s t a t i o n s , Green River, Soo River, Rutherford Creek, Green River at Green Lake, and Birkenhead River. The exponents of the hyd r a u l i c geometry r e l a t i o n s compare favorably with values found by workers i n other areas (Tables V, VII; f i g . 12). The average value of the exponent i n the w=aQ^ r e l a t i o n s h i p , b (0.11), i s lower than that found by some workers but i s s i m i l a r to the average value f o r 158 st a t i o n s i n the United States computed by Leopold, Wolman, and M i l l e r (1964). Mean depth changes i n a manner s i m i l a r to depth changes elsewhere, but v e l o c i t y appears to vary more r a p i d l y than i s usual. Width changes appear to r e f l e c t the l o c a l geology of the sec t i o n to a greater extent than do the other parameters. The lower increases i n width are associated with channels that are downcutting and have no upper l e v e l to t h e i r banks which can be. associated with f l u v i a l a c t i o n . Green River has downcut in t o bedrock along one bank and morainal mater-i a l along the other. The r i v e r i s e s s e n t i a l l y flowing between the walls of a canyon which greatly r e s t r i c t any r a p i d changes i n width. The Rutherford Creek s e c t i o n i s also downcut i n t o g l a c i a l material which r e s t r i c t s i t s width. Below t h i s "canyon" s e c t i o n , the creek spreads out and some b r a i d i n g takes place. Though i t i s not c l e a r where s e c t i o n a l TABLE V AT-A-STATION HYDRAULIC GEOMETRY' 2 2 2 2 S t a t i o n n a b R c f R k m R g h R R u t h e r f o r d Creek 97 9.57 0.08 0.21 0.42 0.33 0.59 0.25 0.59 0.92 3.99 0.41 0.85 -0.86 R u t h e r f o r d C r e e k , 33 1948 t o 1941 R u t h e r f o r d C r e e k , 26 1941 t o 1934 R u t h e r f o r d C r e e k , 38 1934 t o 1924 Green R i v e r a t 113 Green Lake Green R i v e r a t 62 Green L a k e , 1948 t o 1936 Green R i v e r a t 51 Green L a k e , 1936 t o 1920 Soo R i v e r 109 17.87 0.11 0.53 0.23 0.47 0.88 0.24 0.42 0.91 4.11 0.48 0.93 -0.37 Green R i v e r 140 19.70 0.06 0.39 0.26 0.47 0.87 0.20 0.46 0.86 5.11 0.53 0.89 -0.46 B i r k e n h e a d R i v e r 136 22.44 0.12 0.54 0.20 0.36 0.86 0.25 0.48 0.81 4.50 0.48 0.93 -0.59 1 w=aQb v=kQm f f * Qy f ' h d=cQ A=gQ n number o f o b s e r v a t i o n s 2 2 A l l B and R v a l u e s are s t a t i s t i c a l l y d i f f e r e n t from o a t the 99% l e v e l . 11.57 0.06 0.30 0.28 0.40 0.92 0.31 0.54 0.93 3.21 0.46 0.91 -0.68 9.07 0.11 0.74 0.49 0.29 0.76 0.22 0.60 0.94 4.48 0.40 0.87 -0.91 7.89 0.12 0.63 0.56 0.26 0.82 0.23 0.62 0.98 4.40 0.38 0.95 -0.98 11.25 0.18 0.21 0.31 0.34 0.43 0.30 0.48 0.77 3.37 0.52 0.81 -0.62 10.38 0.14 0.52 0.37 0.37 0.71 0.28 0.49 0.78 3.58 0.52 -0.79 -0.60 13.20 0.22 0.38 0.23 0.36 0.60 0.34 0.42 0.82 3.00 0.57 0.91 -0.48 TABLE VI AT-A-STATION HYDRAULIC GEOMETRY OF ADDITIONAL COAST MOUNTAIN SECTIONS Station n a b R 2 c 8GA-56 21 5.88 0.34 0.52 0.21 8GA-49 37 5.91 0.25 0.53 0.22 8MH-108 39 7.34 0.27 0.83 0.30 8GA-57 21 9.39 0.30 0.33 0.23 8MH-76 67 10.29 0.27 0.73 0.28 8MH-66 74 11.08 0.23 0.61 0.26 8GA-10 67 13.65 0.19 0.52 0.29 8GA-30 123 18.96 0.08 0.49 0.59 8GA-17 39 22.65 0.08 0.91 0.51 8GA-54 17 24.50 0.04* 0.04* 0.27 f R k m R g h R y 0. 26 0. 54 0. 81 0. 39 0. 87 1. 25 0. 61 0. 95 -0. 51 0. 19 0. 38 0. 75 0. 57 0. 91 1. 33 0. 43 0. 85 -0. 95 0. 30 0. 51 0. 41 0. 37 0. 77 2. 21 0. 57 0. 75 -0. 65 0. 22* 0. 28 0. 47 0. 48 0. 79 2. 11 0. 52 0. 81 -0. 75 0. 33 0. 70 0. 34 0. 37 0. 70 2. 91 0. 60 0. 83 -0. 42 0. 29 0. 72 0. 35 0. 47 0. 75 2. 87 0. 53 0. 79 -0. 66 0. 33 0. 89 0. 25 0. 48 0. 85 3. 96 0. 53 0. 88 -0. 62 0. 27 0. 63 0. 10 0. 62 0. 82 11. 08 0. 35 0. 76 -0. 97 0. 26 0. 71 0. 09 0. 66 0. 95 11. 70 0. 34 0. 83 -1. 07 0. 39 0. 68 0. 15 0. 57 0. 92 6. 63 0. 43 0. 88 -0. 74 8GA-56 Sentinel Creek above G a r i b a l d i Lake, 1966 to 1908 8GA-49 Mosquito Creek near North Vancouver, 1964 to 1970 8MH-108 Jacobs Creek above Jacobs Lake, 1965 to 1968 8GA-57 Mashiter Creek near Squamish, 1966 to 1969 8MH-76 Kanaka Creek near Webster Corner, 1960 to 1971 8MH-66 North Alouette River near Haney, 1960 to 1971 8GA-10 Capilano River near North Vancouver, 1954 to 1970 8GA-30 Seymour River near North Vancouver, 1945 to 1971 8GA-17 Cheakamus River at G a r i b a l d i , 1961 to 1967 8GA-54 Mamquam River above Mashiter Creek, 1966 to 1968 i b w=aQ d=cQ v=kQ A=gQ m f f r iO/ n number of observations * Not s t a t i s t i c a l l y d i f f e r e n t from 0 at 95% l e v e l * Not s t a t i s t i c a l l y d i f f e r e n t from 0 at 99% l e v e l TABLE VII COMPARISON OF THE EXPONENTS OF AT-A-STATION HYDRAULIC GEOMETRIES1 Streams b f m h Y Green and Birkenhead stations, mean 0.11 0.39 0.49 0.48 -0.58 Additional Coast Mountain stations, mean 0.21 0.32 0.50 0.49 -0.73 ; 2 Mountain streams, mean 0.55 0.45 3 Baffin Island sandurs, mean 0.22 0.31 0.48 0.52 -0.65 4 Midwestern U.S., mean 0.26 0.40 0.34 Average of 158 U.S. streams^ 0.12 0.45 0.43 Brandywine Creek, Pennsylvania 0.04 0.41 0.55 7 Ephemeral streams in semi-arid U.S. 0.29 0.36 0.34 8 Non-cohesive river, theory 0.50 0.23 0.27 Day, 1970 3 Church, 1970 4 Leopold and Maddock, 1953 ^ Leopold, Wolman, and Miller, 1964 6 Wolman, 1955 ' Leopold and Miller, 1956 8 Langbein, 1964 1 b w=aQ d=cQf v=kQ m A=gQ ff«QJ Figure 12, At-A-Station Hydraulic Geometry 1 10 100 200 1 10 100 200 DISCHARGE, m3/s DISCHARGE, m3/s Green River at Nairn Falls Rutherford Creek, 1924 -1934 — — Soo River Green River at Green Lake, 1922-1936 Rutherford Creek, 1934-1940 Birkenhead River Green River at Green Lake, 1936-1948 Rutherford Creek, 1941 -1948 measurements were made, i t may be that the earlier measurements were made in the lower, wider reaches while the last group of measurements were made in the "canyon". The more rapid increases in width are associated with sections where the banks appear to be less dependent on the local geology and more on fluvial deposits. The rate of change of velocity appears to be associated with sectional characteristics. The most rapid increase in velocity is associated with the section where the water surface slope is the steepest, Rutherford Creek. Soo River has the least rapid increase in velocity and the lowest slope. There appears to be a trend for a de-crease in the rate of change of velocity with a decrease in slope. The rate of change of depth does not appear to follow any general trend in sectional characteristics; i t is more dependent upon the way in which width and velocity vary. From the continuity equation i t can be seen that the sum of the exponents of width, depth, and velocity must equal 1. Since velocity and width appear to be more dependent on sectional characteristics, depth must adjust to these values in order to maintain the continuity relation; however, depth is not total-ly dependent on width and velocity, but does affect these parameters. For a i l sections the sum of the exponents is approximately 1. 