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Validation of an urban canyon radiation model for nocturnal long-wave radiative fluxes and the effect.. 1989

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VALIDATION OF AN URBAN CANYON RADIATION MODEL FOR NOCTURNAL LONG-WAVE RADIATIVE FLUXES AND THE EFFECT OF SURFACE GEOMETRY ON COOLING IN URBAN CANYONS By JAMES ADRIAN VOOGT B.Sc, Queen's University, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Geography) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 © James Adrian Voogt, 1 9 8 9 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of t̂ gC^rfyPU-V The University of British Columbia Vancouver, Canada Date S £ P T ~ - 4? . . DE-6 (2/88) 11 ABSTRACT The urban canyon r a d i a t i o n model of A r n f i e l d (1976, 1982) i s v a l i d a t e d u s i n g measurements of long-wave f l u x e s taken w i t h i n a s c a l e urban canyon c o n s t r u c t e d from c o n c r e t e b u i l d i n g b l o c k s . A c u stom-designed t r a v e r s i n g system a l l o w e d m i n i a t u r e r a d i o m e t e r s t o be a u t o m a t i c a l l y moved around the p e r i m e t e r of a canyon c r o s s - s e c t i o n t h e r e b y p r o v i d i n g f o r the v a l i d a t i o n of i n d i v i d u a l model g r i d - p o i n t s . Measured model i n p u t c o n s i s t s of s u r f a c e t e m p e r a t u r e s o b t a i n e d u s i n g f i n e w i r e themocouples, i n c i d e n t long-wave r a d i a t i o n a t the canyon t o p , and e m i s s i v i t y of canyon m a t e r i a l s . T e s t s were conducted t o e s t a b l i s h the e x p e c t e d a c c u r a c y and p r e c i s i o n of the i n p u t d a t a . S u r f a c e t emperature d a t a were f i l t e r e d t o remove a n o i s e component. A p r o b a b l e e r r o r a n a l y s i s of a l l measured model i n p u t and v a l i d a t i o n d a t a i s made. S e n s i t i v i t y t e s t s of the model t o v a r i a t i o n s i n i n p u t d a t a a r e p r e s e n t e d . S u r f a c e temperature i s the dominant c o n t r o l under the c o n d i t i o n s t e s t e d . M o d e l - c a l c u l a t e d v i e w - f a c t o r s a r e shown t o be i n e r r o r f o r a d j a c e n t c o r n e r p o i n t s and a r e r e p l a c e d w i t h v i e w - f a c t o r s c a l c u l a t e d u s i n g e q u a t i o n s d e r i v e d from the N u s s e l t U n i t Sphere method ( S t e y n , p e r s . comm.) V a l i d a t i o n r e s u l t s f o r a range of canyon h e i g h t - t o - w i d t h r a t i o s , m e t e o r o l o g i c a l c o n d i t i o n s and model parameters a r e p r e s e n t e d . E x c e l l e n t agreement between m o d e l l e d and measured f l u x e s i s o b t a i n e d f o r p o i n t s on the canyon f l o o r and t o p . The agreement f o r f l u x e s a t p o i n t s on the canyon w a l l s i s g e n e r a l l y good but i s shown to s u f f e r from e r r o r s in sensor o r i e n t a t i o n r e l a t i v e to the canyon w a l l s . Use of the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n improves model performance s t a t i s t i c s for i n c i d e n t and net long-wave r a d i a t i o n . Four d i f f e r e n t e s t i m a t e s of s u r f a c e temperature are used as model input in p l a c e of the measured v a l u e s to i n v e s t i g a t e the d i f f e r e n c e s i n the model o u t p u t . S u r f a c e t e m p e r a t u r e - b a s e d e s t i m a t e s are found to be s u p e r i o r to those based upon a i r t e m p e r a t u r e . The use of unmodi f i ed s c r e e n - l e v e l a i r temperatures measured at Vancouver A i r p o r t produces the p o o r e s t agreement. The temporal and s p a t i a l v a r i a t i o n of i n - c a n y o n temperatures and r a d i a t i o n are p r e s e n t e d for t h r e e canyon h e i g h t - t o - w i d t h r a t i o s . The canyon geometry i s shown to s i g n i f i c a n t l y reduce the s u r f a c e c o o l i n g on the canyon f l o o r compared to an open s i t e under i d e a l r a d i a t i v e c o o l i n g c o n d i t i o n s . R e s u l t s are compared to p r e v i o u s r e s u l t s from s c a l e models (Oke, 1981) and f i e l d s t u d i e s (Oke and M a x w e l l , 1975; Hogstrdm et al . , 1978). . Atmospher i c c o n t r o l s of i n c i d e n t long-wave r a d i a t i o n , wind speed and d i r e c t i o n are a l s o shown to a f f e c t the observed c o o l i n g . IV TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES L I S T OF FIGURES L I S T OF SYMBOLS ACKNOWLEDGEMENT CHAPTER 1. INTRODUCTION 1 1.1 RESEARCH OBJECTIVES 1 1.2 APPLICATIONS/SIGNIFICANCE OF THE RESEARCH 2 1.3 APPROACHES TO THE STUDY OF THE EFFECTS OF URBAN 4 SURFACE GEOMETRY ON THE CLIMATE WITHIN URBAN AREAS 1.3.1 O b s e r v a t i o n a l S t u d i e s 4 1 .3 .2 S c a l e M o d e l l i n g 6 1 .3 .3 M a t h e m a t i c a l Models 8 1 .4 RESEARCH METHODOLOGY 10 CHAPTER 2. THE CANYON MODEL INSTRUMENTATION, AND 13 TESTS 2.1 INTRODUCTION 13 2.2 SIMULATION METHODOLOGY 13 2.3 THE CANYON MODEL 15 2 .3 .1 G e n e r a l D e s c r i p t i o n 15 2 . 3 . 2 S i t e 16 2 . 3 . 3 M a t e r i a l s 18 I I i v X x i v XX x x i v V 2 . 3 . 4 C o n s t r u c t i o n Technique 18 2 . 3 . 5 Canyon He ight and Width 20 2 . 3 . 6 Canyon Length 22 2.4 CANYON INSTRUMENTATION 23 2 .4 .1 S u r f a c e Temperature 23 2 . 4 . 2 R a d i a t i o n 25 2 . 4 . 3 The Canyon T r a v e r s i n g System 32 2.5 OTHER VARIABLES MEASURED 36 2.6 TESTS 37 2 .6 .1 S u r f a c e E m i s s i v i t y 37 2 . 6 . 2 S u r f a c e Temperature T e s t s . 40 2 . 6 . 3 Radiometer D i s t a n c e Above Face t 43 2 . 6 . 4 T r a v e r s e System 44 CHAPTER 3. THE ARNFIELD MODEL 53 3.1 INTRODUCTION 53 3.2 MODEL FRAMEWORK 54 3 .3 MODEL IMPLEMENTATION 57 3.4 MODIFICATION TO THE ARNFIELD MODEL 59 3 .4 .1 Canyon He ight and Width 61 3 . 4 . 2 V i e w - F a c t o r s 61 3 . 4 . 3 A d d i t i o n a l G r i d - P o i n t s 65 3 .5 SENSITIVITY TESTS 67 3 .5 .1 V i e w - F a c t o r s / G r i d - P o i n t s 67 3 . 5 . 2 S u r f a c e Temperature 73 3 . 5 . 3 R a d i a t i o n 85 3 . 5 . 4 E m i s s i v i t y 87 vi 3 . 5 . 5 Radiance D i s t r i b u t i o n 91 CHAPTER 4. MODEL VALIDATION 94 4.1 INTRODUCTION 94 4.2 MODEL VALIDATION STATISTICS 95 4.3 MODEL VALIDATION USING DATA COLLECTED IN 'POINT' 96 MODE 4.4 MODEL VALIDATION USING AUTOMATICALLY COLLECTED 101 DATA 4.4 .1 August 1/2 101 4 . 4 . 2 August 3/4 108 4 . 4 . 3 August 10/11 112 4 . 4 . 4 August 12/13 116 4.5 SUMMARY OF VALIDATION RESULTS 120 CHAPTER 5. MODEL RESULTS USING DIFFERENT ESTIMATES OF 123 SURFACE TEMPERATURE 5.1 INTRODUCTION 123 5.2 AIR TEMPERATURE 125 5.2 .1 R e s u l t s : Average Canyon A i r Temperature 126 5 .2 .2 R e s u l t s : A i r p o r t A i r Temperature 130 5.3 AVERAGE OR MID-POINT SURFACE TEMPERATURE 130 5 .3 .1 R e s u l t s : F a c e t M i d - P o i n t Temperature 133 5 . 3 . 2 R e s u l t s : Average Face t Temperature 138 5.4 SUMMARY 141 V l l CHAPTER 6. CANYON TEMPERATURES, RADIATION, AND COOLING 151 6.1 INTRODUCTION 151 6.2 SURFACE TEMPERATURE DISTRIBUTIONS 151 6 .2 .1 The D i u r n a l V a r i a t i o n of S u r f a c e H e a t i n g and 151 C o o l i n g i n the Model Canyons 6 . 2 . 2 Temperature D i s t r i b u t i o n s on Canyon F a c e t s 156 6 . 2 . 3 C o o l i n g of Canyon F a c e t s F o l l o w i n g Sunset 160 6.3 AIR TEMPERATURES 165 6 .3 .1 Average Canyon A i r Temperatures 165 6 . 3 . 2 S p a t i a l D i s t r i b u t i o n of Canyon A i r Temperatures 168 6.4 LONG-WAVE RADIATION 171 6.5 SUMMARY OF RESULTS: TEMPORAL AND SPATIAL VARIATION 176 OF IN-CANYON TEMPERATURES AND RADIATION 6.6 CANYON VERSUS OPEN SITE SURFACE COOLING 179 6 .6 .1 Canyon and Open S u r f a c e C o o l i n g : S u r f a c e 181 Geometry C o n t r o l s 6 .6 .2 Canyon and Open S u r f a c e C o o l i n g : Atmospher ic 195 C o n t r o l s 6.7 SUMMARY OF RESULTS: CANYON AND OPEN SURFACE COOLING 197 CHAPTER 7. CONCLUSIONS 199 7.1 ACHIEVEMENT OF THE RESEARCH OBJECTIVES 199 7.2 RECOMMENDATIONS AND FUTURE RESEARCH 200 REFERENCES 203 Vlll APPENDIX A . SPECIFICATION OF SENSOR TRAVERSING SPEED AND 211 DELAY INTERVAL A.1 INTRODUCTION 211 A . 2 CANYON WALLS 214 A . 3 CANYON TOP AND FLOOR 217 A . 4 DELAY INTERVAL 220 A . 5 CONCLUSIONS 221 APPENDIX B . DATA PROCEDURES 222 B . 1 DESCRIPTION OF RECORDED DATA 222 B . 2 ASSIGNMENT OF TRAVERSED DATA TO GRID-POINTS 222 B . 3 SIGNAL FILTERING THE SURFACE TEMPERATURE 223 B . 3 . 1 Methods Used 225 B . 4 EXTRAPOLATION OF ADDITIONAL SURFACE TEMPERATURES 235 ON CANYON WALLS APPENDIX C . CALIBRATION OF THE BARNES PRT-4A INFRARED 240 THERMOMETER APPENDIX D . ERRORS 243 D.1 INTRODUCTION 243 D . 2 SURFACE TEMPERATURE ERRORS 24 4 D . 3 EMI SSIVITY ERRORS 247 D . 4 RADIATION ERRORS 24 9. D . 4 . 1 L i c t E r r o r s 249 D . 4 . 2 L * 0 E r r o r s 250 D . 4 . 3 L * E r r o r s 251 ix D . 4 . 4 L D E r r o r s APPENDIX E. MODEL INPUT APPENDIX F. ADDITIONAL VALIDATION DATA SETS F . 1 August 2/3 F . 2 August 8/9 F . 3 August 1 1/12 F . 4 August 14/15 F . 5 August 22/23 F . 6 August 23/24 APPENDIX G . STATISTICAL INDICIES OF MODEL PERFORMANCE G.1 SUMMARY UNIVARIATE STATISTICS G.2 COEFFICIENTS OF LEAST-SQUARES LINEAR REGRESSION G.3 MEASURES OF ERROR G.4 INDICATORS OF CORRELATION X LIST OF TABLES T a b l e 2.1 T a b l e 2.2 T a b l e 2.3 T a b l e 2.4 T a b l e 2.5 T a b l e 3.1 T a b l e 3.2 T a b l e 3.3 T a b l e 3 .5 T a b l e 3.6 T a b l e 3.7 T a b l e 3.8 T a b l e 4.1 T a b l e 4.2 T a b l e 4.3 T a b l e 4.4 Canyon H/W, Dimens ions , and Number of G r i d - 24 P o i n t s . Comparison of M i n i a t u r e and F u l l - S i z e Net 26 Radiometer S p e c i f i c a t i o n s . E m i s s i v i t y of Canyon S u r f a c e s . 39 S u r f a c e Temperature P r e c i s i o n . 40 Summary of F i x e d V e r s u s T r a v e r s e d Radiometer 49 T e s t s . Components of the R a d i a t i o n Budget C a l c u l - 54 a t e d i n the A r n f i e l d Model wi th O p t i o n s and R e q u i r e d Input F l u x e s . A r n f i e l d Model F o r t r a n S u b r o u t i n e s . 57 Approximate Compile and E x e c u t i o n Times for 58 the A r n f i e l d Model U s i n g an I s o t r o p i c Rad- iance D i s t r i b u t i o n . Tes t 1 of Model S e n s i t i v i t y to the Number of 68 Model G r i d - P o i n t s . Tes t 2 of Model S e n s i t i v i t y to the Number of 68 Model G r i d - P o i n t s . T e s t 1: V i e w - f a c t o r Compar i son . 69 Comparison of V i e w - f a c t o r C a l c u l a t i o n s : 72 W a l l A / F l o o r . Model Performance S t a t i s t i c s : P o i n t Mode 100 V a l i d a t i o n S e t . J u l y 19/20, J u l y 21/22 1988. Model Performance S t a t i s t i c s : August 1/2, 107 I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 1/2, 108 H o u r l y Averaged P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 3 /4 , 109 H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . XI T a b l e 4.5 Comparison of RMSE and d S t a t i s t i c s C a l c u l - a t e d From H o u r l y Averaged Data U s i n g I s o - t r o p i c and the Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n s . T a b l e 4.6 Model Performance S t a t i s t i c s : Aug . 10/11 H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . T a b l e 4.7 Model Performance S t a t i s t i c s : Aug . 12/13, H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . T a b l e 5.1 Summary of Average D i f f e r e n c e s by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 1/2, H/W=2.0) . T a b l e 5.2 Summary of Average L 0 D i f f e r e n c e s by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 1/2, H/W=2.0) . T a b l e 5.3 Summary of Average L * D i f f e r e n c e s by Facet I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 1/2, H/w=2.0) . T a b l e 5.4 Summary of Average L^ D i f f e r e n c e s by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 3 / 4 , H/W=1.0) . 1 1 1 T a b l e 5.5 T a b l e 5.6 T a b l e 5.7 T a b l e 5.8 T a b l e 6.1 T a b l e A.1 T a b l e A . 2 Summary of Average L Q D i f f e r e n c e s by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 3 / 4 , H/W=1.0) . Summary of Average L * D i f f e r e n c e s by Facet I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 3 / 4 , H/W=1.0) . Percentage D i f f e r e n c e of F l u x e s L e a v i n g the Canyon Top f o r V a r i o u s S u r f a c e Temperature A p p r o x i m a t i o n Schemes . (Aug. 1/2 1988 H/W=2.0) Percentage D i f f e r e n c e of F l u x e s L e a v i n g the Canyon Top f o r V a r i o u s Sur face Temperature A p p r o x i m a t i o n Schemes . (Aug. 3/4 1988 H/W=1.0) D a i l y Average M e t e o r o l o g i c a l C o n d i t i o n s 0600-1800 (PDT) A u g u s t , 1988. True and Measured R a d i a t i o n For a Sensor T r a v e r s e d A c r o s s a Canyon W a l l . A t t e n u a t i o n and Lag Time of Canyon F l o o r / T o p Data for V a r i o u s T r a v e r s e T imes . 1 16 120 141 142 143 144 145 1 46 147 1 48 180 215 219 x i i T a b l e A . 3 T a b l e D.1 T a b l e D .2a T a b l e D.2b T a b l e D . 3 a T a b l e D .3b T a b l e D.4 T a b l e D .5 T a b l e D .6a T a b l e D.6b T a b l e D.6c T a b l e E .1 T a b l e F .1 T a b l e F . 2 T a b l e F . 3 T a b l e F . 4 T a b l e F . 5 Adjustment Completed to a Step Change i n R a d i a t i o n U s i n g a M i n i a t u r e Net Rad iometer . E r r o r Summary: S u r f a c e Temperature . E r r o r Summary: E m i s s i v i t y ( T s , T k , T r ) . P r o b a b l e E r r o r A n a l y s i s : E m i s s i v i t y . E r r o r Summary: L ^ c t . P r o b a b l e E r r o r A n a l y s i s : L j c t . E r r o r Summary: L * 0 . E r r o r Summary L * ( T r a v e r s e d ) . E r r o r Summary L 0 ( T r a v e r s e d ) . Probab le E r r o r A n a l y s i s : L Q . T y p i c a l P r o b a b l e E r r o r s : L c . Model Input f or N o c t u r n a l Model Runs . Model Performance S t a t i s t i c s : August 2 / 3 , I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 8 / 9 , I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 11/12, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 14/15, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . Model Performance S t a t i s t i c s : August 22 /23 , I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . 221 245 248 248 249 250 251 252 253 254 255 257 258 259 260 261 262 Xlll T a b l e F . 6 Model Performance S t a t i s t i c s : August 23/24, 263 I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h (1975) Radiance D i s t r i b u t i o n . xiv LIST OF FIGURES F i g u r e 1.1 F i g u r e 2.1 F i g u r e 2.2 F i g u r e 2.3 F i g u r e 2.4 F i g u r e 2.5 F i g u r e 2 .6a F i g u r e 2 .6b F i g u r e 2.7 F i g u r e 2.8 F i g u r e 2.9 F i g u r e 2.10 S i m p l i f i e d diagram of an urban c a n y o n . 1 Map of s tudy l o c a t i o n . 17 The h o l l o w , t w o - c e l l c o n c r e t e b l o c k . 19 I n c r e a s e i n w a l l v i e w - f a c t o r (\//w) f o r a 21 p o i n t l o c a t e d mid-way up the o p p o s i t e c a n - yon w a l l at mid-canyon for i n c r e a s i n g l e n g t h s of w a l l . Rad ius of the area seen by a rad iometer 27 n e c e s s a r y to a c h i e v e a g iven v i e w - f a c t o r (<//) f o r d i f f e r e n t ins trument h e i g h t s above the s u r f a c e . The 'end e f f e c t ' f o r v i e w - f a c t o r s of a t r a - 29 v e r s e d canyon f a c e t as a t r a v e r s e d r a d i o - meter approaches the end of the f a c e t . O b s t r u c t i o n of \ps by a m i n i a t u r e r a d i o - 30 meter l o c a t e d above a canyon s u r f a c e . Same as f o r F i g . 2 .6a but u s i n g a f u l l - 31 s i z e net r a d i o m e t e r . G e n e r a l i z e d drawing of the Canyon T r a v e r s i n g 33 System ( C T S ) . The CTS as mounted i n the model c a n y o n . 35 S u r f a c e temperature v a l i d a t i o n . 42 Change i n long-wave r a d i a t i o n measured as 46 the ins trument r o t a t e s to and from the c a n - yon t o p . F i g u r e 2. 1 1 T e s t s meter between a f o r Lo at f i x e d p o i n t and t r a v e r s e d r a d i o - 8 on the West w a l l . 47 F i g u r e 2. 12 T e s t s meter between a for L 0 a t f i x e d p o i n t and t r a v e r s e d r a d i o - 3 on the West w a l l . 48 F i g u r e 2. 13 T e s t s meter between a f o r L * at f i x e d p o i n t and t r a v e r s e d r a d i o - 7 on the canyon f l o o r . 50 F i g u r e 2. 14 T e s t s meter between a for L * at f i x e d p o i n t and t r a v e r s e d r a d i o - 4 on the canyon f l o o r . 51 F i g u r e 3.1 The canyon c o o r d i n a t e system. 55 XV F i g u r e 3.2 F i g u r e 3.3 F i g u r e 3.4 F i g u r e 3 .5 F i g u r e 3.6 F i g u r e 3.7 F i g u r e 3.8 F i g u r e 3 .9 F i g u r e 3.10 F i g u r e 3.11 F i g u r e 3.12 F i g u r e 3.13 F i g u r e 3.14 F i g u r e 3.15 F i g u r e 4.1 D e f a u l t and a c t u a l g r i d p a t t e r n used i n the 60 A r n f i e l d model and s c a l e canyon . V i e w - f a c t o r of a p a r a l l e l w a l l element and 63 a p e r p e n d i c u l a r w a l l e l ement . T e s t 1 of model s e n s i t i v i t y . 70 Model c o n t r o l r u n . 75 S e n s i t i v i t y of mode l l ed f l u x e s to s u r f a c e 77 temperature changes at p o i n t 5 on the West w a l l . S e n s i t i v i t y of mode l l ed f l u x e s to s u r f a c e 78 temperature changes at p o i n t 1 on the West w a l l . S e n s i t i v i t y of mode l l ed f l u x e s to s u r f a c e 80 temperature changes at p o i n t 11 on the West w a l l . S e n s i t i v i t y of mode l l ed f l u x e s to s u r f a c e 81 temperature changes at p o i n t 5 on the c a n - yon f l o o r . S e n s i t i v i t y of mode l l ed f l u x e s to s u r f a c e 82 temperature changes at p o i n t 1 on the c a n - yon f l o o r . S e n s i t i v i t y of mode l l ed f l u x e s to equa l 84 s u r f a c e temperature changes at a l l p o i n t s . S e n s i t i v i t y of mode l l ed f l u x e s to L i c t . 86 S e n s i t i v i t y of mode l l ed f l u x e s to e. S u r f a c e 89 temperature d i s t r i b u t i o n used i s t y p i c a l f o r the e a r l y e v e n i n g . . S e n s i t i v i t y of mode l l ed f l u x e s to e. S u r f a c e 90 temperature d i s t r i b u t i o n used i s t y p i c a l f o r the l a t e e v e n i n g . S e n s i t i v i t y of mode l l ed f l u x e s to the r a d - 92 i a n c e d i s t r i b u t i o n u s e d . S c a t t e r p l o t s of mode l l ed and measured l o n g - 98 wave f l u x e s from the ' p o i n t t e s t s ' . The i s o t r o p i c r a d i a n c e d i s t r i b u t i o n i s used f o r mode l l ed L ^ . x v i F i g u r e 4.2 F i g u r e 4.3 F i g u r e 4.4 F i g u r e 4.5 F i g u r e 4.6 F i g u r e 4.7 F i g u r e 4.8 F i g u r e 4.9 F i g u r e 5.1 F i g u r e 5.2 F i g u r e 5.3 S c a t t e r p l o t s of mode l l ed and measured l o n g - wave f l u x e s from the ' p o i n t t e s t s ' . The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i - b u t i o n i s used for mode l l ed L ^ . S c a t t e r p l o t s of mode l l ed and measured l o n g - wave f l u x e s from Aug . 1/2 1988, H/W=2.0 u s i n g the complete da ta s e t . The i s o t r o p i c r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d S c a t t e r p l o t s of mode l l ed and measured l o n g - wave f l u x e s from Aug . 1/2 1988, H/W=2.0 u s i n g the complete data s e t . The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used for mode l l ed L j . S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n data by g r i d - p o i n t f or Aug. 1/2 1988, H/W=2.0. The i s o t r o p i c r a d i a n c e d i s t r i b u t i o n i s used for mode l l ed L ^ . S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n da ta by g r i d - p o i n t f or Aug . 1/2 1988, H/W=2.0. The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used for mode l l ed L j S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n da ta by g r i d - p o i n t f o r Aug. 3/4 1988, H/W=1.0. The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used for mode l l ed S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n da ta by g r i d - p o i n t f or Aug . 10/11 1988, H/W=0.67. The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used f o r mode l l ed S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n da ta by g r i d - p o i n t f o r Aug . 12/13 1988, H/W=1.33l The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used for m o d e l l e d M o d e l l e d long-wave f l u x d i f f e r e n c e s o b - t a i n e d u s i n g the average canyon a i r temp- e r a t u r e i n p l a c e of measured s u r f a c e temp- e r a t u r e . Data i s from Aug . 1/2 1988, H/W=2.0. As per F i g . 5.1 for Aug . 3/4 1988, H/W=1.0. M o d e l l e d long-wave f l u x d i f f e r e n c e s ob- t a i n e d u s i n g the a i r p o r t a i r temperature i n p l a c e of measured s u r f a c e t e m p e r a t u r e . Data i s from Aug . 1/2 1988, H/W=2.0. 99 103 104 105 1 06 1 10 1 13 1 17 127 128 131 F i g u r e 5.4 As per F i g . 5.3 for Aug . 3/4 1988, H/W=1.0, 1 32 X V I ] F i g u r e 5.5 F i g u r e 5.6 F i g u r e 5.7 F i g u r e 5.8 F i g u r e 6.1 F i g u r e 6.2 F i g u r e 6.3 F i g u r e 6.4 F i g u r e 6.5 F i g u r e 6.6 F i g u r e 6.7 F i g u r e 6 .B F i g u r e 6.9 F i g u r e 6.10 M o d e l l e d long-wave f l u x d i f f e r e n c e s ob- 134 t a i n e d u s i n g the m i d - p o i n t f ace t t emperature i n p l a c e of measured s u r f a c e t e m p e r a t u r e . Data i s from Aug . 1/2 1988, H/W=2.0. As per F i g . 5.5 f o r Aug . 3/4 1988, H/W=1.0. 135 M o d e l l e d long-wave f l u x d i f f e r e n c e s o b - 139 t a i n e d u s i n g the average f a c e t t emperature i n p l a c e of measured s u r f a c e t e m p e r a t u r e . Data i s from Aug . 1/2 1988, H/W=2.0. As per F i g . 5.7 for Aug . 3/4 1988, H/W=1.0. 140 D i u r n a l v a r i a t i o n of s e l e c t e d s u r f a c e temp- 153 e r a t u r e s i n a canyon wi th H/W=1.0, Aug . 3 /4 . S p a t i a l and temporal v a r i a t i o n of s u r f a c e 157 temperature d i s t r i b u t i o n s i n canyons w i t h H/W=2.0, H/W=1.0 and H/W=0.41. C o o l i n g of s e l e c t e d p o i n t s on canyon f a c e t s ; 162 H/W=2.0 canyon , August 1/2. (a) West w a l l , (b) E a s t w a l l , (c) F l o o r . C o o l i n g of s e l e c t e d p o i n t s on canyon f a c e t s ; 163 H/W=1.0 canyon , August 3 / 4 . (a) West w a l l , (b) E a s t w a l l , (c) F l o o r . C o o l i n g of s e l e c t e d p o i n t s on canyon f a c e t s ; 164 H/W=0.41 canyon , August 22 /23 . (a) West w a l l , (b) E a s t w a l l , (c) F l o o r . Average canyon a i r temperature and a i r temp- 166 e r a t u r e s r e c o r d e d at Vancouver I n t e r n a t i o n a l A i r p o r t f o r (a) Aug . 1/2, (b) Aug . 3 / 4 , (c) Aug . 22 /23 . Average canyon a i r temperature and a i r temp- 167 e r a t u r e s r e c o r d e d at Vancouver I n t e r n a t i o n a l A i r p o r t f o r (a) A u g . 2 / 3 , (b) Aug . 10/11, (c) A u g . 11/12. S p a t i a l and temporal v a r i a t i o n s of a i r temp- 169 e r a t u r e above canyon f a c e t s i n ( top) a canyon wi th H/W=2.0, (middle) H/W=1.0 canyon , and (bottom) 0.41 canyon . S p a t i a l and temporal v a r i a t i o n s of mode l l ed 172 r a d i a t i v e f l u x e s i n an H/W=2.0 canyon . S p a t i a l and temporal v a r i a t i o n s of mode l l ed 173 .^radiative f l u x e s i n an H/W=1.0 canyon . X V I I ] F i g u r e 6.11 F i g u r e 6.12 F i g u r e 6.13 F i g u r e 6.14 F i g u r e 6.15 F i g u r e 6.16 F i g u r e 6.17 F i g u r e 6.18 F i g u r e 6.19 F i g u r e 6.20 F i g u r e 6.21 F i g u r e A.1 F i g u r e A . 2 F i g u r e A . 3 F i g u r e B . 1 a F i g u r e B . l b S p a t i a l and tempora l v a r i a t i o n s of mode l - l e d r a d i a t i v e f l u x e s i n an H/W=0.41 canyon . Bottom: Comparison of canyon and open s i t e c o o l i n g o v e r l a i d w i t h L * Q « L i c t ( top) and wind speed ( u ) , (middle) are a l s o p r e s e n t e d . Data from Aug. 1/2, H/W=2.0 As per F i g u r e 6 .12 . canyon H/W=1.33. As per F i g u r e 6 .12 . canyon H/W=1.0. As per F i g u r e 6 .12 . canyon H/W=1.0. As per F i g u r e 6 .12 . canyon H/W=0.67. As per F i g u r e 6 .12 . canyon H/W=0.41. Data from Aug . 14/15, Data from Aug . 3 / 4 , Data from Aug . 8 / 9 , Data from Aug . 11/12, Data from Aug . 20 /21 , C o o l i n g o f urban and r u r a l s u r f a c e s observed from (a) s c a l e models , (b) the c i t i e s of M o n t r e a l (H/W=3.29), Vancouver (H/W=1.50) and U p p s a l a (H/W=0.76). Temporal changes of heat i s l a n d i n t e n s i t y g e n e r a t e d u s i n g (a) a s c a l e model and (b) observed i n t e n s i t i e s from M o n t r e a l , Vancouver and U p p s a l a . Temporal development of the temperature d i f f e r e n c e between the m i d - p o i n t of the canyon f l o o r and the open c o n c r e t e . S u r f a c e temperature d i f f e r e n c e s between the canyon and open s i t e s at e i g h t hours a f t e r sunset f o r the f i v e H/W t e s t e d . M o d e l l e d d i s t r i b u t i o n s of , Lo and L * over canyon f a c e t s . O c t . 10, 1987. Sensor response to a ramp change i n L Q or L * over a w a l l . Sensor response to a s i n u s o i d a l change i n L 0 or L * . S u r f a c e t e m p e r a t u r e s , even ing of Aug . 1/2. S u r f a c e t e m p e r a t u r e s , even ing of Aug . 25/26 X I X F i g u r e B.2 Power spectrum of r e s c a l e d and d e t r e n d e d data f o r p o i n t s 1 t o 5 on the West w a l l on Aug. 1/2. F i g u r e B .3 Detrended s u r f a c e t emperature of p o i n t 1 on the West w a l l ; A u g . 1 / 2 . F i g u r e B.4 Power spectrum of r e s c a l e d and d e t r e n d e d s u r f a c e temperature shown i n F i g u r e B . 3 . F i g u r e B . 5 T r a n s f e r f u n c t i o n of a low-pass f i l t e r w i t h a s top frequency (SF) of 0.03 f o r d i f f e r e n t t r u n c a t i o n s of the F o u r i e r s e r i e s . F i g u r e B .6 F i l t e r e d s u r f a c e temperature on Aug . 1/2 u s i n g v a r i o u s low-pass f i l t e r s . F i g u r e B .7 Power s p e c t r a l d e n s i t y of s u r f a c e temper- a t u r e ( p o i n t 1 on the West w a l l ) f o r Aug . 1/2 F i g u r e B.8 F i l t e r f u n c t i o n d e r i v e d from an o p t i m a l f i l - t e r compared to a low-pass f i l t e r u s i n g 105 terms and a s top frequency of 0 .015 . F i g u r e B .9 F i l t e r e d s u r f a c e temperature u s i n g an ' o p t i m a l ' l ow-pass f i l t e r . F i g u r e C.1 C a l i b r a t i o n c u r v e f o r Barnes Model PRT-4A I n f r a r e d Thermometer. X X LIST OF SYMBOLS B Rate of change of a c t u a l r a d i a t i o n wi th t ime C C o n s t a n t CTS Canyon t r a v e r s i n g system C ( f ) Measured s i g n a l E A b s o l u t e e r r o r H/W H e i g h t - t o - w i d t h r a t i o IF Canyon f a c e t number IP G r i d - p o i n t number on f a c e t 1 0 Instrument output K Degrees K e l v i n L Lag Lp A n g u l a r ' d i s t r i b u t i o n o f s k y - d e r i v e d long-wave r a d i a t i o n L ^ I n c i d e n t long-wave r a d i a t i o n on a s u r f a c e L 0 O u t g o i n g long-wave r a d i a t i o n from a s u r f a c e L * Net long-wave r a d i a t i o n of a s u r f a c e L * D Net long-wave r a d i a t i o n of the open s u r f a c e L i c t I n c i d e n t long-wave r a d i a t i o n at the p l a n e of the canyon t o p MAE Mean a b s o l u t e e r r o r MBE Mean b i a s e r r o r N ( f ) N o i s e f r e q u e n c i e s 0 Observed mean P P r e d i c t e d mean Pe P r o b a b l e e r r o r Pc Power - s p e c t r a l d e n s i t y XXI Q R a d i a t i o n (measured) Q * Net r a d i a t i o n Q A A d v e c t e d f l u x , a c t u a l r a d i a t i o n (Appendix 1) L a t e n t heat f l u x Q F A n t h r o p o g e n i c heat f l u x Q Q S u b s u r f a c e heat f l u x Q H S e n s i b l e heat f l u x A Q S Change of s t o r a g e heat f l u x R Roof width RMSE Root mean square e r r o r RMSEs S y s t e m a t i c p o r t i o n of the root mean square e r r o r RMSEu U n s y s t e m a t i c p o r t i o n of the root mean square e r r o r RSTOP Convergence c r i t e r i o n for canyon m u l t i p l e r e f l e c t i o n r o u t i n e SF Stop frequency S ( f ) S i g n a l f r e q u e n c i e s T a A i r temperature T c C a v i t y temperature R a d i a t i v e temperature of the sky T r Apparent s u r f a c e temperature measured by an i n f r a r e d thermometer T s S u r f a c e temperature UM Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n f o r s k y - d e r i v e d long-wave r a d i a t i o n a I n t e r c e p t of l e a s t squares r e g r e s s i o n l i n e b S lope of l e a s t squares r e g r e s s i o n , s lope of the UM r a d i a n c e d i s t r i b u t i o n , t o t a l l e n g t h of p lane element f o r v i e w - f a c t o r c a l c u l a t i o n s (Chapter 3.) c I n t e r c e p t of the UM r a d i a n c e d i s t r i b u t i o n x x i i d Index of agreement e E x p o n e n t i a l f Frequency h H e i g h t n Number of da ta C o e f f i c i e n t of d e t e r m i n a t i o n s 0 Observed s t a n d a r d d e v i a t i o n Sp P r e d i c t e d s t a n d a r d d e v i a t i o n t Time x A l o n g - c a n y o n a x i s , o r i g i n i s a t canyon m i d - l e n g t h y D i s t a n c e a r o s s canyon f l o o r from w a l l A , d i s t a n c e from p l a n e element f o r v i e w - f a c t o r c a l c u l a t i o n s y ' D i s t a n c e a c r o s s canyon top from w a l l A z D i s t a n c e up W a l l A , measured from f l o o r z ' D i s t a n c e up W a l l B, measured from f l o o r Z Summation a A t t e n u a t i o n f a c t o r of sensor response 0 A n g l e 6 D i f f e r e n t i a l e E m i s s i v i t y u Thermal admit tance * 3.1415927 a S t e f a n Boltzmann Constant (5.67 x 10~ 8 W m~ 2 K " 4 ) T Time c o n s t a n t i / > s Sky v i e w - f a c t o r XX111 A z i m u t h a l ang le from N o r t h A n g u l a r f requency of o s c i l l a t i o n Degrees C e l c i u s X X I V ACKNOWLEDGEMENTS T h i s t h e s i s r e p r e s e n t s the c u l m i n a t i o n of work conducted over the pas t t h r e e y e a r s . The s u c c e s s f u l c o m p l e t i o n of t h i s p r o j e c t c o u l d not have been o b t a i n e d wi thout the a s s i s t a n c e of a l a r g e number of p e o p l e , each of whom has p l a y e d an important r o l e . To my s u p e r v i s o r s , D r . T . R . Oke and D r . D . G . Steyn I owe many thanks to the support they have shown t h r o u g h o u t . They were a lways a v a i l a b l e to answer my q u e s t i o n s and t h e i r h e l p i s g r a t e f u l l y acknowledged. D r . R . J . Co le a l s o took a keen i n t e r e s t i n the p r o j e c t and p r o v i d e d u s e f u l comments. D r . A . J . A r n f i e l d p r o v i d e d the n u m e r i c a l model which was v a l i d a t e d and a s s i s t e d in the i n i t i a l i n s t a l l a t i o n of the model at U . B . C . H i s on -go ing support has been most h e l p f u l . M r . J . S k a p s k i and M r . H . Kozlow des igned and c o n s t r u c t e d the canyon t r a v e r s i n g sys tem, wi thout which t h i s work would not have been p o s s i b l e . They are to be h i g h l y commended for t h i s f ea t of e n g i n e e r i n g and t h e i r p a t i e n c e in d e a l i n g wi th the many t r y i n g problems which were encountered i n the d es ign and c o n s t r u c t i o n phases . M i s s . S. Tewnion and M r . G . F r i c k s k a performed a d m i r a b l y as f i e l d a s s i s t a n t s d u r i n g the data c o l l e c t i o n p e r i o d and G . F r i s k a a l s o h e l p e d d u r i n g the d e s i g n and t e s t i n g p e r i o d . Many thanks go to my f e l l o w graduate s t u d e n t s , e s p e c i a l l y D r . S. Grimmond and D r . H . C leugh who he lped wi th the i n s t r u m e n t a t i o n in 1987 and 1988, D r . H . P . Schmidt and M. Roth f o r p r o v i d i n g a s s i s t a n c e w i t h my at tempts at F o u r i e r A n a l y s i s , and S. Robeson and J . Schmok who, as my i n i t i a l o f f i c e - m a t e s , broadened my academic h o r i z o n s . A l s o d e s e r v i n g of c r e d i t are my f a m i l y and S. Jenks for t h e i r encouragement and support d u r i n g the p r o j e c t . The f i n a n c i a l a s s i s t a n c e of the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l (NSERC) i n the form of a s c h o l a r s h i p i s g r a t e f u l l y acknowledged. The p r o j e c t was funded by NSERC g r a n t s to D r . Oke. The f i e l d s i t e was made a v a i l a b l e by the Canadian Department of T r a n s p o r t . 1 CHAPTER 1. INTRODUCTION 1.1 RESEARCH OBJECTIVES The p r i m a r y aim of t h i s r e s e a r c h i s the v a l i d a t i o n of a n u m e r i c a l model ( A r n f i e l d , 1976, 1982) which e s t i m a t e s r a d i a t i v e f l u x e s w i t h i n urban c a n y o n s . An urban canyon i s composed of the p r i n c i p a l s u r f a c e e lements ( f a c e t s ) which e x i s t between two b u i l d i n g s , the b u i l d i n g w a l l s (canyon w a l l s ) and ground (canyon f l o o r ) ( F i g u r e 1 ) . The canyon f l o o r i s o f t e n a s t r e e t system c o n s i s t i n g of r o a d s , s i d e w a l k s and v e g e t a t i v e cover which d i f f e r i n t h e i r p r o p o r t i o n s w i t h l o c a t i o n i n the urban a r e a . A canyon- a i r volume i s c o n t a i n e d by the canyon f a c e t s and i s bounded at the t o p by an imag inary h o r i z o n t a l p l a n e a t r o o f - l e v e l and at the ends by v e r t i c a l p l a n e s a t the end of the b u i l d i n g s . Canyon - air volume F i g u r e 1.1 S i m p l i f i e d d iagram of an urban c a n y o n . From Oke (1978) . 