3-2 RESISTANCE PROPERTIES The Darcy-Weisbach nondimensional expression for total flow re-sistance is 2 ff«*gRs/v , where g is gravitational acceleration, s is slope, v is mean velocity and R is hydraulic mean depth which is approximately equal to d, mean depth, in wide channels. From the hydraulic geometry relationships, f f <*((/)• (Q Z)/(Q 2 m) f+z-2m v *Q =Q The exponent, y, of the above r e l a t i o n was computed f o r each gauging sec t i o n . Assuming z equal to zero (sec. 2-2) , the average value of y f o r the f i v e sections i s -0.63. This rate of change i s approximately twice as great as that found f o r streams i n noncohesive materials (Leopold, Wolman, and M i l l e r , 1964). Church (1970) found a s i m i l a r value i n sandur channels on B a f f i n Island; and from the average values of the exponents f o r sections along Brandywine Creek (Wolman, 1955) , the resistance there decreased s l i g h t l y more r a p i d l y (-0.69). The bed and bank materials of the Green and Birkenhead sections are t y p i c a l of those associated with channels with noncohesive boundarie but i t may be that the large s i z e of the material tends to cause the channels' res i s t a n c e to respond i n a manner more t y p i c a l of channels with cohesive boundaries. Sand s i z e material i s the most e a s i l y eroded while smaller ma t e r i a l , such as that found along the banks of Brandywine Creek, and l a r g e r material are much less e a s i l y put i n t o motion by flow-ing water. I t appears that as the sediment s i z e increases beyond sand s i z e , the channel responds i n a manner more t y p i c a l of a cohesive channel. Soo River, where the bed and banks are nearest to sand s i z e , has a channel resistance response most l i k e other channels i n small non-cohesive material while Rutherford Creek, where the l a r g e s t material i s found, has a rate of change i n resistance s i m i l a r to Brandywine Creek. Since the value of y i s derived from observed r e l a t i o n s among v e l o c i t y , depth, and slope rather than d i r e c t l y measured, i t r e f l e c t s the influence of a number of resistance features such as the s i z e of the i n d i v i d u a l roughness elements, bed and bank features such as bars or r i f f l e s , and channel bends (Leopold, Wolman, and M i l l e r , 1964). The 45 s i g n i f i c a n c e o f any one r e s i s t a n c e f e a t u r e can n o t be d e t e r m i n e d from t h i s t y p e o f a n a l y s i s ; however, t h e i r g e n e r a l i n f l u e n c e can be d i s c u s s e d . In f u l l y t u r b u l e n t f l o w , as i s found i n most s t r e a m s , the r e s i s t a n c e due t o t he roughness elements i s a f u n c t i o n o f the s i z e o f the roughness e l e -ments and t h e depth o f f l o w . Wolman (1955) showed t h a t t he t o t a l r e s i s -t a n c e f o r a n a t u r a l s t r e a m b e a r s the same r e l a t i o n t o r e l a t i v e roughness as o c c u r s i n p i p e s o r e x p e r i m e n t a l c h a n n e l s . He found t h a t as the r a t i o o f d e p t h o f f l o w t o bed p a r t i c l e s i z e i n c r e a s e s t he r e s i s t a n c e d e c r e a s e s and t h a t f o r Brandywine Creek the e q u a t i o n i s 1 / f f = 21og(d/D )+1.0, 84 where i s t h e g r a i n d i a m e t e r e q u a l t o o r l a r g e r t h a n 84% o f t h e bed p a r t i c l e s . The s e m i l o g a r i t h m i c form o f the r e l a t i o n s h i p i n d i c a t e s t h a t a s t r e a m w i t h l a r g e roughness elements w i l l have a g r e a t e r d e c r e a s e i n r e s i s t a n c e t h a n w i l l a s t r e a m w i t h s m a l l e r roughness elements f o r a s i m i -l a r change i n f l o w d e p t h . The depth o f f l o w f o r Soo and R u t h e r f o r d i s o f t he same o r d e r o f magnitude (0.1-1 m), b u t t h e s i z e o f t h e dominant bed p a r t i c l e s o f R u t h e r f o r d (400 mm) i s an o r d e r o f magnitude l a r g e r t h a n t h o s e o f Soo (30 mm). T h e r e f o r e a more r a p i d d e c r e a s e i n r e s i s -t a n c e w i t h i n c r e a s i n g d i s c h a r g e would be e x p e c t e d f o r R u t h e r f o r d because o f the s i z e o f the roughness e l e m e n t s . The r a t e o f change o f depth r e l a t i v e t o w i d t h w i l l a l s o be impor-t a n t i n d e t e r m i n i n g the r a t e o f change o f r e s i s t a n c e . In c h a n n e l s w i t h c o h e s i v e banks o r c o a r s e bank m a t e r i a l , w i d t h w i l l t e n d t o i n c r e a s e .more s l o w l y t h a n i n c h a n n e l s w i t h sand s i z e n o n c o h e s i v e b a n k s . With a s i m i -l a r v e l o c i t y change, depth would i n c r e a s e more r a p i d l y i n c h a n n e l s w i t h lower w i d t h i n c r e a s e s , l e a d i n g t o a more r a p i d d e c r e a s e i n r e s i s t a n c e . O t h e r f a c t o r s a r e a l s o o p e r a t i n g t o change the r e s i s t a n c e due t o t h e roughness elements and t h e o t h e r r e s i s t a n c e f e a t u r e s . Church (1970) 46 suggested that the rapid decrease i n resistance observed i n sandur channels may be due to a " l i v e " bed at higher flows, an increase i n " s t r a i g h t through" flow which decreased the influence of the pool and r i f f l e sequence, and a damping o f turbulence by r a p i d l y i n c r e a s i n g sediment discharge. These factors may also be operating i n the sections of Green and Birkenhead Rivers. 3-3 DATA VARIANCE The degree to which the data i s scattered about the mean condition r e -f l e c t e d by the hydraulic geometry equations i s seen v i a the c o e f f i c i e n t s of determination which are derived from the regression analyses and are a measure of the degree to which the equation f i t s the data. The s c a t t e r i s c o n s i s t e n t l y greatest about the width r e l a t i o n s h i p s (Table V). Langbein (1964), i n h i s t h e o r e t i c a l d e r i v a t i o n of the hy d r a u l i c geometry equations, has considered width to be e s s e n t i a l l y independent at a se c t i o n i n cohesive channels and not adjustable i n any short periods of time. The higher variance i n the width r e l a t i o n s and the more d i s t i n c t i v e r e s i d u a l groupings tend to support t h i s consideration. Depth has an intermediate amount of s c a t t e r f o r i t r e f l e c t s the ad-justment of both width and cross s e c t i o n a l area. The p a r t i a l independ-ence of width w i l l tend to increase the variance of the mean depth. Cross s e c t i o n a l area and v e l o c i t y have the l e a s t variance about t h e i r mean condition f o r they are l a r g e l y dependent on discharge and may adjust r a p i d l y over a short p e r i o d of time. Cross s e c t i o n a l area can adjust to fl u c t u a t i o n s i n discharge and width by changes i n the water depth. I f the channel i s s l i g h t l y wider a f t e r a high flow than i s usual, a decrease i n mean depth can maintain e s s e n t i a l l y the same cross s e c t i o n a l area. V e l o c i t y can also adjust r a p i d l y to changes i n discharge; and i f the 47 cross sectional area is able to adjust in the same way for varying dis-charges, velocity must also vary only slightly about its mean rate of adjustment. When the residual values of the regressions were listed as an ordered sequence in time, two types of distinctive groupings of posi-tive and negative residual values appeared. The first type of grouping appears to reflect distinct changes in the cross section at which the measurements were made. For Green River at Green Lake two sub-groupings of residuals were present with the break in groupings appearing in 1936 (fig. 13). In that year the gauging section was relocated a short distance down-stream. The width residuals in each group are larger and more distinc-tively grouped than are the cross sectional area residuals. This pattern would tend to indicate the stronger dependence of sectional area on the discharge of the stream. Rutherford Creek shows a similar, though less distinctive, residual grouping. Three sub-groupings appear, one of which may be a result of the second highest recorded discharge which occurred in 1940. It is not clear whether or not the actual gauging section was ever moved. The data for these two sections were divided on the basis of the residual groupings and separate hydraulic geometries were computed. Groupings once again appeared in the residuals when they were listed as an ordered sequence in time, but the pattern observed was similar to the pattern seen in the data for the three other sections. This second type of grouping showed a number of positive residuals followed by a number of negative ones (fig. 14). This pattern existed in a l l relations though i t was not distinctive throughout. It suggests that the data do not fluctuate about the regression lines in a totally random way. Rather •12 10 + Figure 13 Residuals of Sectional Area and Width Green River at Green Lake 8 6 4 2 co _i < i 0-w HI tr 2 — * — + + + 4 6 8 -10 -•12 1923 1928 1933 1938 DATE OF MEASUREMENTS 1943 1948 • Width Residuals, m + Area Residuals, m 2 Figure 14 Residuals of Sectional Width • 12 Birkenhead River 10 • 8 6 ' • • • 4 • • • 2 0 -« • * • • • i . • • • ' . • • • • • .• • • • • ' « . 2 • • • • • • • • 4 -• 6 • * 8 • 10 -12 • 1945 1950 1955 1960 1965 1970 DATE OF MEASUREMENTS t h e s e c t i o n s a p p e a r t o f l u c t u a t e s l o w l y a bout t h e mean c o n d i t i o n ex-p r e s s e d by t h e r e g r e s s i o n l i n e , a t y p e o f b e h a v i o r w h i c h L a n g b e i n and L e o p o l d (1964) have s u g g e s t e d i s t y p i c a l o f a c h a n n e l w h i c h has r e a c h e d e q u i l i b r i u m . Due t o d i s t u r b a n c e s i n t r o d u c e d i n t o t h e c h a n n e l , t h e h y -d r a u l i c p a r a m e t e r s a r e s e l d o m a t t h e e x a c t s t a t e o f e q u i l i b r i u m . I n -s t e a d t h e d i s t u r b a n c e s cause a d j u s t m e n t s among th e p a r a m e t e r s so t h a t t h e y a r e c o n t i n u a l l y f l u c t u a t i n g a b o u t some mean c o n d i t i o n . The hy-. d r a u l i c geometry r e l a t i o n s can be t a k e n as t h e mean c o n d i t i o n , and t h e o b s e r v e d v a l u e s o f t h e p a r a m e t e r s can be s e e n t o f l u c t u a t e about t h e r e l a t i o n s t h r o u g h t i m e i n an a p p a r e n t l y nonrandom manner. 3-4 ADDITIONAL COAST MOUNTAIN SECTIONS I n a d d i t i o n t o t h e f i v e s e c t i o n s w i t h i n t h e Green and B i r k e n h e a d b a s i n s , t e n o t h e r s e c t i o n s on s t r e a m s i n t h e c o a s t a l m o u n t a i n s o f s o u t h -w e s t e r n B r i t i s h C o l u m b i a were c o n s i d e r e d i n o r d e r t o see i f t h e Green and B i r k e n h e a d s e c t i o n s were c h a r a c t e r i s t i c o f t h e a r e a ( T a b l e V I , f i g s . 