2 Urban canyons (p lus the i n t e r v e n i n g r o o f s ) are the b a s i c c o m b i n a t i o n of v e r t i c a l and h o r i z o n t a l s u r f a c e s i n urban areas w h i c h , t o g e t h e r can be used to r e p r e s e n t the t r u e c i t y s u r f a c e . V a l i d a t i o n of t h i s model f o r the e s t i m a t i o n of urban s u r f a c e a l b e d o s has been r e c e n t l y completed by A r n f i e l d (1988) . The model v a l i d a t i o n h e r e i n encompasses v a r i o u s r a t i o s of b u i l d i n g h e i g h t to canyon width (H/W) f o r a s i n g l e canyon o r i e n t a t i o n , but i s r e s t r i c t e d to n o c t u r n a l long-wave f l u x e s . T h i s work w i l l f u r t h e r d e f i n e the a b i l i t y of the model to e s t i m a t e long-wave f l u x e s . The second o b j e c t i v e i s to t e s t the e f f e c t of u s i n g v a r i o u s a p p r o x i m a t i o n s to the f u l l model input d a t a . These t e s t s are of p r a c t i c a l use because data at the s c a l e and r e s o l u t i o n used f o r model v a l i d a t i o n are not n o r m a l l y a v a i l a b l e . A t h i r d o b j e c t i v e i s to examine the e f f e c t s of urban s u r f a c e geometry upon the c o o l i n g of urban a r e a s , e s p e c i a l l y , the i n - canyon s p a t i a l and t empora l v a r i a t i o n s of long-wave r a d i a t i o n and a i r and s u r f a c e t e m p e r a t u r e s . 1.2 APPLICATIONS/SIGNIFICANCE GF THE RESEARCH Model v a l i d a t i o n i s , by i t s e l f , a very important p a r t of the s c i e n t i f i c p r o c e s s of e x p e r i m e n t a t i o n ( F l u e c k , 1978; I d s o , 1987). Too o f t en i n the d i s c i p l i n e of urban c l i m a t o l o g y v a l i d a t i o n s t u d i e s have not kept pace wi th new m o d e l l i n g e n d e a v o u r s , r e s u l t i n g i n a s i t u a t i o n where many u n t r i e d models e x i s t , and t h e i r u s e f u l n e s s as p r e d i c t i v e t o o l s i s thereby 3 l i m i t e d . T h i s r e s e a r c h u t i l i z e s a measurement programme t a i l o r e d s p e c i f i c a l l y for model v a l i d a t i o n . I t a l l o w s v a l i d a t i o n of model p r e d i c t i o n s for i n d i v i d u a l g r i d - p o i n t s w i t h i n the canyon over s h o r t t ime p e r i o d s , and for both s p a t i a l l y and t e m p o r a l l y averaged d a t a . The a v a i l a b i l i t y of a complete and a c c u r a t e data se t f or model i n p u t , makes i t p o s s i b l e to t e s t a p p r o x i m a t i o n schemes which may be used in p l a c e of the f u l l model input d a t a . T h i s i s an improvement upon model s e n s i t i v i t y a n a l y s e s which cannot be done wi thout f i r s t hav ing c o r r e c t l y measured the t r u e model input c o n d i t i o n s . One of the o b j e c t i v e s of urban c l i m a t o l o g y i s to have f i n d i n g s a p p l i e d to the p r o c e s s of urban p l a n n i n g and d e s i g n (Oke, 1984). Urban s u r f a c e geometry i s one parameter of urban d e s i g n which has been shown to have v a r i o u s c l i m a t i c impacts upon the o b j e c t s and organisms l o c a t e d w i t h i n the urban geometry. Oke (1988) d e s c r i b e s urban geometr ies d e f i n e d only by the parameters H/W and b u i l d i n g d e n s i t y which are best s u i t e d to maximize s h e l t e r , p o l l u t a n t d i s p e r s i o n , urban warmth, and s o l a r a c c e s s . The p r e s e n t work p r o v i d e s f u r t h e r i n s i g h t i n t o the r e d u c t i o n of c o o l i n g i n urban areas a r i s i n g from a l t e r e d s u r f a c e geometry (H/W) under d i f f e r e n t weather c o n d i t i o n s . W h i l e the focus of t h i s r e s e a r c h i s upon m i c r o s c a l e canyon p r o c e s s e s , the knowledge of c o n d i t i o n s w i t h i n urban canyons a i d s i n the r e s e a r c h of p r o c e s s e s o c c u r r i n g at l a r g e r s c a l e s in the a t m o s p h e r i c boundary l a y e r over urban a r e a s . Of p a r t i c u l a r importance i s the a b i l i t y to a c c u r a t e l y model the long-wave f l u x e s l e a v i n g the urban canyons which form the lower boundary 4 c o n d i t i o n s of the atmosphere above . Thus , t h i s r e s e a r c h can c o n t r i b u t e to the u n d e r s t a n d i n g of p r o c e s s e s o c c u r r i n g at the m e s o - s c a l e . 1.3 APPROACHES TO THE STUDY OF THE EFFECTS OF URBAN SURFACE GEOMETRY ON THE CLIMATE WITHIN URBAN AREAS T h i s s e c t i o n b r i e f l y rev iews the three major approaches which have been used to i n v e s t i g a t e the e f f e c t s of urban s u r f a c e geometry on the c l i m a t e w i t h i n urban a r e a s . Examples are rev iewed and the approaches e v a l u a t e d w i t h r e s p e c t to t h e i r s u i t a b i l i t y towards a c h i e v i n g the presen t o b j e c t i v e s . The r e s e a r c h methodology adopted i s o u t l i n e d i n S e c t i o n 1.4. 1.3.1 O b s e r v a t i o n a l S t u d i e s Methods of o b s e r v a t i o n i n urban c l i m a t o l o g y have been rev iewed by Oke (1984) who deve loped a t w o - t i e r c l a s s i f i c a t i o n system based upon two se t s of c o n t r o l s u n d e r l y i n g the urban c l i m a t e ; t u r b u l e n t boundary l a y e r s and urban m o r p h o l o g i c a l u n i t s . The major d i s t i n c t i o n l i e s between the urban canopy l a y e r ( U C L ) , dominated by the i n d i v i d u a l roughness e lements and t h e i r s u r f a c e p r o p e r t i e s , and the urban boundary l a y e r (UBL) which r e p r e s e n t s the i n t e g r a t e d e f f e c t s of the UCL w i t h i n a t u r b u l e n t l a y e r combining the s u r f a c e and mixed l a y e r s (Oke, 1976, 1984). 5 O b s e r v a t i o n a l s t u d i e s have been based upon e i t h e r i n t e n s i v e • measurement i n a s i n g l e urban canyon or o b s e r v a t i o n s taken from a number of urban canyons w i t h d i f f e r e n t p r o p e r t i e s . Examples of the f i r s t type i n c l u d e : the a lbedo of a canyon system (Nunez, 1975), the long-wave r a d i a t i v e f l u x d i v e r g e n c e and n o c t u r n a l c o o l i n g w i t h i n a canyon (Nunez and Oke 1976), and the energy ba lance of an urban canyon (Nunez and Oke 1977), a l l based upon measurements o b t a i n e d i n a s i n g l e canyon in Vancouver , B . C . A recent s tudy by Nakamura and Oke (1988) focussed upon the temporal development of wind , temperature and s t a b i l i t y c o n d i t i o n s i n an urban canyon . Research u s i n g measurements from a range of canyons i n c l u d e the v a s t number of urban heat i s l a n d s t u d i e s which i n c l u d e measurements of a i r a n d / o r s u r f a c e temperatures w i t h i n the urban canopy l a y e r . Two recent examples which have focussed upon the r o l e of the urban s u r f a c e geometry i n m o d i f i y i n g the observed a i r and s u r f a c e temperatures i n urban canyons are B a r r i n g et. al . (1985) i n Malmo, Sweden and Yamashita et al. (1986) i n the Tama R i v e r B a s i n (Tokyo suburbs) i n J a p a n . The m i c r o c l i m a t e of s idewalk s u r f a c e s w i t h i n a range of urban canyons has a l s o been examined ( T u l l e r , 1973). The use of a s i n g l e canyon means the r e s u l t s are on ly s t r i c t l y a p p l i c a b l e to t h a t s i t e , and c a u t i o n must be e x e r c i s e d when t r y i n g to g e n e r a l i z e . The g e n e r a l r e l a t i o n s h i p s which have been demonstrated between s u r f a c e geometry and a i r and s u r f a c e temperature must be c a r e f u l l y a n a l y z e d wi th r e s p e c t to the c o n d i t i o n s under which they were ga thered and the presence of 6 other i n f l u e n c i n g f a c t o r s , n e v e r t h e l e s s , o b s e r v a t i o n a l s t u d i e s have been u s e f u l . 1 .3.2 S c a l e M o d e l l i n g S c a l e m o d e l l i n g can overcome some of the l i m i t a t i o n s imposed by the n a t u r a l v a r i a b i l i t y i n h e r e n t i n f i e l d o b s e r v a t i o n s ( e . g . canyon d imens ions and p r o p e r t i e s and weather ) . L o g i s t i c a l c o n s t r a i n t s i n v o l v e d in i n s t r u m e n t i n g and u s i n g ' ' r e a l ' urban canyons may a l s o be a v o i d e d . For the case of s u r f a c e geometry e f f e c t s upon the canyon temperature and r a d i a t i v e b a l a n c e , use of a s c a l e d model a l l o w s a range of s u r f a c e geometr ies to be t e s t e d under d i f f e r e n t c o n d i t i o n , a f l e x i b i l i t y not u s u a l l y a v a i l a b l e in f i e l d s t u d i e s . For the r e s u l t s of a s c a l e model to a p p l y to the r e a l wor ld however, the model must meet c e r t a i n s c a l i n g c r i t e r i a for the v a r i a b l e s of i n t e r e s t . For t h i s r e a s o n , s c a l e m o d e l l i n g w i t h i n canyons has c o n c e n t r a t e d upon r a d i a t i v e t r a n s f e r p r o c e s s e s in which the t u r b u l e n t f l u x e s can be n e g l e c t e d . S e v e r a l examples of s c a l e m o d e l l i n g of canyon p r o c e s s e s e x i s t . Oke (1981) used a s c a l e model b u i l t of plywood and e n c l o s e d by a p o l y e t h y l e n e v t e n t ' " t o r e p r e s e n t r u r a l and urban s u r f a c e s of v a r y i n g geometry i n a s t u d y of the n o c t u r n a l r a d i a t i v e c o o l i n g . S i m u l a t i o n c o n d i t i o n s were r e s t r i c t e d to calm and c l o u d l e s s n i g h t s d u r i n g which a n t h r o p o g e n i c heat r e l e a s e was of n e g l i g i b l e i m p o r t a n c e . Comparison of model r e s u l t s wi th f i e l d o b s e r v a t i o n s showed good agreement. Changing the geometry of the 7 urban model by a l t e r i n g the h e i g h t to width r a t i o (H/W), r e s u l t e d i n a s lowing of the r a t e of temperature decrease as H/W i n c r e a s e d . H igher H/W r a t i o s a l s o l e d to an i n c r e a s e of the t ime taken f o r the urban heat i s l a n d i n t e n s i t y to r e a c h a maximum. The importance of the sky-v iew f a c t o r (^ s) in d e c r e a s i n g the f l u x of net long-wave r a d i a t i o n a t the f l o o r of the canyon and thus on c o o l i n g was n o t e d . R e s u l t s of a thermal admit tance exper iment p a r a l l e l e d f i e l d o b s e r v a t i o n s , w i th the c o n c r e t e c i t y g e n e r a t i n g a l a r g e heat i s l a n d . In a s i m i l a r s t u d y , Johnson and Watson (1987) used s c a l e m o d e l l i n g t e c h n i q u e s to v a l i d a t e a s imple n u m e r i c a l model deve loped to approximate the c o o l i n g or h e a t i n g of a t y p i c a l urban s u r f a c e under c o n d i t i o n s when r a d i a t i v e heat t r a n s f e r i s the dominant heat t r a n s f e r mechanism. T h e i r model was s imply two sheets of plywood g l u e d toge ther and ins t rumented w i t h thermocouples on each s u r f a c e and between the l a y e r s . The plywood shee t , r e p r e s e n t i n g the f l o o r of the canyon , was then p l a c e d on the open top of a ches t f r e e z e r , whose w a l l s take on the r o l e of the canyon w a l l s , and the bottom of which served as the n i g h t sky ( r a d i a t i v e s i n k ) . The f r e e z e r s i d e s and bottom were a l s o ins t rumented w i t h t h e r m o c o u p l e s . Comparison of mode l l ed and observed temperatures e x h i b i t e d good agreement. A i d a (1982) used c o n c r e t e b l o c k s t o . r e p r e s e n t the urban system i n order to s tudy the e f f e c t of s u r f a c e geometry on the f o r m a t i o n of heat i s l a n d s through the d i f f e r e n t i a l a b s o r p t i o n of short -wave r a d i a t i o n . The model was c o n s t r u c t e d outdoors and measurements were taken d u r i n g c l e a r m e t e o r o l o g i c a l c o n d i t i o n s 8 i n d i f f e r e n t seasons . A l i m i t e d number of s imple s u r f a c e geometr ies were examined. The c o n c l u s i o n s i n d i c a t e d tha t the s u r f a c e geometry a lone c o u l d account for an anomolous a b s o r p t i o n of s o l a r r a d i a t i o n , w i t h the major c o n t r o l b e i n g the r a t i o of r e l a t i v e canyon area to the e n t i r e model a r e a . Deeper canyons , ( those wi th a g r e a t e r H/W r a t i o ) , e x h i b i t e d s m a l l e r v a l u e s of a lbedo per canyon top a r e a . D i u r n a l and s e a s o n a l v a r i a t i o n s of the a lbedo were observed i n r e l a t i o n to the z e n i t h angle of the Sun. The r e s u l t s from t h i s s tudy have r e c e n t l y been used ( A r n f i e l d , 1988) in a v a l i d a t i o n of the a lbedo e s t i m a t e s produced by the A r n f i e l d (1976, 1982) model . 1.3.3 M a t h e m a t i c a l Models Due to the c o n s i d e r a b l e r e s o u r c e s needed f o r f u l l - s c a l e o b s e r v a t i o n a l s t u d i e s and the problem of s i m i l t u d e a s s o c i a t e d wi th s c a l e m o d e l l i n g , the use of s t a t i s t i c a l and n u m e r i c a l models has i n c r e a s i n g l y become the methodology of c h o i c e for many urban c l i m a t o l o g i s t s . The use of mathemat ica l models makes i t p o s s i b l e to t e s t d i f f e r e n t boundary c o n d i t i o n s and a l l o w the parameters of i n t e r e s t to v a r y . T h i s i s o therwise d i f f i c u l t or i m p o s s i b l e to a c h i e v e . E x t e n s i v e reviews of urban c l i m a t e m o d e l l i n g have been undertaken by B o r n s t e i n (1986) , B o r n s t e i n and Oke (1981) , Landsberg (1981), and Oke (1974, 1979). Todhunter and T e r j u n g (1988) have r e c e n t l y p r e s e n t e d an i n - d e p t h comparison of three urban c l i m a t e models and t h e i r a b i l i t y to 9 model the effects of urban geometry upon the surface energy budget. Urban climate models may be subdivided into two main classes following the d i f f e r e n t i a t i o n of the boundary layer over c i t i e s by Oke (1976) into the urban canopy layer (UCL) and the urban boundary layer (UBL). Those models which describe the climate of the urban canyon are one type of UCL model and are the subject of interest here. Canyon models evaluate terms of the surface energy balance, which, in the absence of advection, may be written: Q F + K* + L* = Q* + Q p = Q H + Q E + Q G + AQg (1.1) where the f i r s t two terms make up the surface radiation balance (Q*) including the net short-wave (K*) and net-longwave radiation (L*). Q F i s the anthropogenic heat flux, Q H and Q E are the turbulent fluxes of sensible and latent heat respectively, Q G i s the subsurface heat flux, and AQS i s the change of energy storage by the system. The models vary in the i r complexity and emphasis; some concentrate solely upon ra d i a t i v e parameters (eg. A r n f i e l d (1976, 1982), while others include a l l terms. The canyon radiation models are based upon the r e l a t i v e l y straight forward p r i n c i p l e s of radiation geometry, multiple r e f l e c t i o n s , surface properties and the measurement or modelling of incident radiative fluxes. The canyon models are important because they provide results for use in larger scale models of the urban 10 boundary l a y e r and r e s o l v e m i c r o s c a l e v a r i a t i o n s of the energy b a l a n c e . They a l s o a l l o w the development of p l a n n i n g and b u i l d i n g d e s i g n s t r a t e g i e s which i n c o r p o r a t e c l i m a t i c a s p e c t s . Canyon models have been deve loped by: A i d a and Gotoh (1982) , A r n f i e l d (1976, 1982), B r u h l and Zdunkowski (1983) , Nunez (1975), S i e v e r s and Zdunkowski (1985), T e r j u n g and L o u i e (1973, 1974), T e r j u n g and O'Rourke (1980a,b , 1981), Verseghy (1987) , and Zdunkowski and B r u h l (1983) . The model deve loped by S i e v e r s and Zdunkowski (1986) i n c o r p o r a t e s energy ba lance parameters in a t w o - d i m e n s i o n a l n u m e r i c a l s i m u l a t i o n scheme to compute the a i r flow over urban s u r f a c e s and r e p r e s e n t s the s t a t e - o f - t h e - a r t urban c l i m a t e model . 