1 5 a - d ) . F o r t h e s e s t a t i o n s , v e l o c i t y p r o v e d t o be t h e most r a p i d l y v a r y i n g p a r a m e t e r i n a l l c a s e s , v a r y i n g s l i g h t l y more r a p i d l y on t h e a v e r a g e t h a n w i t h i n t h e Green and B i r k e n h e a d where v e l o c i t y was a l s o t h e most r a p i d l y v a r y i n g p a r a m e t e r f o r t h r e e o f t h e f i v e s e c t i o n s . E x c e p t f o r t h e r e s u l t s o f Church (1970) and Day (1970) t h i s dominance o f v e l o c i t y v a r i a t i o n i n n o n c o h e s i v e m a t e r i a l s had n o t been r e p o r t e d b e f o r e i n t h e l i t e r a t u r e . W i d t h changes were more r a p i d f o r t h e a d d i t -i o n a l s t a t i o n s w h i l e d e p t h changed l e s s r a p i d l y , l e a d i n g t o a r a t e o f change i n c r o s s s e c t i o n a l a r e a o n l y s l i g h t l y l e s s t h a n f o r t h e Green and B i r k e n h e a d s e c t i o n s . Changes i n t h e r e s i s t a n c e f a c t o r were a l s o h i g h f o r t h e s e s t r e a m s w i t h an a v e r a g e v a l u e o f -0.75 and a range o f v a l u e s s i m i l a r t o t h e Green and B i r k e n h e a d r a n g e . The s c a t t e r o f t h e Figure 15a At-A-Station Hydraulic Geometry of Ten Streams in Southwestern British Columbia Sectional Width T — I I I 11 11| 1 — i — i I i i 111 • 100F = 101 I i i i 111 - T — i — i i i i 11 0 3 M r n ^ I I I I I I I I ' » I I i M I i i i i i i i i I i | | I M i l l .01 Capilano, Co Cheakamus, Cs Jacobs, Js • • Kanaka, Ka Mamquam, Mm — — — — Regression Lines Limited to Domain of Data 1 DISCHARGE, m3/s Mashiter, Mr Mosquito, Mo North Alouette, NA Sentinel, SI Seymour, Sr 10 100 200 Figure 15c, Mean Sectional Velocity 01 .1 1 10 100 200 DISCHARGE, m3/s Figure 15d, Sectional Area 1 10 100 200 DISCHARGE. m3/s 55 data was also s i m i l a r to that found at the Green and Birkenhead s e c t i o n s . Width and mean depth continue to have the greatest s c a t t e r though width 2 d i d not always have the lowest R . At a l l sections r e s i d u a l groupings of the second type were found. These h y d r a u l i c geometry r e l a t i o n s are s i m i l a r to those found f o r the Green and Birkenhead sections and would tend to i n d i c a t e that these are t y p i c a l of the regional r e l a t i o n s within the Coast Mountains. There i s a considerable range i n the values of the exponents, and adjustment of the hydraulic parameters i n the study area would appear to be s l i g h t l y d i f f e r e n t from other regions with lower rates of width adjustment, a dom-inant rate of v e l o c i t y adjustment and a high rate of decrease i n r e s i s -tance (Table v u ) . 56 Chapter 4 DOWNSTREAM HYDRAULIC GEOMETRY 4-1 FLOW FREQUENCY AND BANKFULL DISCHARGE Downstream hydraulic geometry considers the changes i n the hydraul-i c parameters as the discharge increases downstream. In most streams, discharge w i l l increase downstream because the s i z e of the c o n t r i b u t i n g basin i s increasing. To accommodate t h i s i n c r e a s i n g flow the channel must also adjust. Since the downstream r e l a t i o n s are only concerned with changes i n the channel form caused by changes i n s e c t i o n l o c a t i o n within the system, the time v a r i a t i o n of flow must be eliminated. This can be done by considering flows of equal frequency at each section (Leopold and Maddock, 1953). I t would be d e s i r a b l e to analyze a system wi t h i n which a flow with a two year recurrence i n t e r v a l a f f e c t s each channel s e c t i o n at essent-i a l l y the same time, but t h i s i s r a r e l y the case. Therefore, flows occurring at d i f f e r e n t times but with the same recurrence i n t e r v a l are considered s i m i l a r i n so f a r as channel forming c h a r a c t e r i s t i c s are concerned. Any frequency of flow could be considered for the downstream a n a l y s i s , but each w i l l give s l i g h t l y d i f f e r e n t r e s u l t s (Wolman, 1955). Since the parameters being considered r e l a t e the shape of the channel to discharge, a frequency of flow which f i l l s the e n t i r e channel would appear to be a meaningful flow to consider. B a n k f u l l discharge meets the above consideration and appears to be one of the important channel forming flows (Leopold, 1962) though other flows are also as important (Dury, 1969). The frequency o f b a n k f u l l flow has been computed f o r many sections on a wide v a r i e t y of streams and has been found to have a recurrence i n t e r v a l between one and three years (cf. K i l p a t r i c k and Barnes, 1964). The climate of the basins considered accounts for some of the v a r i a -b i l i t y i n b a n k f u l l recurrence from region to region though K i l p a t r i c k and Barnes found a range i n b a n k f u l l recurrence between basins i n the southern piedmont of the United States. The flo o d s e r i e s considered i s also important. Nixon (1959), working i n England and Wales, found a recurrence i n t e r v a l of only s i x months for b a n k f u l l discharge. The value i s low because he considered a l l floods above an a r b i t r a r y l e v e l rather than only the annual peak flow. Henderson (1966) shows that the recurrence i n t e r v a l of 1.4 years computed by Leopold and Wolman (1957) i n the southwestern United States would be nine months i f a f l o o d s e r i e s such as that used by Nixon had been used instead of the annual flow. Others such as Brush (1961), working i n c e n t r a l Pennsylvania, and M i l l e r (1958), working i n the mountains of New Mexico, have found that the re-currence i n t e r v a l i s 2.33 years. Another f a c t o r which influences the frequency of b a n k f u l l flow i s the l o c a t i o n of the section within the channel system. There.is evidenc to suggest that b a n k f u l l may occur more frequently i n the headwaters of a stream (Dury, 1961). When the main channel reaches b a n k f u l l , i t i s l i k e l y that a heavy input has been experienced throughout the e n t i r e basin and that many of the t r i b u t a r i e s have also reached b a n k f u l l . How-ever, a l o c a l i z e d input, such as a thunderstorm or l o c a l l y r a pid snow melting, may b r i n g a t r i b u t a r y to b a n k f u l l without a large e f f e c t on the main stream. Within the study area two reaches, Green River above Green Lake and lower Poole Creek, were observed to be at or very near b a n k f u l l on three occasions during the spring and summer of 1970. The Green s e c t i o n was also observed