1.4 RESEARCH METHODOLOGY The r e s e a r c h o b j e c t i v e s r e q u i r e the a c c u r a t e s p e c i f i c a t i o n of model i n p u t data and a measured data set which may be used to compare to the model output for v a l i d a t i o n p u r p o s e s . A degree of f l e x i b i l i t y in m o d i f y i n g canyon d imens ions i s d e s i r e d i n o r d e r to examine a range of H/W and t h e i r e f f e c t upon the c o o l i n g of the canyon . The approach adopted here combines e lements of s t u d i e s performed i n both reduced s c a l e and f u l l - s c a l e urban canyons . A s i m p l i f i e d model of an urban canyon was des igned and c o n s t r u c t e d at a reduced s c a l e u s i n g the s i n g l e geometr ic s c a l i n g c r i t e r i a h e i g h t to width r a t i o (H/W) to p r e s e r v e l e n g t h s c a l e s . The model was c o n s t r u c t e d outdoors u t i l i z i n g n a t u r a l d a y l e n g t h and m e t e o r o l o g i c a l c o n d i t i o n s . The reduced s c a l e of 11 the s t r u c t u r e a l l o w e d d i f f e r e n t H/W r a t i o s to be t e s t e d and overcame some of the l o g i s t i c a l problems commonly encountered by o b s e r v a t i o n a l s t u d i e s . The o v e r a l l s c a l i n g of the model i s such t h a t e x t e n s i v e w i t h i n - c a n y o n measurements can be made, a p r o c e d u r e at tempted p r e v i o u s l y o n l y in s i n g l e , f u l l - s i z e c a n y o n s . T h u s , i n a spectrum of model s i z e s t h i s canyon o c c u p i e s an i n t e r m e d i a t e p o s i t i o n between r e a l canyons and those d e s c r i b e d i n S e c t i o n 1 .3 .2 . The model canyon i s not a t r u e s c a l e model i n the manner which most d i m e n s i o n a l a n a l y s t s d e f i n e the term ( S c h u r i n g , 1977; S k o g l u n d , 1967; V e n i k o v , 1969) because f u l l dynamic s i m i l a r i t y i s not a c h i e v e d . Canyons of s i m i l a r H/W e x i s t i n the r e a l wor ld w i t h very d i f f e r e n t a b s o l u t e d i m e n s i o n s , thus no s i n g l e p r o t o t y p e e x i s t s upon which the model i s based . S t r i c t adherance to s c a l i n g c r i t e r i a i s not viewed as a l i m i t a t i o n to the s u c c e s s f u l v a l i d a t i o n of the A r n f i e l d model , the a c c u r a t e p r e s c r i p t i o n of input v a r i a b l e s and the a v a i l a b i l i t y of a matching measured data set d e f i n e a complete v a l i d a t i o n s e t . To i n v e s t i g a t e the r o l e of s u r f a c e geometry upon the c o o l i n g of urban s u r f a c e s , s c a l i n g l i m i t a t i o n s become more important and are o u t l i n e d i n Chapter 2. E x t e n s i v e i n s t r u m e n t a t i o n of the canyon was made u s i n g a c o m b i n a t i o n of s t a t i o n a r y and t r a v e r s i n g s e n s o r s . The l a t t e r were used to o b t a i n two of the three components of the long-wave r a d i a t i v e b a l a n c e . A f u l l y automated t r a v e r s i n g system r e f e r r e d to as the canyon t r a v e r s i n g system (CTS) was e s p e c i a l l y c o n s t r u c t e d f o r t h i s purpose . A second, open, s i t e was 12 i n s t r u m e n t e d near the canyon to a l l o w comparison of canyon c o o l i n g w i t h s u r f a c e s h a v i n g a p lane s u r f a c e geometry. 13 CHAPTER 2. THE CANYON MODEL, INSTRUMENTATION, AND TESTS 2.1 INTRODUCTION T h i s c h a p t e r d e s c r i b e s the canyon model , i t ' s i n s t r u m e n t a t i o n , v a r i o u s t e s t s to a s c e r t a i n i t ' s performance and any l i m i t a t i o n s to the data c o l l e c t e d . F o l l o w i n g a g e n e r a l d e s c r i p t i o n of the canyon model , the f i e l d s i t e , canyon d i m e n s i o n s , m a t e r i a l s used , and c o n s t r u c t i o n methodology are d e s c r i b e d i n g r e a t e r d e t a i l . The canyon i n s t r u m e n t a t i o n i s o u t l i n e d in S e c t i o n 2.4 and t e s t s conducted i n r e l a t i o n to the s u r f a c e t e m p e r a t u r e , s u r f a c e e m i s s i v i t y and t r a v e r s i n g speeds are d e s c r i b e d i n S e c t i o n 2 . 6 . A d e t a i l e d e r r o r a n a l y s i s of measured input i s p r e s e n t e d i n Appendix D. 2.2 SIMULATION METHODOLOGY The c o n d i t i o n s to be s i m u l a t e d are set a c c o r d i n g to the r e s e a r c h o b j e c t i v e s o u t l i n e d i n Chapter 1 and form g u i d e l i n e s f o r the d e s i g n of a s u i t a b l e canyon model as w e l l as important l i m i t a t i o n s when the performance of the model i s a s s e s s e d . The c o n d i t i o n s to be s i m u l a t e d i n o r d e r to ach i eve each of the main o b j e c t i v e s are d e s c r i b e d below. To v a l i d a t e the n i g h t - t i m e long-wave r a d i a t i v e f l u x e s p r e d i c t e d by the A r n f i e l d model r e q u i r e s o n l y the s i m u l a t i o n of 14 urban s u r f a c e geometry and n o c t u r n a l c o n d i t i o n s . The s i m u l a t e d urban s u r f a c e geometry conforms to the requ irements of the A r n f i e l d model (see Chapter 3) by u s i n g the s c a l i n g H/W (he ight to width r a t i o ) of urban canyons to g ive a number r e p r e s e n t i n g the 'openness ' of the c a n y o n . Large v a l u e s of H/W i n d i c a t e a narrow, deep geometry. To match n o c t u r n a l m e t e o r o l o g i c a l c o n d i t i o n s , the model i s opera ted o u t - o f - d o o r s at n i g h t . With the s a t i s f a c t i o n of these two c r i t e r i a , no f u r t h e r r e s t r i c t i o n s e x i s t f o r model v a l i d a t i o n . I n v e s t i g a t i o n of s u r f a c e geometry e f f e c t s on the long-wave r a d i a t i v e ba lance and s u r f a c e c o o l i n g of canyon f a c e t s r e q u i r e s the same s i m u l a t i o n c o n d i t i o n s neces sary for model v a l i d a t i o n . In a d d i t i o n , the assumption that an thropogen ic heat i s of n e g l i g b l e importance i s made. The c o n t r o l by s u r f a c e geometry on s u r f a c e c o o l i n g and r a d i a t i o n i s best expressed under c a l m , c l e a r and dry c o n d i t i o n s which maximize r a d i a t i v e exchange and min imize the t u r b u l e n t f l u x e s of s e n s i b l e and l a t e n t h e a t , (Q H and 0_E r e s p e c t i v e l y ) . These i d e a l c o n d i t i o n s , a l o n g wi th the absence of a d v e c t i o n have been adopted by Oke (1981) in h i s s tudy of the e f f e c t s of s u r f a c e geometry upon the format ion of the urban heat i s l a n d . Use of an o u t s i d e l o c a t i o n p r e c l u d e s the complete e l i m i n a t i o n of these i n f l u e n c e s in t h i s s t u d y . The data c o l l e c t e d may, however, be a n a l y z e d to i n v e s t i g a t e the degree to which i t meets these a s sumpt ions , based upon r e c o r d e d l o c a l m e t e r o l o g i c a l c o n d i t i o n s . On t h i s b a s i s , data may be grouped f o r c o m p a r i s o n . 15 S i m i l a r i t y w i t h r e a l - w o r l d canyons i s l i m i t e d to the geo- m e t r i c s i m i l a r i t y o b t a i n e d through the use of the H/W r a t i o . Canyon m a t e r i a l s and c o n s t r u c t i o n t e c h n i q u e s have not been ex- p l i c i l t y s c a l e d , a l t h o u g h they bear some resemblance to those found i n r e a l , f u l l - s c a l e canyons . Proces se s g o v e r n i n g r a d i a t i v e t r a n s f e r have not been s c a l e d . Oke (1981) c o n s i d e r e d the c h a r - a c t e r i s t i c l e n g t h s c a l e of r a d i a t i v e t r a n s m i s s i o n to be neg- l i g i b l e compared to tha t of h i s s m a l l e r model . The path l e n g t h a v a i l a b l e for gaseous a b s o r b t i o n and r e - e m i s s i o n of r a d i a t i o n w i t h i n the canyon i s l e s s than t h a t a v a i l a b l e i n the r e a l wor ld and i s not s c a l e d . The time a v a i l a b l e for canyon h e a t i n g and c o o l i n g i s governed by a s t r o n o m i c a l and m e t e o r o l o g i c a l f a c t o r s . 2.3 THE CANYON MODEL 2 .3 .1 G e n e r a l D e s c r i p t i o n The model canyon i s composed of two i d e n t i c a l w a l l s a p p r o x i m a t e l y 10 m l o n g and 1 m h i g h o r i e n t e d N o r t h - S o u t h on a base of poured c o n c r e t e . Canyon width i s v a r i e d from 0.5 m to 1.5 a c c o r d i n g to the H/W r a t i o d e s i r e d . Each w a l l i s c o n s t r u c t e d u s i n g a s tack of f i v e o v e r l a p p i n g l a y e r s of 200x200x400 mm two- c e l l , ho l low c o n c r e t e b l o c k s , capped by a s i n g l e l a y e r of 200x200x50 mm s o l i d c o n c r e t e s l a b s wi th the e x c e p t i o n of the s m a l l e s t H/W, (0.41) which used o n l y 3 l a y e r s of b l o c k s . No s u p p o r t i n g framework i s used and the b l o c k s are not j o i n e d by 16 any bonding m a t e r i a l . A 25 mm l a y e r of ex truded p o l y s t y r e n e i n s u l a t i o n i s taped to the e x t e r i o r of the w a l l s to a i d in i s o l a t i n g the i n n e r , a c t i v e canyon s u r f a c e s of i n t e r e s t . The canyon t r a v e r s i n g system ( C T S ) , d e s c r i b e d i n S e c t i o n 2 . 4 . 3 , i s l o c a t e d j u s t N o r t h of the canyon m i d - l e n g t h p o i n t , so tha t the t r a v e r s e d i n s t r u m e n t s and thermocouples were c e n t r e d on the canyon m i d - l e n g t h . The f o l l o w i n g s e c t i o n s p r o v i d e g r e a t e r d e t a i l r e g a r d i n g s p e c i f i c a s p e c t s of the canyon model , the CTS and the canyon i n s t r u m e n t a t i o n . 2 . 3 . 2 S i t e The model s i t e i s a p a r k i n g compound a d m i n i s t e r e d by the Department of T r a n s p o r t a t Vancouver I n t e r n a t i o n a l A i r p o r t . The a i r p o r t i s l o c a t e d on Sea I s l a n d i n the Lower M a i n l a n d r e g i o n of B r i t i s h C o l u m b i a , Canada ( F i g u r e 2 . 1 ) . The s i t e p r o v i d e d the n e c e s s a r y s e c u r i t y , open exposure and a f l a t , c o n c r e t e base for the canyon model . The c o n c r e t e base was f o r m e r l y the foundat ion of a l a r g e b u i l d i n g . A l a r g e hangar to the Northwest of the s i t e c a s t s shadows over the model canyon be fore the time of a c t u a l s u n s e t . Sky v i e w - f a c t o r ( $ s ) e s t i m a t e s f o r the s i t e , u s i n g the method d e s c r i b e d by Steyn (1980) , were i n excess of 0 .99 . 17 18 2 . 3 . 3 M a t e r i a l s To v a l i d a t e t h e A r n f i e l d model the m a t e r i a l s used f o r the canyon f a c e t s a r e i r r e l e v a n t as l o n g as they a r e of known e m i s s i v i t y and s u r f a c e temperature and are homogeneous i n the a l o n g - c a n y o n d i r e c t i o n . T w o - c e l l ho l low c o n c r e t e b l o c k s ( F i g u r e 2.2) were s e l e c t e d f o r use i n the canyon model because: o n l y a s m a l l number a r e n e c e s s a r y to c o n s t r u c t the canyon , they can e a s i l y be hand led by a s i n g l e person a n d , when made i n t o a w a l l , are s t a b l e enough t o not r e q u i r e any s u p p o r t i n g framework. They pos se s sed the f u r t h e r advantages of ready a v a i l a b i l i t y , low c o s t , and d u r a b i l i t y . The s o l i d c o n c r e t e s l a b s used to cap the w a l l s are .manufactured u s i n g m a t e r i a l s s i m i l a r to those used i n the b l o c k s . The i n s u l a t i o n i s an e x t r u d e d p o l y s t y r e n e ' S t y r o f o a m - SM B r a n d ' a v a i l a b l e i n .0.6" m x 2.4 m s h e e t s . The i n s u l a t i o n f u n c t i o n s i n two ways: i t reduces heat l o s s t h rou gh the backs of the canyon w a l l s a t n i g h t , and i t ' s l i g h t c o l o u r i s a good r e f l e c t o r of i n c i d e n t short -wave r a d i a t i o n on the o u t s i d e of the canyon w a l l s - d u r i n g t^ve d a y . 2 . 3 . 4 C o n s t r u c t i o n T e c h n i q u e S t a n d a r d o v e r l a p p i n g b r i c k c o n s t r u c t i o n i s used to b u i l d the w a l l s . No adhesiv-es are used because t e a r down and r e b u i l d i n g of the w a l l i s - n e c e s s a r y t o generate new H/W r a t i o s . The s tyrofoam i n s u l a t i o n i s t a p e d to the e x t e r i o r of the canyon w a l l s u s i n g 19 38 < » 49 SIDE VIEW 32 27 36 « ;> 193 END VIEW 32 193 T O P VIEW 395 F i g u r e 2.2 The h o l l o w , t w o - c e l l c o n c r e t e b l o c k . Dimens ions i n m i l l i m e t r e s . 20 t w o - s i d e d c a r p e t or duct t a p e . Two p i e c e s of lumber 50x100 mm and a p p r o x i m a t e l y 2 m i n l e n g t h were l a i d a c r o s s the canyon ends on the cement base and taped to the cement to p r e v e n t the a c c u m u l a t i o n of d i r t i n s i d e the c a n y o n . 2 . 3 . 5 Canyon He ight and Width Canyon h e i g h t and width are c o n s i d e r e d t o g e t h e r because t h e i r r a t i o s e t s the s c a l i n g parameter H/W. Canyon H/W i s v a r i e d i n a l l but one case (0.41) by. v a r y i n g the width r a t h e r than the h e i g h t . T h i s method i s adopted because the CTS a l l o w s more v a r i a b i l i t y i n the l e n g t h , r a t h e r than the h e i g h t of t r a v e r s e . Maximum canyon width i s l i m i t e d to 1.9 m, l i m i t e d by the l e n g t h of the h o r i z o n t a l d r i v e screw of the CTS ( F i g u r e 2 . 7 ) . The minimum w i d t h i s 0.5 m and i s set by the arc of r o t a t i o n r e q u i r e d by the t r a v e r s i n g ins truments ( S e c t i o n 2 . 4 . 3 ) . Two canyon h e i g h t s are used; 1 m and 0.62 m. The h e i g h t i s v a r i e d by the a d d i t i o n or removal of an e q u a l number of l a y e r s of b l o c k s from each w a l l . The 1 m h e i g h t was used wi th H/W r a t i o s of 2 . 0 , 1 .33, 1 .0 , and 0 .67 . The 0.62 m (3 rows of b l o c k s ) w a l l h e i g h t was used i n c o n j u n c t i o n w i t h a canyon width of 1.5 m to a c h i e v e the minimum H/W of 0 . 4 1 . The c o m b i n a t i o n s of H and W s e l e c t e d may be ach ieved , u s i n g o n l y two s p a c i n g s of measurement p o i n t s f o r model input on the f l o o r , 0.1 m and 0.15 m ( T a b l e 2 . 1 ) . T h i s r educes the number of t imes thermocouples must be r e c o n s t r u c t e d and r e a f f i x e d to the canyon f l o o r . 21 F i g u r e 2.3 I n c r e a s e i n w a l l v i e w - f a c t o r (^w) f o r a p o i n t l o c a t e d mid-way up the o p p o s i t e canyon w a l l at mid-canyon f o r i n c r e a s i n g l e n g t h s of w a l l . 22 2 . 3 . 6 Canyon Length The t h e o r y of the A r n f i e l d model (Chapter 3) assumes an i n f i n i t e l y l o n g canyon; an assumption which must be r e l a x e d in both the n u m e r i c a l implementat ion of the model and i n the model c a n y o n . A r n f i e l d (1976) suggests an a p p r o p r i a t e approx imat ion to i n i f i n i t y i s a c h i e v e d f o r canyon l e n g t h s of 8xH or W, which ever i s the l a r g e r . T h i s v a l u e i s d e r i v e d from an a n a l y s i s des igned to f i n d the l e n g t h of a p e r f e c t l y r e f l e c t i n g canyon tha t i s n e c e s s a r y to produce convergence to a canyon r e f l e c t i v i t y va lue of 1.0. A s i m i l a r measure of i n i f i n i t y may be d e r i v e d from v iew- f a c t o r geometry. For a g i v e n canyon width and w a l l h e i g h t the v i e w - f a c t o r of one w a l l may be c a l c u l a t e d for a p o i n t l o c a t e d at canyon m i d - l e n g t h h a l f way up the o ther w a l l . I f t h i s c a l c u l a t i o n i s r epea ted f o r a number of i n c r e a s e d w a l l l e n g t h s the d i f f e r e n c e s in the v i e w - f a c t o r s o b t a i n e d can be p l o t t e d , as i n F i g u r e 2 . 3 , to show the i n c r e a s e i n v i e w - f a c t o r a c h i e v e d w i t h l o n g e r w a l l s . The r e s u l t s for the H/W=1.0, 2.0 and 0.41 canyons show tha t when the 1.0 canyon reaches a l e n g t h of 5 m the i n c r e a s e i n v i e w - f a c t o r i s l e s s than 0 . 1 , and a f t e r 8 m the i n c r e a s e i s l e s s than 0 .001 . These r e s u l t s t h e r e f o r e c o n f i r m the f i n d i n g s of A r n f i e l d (1976) . The model canyon l e n g t h was set a t 10.2m and was not v a r i e d wi th the H/W r a t i o used . T h i s r e s u l t s i n a v a r y i n g a p p r o x i m a t i o n t o i n f i n i t y for the canyon l e n g t h , w i th a 23 v i e w - f a c t o r change of l e s s than 0.0005 for the 2.0 canyon and 0.001 f o r the 0.41 canyon . 2.4 CANYON INSTRUMENTATION 2.4 .1 S u r f a c e Temperature The method of s u r f a c e temperature measurement i s adopted from F a i r e y and Kalaghchy (1982) . They c o n s t r u c t thermocouples i n c o r p o r a t i n g a l o o p , or a r c , of wire between the j u n c t i o n and the i n s u l a t e d l eads to measure the s u r f a c e temperature of ho l low c o n c r e t e b l o c k s and frame w a l l s . The use of the arc method g i v e s a good a p p r o x i m a t i o n to the t r u e s u r f a c e temperature wi thout r e q u i r i n g the thermocouple j u n c t i o n s and l e a d wires to be imbedded i n the s u r f a c e m a t e r i a l ( F a i r e y and K a l a g h c h y , 1982). The arc of wire reduces c o n d u c t i o n e r r o r s which occur due to the h i g h c o n d u c t i v i t y of the thermocouple l eads when o n l y the j u n c t i o n i s a t t a c h e d to the s u r f a c e . The c h o i c e of arc s i z e i s r e l a t e d to the wire gauge, the temperature d i f f e r e n c e a c r o s s the s u r f a c e boundary and the c o n d u c t i v i t y of the s u r f a c e m a t e r i a l ( F a i r e y and K a l a g h c h y , 1982). S u r f a c e temperature measurements on the canyon f a c e t s are made u s i n g thermocouple a r c s c o n s t r u c t e d from 30 awg c o p p e r - c o n s t a n t a n thermocouple wire w i th double i n s u l a t e d f i b r e g l a s s c o a t i n g . A l l thermocouple a r c s are c o n s t r u c t e d u s i n g the same c y l i n d r i c a l form so that the d iameter remains cons tant 24 ( a p p r o x i m a t e l y 10 mm). L a r g e r s i z e s are d i f f i c u l t to a t t a c h to the rough s u r f a c e of the b l o c k . . J u n c t i o n s are t i g h t l y t w i s t e d and trimmed to a l e n g t h of 2 mm p r i o r to s o l d e r i n g . The e n t i r e a r c i s a t t a c h e d to the s u r f a c e u s i n g a f a s t bonding t w o - p a r t a d h e s i v e . A d d i t i o n a l bonding i s a c h i e v e d by a p p l y i n g a t h i n c o a t i n g of the adhes ive over the e n t i r e a r c . W h i l e the second coat of adhes ive i s s t i l l t acky a d u s t i n g of powdered cement i s a p p l i e d to keep the s u r f a c e r a d i a t i v e p r o p e r t i e s s i m i l a r . Thermocouples are a f f i x e d i n the same r e l a t i v e p o s i t i o n on each b l o c k to min imize b i a s e s due to temperature v a r i a t i o n s a c r o s s the s u r f a c e of the b l o c k a r i s i n g from i t ' s i n t e r n a l a r c h i t e c t u r e ( S e c t i o n 2 . 6 . 2 ) . On the w a l l s , the i n s u l a t e d l eads are l e d away from the measurement p o i n t h o r i z o n t a l l y a l on g the b l o c k face to the neares t space between b l o c k s and then to the back of the w a l l . T a b l e 2.1 Canyon H/W, D imens ions , and Number of G r i d - P o i n t s . H/W H (m) W (m) G r i d - p o i n t s on G r i d - p o i n t s on Each W a l l F l o o r 2 .0 1.0 0.5 10 5 1 .33 1.0 0.75 10 5 1 .0 1.0 1.0 10 10 1 .67 1.0 1.5 10 10 0.41 0.62 1.5 6 10 25 The number of measurement g r i d - p o i n t s v a r i e s w i t h the a b s o l u t e d imens ions and H/W of the canyon . T a b l e 2.1 summarizes the canyon H/W, a b s o l u t e d imens ions and number of thermocouples on each f a c e t . 2 . 4 . 2 R a d i a t i o n E s t i m a t e s of long-wave r a d i a t i o n are needed for two purposes ; model input and model v a l i d a t i o n . Model input r e q u i r e s the l o n g - wave i r r a d i a n c e i n c i d e n t on a h o r i z o n t a l p lane s i t u a t e d c o i n c i d e n t w i th the canyon t o p C L i c t r ) • Model v a l i d a t i o n r e q u i r e s the t h r e e long-wave f l u x d e n s i t i e s at each canyon s u r f a c e ; i n c i d e n t ( L ^ ) , ou tgo ing C L 0 ) . , and net long-wave r a d i a t i o n ( L * ) . Measurement of two of the t h r e e f l u x e s a l l o w s the c a l c u l a t i o n of the t h i r d by r e s i d u a l . L ^ c t i s measured u s i n g an E p p l e y PIR pyrgeometer l o c a t e d on the t o p of the west canyon w a l l . Two m i n i a t u r e net r a d i o m e t e r s (Swis s t eco Model S1 Minor Mk 2) are mounted on the t r a v e r s i n g system (see 2 .4 .3 ) to measure r a d i a t i v e f l u x e s w i t h i n the c a n y o n . One of the i n s t r u m e n t s i s eguipped wi th a b lackbody c a v i t y to measure L D from the canyon w a l l s and f l o o r . The second i s o p e r a t e d i n a net r a d i a t i o n x - o n i i g u r a t i o n . L Q i s measured in p r e f e r e n c e to because the h e i g h t of the ins trument dome i s l e s s than t h a t of the b l a c k b o d y c a v i t y , and t h e r e f o r e a l l o w s the i n s t r u m e n t to be l o c a t e d c l o s e r t o the s u r f a c e . 26 The instrument s p e c i f i c a t i o n s of the more commonly used f u l l - size Swissteco S1 net radiometers are compared with the miniature version in Table 2.2 Table 2.2 Comparison of Miniature and F u l l - S i z e Net Radiometer Spe c i f i c a t i o n s . Source: Fritschen and Gay (1979). Parameter Miniature F u l l Size Diameter of sensing surface Instrument diameter Height of dome above sensing surface S e n s i t i v i t y (typical) Accuracy of Calibration Response Time 1 6 mm 22 mm 1 0 mm 0.006 mV W"1m2 +/- 2.5% 98% in 25 sec 50 mm 95 mm 30 mm 0.04 mV W_1m2 +/- 2.5% 98% in 25 sec The use of miniature net radiometers for within-canyon measurements offers several advantages over the standard model. The l i g h t e r weight of the miniaturized instrument reduces the power and strength requirements of the CTS. The smaller instrument domes allow the radiometers to be located much closer to the surface. Figure 2.4 presents the radius of the area seen by a radiometer at a given height that i s necessary to achieve a given view-factor. It emphasizes that a .smaller area i s needed to achieve a given view-factor (Figure 2.4). This i s an important consideration because the A r n f i e l d model predicts the flux density for a s p e c i f i c point on the surface while the 27 0.5 r o.o h 1 — 1 1_ 10 20 30 40 Radiometer Height Above Surface (mm) F i g u r e 2.4 Radius of the a r e a seen a c h i e v e a g i v e n v i e w - f a c t o r (\p) f o r above the s u r f a c e . by a rad iometer n e c e s s a r y to d i f f e r e n t i n s t r u m e n t h e i g h t s 28 r a d i o m e t e r measures the average f l u x d e n s i t y for the area v i e w e d . The v i e w - f a c t o r of the s u r f a c e becomes c r i t i c a l near the ends of the f a c e t s , p a r t i c u l a r l y the tops of the canyon w a l l s . Wi th i n c r e a s i n g ins trument d i s t a n c e above the canyon f a c e t (measured p e r p e n d i c u l a r to the f a c e t ) i n these r e g i o n s an i n c r e a s i n g p r o p o r t i o n of the i n s t r u m e n t ' s v i e w - f a c t o r w i l l be o c c u p i e d by a f a c e t o ther than that b e i n g t r a v e r s e d , o r , i n the case of the tops of the w a l l s , the n i g h t s k y . F i g u r e 2.5 i l l u s t r a t e s t h i s 'end e f f e c t ' f o r a g i v e n rad iometer d i s t a n c e above a f a c e t . The lower l i m i t to the d i s t a n c e above the s u r f a c e at which the rad iometer may be p l a c e d i s de termined by the roughness of the s u r f a c e and the h e i g h t to which the ins trument domes p r o j e c t above the s e n s i n g s u r f a c e . Another f a c t o r i s the o b s t r u c t i o n of (//s (and w a l l and f l o o r v i e w - f a c t o r s ) f o r the f a c e t be ing measured. The placement of a rad iometer c l o s e to the s u r f a c e obscures a p o r t i o n of the v i e w - f a c t o r f o r tha t p o i n t ( F i g u r e s 2 . 6 a , b ) and r e p l a c e s the sources of sky , w a l l s and f l o o r w i t h Lj e m i t t e d from the r a d i o m e t e r . T h i s p o s s i b i l i t y i s e x p l o r e d by Idso and Coo ley (1972) . They f i n d no g r a d i e n t s in s u r f a c e temperature beneath a f u l l - s i z e r a d i o m e t e r l o c a t e d 0.2 m above a gras s s u r f a c e . The t e s t s are r e p e a t e d here (see 2 . 6 . 3 below) f o r m i n i a t u r e and f u l l - s i z e r a d i o m e t e r s mounted 50 mm above an open s u r f a c e . No d i s c e r n i b l e s u r f a c e temperature changes were d e t e c t e d u s i n g a narrow-view i n f r a r e d thermometer (Barnes Model PRT 4 A ) . 29 1.0 _ 0.9 o u o •D e m o 0.8 > o o 0.7 *o o u. I > Height ot Trov»r»« (mm) 50 40 30 20 10 < 0.6 0.5 0.5 0.6 0.7 0.8 0.9 Distance From Start ot Traverse (m) 1.0 F i g u r e 2 .5 The ' e n d e f f e c t ' f o r v i e w - f a c t o r s of a t r a v e r s e d canyon f a c e t as a t r a v e r s e d r a d i o m e t e r approaches the end of the f a c e t . The e f f e c t i s d i m i n i s h e d a t lower t r a v e r s e h e i g h t s . 30 10 20 30 40 Radiometer Height Above Surface (mm) F i g u r e 2 .6a O b s t r u c t i o n of ^ 5 by a m i n i a t u r e r a d i o m e t e r l o c a t e d above a canyon s u r f a c e . V i e w - f a c t o r s are c a l c u l a t e d f o r a p o i n t d i r e c t l y beneath the r a d i o m e t e r ( l i m i t i n g case ) and f o r the c i r c u l a r a r e a s which make up the s u r f a c e v i e w - f a c t o r of the i n s t r u m e n t . 31 Radiometer Height Above Surface (mm) F i g u r e 2 .6b Same as f o r F i g . 2 .6a but u s i n g a f u l l - s i z e net r a d i o m e t e r . 32 A f i n a l advantage to l o c a t i n g the rad iometer c l o s e to the s u r f a c e i s the r e d u c t i o n of e r r o r s due to e m i s s i o n by the i n t e r v e n i n g a i r l a y e r (Idso and C o o l e y , 1971). T h i s might be important i n the e a r l y even ing when the s u r f a c e temperature i s s i g n i f i c a n t l y d i f f e r e n t from a i r t e m p e r a t u r e . 2 . 4 . 3 The Canyon T r a v e r s i n g System The Canyon T r a v e r s i n g System (CTS) i s a cus tom-des igned and b u i l t apparatus used to t r a v e r s e r a d i o m e t e r s around a canyon c r o s s - s e c t i o n . In the c o n f i g u r a t i o n employed, the t r a v e r s e d sensors i n c l u d e d two m i n i a t u r e net r a d i o m e t e r s , one of which was equipped w i t h a u n i - d i r e c t i o n a l c a p , and a 30 awg t y p e - T u n s h i e l d e d thermocouple f o r m o n i t o r i n g a i r t e m p e r a t u r e . F i g u r e 2.7 shows a g e n e r a l i z e d s i d e - v i e w of the CTS (top) and an end- view of the v e r t i c a l t r a v e r s i n g assembly (bot tom) . (The numbered l a b e l s c o r r e s p o n d to those used i n the f o l l o w i n g d e s c r i p t i o n ) . The CTS uses two e l e c t r i c motors (1 ,2) to p r o v i d e power for h o r i z o n t a l (3) and v e r t i c a l (4) b o l t screws which d r i v e the ins trument c a r r i a g e (5) around the canyon c r o s s - s e c t i o n . A t h i r d motor ( 6 ) , mounted on the c a r r i a g e , i s used to r o t a t e the r a d i o m e t e r s ( 7 ) . C o n t r o l and power are p r o v i d e d v i a a p a n e l mounted beh ind the E a s t w a l l of the canyon . The d r i v e screws are mounted i n a framework (8) b o l t e d to the ground o u t s i d e of the canyon and s t i f f e n e d by a guide wire (9) r u n n i n g through the w a l l s a c r o s s the bottom of the canyon . E l e c t r o - m e c h a n i c a l l i m i t swi tches ( 1 0 ) . g o v e r n the l e n g t h of t r a v e l of the ins trument 33 F i g u r e 2.7 G e n e r a l i z e d drawing of the Canyon T r a v e r s i n g System ( C T S ) . T o p : s i d e v i ew , bot tom: end-view of the v e r t i c a l t r a v e r s i n g a s s e m b l y . L a b e l s : 1 - h o r i z o n t a l d r i v e m o t o r , 2 - v e r t i c a l d r i v e motor , 3 - h o r i z o n t a l b o l t screw, 4 - v e r t i c a l b o l t screw, 5 - i n s t r u m e n t c a r r i a g e , 6 - in s t rument r o t a t i o n motor , 7 - m i n i a t u r e r a d i o m e t e r s , 8 - support f rame , 9 - g u i d e - w i r e , 10 - l i m i t s w i t c h e s , 11 - ins trument c a r r i a g e a r m . 34 c a r r i a g e and are used to t r i p the r o t a t i o n and d e l a y sequence of the i n s t r u m e n t s . The r a d i o m e t e r s are mounted on an arm (11) e x t e n d i n g outwards from the t r a v e r s i n g c a r r i a g e to reduce o b s t r u c t i o n of the i n s t r u m e n t ' s v i e w - f a c t o r by the C T S . Manual and automat ic modes of o p e r a t i o n are a v a i l a b l e . Manual o p e r a t i o n p r o v i d e s i n d i v i d u a l c o n t r o l over each motor so tha t the ins trument c a r r i a g e may be p l a c e d anywhere i n the canyon c r o s s - s e c t i o n boundar ie s d e f i n e d by the l i m i t s w i t c h e s . In the automat ic mode the ins t ruments t r a v e r s e around the canyon c r o s s - s e c t i o n and automat ic ins trument r o t a t i o n and d e l a y s are engaged. T h i s i s the p r i m a r y mode for da ta c o l l e c t i o n . F i g u r e 2.8 d e p i c t s the CTS i n o p e r a t i o n i n the f i e l d d u r i n g 1988. The d i s t a n c e between the rad iometer s e n s i n g s u r f a c e and the canyon f a c e t be ing measured averages 30 - 40 mm for the w a l l s , and 20 mm f o r the canyon f l o o r . At the canyon top the i n s t r u m e n t s were set c o i n c i d e n t w i t h the p lane a c r o s s the canyon t o p . I r r e g u l a r i t i e s i n the l e v e l of the cement base upon which the canyon w a l l s were c o n s t r u c t e d r e s u l t e d i n the t r a v e r s i n g system not p r o v i d i n g a p e r f e c t l y p a r a l l e l t r a v e r s e to a l l the canyon f a c e t s . The a p p r o p r i a t e speed of t r a v e r s e was c a l c u l a t e d from i n f o r m a t i o n on the response c h a r a c t e r i s t i c s of the r a d i o m e t e r s and m o d e l l e d p r e d i c t i o n s of the r a d i a t i o n d i s t r i b u t i o n a c r o s s the canyon f a c e t s . U s i n g F r i t s c h e n and Gay (1979), the response of an ins t rument wi th a known time c o n s t a n t to e i t h e r a ramp or s i n u s o i d a l change may be approx imated by a d i f f e r e n t i a l e q u a t i o n i f the ins trument beg ins i n e q u i l i b r i u m wi th the t r u e v a l u e . The F i g u r e 2.8 The CTS as mounted i n the model c a n y o n . 36 d i f f e r e n c e between the t r u e and measured r a d i a t i o n can then be de termined for d i f f e r e n t t r a v e r s i n g speeds (see Appendix A for d e t a i l s ) . From t h i s a n a l y s i s i t i s e s t i m a t e d t h a t a t r a v e r s e t ime of a p p r o x i m a t e l y 5.5 mm s~ 1 i s adequate . I n c l u s i o n of a d e l a y of 27 seconds be fore b e g i n n i n g the t r a v e r s e of each f a c e t f u l f i l l s the requirement tha t the ins trument beg in the t r a v e r s e i n e q u i l i b r i u m w i t h the e n v i r o n m e n t . Appendix A p r o v i d e s f u r t h e r d e t a i l s . The sensor p o s i t i o n may be determined at any p o i n t though knowledge of three parameters : t r a v e r s e speed, s t a t e of the t r a v e r s e (de lay or t r a v e r s i n g ) , and s t a r t l o c a t i o n . The s t a t e of the t r a v e r s e i s i n d i c a t e d by a v o l t a g e at the c o n t r o l pane l (h igh when the t r a v e r s i n g system i s i n a d e l a y and low o t h e r w i s e ) ; a number r e p r e s e n t i n g the s t a t e of the t r a v e r s e i s then r e c o r d e d a long w i t h the sesnor outputs (See Appendix B for more on the assignment of r e c o r d e d v a l u e s to p a r t i c u l a r g r i d - p o i n t s ) . 2 .5 OTHER VARIABLES MEASURED In a d d i t i o n to the i n s t r u m e n t a t i o n a l r e a d y documented, a number of o ther v a r i a b l e s were r e c o r d e d in or near the model c a n y o n . These i n c l u d e d : the s u r f a c e temperature of the open cement s u r f a c e measured u s i n g a s i n g l e arc thermocouple , the net r a d i a t i o n ( L * 0 ) of the open cement at 1m (Swiss teco Model S1 net p y r r a d i o m e t e r ) , and wind speed at 1 m (MET-ONE Model 014A 3-cup 37 anemometer). I n f o r m a t i o n on c l o u d cover and g e n e r a l s u r f a c e m e t e o r o l o g i c a l c o n d i t i o n s was o b t a i n e d from the Atmospher ic Environment S e r v i c e (AES) r e c o r d s f o r the Vancouver I n t e r n a t i o n a l A i r p o r t S i t e . 