to be near b a n k f u l l i n the s p r i n g of 1971. On only one of these occasions were the other streams i n the area near b a n k f u l l . At the other times the flow i n the other t r i b u -t a r i e s was only moderately high. Both reaches have a low slope r e l a t i v e to t h e i r upstream reaches and both are near t h e i r l o c a l base l e v e l . The apparently high frequency with which these sections approach or reach b a n k f u l l may be a r e s u l t of these f a c t o r s . In s p i t e of the deviations from a constant recurrence i n t e r v a l , b a n k f u l l flow s t i l l has the great advantage that c l e a r evidence of t h i s flow i s generally l e f t and can be i d e n t i f i e d during times of lower flows (Leopold and f k i b i t z k e , 1967). I t would be i m p r a c t i c a l to attempt to measure the hydraulic parameters at a high flow at a large number of sections f or the duration of these flows i s r e l a t i v e l y short. Instead the b a n k f u l l width and b a n k f u l l cross s e c t i o n a l area can be measured from evidence of b a n k f u l l flow. Ban k f u l l mean depth i s computed from the r e l a t i o n , d=A/w, and b a n k f u l l mean v e l o c i t y can be found from the c o n t i n u i t y eauation, v=Q, , _ „„/A (sec. 2-3). b a n k f u l l 4-2 HYDPAULIC EQUATIONS Width shows the most rapid increase within the channel systems o f Green and Birkenhead Rivers while mean depth increases l e s s r a p i d l y (Table VIII, f i g . 16). Mean v e l o c i t y tends to decrease slowly i n the downstream d i r e c t i o n , and slope and the Darcy-Weisbach resistance f a c t o r decrease r a p i d l y . The increase i n width i s more rapid than i n other areas (Table IX). Leopold and Maddock (1953) and Brush (1961) d i d f i n d values as high as these, but t h e i r average values (0.50 and 0.55) are TABLE V I I I DOWNSTREAM HYDRAULIC GEOMETRY' Stream n a b R2 c f R2 k m R2 Green R i v e r 6 1.94 0.60* 0.70* 0.23 0.40* * 0.42* * 2.16 -0.01* * * 0.00* * B i r k e n h e a d R i v e r 10 1.00 0.79 0.94 0.17 0.44 0.94 6.04 -0.23* 0.48* Stream g h R2 t z R2 y Green R i v e r 0.45 l.ooj 0.63*- 1.19 -1.295 0.69* -0.87 B i r k e n h e a d R i v e r 0.22 1.17 0.89 2.93 -1.51 0.60 -0.61 1 ^b w=aQ , ~m v=kQ z s=tQ d=cQf A=gQh f f « QY n number o f o b s e r v a t i o n s * Not s t a t i s t i c a l l y d i f f e r e n t from 0 a t the 99% l e v e l * Not s t a t i s t i c a l l y d i f f e r e n t from 0 a t the 95% l e v e l * * Not s t a t i s t i c a l l y d i f f e r e n t from 0 a t the 90% l e v e l TABLE IX COMPARISON OF THE EXPONENTS OF DOWNSTREAM HYDRAULIC GEOMETRIES 1 Streams b f m z Green River 0.60 0.40 -0.01 -1.29 Birkenhead River 0.79 0.44 -0.23 -1.51 2 Brandywine Creek, Pennsylvania 0.42 0.45 0.05 -1.07 3 Appalachian streams 0.55 0.36 0.09 4 Average values, midwestern U.S. 0.50 0.40 0.10 -0.75 Ephemeral streams i n semiarid U.S."* 0.50 0.30 0.20 -0.95 Appalachian plateau streams^" 0.36 0.20 0.44 Theory^ 0.53 0.10 0.37 -0.75 1 n b 2 w=aQ Wolman, 1955 d=cQf 3 Brush, 1961 , m v=kQ 4 Leopold and Maddock, 1953 s = t Q 5 Leopold and M i l l e r , 1956 6 Coates, 1969 7 Langbein, i n Leopold, Wolman, and M i l l e r , 1964 Figure 16 Downstream Hydraulic Geometry 10 100 200 DISCHARGE, rrrVs 62 well below the average value of 0.71 for Green and Birkenhead Rivers. This r a p i d increase i n width i s probably a r e s u l t of the changing nature of the banks downstream. In t h e i r upstream reaches, channels are con-t r o l l e d by large bed and bank material while downstream reaches are l e s s confined because of smaller material which can be moved more often by the streams. Depth changes s l i g h t l y more r a p i d l y than i n most other areas con-sidered. Nevertheless, channels i n t h i s area appear to have a more rap-i d l y i n c r e a s i n g width to depth r a t i o than i n other areas i n d i c a t i n g t h a t the headwater channels are r e l a t i v e l y narrow and deep while downstream reaches are r e l a t i v e l y wide and shallow. This too i s a r e s u l t of the changing nature of the banks. The r a p i d decrease i n resistance i s p a r t l y a r e s u l t of the change i n the s i z e of the bed and bank mat e r i a l . I t i s also a r e s u l t of the increase i n depth. This leads to an increase i n the r a t i o o f depth of flow to p a r t i c l e s i z e which leads to a decrease i n resistance (sec. 3-2). V e l o c i t y shows the greatest departure from previous studies. For each set of data, v e l o c i t y shows a decrease i n the downstream d i r e c t i o n . Brush (1961) d i d f i n d that several of the mountain streams he studied also showed a decrease i n v e l o c i t y , but most other workers have found that v e l o c i t y increases. The average value of the v e l o c i t y exponent f o r the Green and Birkenhead i s -0.12; a l l other workers have found that the average change of v e l o c i t y i n an area i s p o s i t i v e . There are two po s s i b l e reasons why the change of v e l o c i t y i s nega-t i v e . I t may be a r e s u l t of the p h y s i c a l c h a r a c t e r i s t i c s of the basins, or i t may simply be a r e s u l t of measurement e r r o r s . A t h i r d a l t e r n a t i v e i s that the v e l o c i t y at b a n k f u l l remains e s s e n t i a l l y constant. Since b a n k f u l l v e l o c i t y i s derived from the r e l a t i o n , v=Q/A, i t w i l l r e f l e c t e rrors i n the estimation and measurement of A and the e s t i -mation of Q (sec. 4-3). These errors may be s u f f i c i e n t to obscure the 2 true r e l a t i o n s h i p . The c o e f f i c i e n t of determination, R , i s 0.00 f o r Green River and 0.48 f o r Birkenhead River, and these low values may re-f l e c t the errors of measurement. 