2.6 TESTS 2 .6 .1 S u r f a c e E m i s s i v i t y The A r n f i e l d model r e q u i r e s v a l u e s of s u r f a c e r a d i a t i v e p r o p e r t i e s for each g r i d - p o i n t . Under the r e s t r i c t i o n of n o c t u r n a l c o n d i t i o n s , the on ly parameter to be determined i s the s u r f a c e e m i s s i v i t y of the canyons w a l l s and f l o o r . Measurements were made u s i n g the procedure o u t l i n e d by D a v i e s et al . (1971) i n which e m i s s i v i t y (e) i s de termined from e = ( T r 4 - T k 4 ) / ( T s 4 - T k 4 ) (2 .1) where T r i s the apparent s u r f a c e temperature measured d i r e c t l y by an i n f r a r e d thermometer, c T r 4 = eaT s 4 + (1-e) L i (2.2) Tj,. i s the r a d i a t i v e temperature of the sky , and T s i s the t r u e s u r f a c e t e m p e r a t u r e . T h i s e q u a t i o n assumes tha t the e m i s s i v i t y i s c o n s t a n t w i t h wavelength and t h a t the range of temperatures 38 i s s m a l l so that the f i l t e r c o e f f i c i e n t s d e f i n i n g the f r a c t i o n of r a d i a t i o n t r a n s m i t t e d by the f i l t e r are equa l (Davies et a l . , 1971). Temperature measurements were made u s i n g a Barnes I n f r a r e d Thermometer (Model PRT-4A) which has a bandpass f i l t e r of 8.0 to 14.0 Mm. The output from the thermometer was sampled every second and r e c o r d e d as a v o l t a g e on a Campbel l S c i e n t i f i c CR 21X d i g i t a l r e c o r d e r . U s i n g a c a l i b r a t i o n curve de termined p r e v i o u s l y (see Appendix C) the v o l t a g e s were c o n v e r t e d to e q u i v a l e n t b lackbody t e m p e r a t u r e s . E s t i m a t e s of T^ were made f o l l o w i n g the approach of Lorenz (1966) i n which samples of T^ are taken from v a r i o u s z e n i t h ang le s and az imuths to determine an o v e r a l l v a l u e of T ^ . Measurements were made under 10/10 low to medium l e v e l c l o u d so tha t v a r i a t i o n i n T^ w i t h z e n i t h ang le was m i n i m i z e d . T s was measured by p l a c i n g an aluminum cone w i t h a p o l i s h e d inner s u r f a c e over the c o n c r e t e and t a k i n g a r e a d i n g w i t h the IR thermometer through an a p e r t u r e at the apex of the cone . When the temperature of the cone i s equa l to the s u r f a c e t emperature , the cone changes the e f f e c t i v e s u r f a c e e m i s s i v i t y to u n i t y so that i t behaves as a b l a c k b o d y ; the second term of (4 .2) d i s a p p e a r s and T r = T s . Fuchs and Tanner (1966) have shown tha t the method i s not s e n s i t i v e to d i f f e r e n c e s between the s u r f a c e temperature and tha t of the inner s u r f a c e of the cone . Measurements taken under c loudy c o n d i t i o n s or at n i g h t min imize a l t e r a t i o n s i n the s u r f a c e r a d i a t i v e ba lance and T s which occur when the cone i s p l a c e d over the s u r f a c e . Record ing T s 39 immediate ly a f t e r placement of the cone upon the s u r f a c e , a l s o m i n i m i z e s e r r o r s . Dav ie s e t . a l (1971) observed a p e r i o d of 20 seconds when the s u r f a c e temperature remained c o n s t a n t ; f or Fuchs and Tanner (1966) the p e r i o d was 5 to 15 seconds at n i g h t . The e x p e r i m e n t a l l y de termined v a l u e s of s u r f a c e e m i s s i v i t y are l i s t e d in T a b l e 2 . 3 . T a b l e 2.3 E m i s s i v i t y of Canyon S u r f a c e s . Concre te B l o c k s C o n c r e t e Base n e s 0 n e s 0 13 0.964 0.015 47 0.954 0.017 The range of v a l u e s for the e m i s s i v i t y of c o n c r e t e p u b l i s h e d i n the l i t e r a t u r e range from 0.71 - 0.90 (Oke, 1978), 0.85 - 0.95 (ASHRAE, 1981), and 0.98 Verseghy (1987) . To determine i f the e m i s s i v i t i e s of the b l o c k s and base are s i g n i f i c a n t l y d i f f e r e n t a T - t e s t between the two sample means was conducted at the 0.01 c o n f i d e n c e l e v e l . The n u l l h y p o t h e s i s was taken to be no d i f f e r e n c e between the means. The T - s t a t i s t i c c a l c u l a t e d was 1.96, wh ich , at the 0.01 l e v e l of s i g n i f i c a n c e , r e s u l t s in the acceptance of the n u l l h y p o t h e s i s . 40 2 . 6 . 2 S u r f a c e Temperature S e v e r a l t e s t s to a s c e r t a i n the p r e c i s i o n and a c c u r a c y of the temperature measurements were made. P r i o r to the attachment of any thermocouples to canyon s u r f a c e s twenty a r c thermocouples were t e s t e d in the l a b o r a t o r y . The thermocouples were c a r e f u l l y c o n s t r u c t e d u s i n g the method o u t l i n e d i n 4 .2 .1 to be as s i m i l a r as p o s s i b l e . They were p l a c e d i n a s e a l e d c a r d b o a r d box i n the darkened l a b o r a t o r y temperatures sampled every 30 seconds and averaged over 10 minute i n t e r v a l s . The s tandard d e v i a t i o n a f t e r the f i r s t 20 minutes was l e s s than 0 . 0 3 ° C . Next , a t e s t of the p r e c i s i o n of the s u r f a c e temperature measurements was conducted on the open c o n c r e t e s u r f a c e of the s i t e . F i v e arc thermocouples were a t t a c h e d to the c o n c r e t e in the method d e s c r i b e d i n 4 . 2 . 1 . The s u r f a c e temperature measured by the thermocouples i s p r e s e n t e d for s e l e c t e d t imes i n T a b l e 2 . 4 . T a b l e 2.4 S u r f a c e Temperature P r e c i s i o n Time T1 T2 T3 T4 T5 T a v g s range 1500 45.01 44.48 44.44 44.57 44.92 44.68 0.26 0.57 2245 23.95 23.90 24.03 23.90 23.77 23 .91 0.09 0.26 0035 21.53 21.53 21.64 21.56 21.47 21.55 0.06 0.17 0425 19.33 19.33 19.38 19.31 19.24 19.32 0.05 0.14 41 The r e s u l t s i n d i c a t e g r e a t e r p r e c i s i o n in the e v e n i n g , p r i m a r i l y due to reduced r a d i a t i o n e r r o r s . O v e r a l l , adherance to c a r e f u l mounting p r o c e d u r e s appears to a l l o w p r e c i s e s u r f a c e temperature measurements to be made u s i n g t h i s method. A measure of the a c c u r a c y of s u r f a c e temperature measurements was d e s i r a b l e both to c o n f i r m the a c c u r a c y of the mounting method and to overcome the l i m i t a t i o n of the o r i g i n a l study by F a i r e y and Kalaghchy (1982) . Two separate v a l i d a t i o n s of the s u r f a c e temperature a c c u r a c y have been made ( F i g u r e 2 . 9 ) . In the f i r s t , an aluminum cone was used to cover a g r o u p . o f thermocouples a t t a c h e d to the open c o n c r e t e s u r f a c e and i n c r e a s e the e f f e c t i v e s u r f a c e e m i s s i v i t y to u n i t y . The r e a d i n g s from an • IR thermometer taken from the apex of the cone were compared w i t h the average s u r f a c e temperature r e c o r d e d from the t h e r m o c o u p l e s . The second method e n t a i l e d r e a r r a n g i n g e q u a t i o n (2 .2) and s o l v i n g i t t o f i n d T s : T s = [ ( o T r 4 - (1-e) / eo ]1 / 4 (2 .3) The e x p e r i m e n t a l l y de termined e m i s s i v i t y of the c o n c r e t e s u r f a c e was used and L^ was measured by a m i n i a t u r e net rad iometer equ ipped w i t h a u n i - d i r e c t i o n a l cap over the lower s u r f a c e . T r was o b t a i n e d from an IR thermometer mounted above the s u r f a c e . The r e s u l t s i n F i g u r e 2.9 i n d i c a t e tha t the f i r s t method s l i g h t l y (0 .5 °C) underes t imates the a r c temperature over the 42 20 - L 21 22 23 Surface Temperature (°C) 24 25 _i_ 26 F i g u r e 2 .9 v a l i d a t i o n , S u r f a c e t emperature v a l i d a t i o n . Squares - cone t r i a n g l e s - r a d i a t i v e energy b a l a n c e v a l i d a t i o n , 43 range of temperatures t e s t e d . The second method produced very good agreement. E a r l i e r , i t was s t a t e d tha t thermocouples were p l a c e d i n the same r e l a t i v e p o s i t i o n on each b l o c k i n an attempt to prevent anomal ies from s u r f a c e temperature d i s t r i b u t i o n s due to the c o n s t r u c t i o n of the c o n c r e t e b l o c k s . Measurement of s u r f a c e temperature d i s t r i b u t i o n s a c r o s s the face of c o n c r e t e b l o c k s wi th no b a c k i n g i n s u l a t i o n on the w a l l s (but p a i n t e d white) r e v e a l e d r a d i a t i v e temperature d i f f e r e n c e s of up to 0.6 degrees in the e a r l y evening between p o r t i o n s of the b r i c k which have a s o l i d c o n c r e t e core and over the ho l low c e l l . The chosen placement of the thermocouples over the middle of the c e l l may t h e r e f o r e induce a b i a s e d underes t imate of s u r f a c e temperature r e l a t i v e to that sensed r a d i a t i v e l y by a rad iometer l o c a t e d above the s u r f a c e . However, g i v e n the g r e a t e r p r o p o r t i o n of s u r f a c e a r e a u n d e r l a i n by a ho l low c e l l , the c h o i c e i s l o g i c a l . The placement of i n s u l a t i o n beh ind the w a l l s may reduce the s u r f a c e temperature changes over the b lock f a c e , p a r t i c u l a r l y for the west w a l l , s i n c e h e a t i n g from behind the w a l l near sunset w i l l be r e d u c e d . 2 . 6 . 3 Radiometer D i s t a n c e Above F a c e t Given the d i f f e r e n c e s in d i s t a n c e s and of radiometer s i z e s used i n the model canyon from those used i n the a n a l y s i s of Idso and Coo ley (1972), t e s t s were conducted to determine i f any s u r f a c e temperature g r a d i e n t induced by the presence of a 44 r a d i o m e t e r p l a c e d c l o s e above the s u r f a c e c o u l d be d e t e c t e d . A w o r s t - c a s e s c e n a r i o was c o n s t r u c t e d : the m i n i a t u r e and f u l l - s i z e net r a d i o m e t e r s were p l a c e d c l o s e to the s u r f a c e over an open s i t e f o r an extended p e r i o d under c l o u d l e s s sky c o n d i t i o n s , and s u r f a c e temperature t r a n s e c t s were taken u s i n g a narrow-view IR thermometer . Even u s i n g the f u l l - s i z e rad iometer no changes were d e t e c t e d . W i t h i n a canyon the e f f e c t i s s m a l l because the \ps f or p o i n t s i s a l r e a d y v e r y much reduced and the d i f f e r e n c e s in r a d i a t i v e temperature between the radiometer and canyon s u r f a c e s are m i n o r . H i g h s u r f a c e e m i s s i v i t i e s mean that even minor changes in the i n c i d e n t f l u x r e s u l t i n s u r f a c e temperature changes which are sensed by the r a d i o m e t e r , because the long-wave r e f l e c t i v i t y of the s u r f a c e i s low. The g r e a t e r area viewed by the rad iometer when compared to the narrow view IR thermometer used to search f o r s u r f a c e temperature changes w i l l decrease the e f f e c t of any change o c c u r r i n g i n a s m a l l a r e a . T h e r e f o r e , the use of m i n i a t u r e net r a d i o m e t e r s o f f s e t s to some degree the reduced i n c u r r e d by u s i n g lower t r a v e r s i n g h e i g h t s . C o n t i n u a l t r a v e r s i n g of the r a d i o m e t e r s p r o b a b l y renders any e f f e c t n e g l i g i b l e . 2 . 6 . 4 T r a v e r s e System T e s t s of the CTS were performed to determine i f the d e l a y i n t e r v a l or t r a v e r s e speed (Appendix A) are a p p r o p r i a t e under f i e l d c o n d i t i o n s . 45 C o n f i r m a t i o n of the d e l a y i n t e r v a l can be e a s i l y accompl i shed by p l o t t i n g the ins trument output from the rad iometer a f t e r a r o t a t i o n has o c c u r r e d and whi l e the CTS i s i n a d e l a y mode. I f the t r a c e i n d i c a t e s l i t t l e or no change by the end of the d e l a y i n t e r v a l , then the l e n g t h of the d e l a y i s adequate . The s t ep change i n r a d i a t i o n between f a c e t s w i l l be most c l e a r l y e v i d e n t as the ins trument t r a v e r s e changes to and from the canyon top because of the c o n t r a s t between the sky and canyon t e m p e r a t u r e s . F i g u r e 2.10 i l l u s t r a t e s the nature of the change in r a d i a t i o n measured a f t e r these r o t a t i o n s . To t e s t i f the r a d i o m e t e r s a c c u r a t e l y measure the changing r a d i a t i o n whi l e t r a v e r s i n g , a s i n g l e net rad iometer was set in a f i x e d p o s t i o n over a canyon f a c e t and the r e a d i n g s compared to those of the t r a v e r s e d ins truments as they passed the p o i n t . The f i x e d rad iometer may be c o n s i d e r e d to g ive the ' b e s t ' e s t i m a t e . By n e c e s s i t y , the f i x e d rad iometer was o f f s e t from the t r a c k of the t r a v e r s e d r a d i o m e t e r s so that the areas viewed by the two i n s t r u m e n t s d i f f e r s l i g h t l y . T a b l e 2.5 summarizes the r e s u l t s of these t e s t s completed u s i n g t h i s method and F i g u r e s 2.11 - 2.14 show p l o t s of the f i x e d rad iometer output (open c i r c l e s ) and the t r a v e r s e data ( c r o s s e s ) . The e r r o r bars on the t r a v e r s e da ta are the s t a n d a r d d e v i a t i o n s from the average v a l u e f o r a l l samples taken w i t h i n the g r i d - p o i n t boundar i e s of the p o i n t b e i n g t e s t e d . The e r r o r bars f o r the f i x e d rad iometer data r e p r e s e n t the s t a n d a r d d e v i a t i o n from the average v a l u e over the t ime taken f o r the t r a v e r s e . The f i x e d rad iometer output i s a one minute a v e r a g e , thus the s t a n d a r d d e v i a t i o n s are g e n e r a l l y 46 4 2 0 ) E 5 O 4 0 0 f O QL a > § 380 f I •> c o D 3 6 0 f 3 4 0 Sky-to-wal l transition Woll-to-sky transition — B —a 1 0 1 5 Time From Start of Delay (s) 2 0 2 5 F i g u r e 2.10 Change i n long-wa i n s t r u m e n t r o t a t e s to and from at the canyon t o p , L Q from the ve r a d i a t i o n measured as the the canyon t o p . (L{ i s measur w a l l s ) . 47 39S 398 400 402 404 406 408 L. (W m-») 410 400 405 410 L» (W rn"*) 415 F i g u r e 2.11 T e s t s between a f i x e d ( c r o s s e s ) and t r a v e r s e d ( s q u a r e s ) r a d i o m e t e r f o r Lo a t p o i n t 8 on the West w a l l . E r r o r b a r s a r e s t a n d a r d d e v i a t i o n s of a l l samples t a k e n over each g r i d - p o i n t d u r i n g the t r a v e r s e . 48 to • l — 6—' 9 -a— * B i B — i • * 7 c e 6 <—B—i Po in t S i B -— I Fo ei  4 • • • < 3 2 i i 1 1 i • i 400 404 408 412 L . (W m-*) 10 • - B - • • i 8 i-e-i 6 4 i-e-i 2 © 1 390 395 400 405 410 L . (W m-») 415 10 - - B - i i—e- 8 I - B - I 6 4 t - B - 2 I • i •&< 390 395 400 405 410 415 « , (W m-») F i g u r e 2.12 T e s t s between a f i x e d ( c r o s s e s ) and t r a v e r s e d ( s q u a r e s ) r a d i o m e t e r f o r L Q a t p o i n t 3 on the West w a l l . 49 s m a l l . The l a r g e e r r o r bars f o r the t r a v e r s e d i n s t r u m e n t , r e s u l t from n o i s e i n the d a t a , the o r i g i n of which i s d i s c u s s e d in Appendix B . No f i l t e r i n g of the data was performed i n t h i s a n a l y s i s. T a b l e 2.5 Summary of F i x e d Versus T r a v e r s e d Radiometer T e s t s . Date Time F a c e t P o i n t F l u x F i g u r e J u l y 20/21 0100-0230 3 7 L * 2.13 3 4 L * 2.14 J u l y 21/22 0100-0300 1 8 L 0 2.11 1 3 L Q 2.12 The c o m p a r i s i o n between the f i x e d and t r a v e r s e e s t i m a t e s of L 0 f o r W a l l A ( F i g u r e s 2 .11 , 2.12) i n d i c a t e a s l i g h t o v e r e s t i m a t e by the t r a v e r s e d r a d i o m e t e r . The t e s t for g r i d - p o i n t 3, which i n v o l v e s a g r e a t e r l e n g t h of t r a v e r s e a l s o i n d i c a t e s an o v e r e s t i m a t e of between 2-3 W m ~ 2 . G iven the d i r e c t i o n of t r a v e r s e (towards the base of the w a l l ) , and the expec ted d i s t r i b u t i o n of r a d i a t i o n d u r i n g t h i s p e r i o d of the e v e n i n g (see F i g u r e A . 1 ) , the t e s t s do not produce ev idence of a l a g i n the sensor r e s p o n s e . A l a g i n response to a ramp change ( F i g u r e A . 2 ) would produce a lower v a l u e f o r the t r a v e r s e d sensor compared to the f i x e d rad iometer whereas the t e s t s i n d i c a t e the o p p o s i t e . I t i s p o s s i b l e w i th p o i n t 3 that an S- shaped d i s t r i b u t i o n of r a d i a t i o n (as i l l u s t r a t e d i n A . 1 ) c o u l d produce an o v e r e s t i m a t e of the f l u x d e n s i t y as the t r a v e r s e d ins t ruments l a g in t h e i r response to the decreased change in L Q . 50 4 5 6 7 Facet Point on Floor 8 9 4 5 6 7 Facet Point on Floor 4 5 6 7 Facet Point on Floor F i g u r e 2.13 T e s t s between a f i x e d ( c r o s s e s ) and t r a v e r s e d ( squares ) r a d i o m e t e r for L * a t p o i n t 7 on the canyon f l o o r . 51 2.0 2.5 3.0 3.5 4.0 • Facet Point on Floor 4.5 5.0 2.0 2.5 3.0 3.5 4.0 Facet Point on Floor 4.5 5.0 2.0 2.5 3.0 3.5 4.0 Facet Point on Floor 4.5 5.0 F i g u r e 2 14 ^Tests between a f i x e d ( c r o s s e s ) and t r a v e r s e d ( s q u a r e s ) r a d i o m e t e r f o r L * a t p o i n t 4 on the canyon f l o o r . 52 The chances of t h i s o c c u r r i n g at p o i n t 7 are much l e s s . I t i s more l i k e l y that the r e s u l t s are due t o : (a) minor d i f f e r e n c e s i n the ang le of the two r a d i o m e t e r s r e l a t i v e to the s u r f a c e they view a n d / o r (b) the o f f s e t of the f i x e d rad iometer from the l i n e of t r a v e r s e . Comparison of L * between the f i x e d and t r a v e r s e d r a d i o m e t e r s ( F i g u r e s 2.13 and 2.14) shows n e g l i g i b l e d i f f e r e n c e s between the two i n s t r u m e n t s . U n f o r t u n a t e l y the l a r g e s t a n d a r d d e v i a t i o n of the net r a d i a t i o n from the t r a v e r s e d r a d i o m e t e r s d e t r a c t from the t e s t . A f i n a l t e s t conducted p r i o r to data c o l l e c t i o n was the use of the CTS i n manual mode r a t h e r than automat ic mode. The r e s u l t s from t h i s t e s t are d e s c r i b e d in Chapter 4. 53 CHAPTER 3. THE ARNFIELD MODEL 3.1 INTRODUCTION The n u m e r i c a l model s e l e c t e d f o r v a l i d a t i o n i s tha t d e s c r i b e d by A r n f i e l d (1976, 1982), r e f e r r e d to h e r e a f t e r as the A r n f i e l d mode l . The model can be used to c a l c u l a t e e s t i m a t e s of the r a d i a t i o n budget components at a l l s u r f a c e s w i t h i n the u n i t of an urban canyon , i n c l u d i n g the canyon t o p , f or a g i v e n geometr ic c o n f i g u r a t i o n and d i s t r i b u t i o n of s u r f a c e m a t e r i a l s . U s i n g v iew- f a c t o r geometry and the assumption tha t canyon f a c e t s are L a m b e r t i a n r e f l e c t o r s , m u l t i p l e r e f l e c t i o n events are i n c l u d e d i n the m o d e l l i n g methodology. T a b l e 3.1 l i s t s the r a d i a t i o n budget components which may be c a l c u l a t e d u s i n g the model , v a r i o u s o p t i o n s a v a i l a b l e , and the necessary input f l u x e s . Only a b r i e f account of the model i s p r e s e n t e d h e r e . For a more complete e x p l a n a t i o n of the c o m p u t a t i o n a l methods, r e f e r to A r n f i e l d (1976) . T h i s c h a p t e r d e s c r i b e s the framework of the model , i t ' s implementa t ion and the r e q u i r e d input d a t a . M o d i f i c a t i o n s made to the o r i g i n a l model are d e s c r i b e d . S e n s i t i v i t y t e s t s to the major i n p u t parameters are p r e s e n t e d in S e c t i o n 3 . 5 . 54 T a b l e 3 . 1 . Components of the R a d i a t i o n Budget C a l c u l a t e d in the A r n f i e l d Model w i th O p t i o n s and R e q u i r e d Input F l u x e s . C a l c u l a t e d O p t i o n s R e q u i r e d Input Absorbed G l o b a l i s o t r o p i c d i s t . of S o l a r R a d i a t i o n d i f f u s e s o l a r o v e r c a s t sky r a d i a n c e d i s t r i b u t i o n of Steven and Unsworth (1980) c l e a r sky r a d i a n c e d i s t r i b u t i o n of Steven and Unsworth (1979) i s o t r o p i c r a d i a n c e d i s t r i b u t i o n r a d i a n c e d i s t r i b u t i o n of Unsworth and M o n t e i t h (1975) Long-Wave Rad. from the sky Long-Wave Rad. at a canyon f a c e t Long-wave r a d . e m i t t e d by canyon f a c e t s Net A l l - W a v e R a d i a t i o n s t a n d a r d i r r a d . of d i r e c t s o l a r s t a n d a r d i r r a d . of d i f f u s e s o l a r s t d . i r r a d . of long-wave r a d . s t d . i r r a d . of long-wave r a d . s t d . i r r a d . of long-wave r a d . 3.2 MODEL FRAMEWORK In the model , an urban canyon i s r e p r e s e n t e d as an i n f i n i t e l y l o n g , r e c t a n g u l a r space bounded by w a l l s of h e i g h t ' H ' and a f l o o r w i t h width 'W' ( F i g u r e 3 . 1 ) . T h i s p e r m i t s a s i m p l i f i e d r e p r e s e n t a t i o n of the urban s u r f a c e as a s e r i e s of urban canyons s e p a r a t e d by r o o f s of width 'R' a p a r t ( A r n f i e l d , 1982; S i e v e r s and Zdunkowski , 1986). For the purposes of the present 55 F i g u r e 3.1 Canyon c o o r d i n a t e system. The f a c e t and p o i n t numbering c o n v e n t i o n s a r e taken from A r n f i e l d (1976, 1982) and a r e : West w a l l - W a l l A or IF 1, E a s t w a l l - W a l l B or IF 2, F l o o r - IF 3, Top - IF 4. P o i n t s are numbered upwards on the w a l l s from IP 1 (base) to IP 10 ( t o p ) . P o i n t s on f l o o r and top a r e number i n i n c r e a s i n g o r d e r from w a l l A (IF 1) to w a l l B (IF 2 ) . 56 v a l i d a t i o n , o n l y a s i n g l e c a n y o n i s m o d e l l e d and t h e r o o f a r e a i s not i n c o r p o r a t e d . The c a n y o n i s a l i g n e d a t an a z i m u t h 0 (0 = 0 o r 360° f o r t h e model c a n y o n ) f r o m N o r t h and t h e w a l l s i d e n t i f i e d as A and B. W a l l A i s d e f i n e d as t h a t w h i c h would be i r r a d i a t e d by a p o i n t s o u r c e l o c a t e d a t an a z i m u t h a, s u c h t h a t 0 < a < ( 0 + 7 r ) . In t h e model c a n y o n c o n s t r u c t e d , W a l l A i s synonmous w i t h t h e West w a l l o r I F 1 ( F a c e t number 1) ( t h e w a l l w i t h t h e i n n e r s u r f a c e o r i e n t e d E a s t ) . W a l l B i s t h e E a s t ( o r W e s t - f a c i n g w a l l ) , and has a f a c e t number of 2. Where p o s s i b l e , t h e w a l l s w i l l be r e f e r r e d t o as E a s t and West, even t h o u g h d i r e c t i o n a l i t y may be of no c o n s e q u e n c e , as i n most of t h e s e n s i t i v i t y t e s t s . P o i n t s w i t h i n t h e c a n y o n may be u n i q u e l y i d e n t i f i e d by means of a c o o r d i n a t e s y s t e m i n w h i c h x d e n o t e s d i s t a n c e a l o n g t h e c a n y o n a x i s f r o m th e o r i g i n a t c a n y o n m i d - l e n g t h ; y and y' r e p r e s e n t d i s t a n c e s a c r o s s t h e c a n y o n f l o o r ( I F 3) and t o p ( I F 4) measured from w a l l A; and z and z ' a r e t h e v e r t i c a l c o o r d i n a t e s measured upwards on w a l l s A and B r e s p e c t i v e l y . F i g u r e 3.1 a l s o i l l u s t r a t e s t h e f a c e t and p o i n t n u m b e ring c o n v e n t i o n u t i l i z e d by t h e m o d e l . G e o m e t r i c a l and r a d i a t i v e p r o p e r t i e s a r e assumed i n v a r i a n t i n t h e a l o n g - c a n y o n d i m e n s i o n . V a l u e s f o r t h e e m i s s i v i t y and a l b e d o o f c a n y o n m a t e r i a l s a r e a s s i g n e d t o i n d i v i d u a l g r i d - p o i n t s and t h e r e f o r e r e p r e s e n t i n f i n i t e l y l o n g h o r i z o n t a l s t r i p s r u n n i n g t h e l e n g t h o f t h e c a n y o n f l o o r and w a l l s . 57 3.3 MODEL IMPLEMENTATION The A r n f i e l d model c o n s i s t s of a set of four FORTRAN s u b r o u t i n e s , c a l l e d by a u s e r - w r i t t e n main program which d e f i n e s c o n s t a n t s , i n i t i a l i z e s v a r i a b l e s and a r r a y s and performs the neces sary i n p u t / o u t p u t o p e r a t i o n s . T a b l e 3.2 l i s t s the s u b r o u t i n e s and b r i e f l y d e s c r i b e s the purpose of e a c h . One of the input data f i l e s c o n t a i n s measured va lues of L 0 and L * for s p e c i f i e d canyon p o i n t s . Us ing i n f o r m a t i o n for a p o i n t which has been v a l i d a t e d for a p a r t i c u l a r a v e r a g i n g p e r i o d , the main program output s the d a t e , t ime , f a c e t and p o i n t numbers, and measured and mode l l ed data for the p o i n t . T a b l e 3.2 A r n f i e l d Model F o r t r a n S u b r o u t i n e s S u b r o u t i n e Name F u n c t i o n CARABU CAnyon RAdia t ion Budget RADIOS RADiat ion on I n c l i n e d O b s t r u c t e d S u r f a c e s CNRDF2 CaNyon R a D i a t i o n v e r s i o n 2 SIMPS SIMPSon's r u l e i n t e g r a t ion Main s u b r o u t i n e : other s u b r o u t i n e s are c a l l e d from CARABU. Uses input data to determine the canyon r a d - i a t i o n budget . C a l c u l a t e s view f a c t o r s i f r e q u i r e d . A g e n e r a l r o u t i n e to c a l c u l a t e the i r r a d i a n c e on a p lane of g iven o r i e n t a t i o n (from both the upper and lower h e m i s p h e r e s ) , i n c l u d i n g o b s t r u c t e d and u n o b s t r u c t e d p o r t i o n s . C a l c u l a t e s m u l t i p l e exchanges of r a d i a t i o n w i t h i n the canyon . Performs S impson's r u l e i n t e g r a - t i o n of a f u n c t i o n d e f i n e d by a t a b l e of e q u i - s p a c e d v a l u e s . 58 Model input i s described in Appendix E, Table E.1. Two cases require approximations to be made to the model theory. The f i r s t i s the assumption of an i n f i n i t e l y long canyon. The t h e o r e t i c a l l i m i t of i n f i n i t y i s replaced by the canyon half-length, measured from the canyon midpoint to one end; ( i . e . one half the t o t a l canyon length). The second approximation concerns the upper l i m i t of integration for multiple r e f l e c t i o n s within the canyon. In t h i s instance, a c r i t i c a l value for the difference between successive c a l c u l a t i o n s of the flux leaving the canyon top i s specified by the user; when the difference f a l l s below the set value, i t e r a t i o n s cease. The model has been implemented on a personal computer (IBM- Compatible PC-AT 286) using WATFOR-77 (Coschi and Schueler, 1985). Compile and execution times for the model using a nocturnal data set with an isotropic radiance d i s t r i b u t i o n are presented in Table 3.3. Table 3.3 Approximate Compile and Execution Times for the A r n f i e l d Model Using an Isotropic Radiance D i s t r i b u t i o n . Computer Compile Time Execut ion Times (sec) (sec) F i r s t Subsequent Point Points IBM PC AT-286 12.2 68.6 4.4 8087 equipped 59 When the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s u s e d , compi l e t imes are s i m i l a r and e x e c u t i o n t imes are e q u i v a l e n t to the f i r s t p o i n t of the i s o t r o p i c runs when the o p t i c a l water d e p t h , u , changes . 3.4 MODIFICATIONS TO THE ARNFIELD MODEL A number of m o d i f i c a t i o n s have been made to the A r n f i e l d model to ensure i t matches the set up of the s c a l e canyon as c l o s e l y as p o s s i b l e , e s p e c i a l l y when the s u r f a c e temperature measurement l o c a t i o n s do not r e p r e s e n t a r e g u l a r g r i d . The A r n f i e l d model i m p l i c i t l y r e q u i r e s tha t the l o c a t i o n s of the s u r f a c e temperature measurements be at r e g u l a r i n t e r v a l s a c r o s s each f a c e t . The number of g r i d - p o i n t s was set equa l to the d imens ion of the f a c e t and the p o i n t s d i s t r i b u t e d a t e q u a l i n t e r v a l s a c r o s s i t . The h e i g h t and width are o n l y r e l a t e d to the a c t u a l d imens ions of a canyon through the H/W r a t i o and have no meaning on t h e i r own. F i g u r e 3.2 i l l u s t r a t e s the r e g u l a r g r i d p a t t e r n assumed by the A r n f i e l d model ( cros se s ) and how the s c a l e canyon d e v i a t e s from i t ( d o t s ) . The a d d i t i o n of the capp ing b l o c k s on the tops of the w a l l s upsets the r e g u l a r g r i d e s t a b l i s h e d over the remainder of the b r i c k s . The a d d i t i o n a l h e i g h t from the c a p p i n g b l o c k a l s o causes the g r i d s p a c i n g on the canyon f l o o r to be s l i g h t l y d i f f e r e n t from that on the w a l l s . To overcome the i r r e g u l a r i t i e s imposed by the sampl ing g r i d of thermocouples , a number of changes were made to the model , p r i m a r i l y a f f e c t i n g 60 -* *r- -* —*- "* X * * - ^n^L Z ^ l l t ^ l l ^ ^ ^ T i d pattern 61 the s p e c i f i c a t i o n of the h e i g h t , w idth and canyon v i e w - f a c t o r a r r a y s . 3 .4 .1 Canyon He ight and Width Canyon h e i g h t and width can no longer be d i r e c t l y t i e d to the s p a c i n g of g r i d - p o i n t s as i n t h e - o r i g i n a l model f o r m u l a t i o n . In the m o d i f i e d model , an i n t e g e r parameter f o r the he ight and w i d t h i s s p e c i f i e d to c o n t r o l the number of g r i d - p o i n t s to be used f o r the canyon f a c e t s . A second c o n s t a n t i s set w i t h i n the CARABU s u b r o u t i n e to r e p r e s e n t the a c t u a l canyon h e i g h t and width f o r use i n subsequent v i e w - f a c t o r and geometry c a l c u l a t i o n s . F i n a l l y , to address the change i n g r i d s p a c i n g between the f l o o r and w a l l s , the g r i d - p o i n t ' c o o r d i n a t e s ' are read i n from an e x t e r n a l f i l e . These are d i s t a n c e s measured upwards from the f l o o r for the w a l l s , and a c r o s s the canyon from w a l l A f o r the f l o o r and canyon t o p . The u n i t s of the c o o r d i n a t e s are a r b i t r a r y , but must be c o n s i s t e n t w i th the v a l u e s s p e c i f i e d f o r the canyon h e i g h t and w i d t h . 3 . 4 . 2 V i e w - F a c t o r s A d d i t i o n a l v i e w - f a c t o r a r r a y s are d imens ioned for the s p e c i f i c a t i o n of w i t h i n - c a n y o n v i e w - f a c t o r s . Due to the d i f f e r e n c e i n s p a c i n g of p o i n t s on the w a l l s and f l o o r , v i ew- f a c t o r r e l a t i o n s h i p s between p a i r s of p e r p e n d i c u l a r f a c e t s are no l o n g e r r e c i p r o c a l as assumed i n the o r i g i n a l model; (the 62 r e c i p r o c i t y between p a r a l l e l f a c e t s i s m a i n t a i n e d ) . A l l canyon v i e w - f a c t o r s have been c a l c u l a t e d u s i n g e q u a t i o n s for an i n f i n i t e s i m a l s u r f a c e element p a r a l l e l and p e r p e n d i c u l a r to a p lane of g iven l e n g t h and width (he igh t ) d e r i v e d u s i n g N u s s e l t ' s "Unit Sphere Method" ( S i e g e l and H o w e l l , 1972; Steyn and L y o n s , 1 985) . The v i e w - f a c t o r \p, f o r an element p e r p e n d i c u l a r to a p lane s u r f a c e from a p o i n t l o c a t e d at x=0 (canyon m i d - l e n g t h ) i s ^p 0 ,y = ( 1 / 2 T T ) [ 2 t a n " 1 A - B t a n " 1 C ] (3 .1) w i t h A = b/2y B = 2 y / ( h 2 + y 2 ) C = b / 2 ( h 2 + y 2 ) where y i s the d i s t a n c e p e r p e n d i c u l a r to the e lement , b = t o t a l l e n g t h of the p lane e lement , and h = h e i g h t of the p lane e lement . The s u b s c r i p t p denotes the v i e w - f a c t o r of the p lane e lement . P e r p e n d i c u l a r v i e w - f a c t o r s for each g r i d - p o i n t are de termined by c a l c u l a t i n g the v i e w - f a c t o r of the p a r t i a l h e i g h t of w a l l de termined from the sum of g r i d - p o i n t h e i g h t s up to and i n c l u d i n g the p o i n t of i n t e r e s t and then s u b t r a c t i n g the v iew- f a c t o r for the p a r t i a l h e i g h t of w a l l de termined by a l l g r i d - 63 64 p o i n t s l o c a t e d below the p o i n t of i n t e r e s t . For example, the v i e w - f a c t o r of g r i d - p o i n t 8 on w a l l A for a p o i n t on the f l o o r i s c a l c u l a t e d u s i n g (1) w i t h h e q u a l to the sum of g r i d - p o i n t h e i g h t s from p o i n t 1 to 8 and s u b t r a c t i n g the v i e w - f a c t o r c a l c u l a t e d u s i n g h set to the sum of h e i g h t s of p o i n t s 1 to 7. In mathemat ica l n o t a t i o n , t h i s procedure may be expressed as j J-1 <//(j) = ^ I gpht - </> I gpht (3 .2) gp=1 gp=1 where the v i e w - f a c t o r of g r i d - p o i n t j (\Mj)) i s c a l c u l a t e d from i n f o r m a t i o n on the g r i d - p o i n t h e i g h t (gpht) f o r each g r i d - p o i n t ( g p ) . For an element p a r a l l e l to a p l a n e s u r f a c e e lement , 0,y = (1 /2ir) [ D ( 2 t a n ~ 1 E ) + F ( 2 t a n ~ 1 G ) ] (3 .3) w i t h D = l / ( l + y 2 / h 2 ) E = ( b / 2 ) / ( y 2 + h 2 ) F = { 1 / t i + y 2 / ( b / 2 ) 2 ] } G = h / [ y 2 + ( b / 2 ) 2 ] 65 and y i s a g a i n measured p e r p e n d i c u l a r from the p lane e lement . F o r a g r i d - p o i n t d i r e c t l y o p p o s i t e , h i s set at o n e - h a l f the g r i d - p o i n t h e i g h t and the v i e w - f a c t o r i s m u l t i p l i e d by two. For p o i n t s above or below the p o i n t of i n t e r e s t , a method s i m i l a r to t h a t employed in the p e r p e n d i c u l a r v i e w - f a c t o r c a l c u l a t i o n i s used . (See F i g u r e 3 . 3 ) . 3 .4 .3 A d d i t i o n a l G r i d - P o i n t s The a d d i t i o n a l area which the capp ing b l o c k s add to the r e g u l a r g r i d of thermocouples on the w a l l s may be accounted for i n one of two ways. The a r e a may be added to that r e p r e s e n t e d by the t e n t h g r i d - p o i n t of each w a l l . In t h i s c a s e , the temperature measured at t h i s p o i n t would r e p r e s e n t the a r e a from the top of the w a l l to a p o i n t midway down the topmost ho l low c o n c r e t e b l o c k . The second method i s to m a i n t a i n a r e g u l a r g r i d for the ten thermocouples and c r e a t e an e l e v e n t h ' f a k e ' thermocouple l o c a t e d at the m i d - p o i n t of the e x t r a area added by the capp ing b l o c k . S ince no measured temperature i s a v a i l a b l e for t h i s p o i n t , i t must be e s t i m a t e d by some means. Each method has advantages and d i s a d v a n t a g e s . The f i r s t scheme does not r e q u i r e the e s t i m a t i o n of a s u r f a c e t emperature , but i n c r e a s e s the area which the t e n t h thermocouple r e p r e s e n t s , i n an area where r e l a t i v e l y h i g h temperature g r a d i e n t s are e x p e c t e d . Given the f a c e t temperature d i s t r i b u t i o n s of 6 .2 .2 the a d d i t i o n of the capp ing b l o c k area to g r i d - p o i n t 10 would r e s u l t 66 i n a b i a s towards o v e r - e s t i m a t i o n of the temperature for the g r i d - p o i n t a r e a . S e n s i t i v i t y t e s t s of s u r f a c e temperature ( S e c t i o n 3 . 