2 The low R values may also r e s u l t from the fac t that no systematic change i n v e l o c i t y takes place. I t has been suggested (Dury, 1969) that the change i n v e l o c i t y i s i n f a c t zero i n the downstream d i r e c t i o n . Neither the c o r r e l a t i o n s between v e l o c i t y and discharge nor the slopes of the r e l a t i o n s h i p are s t a t i s t i c a l l y d i f f e r e n t from zero (Table V I I I ) . This i s the case f o r the Birkenhead only at the 99% s i g n i f i c a n c e l e v e l ; but since the data so poorly f i t the assumptions of the s t a t i s t i c a l model (sec. 2-5) , the s i g n i f i c a n c e l e v e l should.be considered lower 2 (Yamane, 196"). S i m i l a r l y low R values were found when regressions were run on one set of data c o l l e c t e d by M i l l e r (1958) and three sets c o l l e c t e d by Brush (1961) (Table IV). Neither the c o r r e l a t i o n s nor the slopes of these r e l a t i o n s were s t a t i s t i c a l l y d i f f e r e n t from zero. 2 These low R values tend to r e i n f o r c e the impression i n the l i t e r a t u r e of c o n s i s t e n t l y greatest s c a t t e r about the downstream ve l o c i t y - d i s c h a r g e r e l a t i o n s h i p s and would suggest that there i s i n f a c t no systematic change i n v e l o c i t y . F i n a l l y , the p h y s i c a l c h a r a c t e r i s t i c s of the basins may lead to an actual decrease i n v e l o c i t y . The average decrease i n water surface slope i n the downstream d i r e c t i o n i s -1.40. This decrease i n slope i s much more rapid than i n other areas (Table IX) and i s a r e s u l t of g l a c i a t i o n and the p a r t i a l independence of slope from discharge. Though 64 the slope in the lower reaches is probably partly dependent on discharge, the headwater reaches can be assumed to be independent (Day, 1 9 7 0 ) . Leo-pold (1953) suggested that the observed increase in velocity downstream was a result of the increase in depth and the decrease in resistance. These changes were sufficiently great to compensate for the decrease in slope which would tend to reduce velocity. In this area the rapid decrease in slope would appear to be sufficiently greater than the decreasing re-sistance and increasing depth to cause a reduction in velocity downstream. 4-3 DATA VARIANCE The variance of the data about the regression lines indicates that the hydraulic geometry equations do not completely describe the change of the dependent parameters with discharge. There are several sources of variance which lead to deviations of the observed values from the regres-sion line. Measurement errors are one source, but this should be rela-tively small. A larger source is the estimations which must be made. Deviations which exist in the actual values of the hydraulic parameters also contribute to overall variation. Interdependence among these para-meters is high, and a change in one will cause changes in each of the others. Adjustments in the system are not instantaneous, and the system is seldom at its mean condition. Since this condition can not be pre-cisely determined, a certain degree of inherent variance must be ex-pected (Langbein and Leopold, 1964). Natural inhomogeneities, such as tree falls or bank slumps, provide another source which may be indis-tinguishable from the inherent variance. It is not possible to divide the variance among the different sources, but an intuitive impression of the degree to which each factor may contribute to the overall variation can be established. The largest 65 2 scatter (lowest R ) occurs about the velocity-discharge relations. A large part of this may be a result of the estimations which must be made in order to determine the value of velocity. The error in discharge (sec. 2-3) is reflected in both the dependent and independent variables of the regression and would tend to increase the scatter. In the other regressions, the discharge error is only reflected in the independent variable, discharge. Though this large possible error in discharge, and hence velocity, may account for a large portion of the scatter about the velocity-discharge relations for Green and Birkenhead Rivers, other studies where the error in discharge estimates was smaller also have large scatter. Another cause of the large variance in the velocity relationships is that velocity is the most sensitive of the hydraulic parameters. It may adjust almost instantly to disturbances in the channel while adjustment of the other parameters will take longer. For this reason, velocity will fluctuate rapidly about its mean with a large relative range of fluctuation. The partial independence of slope has a greater effect on velocity than on the other parameters. Width shows the best f i t for the two sets of data. This also appears to be true in most other studies of downstream hydraulic geometry. Width variance is least because i t is the least sensitive of the parameters and can be measured with the least error. Evidence of bankfull flow is rela-tively clear so that only a small error will result from its estimation. Depth gives the next best f i t for the study area and in general for the data from other areas. Measurement error will be larger for a small error in both sectional area and width can lead to a larger error in depth. The assumption that the water surface is level between the banks may not be correct, and some scour or deposition may take place at bank-f u l l flow. Adjustments in the channel at lower flows will result in a larger error in the determination of depth. The position of the cross section within the pool and r i f f l e sequence will also lead to a larger variance in depth. The variance in slope is only slightly greater than the variance in depth. This would seem to be largely a result of natural inhomo-geneities (the partial independence of slope) between the different sections. Variance in the Green River relations is consistently greater than in the Birkenhead River relations. At the 99% significance level none of the slopes or the correlations of the Green relations are different from zero. Though this might suggest that the Green is very different from the Birkenhead and that the hydraulic geometry relations do not hold, i t is more likely a result of the small number of sections measured. 67 C h a p t e r 5 QUASI-EQUILIBRIUM IN COAST MOUNTAIN STREAMS 5-1 INTRODUCTION A s t r e a m w h i c h has r e a c h e d a s t a t e o f q u a s i - e q u i l i b r i u m i s n e i t h e r r a p i d l y a g g r a d i n g n o r d e g r a d i n g , and t h e h y d r a u l i c p a r a m e t e r s a r e ad-j u s t e d so t h a t a change i n one w i l l be n e g a t e d by a d j u s t m e n t s i n t h e o t h e r p a r a m e t e r s ( M a c k i n , 1 9 4 8 ) . Such a s t r e a m can be v i e w e d as an open s y s t e m w h i c h has r e a c h e d a s t e a d y s t a t e w i t h m a t e r i a l and en e r g y p a s s i n g t h r o u g h i t b u t w i t h no s i g n i f i c a n t changes t a k i n g p l a c e w i t h i n t h e s y s t e m ( C h o r l e y , 1 9 6 2 ) . T h i s c o n d i t i o n may a p p l y t o an e n t i r e c h a n n e l s y s t e m , o r i t may a p p l y t o s e v e r a l s e c t i o n s w i t h i n t h e s y s t e m . I f a q u a s i - e q u i l i b r i u m has been e s t a b l i s h e d w i t h i n t h e s y s t e m , i t w i l l be r e f l e c t e d n o t o n l y i n t h e downstream c h a n n e l a d j u s t m e n t s b u t a l s o i n the s e c t i o n a l a d j u s t m e n t s . 5-2 QUASI-EQUILIBRIUM AND COAST MOUNTAIN STREAMS Though t h e o r e t i c a l v a l u e s f o r t h e h y d r a u l i c geometry o f a s t r e a m a t e q u i l i b r i u m have been d e r i v e d ( L e o p o l d and L a n g b e i n , 1962; L a n g b e i n , 1 9 6 4 ) , i t i s u n l i k e l y t h a t t h e s e r e s u l t s w i l l a p p l y t o C o a s t M o u n t a i n s t r e a m s . The a s s u m p t i o n was made i n t h e t h e o r e t i c a l d e r i v a t i o n t h a t t h e s t r e a m s had s e l f - f o r m e d a l l u v i a l c h a n n e l s . T h i s i s h a r d l y t h e c a s e i n t h e C o a s t M o u n t a i n s where s l o p e i s c l e a r l y imposed on t h e u p l a n d c h a n n e l s and w i d t h and r e s i s t a n c e a r e p r o b a b l y p a r t l y imposed and where s l o p e i s a t l e a s t p a r t l y imposed on t h e c h a n n e l s i n t h e main v a l l e y s . F o r t h i s r e a s o n i t i s u n l i k e l y t h a t any s e l f - i m p o s e d e q u i l i b r i u m e x i s t s 68 throughout e i t h e r the Green or Birkenhead system. The lack of a self-imposed equilibrium, however, does not imply that r a p i d changes are taking place within each system or that a quasi-e q u i l i b r i u m does not e x i s t - The s t a b i l i t y of the a t - a - s t a t i o n h y d r a u l i c geometry r e l a t i o n s over a reasonably long period of record suggest that these sections are s t a b l e . In a d d i t i o n , the f l u c t u a t i o n s of the parameters about the mean condition expressed by the hydraulic geometry r e l a t i o n s occur i n a seemingly systematic manner over time. I f these channel sections are at equilibrium, i t would imply that the reaches wi t h i n which they are located are also at e q u i l i b r i u m . An e q u i l i b r i u m has been es-t a b l i s h e d which accommodates the imposed conditions. Though a qua s i - e q u i l i b r i u m may e x i s t f o r reaches within each basin, i t i s u n l i k e l y that one has been established f o r each complete system. Water f a l l s , which are common on these streams, are c l e a r l y a non-equi-l i b r i u m feature. Reports of r a p i d degradation along some reaches have been made, and evidence of aggradation and degradation e x i s t s along other reaches. The change i n the conditions imposed on the channels would also suggest that d i f f e r e n t channel adjustments take place i n d i f f e r e n t areas. Instead o f one type of channel adjustment which brings an e q u i l i b r i u m t c the e n t i r e channel system, i t would appear that a number of d i f f e r e n t e q u i l i b r i u m reaches have been formed which have not yet been integrated i n t o one balanced system. This s i t u a t i o n e x i s t s because of the strong influence that g l a c i a l l y derived features s t i l l exert over the drainage system (sec. 1-3) and because of the post-Pleistocene change i n the im-posed conditions. I t i s not c l e a r how many d i f f e r e n t types of e q u i l i b r i u m reaches are i n the area. There may be only two, one of which i s found i n the upland channels and one which i s located i n the main v a l l e y channels. A t a l l even t s , the evidence o f the h y d r a u l i c geometry suggests streams i n t h i s r e g i o n have e s t a b l i s h e d reaches which ad jus t i n an o r d e r l y f a s h i o n to the g e o l o g i c c o n d i t i o n s imposed upon them. PHOTOGRAPH 1 Soo River, looking upstream near gauging section PHOTOGRAPH 2 Rutherford Creek, looking upstream near gauging s e c t i o n PHOTOGRAPH 3 Birkenhead River, looking upstream at gauging section and downstream section during near bankfull discharge PHOTOGRAPH 4 Birkenhead Lake River between Birkenhead Lake and confluence with Birkenhead River, looking upstream at downstream section 72 PHOTOGRAPH 5 Birkenhead River above confluence with Tenas Creek, looking downstream at downstream section PHOTOGRAPH 6 Owl Creek, looking upstream at downstream section PHOTOGRAPH 7 Tenas C r e°k , l o o k i n g upstream a t downstream s e c t i o n PHOTOGRAPH 9 Green River above Green Lake, looking downstream at downstream section PHOTOGRAPH 10 21 Mile Creek, looking downstream at downstream section 75 BIBLIOGRAPHY Brush, Lucien M., 1961, Drainage Basins, Channels, and Flow C h a r a c t e r i s t i c s of Selected Streams i n Central Pennsylvania, United States Geologic Survey P r o f e s s i o n a l Paper 282-F. Chorley, R. J . , 1962, Geomorphology and General Systems Theory, United States Geologic Survey P r o f e s s i o n a l Paper 500-B. Church, M. A., 1970, B a f f i n Island Sandar, A Study of A r c t i c F l u v i a l  Environments, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia. Coates, D. R. , 1969, "Hydraulic Geometry i n a Glaciated Region", Paper Annual Meeting, American Geophysical Union. Day, T. J . , 1970, The Channel Geometry of Mountain Streams, M.A. Thesis, U n i v e r s i t y of B r i t i s h Columbia. Dury, G. H., 1961, "Bankfull Discharge: An Example of I t s S t a t i s t i c a l Relationships", B u l l e t i n of the International A s s o c i a t i o n of  S c i e n t i f i c Hydrology, Vol. 6, No. 3, pp. 48-55. Dury, G. H., 1969, Perspectives on Geomorphic Processes, Resource Paper No., 3, A s s o c i a t i o n of American Geographers. Dury, G. H., 1969, "Hydraulic Geometry", i n Water, Earth, and Man (ed. R. J . Chorley), Methuen and Co. Ltd., pp. 319-329. G i l b e r t , R. , 1972, "Observations on Sedimentation at L i l l o o e t Delta, B r i t i s h Columbia", Chapter 4-6 i n Mountain Geomorphology (eds. H. 0. Slaymaker and H. J . McPherson), Tantalus Press. Henderscn, F. M., 1966, Open Channel Flow, Macmillan, Inc. I n g l i s , S i r C. C., 1949, "The Behaviour and Control of Rivers and Canals", Research P u b l i c a t i o n , Poona (India), No. 13. Kennedy, R. G., 1895, "The Prevention of S i l t i n g i n I r r i g a t i o n Canals", Proceedings of the I n s t i t u t e of C i v i l Engineers, Vol. 119. K i l p a t r i c k , F. A. and H. H. Barnes, J r . , 1964, Channel Geometry of  Piedmont Streams as Related to Frequency of Floods, United States Geologic Survey P r o f e s s i o n a l Paper 422-E. Krumbein, W. C. and F. A. G r a y b i l l , 1965, An Introduction to S t a t i s t i c a l  Models i n Geology, McGraw-Hill, Inc. Lacey, G., 1929, "Stable Channels i n Alluvium", Proceedings of the  I n s t i t u t e of C i v i l Engineers, Vol. 229. 76 L a n g b e i n , W. B., 1964, "Geometry o f R i v e r C h a n n e l s " , 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 , A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , V o l . 90, No. HY2, P r o c . P a p e r 3846, pp. 301-312. L a n g b e i n , W. B. and L. B. L e o p o l d , 1964, " Q u a s i - e q u i l i b r i u m S t a t e s i n C h a n n e l M o r p h o l o g y " , A m e r i c a n J o u r n a l o f S c i e n c e , V o l . 262, pp. 782-794. L e o p o l d , L. B., 1953, "Downstream Changes o f V e l o c i t y i n R i v e r s " , A m e r i c a n J o u r n a l o f S c i e n c e , V o l . 251, pp. 606-624. L e o p o l d , L. B., 1962, " R i v e r s " , A m e r i c a n S c i e n t i s t , V o l . 50, pp. 511-537. L e o p o l d , L. B. and T. Maddock, J r . , 1953, The H y d r a u l i c Geometry o f  S t r e a m C h a n n e l s and Some P h y s i o g r a p h i c I m p l i c a t i o n s , U n i t e d S t a t e s G e o l o g i c S u r v e y P r o f e s s i o n a l P a p e r 252. L e o p o l d , L. B. and J . P. M i l l e r , 1956, Ephemeral S t r e a m s - H y d r a u l i c  F a c t o r s and T h e i r R e l a t i o n t o t h e D r a i n a g e N e t , U n i t e d S t a t e s G e o l o g i c S urvey P r o f e s s i o n a l P a p e r 282-A. L e o p o l d , L. B. and M. G. Wolman, 1957, R i v e r C h a n n e l P a t t e r n s ; B r a i d e d , M e a n d e r i n g , and S t r a i g h t , U n i t e d S t a t e s G e o l o g i c S u r v e y P r o f e s s i o n a l P a p e r 282-B. L e o p o l d , L. B. and W. B. L a n g b e i n , 1962, The Concept o f E n t r o p y i n L a n d s c a p e E v o l u t i o n , U n i t e d S t a t e s G e o l o g i c S u r v e y P r o f e s s i o n a l P a p e r 500-A. L e o p o l d , L. B., M. G. Wolman, and J . P. M i l l e r , 1964, F l u v i a l P r o c e s s e s  i n Geomorphology, W. H. Freeman and Co. L e o p o l d , L. B. and H. E. S k i b i t z k e , 1967, " O b s e r v a t i o n s on Unmeasured R i v e r s " , G e o g r a f i s k a A n n a l e r , V o l . 49, pp. 247-255. M a c k i n , J . H., 1948, "Concept o f t h e Graded R i v e r " , B u l l e t i n o f t h e  G e o l o g i c S o c i e t y o f A m e r i c a , V o l . 59, pp. 463-512. Mathews, W. H., 1958, "Geology o f t h e Mount G a r i b a l d i Map-Area, S o u t h w e s t e r n B r i t i s h C o l u m b i a , Canada; P a r t 2: Geomorphology and Q u a t e r n a r y V o l c a n i c R o c k s " , B u l l e t i n o f t h e G e o l o g i c  S o c i e t y o f A m e r i c a , V o l . 69, pp. 179-198. M i l l e r , J . P., 1958, H i g h M o u n t a i n S t r e a m s : E f f e c t s o f G e o l o g y on  C h a n n e l C h a r a c t e r i s t i c s and Bed M a t e r i a l , Memoir 4, New M e x i c o I n s t i t u t e o f M i n i n g and T e c h n o l o g y , S o c o r r o , New M e x i c o . M o n t h l y R e c o r d , 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 n Canada, Department o f T r a n s p o r t , M e t e o r o l o g i c a l B r a n c h . N i x o n , M., 1959, "A Study o f t h e B a n k f u l l D i s c h a r g e s o f R i v e r s i n E n g l a n d and W a l e s " , I n s t i t u t e o f C i v i l E n g i n e e r s P r o c e e d i n g s , P a p e r No. 6322, pp. 157-174. 77 Ryder, J. M., 1972, "Pleistocene Chronology and Glacial Geomorphology Studies in Southwestern British Columbia", Chapter 2-4 in Mountain Geomorphology (eds. H. 0. Slaymaker and H. J. McPherson), Tantalus Press. Scheidegger, A. E. and W. B. Langbein, 1966, Probability Concepts in  Geomorphology, United States Geologic Survey Professional Paper 500-C. Shreve, R.L., 1966, "Statistical Law of Stream Numbers", Journal of  Geology, Vol. 74, pp. 17-37. Slaymaker, H. 0. and R. Gilbert, 1972, "Geomorphic Processes and Land Use Changes in the Coast Mountains of British Columbia", Proceedings  of the International Geomorphology Symposium, Liege-Caen, 1971, (ed. A. Pissart). Strahler, A. N., 1952, "Dynamic Basis of Geomorphology", Bulletin of the  Geologic Society of America, Vol. 63, pp. 923-938. Strahler, A. N., 1952, "Hypsometric (Area-Altitude) Analysis of Erosional Topography", Bulletin of the Geologic Society of America, Vol. 63, pp. 1117-1142. Strahler, A. N., 1956, "Quantitative Slope Analysis", Bulletin of the  Geologic Society of America, Vol. 67, pp. 571-596. Strahler, A. N., 1957, "Quantitative Analysis of Watershed Geomorphology"; Transactions of the American Geophysical Union, Vol. 38, pp. 913-920. Wolman, M. G., 1955, The Natural Channel of Brandywine Creek, Pennsylvania, United States Geologic Survey Professional Paper 271. Yamane, T., 1967, Statistics; An Introductory Analysis, Harper and Row, Inc. 

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