5 . 2 ) i n d i c a t e e r r o r s i n the s p e c i f i c a t i o n of s u r f a c e temperature f o r c o r n e r g r i d - p o i n t s would r e s u l t i n model e r r o r s a t the p o i n t i n e r r o r and at the f i r s t p o i n t on the p lane of the canyon top where the v i e w - f a c t o r f o r the t e n t h p o i n t on the w a l l s i s s i g n i f i c a n t . E f f e c t s on o t h e r p o i n t s would be m i n o r . R e p r e s e n t i n g the e x t r a area by the a d d i t i o n of an e l e v e n t h p o i n t a l l o w s the measured temperature at the t e n t h g r i d - p o i n t to r e p r e s e n t the same area as a l l the o ther p o i n t s on the w a l l s . S ince the added p o i n t i s never v a l i d a t e d on i t s own, e r r o r s in the e s t i m a t i o n of the temperature f o r the p o i n t would o n l y be r e f l e c t e d i n the v a l i d a t i o n of p o i n t s on the canyon top neares t the w a l l s . The d i sadvantage of h a v i n g to e s t i m a t e a temperature f o r the a d d i t i o n a l p o i n t i s o f f s e t by the advantage that no f u r t h e r e r r o r s w i l l be i n c u r r e d i n the temperature r e p r e s e n t a t i o n of the t e n t h p o i n t . The net e f f e c t s h o u l d be that the a d d i t i o n of an e l e v e n t h p o i n t w i l l reduce the number of p o i n t s i n e r r o r from two to one. An added advantage of the second method i s tha t the t r a v e r s e path i n c l u d e s the t e n t h g r i d - p o i n t on the w a l l s but not the f i r s t or l a s t p o i n t on the canyon t o p . T h i s i s a r e s u l t of the a r c n e c e s s a r y f o r the rad iometers to t u r n between f a c e t s and the added h e i g h t of the c a p p i n g b lock on the w a l l s . Thus , those p o i n t s which w i l l i n c u r the maximum e r r o r due to the a d d i t i o n of an e l e v e n t h p o i n t on the w a l l s are not i n c l u d e d i n the t r a v e r s e d 67 d a t a . Given these advantages , the method of c r e a t i n g an e x t r a g r i d - p o i n t on the w a l l s was s e l e c t e d . P r o c e d u r e s for e s t i m a t i n g the temperature of the e l e v e n t h p o i n t are d e s c r i b e d i n Appendix B . 3.5 SENSITIVITY TESTS 3 .5 .1 V i e w - F a c t o r s / G r i d - P o i n t s R a d i a t i o n e s t i m a t e s w i t h i n complex geometr ic s e t t i n g s are s t r o n g l y dependent upon the a c c u r a c y of v i e w - f a c t o r c a l c u l a t i o n s . W i t h i n the A r n f i e l d model , v i e w - f a c t o r s are d e f i n e d f o r p a i r s of canyon f a c e t s and s t o r e d as a r r a y s . Each g r i d - p o i n t on a canyon f a c e t has a v i e w - f a c t o r a s s o c i a t e d w i t h every g r i d - p o i n t on the a s s o c i a t e d p a i r e d f a c e t . Each p o i n t r e p r e s e n t s an a r e a , which i n c r e a s e s or d e c r e a s e s as the d e n s i t y of g r i d - p o i n t s decreases or i n c r e a s e s . For a canyon of g i v e n d i m e n s i o n s , a h i g h e r d e n s i t y of g r i d - p o i n t s w i l l r e s u l t i n a g r e a t e r a b i l i t y to r e s o l v e the r a d i a t i o n b a l a n c e of the s u r f a c e . D u r i n g daytime t h i s becomes important as shadows extend a c r o s s the canyon f a c e t s . The A r n f i e l d model p r o v i d e s f o r the c a l c u l a t i o n of canyon v i e w - f a c t o r s i n t e r n a l l y , or as an o p t i o n , to be read i n from an e x t e r n a l f i l e . A number of t e s t s have been conducted on the s e n s i t i v i t y of the model to the number of g r i d - p o i n t s ( A r n f i e l d , p e r s . comm., 1987). Hav ing s u b s t a n t i a l l y a l t e r e d the v i e w - f a c t o r c a l c u l a t i o n 68 p r o c e d u r e and the g r i d - p o i n t s t r u c t u r e of the canyon , s e v e r a l t e s t s were r e p e a t e d for the model as i n i t i a l i z e d f o r the e x p e r i m e n t a l c a n y o n . The t e s t s are des igned to i l l u s t r a t e the model response to a set of input c o n d i t i o n s for which the t h e o r e t i c a l l y t r u e v a l u e i s known. T a b l e 3 .5 T e s t 1 of Model S e n s i t i v i t y to the Number of Model G r i d - P o i n t s . T h i s t e s t c a l c u l a t e s L c (W m~ 2) at the canyon top for the f o l l o w i n g model input parameters : L i c t : 100 W m 2 S u r f a c e Temp.: 283 K D i s t r i b u t i o n : i s o t r o p i c S u r f a c e E m i s s i v i t y : 0.0 E x p e c t e d R e s u l t : L 0 = 100 W m~ 2 O r i g i n a l S c a l e Canyon M i n . Canyon Dimension Lo Lo 10 104.2 99. 1 20 102.3 n /a 50 100.1 n / a 100 99.7 n / a T a b l e 3.6 T e s t 2 of Model S e n s i t i v i t y to the Number of Model G r i d - P o i n t s . Model Input Parameters : L ^ : 363.7 W m~ 2 ( e q u i v . b l a c k body emi t tance at 283 K) S u r f a c e Temperature: 283 K D i s t r i b u t i o n : i s o t r o p i c S u r f a c e E m i s s i v i t y : 0.5 Expec ted R e s u l t : L * = 0 W m~ 2 M i n . Canyon Dimension O r i g i n a l L * S c a l e Canyon L * 10 - 8 . 7 1 .3 20 - 4 . 6 n /a 50 - 1 . 6 n / a 100 - 0 . 8 n /a 69 The r e s u l t s in T a b l e s 3 .5 and 3.6 i n d i c a t e a s i g n i f i c a n t d i f f e r e n c e between the two canyon mode l s . The o n l y major changes made i n the model were the v i e w - f a c t o r s p e c i f i c a t i o n and the a d d i t i o n of the e l e v e n t h g r i d - p o i n t on the w a l l s . Each of these may a f f e c t e s t i m a t e s of the f l u x e s a t the canyon t o p . F o r a s t r i c t compar i son of model output as i t v a r i e s w i t h the method of c a l c u l a t i n g v i e w - f a c t o r s , T e s t 1 was r e p e a t e d f i r s t u s i n g the d e f a u l t v i e w - f a c t o r s c a l c u l a t e d by the program f o r a 10 by 10 c a n y o n , and s e c o n d l y u s i n g the v i e w - f a c t o r s c a l c u l a t e d from the e q u a t i o n s p r o v i d e d by Steyn (1987 p e r s . comm.) . T a b l e 3.7 T e s t 1: V i e w - f a c t o r C o m p a r i s o n . G r i d - p o i n t s D e f a u l t N u s s e l t V i e w - f a c t o r s Sphere L D (W m~ 2) 10 104.2 98.6 (Average over 20 102.3 99.4 t o p ; W m 2 ) The r e s u l t s i n T a b l e 3.7 and F i g u r e 3.4 suppor t the h y p o t h e s i s t h a t the method of c a l c u l a t i n g the v i e w - f a c t o r s i s the major d i f f e r e n c e in the t e s t s . The p l o t s of F i g u r e 3.4 are d e s i g n e d so t h a t the axes r e p r e s e n t the canyon f a c e t s , w i t h the s c a l i n g of the da ta per formed a u t o m a t i c a l l y , g i v e n the range of da ta i n p u t . T h e r e f o r e , s c a l i n g v a r i e s between f a c e t s and between p l o t s and r e f e r e n c e to the s c a l e s i s n e c e s s a r y to a v o i d v i s u a l m i s - i n t e r p r e t a t i o n . Note the d i f f e r e n c e i n L Q between the Focet Point: Top Lo Focet Point: Floor L o S c o ^ u rn " 0 . 0 ~ i — i — r 2 . 0 I 4 . 0 Focet Point: T o p 1 2 3 4 5 6 7 8 9 A A A A A A A A A -ftl—y y — ^ — y y — ^ — ^ — ^ 3 4 5 6 7 8 Focet Point: Floor 9 10 5 . 0 ^ 1 3 . 0 L . i n . 1.0 F i g u r e 3.4 T e s t 1 of model s e n s i t i v i t y , c i r c l e s - o r i g i n a l model v i e w - f a c t o r s , t r i a n g l e s - N u s s e l t Sphere v i e w - f a c t o r s . F l u x e s i n W m ~ . 71 o r i g i n a l and the m o d i f i e d models i s g r e a t e r in each case than the d i f f e r e n c e between the two m o d i f i e d models . As f u r t h e r c o m f i r m a t i o n , the t e s t was repeated f o r a 20 x 20 u n i t canyon . The i n c r e a s e i n a c c u r a c y due to the added g r i d - p o i n t s u s i n g the e x t e r n a l l y read v i e w - f a c t o r s i s l e s s than that a c h i e v e d when u s i n g the d e f a u l t v i e w - f a c t o r s p r o v i d e d w i t h i n the program. The apparent d i s c r e p a n c y i n the r e s u l t s due to the v iew- f a c t o r c a l c u l a t i o n method has been e x p l a i n e d by A r n f i e l d (1988 p e r s . comm.) as r e s u l t i n g from the f i n i t e d i f f e r e n c e a p p r o x i m a t i o n of the f l u x ( Q ) t r a v e l l i n g from one g r i d - p o i n t to another Q = (NT C O S ^ cos/32 AT A 2 ) / r 2 (3 .4) where 0̂  and 02 are the ang le s between the p e r p e n d i c u l a r to each g r i d - p o i n t , the A ' s are the areas a s s o c i a t e d w i t h the g r i d - p o i n t , r i s the l e n g t h between the p o i n t s and N the r a d i a t i v e f l u x . In cases where r i s not a good r e p r e s e n t a t i o n of the paths which r a d i a t i o n may take between the elements e r r o r s i n c r e a s e . T h i s o c c u r s when r i s s m a l l . In the v i e w - f a c t o r a r r a y s , r reaches a minimum i n any p a i r of p e r p e n d i c u l a r canyon f a c e t s , where two g r i d - p o i n t areas a d j o i n ; i e . the c o r n e r e l ements . The e r r o r s in the c o r n e r e lements are shown c l e a r l y i n F i g u r e 3.4 which d i s p l a y s the e n t i r e Tes t 1 r e s u l t s f or the o r i g i n a l and m o d i f i e d models . The r e s u l t s i n d i c a t e the m o d i f i e d model tends to u n d e r e s t i m a t e L 0 and in c o r n e r e lements . T a b l e 3.8 i l l u s t r a t e s the d i f f e r e n c e s i n the v i e w - f a c t o r s c a l c u l a t e d u s i n g 72 the two ca l c u l a t i o n methods for grid-points on wall A for the grid-point on the floor nearest wall A. Table 3.8 Comparison of View-factor Calculations: Wall A/Floor. Point on Wall A r n f i e l d Model Nusselt Sphere Difference (Percent) 1 • 0.35355246 0.27639282 21 .8 2 0.09486705 0.10233772 -7.3 3 0.03771256 0.03906620 -3.5 4 0.01979602 0.02017921 -1.9 5 0.01211674 0.01226169 -1.2 6 0.00815845 0.00822449 -0.8 7 0.00585961 0.00589380 -0.6 8 0.00440878 0.00442821 -0.4 9 0.00343535 0.00344712 -0.3 10 0.00275088 0.00275850 -0.3 The adverse e f f e c t of using the model calculated view-factors was offset by using a greater number of grid-points per facet (note the increased accuracy in the tests of Tables 3.5 and 3.6 as the number of points i s increased). The over-estimation that results from using t h i s method of calc u l a t i o n also compensates for the under-estimation a r i s i n g from the truncated number of multiple r e f l e c t i o n s allowed (A r n f i e l d , 1988 pers. comm.). The advantage of using the view-factor equations derived from the Nusselt Sphere method i s higher accuracy at lower grid-point resolutions. Repeating Test 1 with a higher grid-point density with t h i s method provides only a s l i g h t increase in accuracy (Table 3.7). It i s concluded that both methods of cal c u l a t i n g the view-factors suffer from inaccuracies when the elements are close together, but the equations derived from the Nusselt 73 Sphere Method are b e t t e r than those a v a i l a b l e in the o r i g i n a l program code . 3 . 5 . 2 S u r f a c e Temperature S u r f a c e temperature measurements are the most numerous of the input parameters to the A r n f i e l d model and thereby c o n s t i t u t e the g r e a t e s t p o s s i b l e source ( n u m e r i c a l l y ) of measurement e r r o r . At n i g h t , under c l e a r and calm c o n d i t i o n s , s u r f a c e temperature i s the dominant c o n t r o l of the s u r f a c e r a d i a t i v e b a l a n c e . S u c c e s s f u l m o d e l l i n g of the r a d i a t i v e ba lance w i t h i n the canyon under these c o n d i t i o n s w i l l r e q u i r e a c c u r a t e measurement of the s u r f a c e t e m p e r a t u r e s . The o b j e c t i v e of the temperature s e n s i t i v i t y t e s t s i s to examine the e f f e c t of d i f f e r e n c e s i n measured s u r f a c e t emperature from the ' t r u e ' v a l u e upon model output f o r the long-wave r a d i a t i v e f l u x e s . V a r i a t i o n s i n measured s u r f a c e temperature may a r i s e due to d i f f e r e n c e s i n the attachment of i n d i v i d u a l thermocouples to the canyon f a c e t s . These e r r o r s may be l o c a l , a f f e c t i n g on ly a few i s o l a t e d thermocouple s , or more g e n e r a l , r e s u l t i n g i n an o v e r a l l u n d e r - or o v e r e s t i m a t i o n of the s u r f a c e t empera ture . The s e n s i t i v i t y t e s t s of s u r f a c e temperature have been d e s i g n e d to c o v e r both s c e n a r i o s . Model parameters f o r the temperature s e n s i t i v i t y t e s t s were set as f o l l o w s : I n c i d e n t long-wave r a d i a t i o n at the canyon top 74 was set at 363.7 W m ~ 2 , the e q u i v a l e n t b lackbody r a d i a t i o n for a s u r f a c e temperature of 283K. The e x p e r i m e n t a l l y determined e m i s s i v i t i e s f o r the f l o o r and w a l l s were used . RSTOP, the convergence c r i t e r i o n for the canyon m u l t i p l e r e f l e c t i o n r o u t i n e , was set a t 0 .0333. S i n c e i t i s c a l l e d up to three t imes the t o t a l e r r o r r e s u l t i n g from the m u l t i p l e r e f l e c t i o n p r o c e s s shou l d be l e s s than 0.1 W m ~ 2 . T h i s i s low enough not to be a l i m i t i n g f a c t o r . The s u r f a c e temperature i n p u t , was v a r i e d in the f o l l o w i n g ways: (1) the s u r f a c e temperature was f i x e d at 283K for a l l f a c e t s (the c o n t r o l run) (2) the s u r f a c e temperature of a s i n g l e p o i n t on a w a l l was v a r i e d (IF 1 ,IP 5) to determine the e f f e c t s on the mode l l ed v a l u e s at a l l o ther p o i n t s (3) the s u r f a c e temperature of the e x t r a p o l a t e d p o i n t on the w a l l s (IP 11) i s i n c r e a s e d to determine the s e n s i t i v i t y to the e s t i m a t i o n of t h i s temperature (4) the s u r f a c e temperature of a p o i n t on the f l o o r was v a r i e d i n a manner s i m i l a r to (2) (5) a l l s u r f a c e temperatures were i n c r e a s e d e q u a l l y to i n v e s t i g a t e the e f f e c t s of a g e n e r a l o v e r - e s t i m a t i o n . F i g u r e 3 .5 p r e s e n t s the output of the model c o n t r o l r u n . The v a l u e s of L D v a r y between 363.8 and 363.6 W m~ 2 for p o i n t s on the canyon w a l l s (data for the e l e v e n t h p o i n t are o m i t t e d ) . For p o i n t s a c r o s s the top L 0 i s c o n s t a n t at 362.8 Wm~ 2 . The v a l u e s of v a r y s l i g h t l y from p o i n t to p o i n t ; t h i s i s a t t r i b u t e d to Focet Point: T o p Facet Point: Top F i g u r e 3.5 Model c o n t r o l r u n . F l u x e s i n W m 76 the manner in which the v i e w - f a c t o r s have been c a l c u l a t e d . The behav iour of L * i s a f f e c t e d by the i n s t a b i l i t y i n the v a l u e s of L ^ . In the o ther comparisons of model o u t p u t , o n l y the d i f f e r e n c e s from the c o n t r o l run are t a b l e d . F i g u r e 3.6 i l l u s t r a t e s the r e s u l t s of i n c r e a s i n g the s u r f a c e temperature at the m i d - p o i n t of the West w a l l . The c a l c u l a t e d d i f f e r e n c e s i n d i c a t e major i n c r e a s e s i n the model output of L Q at the p o i n t at which the s u r f a c e temperature was i n c r e a s e d , a r e s u l t which i s to be e x p e c t e d . A secondary area of i n f l u e n c e i s found a c r o s s the canyon top where L Q v a l u e s are i n c r e a s e d s l i g h t l y . Minor v a r i a t i o n s i n the model led f l u x e s occur at p o i n t s on the w a l l o p p o s i t e the a f f e c t e d p o i n t and on the f l o o r , r e s u l t i n g from m u l t i p l e r e f l e c t i o n s . i n c r e a s e s s l i g h t l y for p o i n t s on the f l o o r and near the top of the E a s t w a l l . L * becomes n e g a t i v e a c r o s s the top i n accordance w i t h the i n c r e a s e in the L Q and p o s i t i v e in response to i n c r e a s e s in L^ for the f l o o r and w a l l B. F i g u r e 3.7 p r e s e n t s the r e s u l t s when the temperature of p o i n t 1 (IP 1) on the f l o o r i s i n c r e a s e d . A g a i n , the major i n c r e a s e in L Q i s at the p o i n t of the anomaly, w i t h minor i n c r e a s e s a c r o s s the canyon t o p , e s p e c i a l l y towards the Eas t w a l l , and at the . p o i n t on the f l o o r neares t the a f f e c t e d w a l l . L^ i s i n c r e a s e d s t r o n g l y for the p o i n t s on the f l o o r a d j a c e n t to the West w a l l . I n c r e a s e s are a l s o noted for the lower p o r t i o n s of the o p p o s i t e w a l l . L * i n c r e a s e s p o s i t i v e l y for the o p p o s i t e w a l l and the f l o o r and decreases a c r o s s the canyon t o p . 77 Facet Point: Top Surface Temperature Increased By. O 0.2 K A 0.5 K + 1.0 K X 2.0 K O 3.0 K • 5.0 K F i g u r e 3 .6 S e n s i t i v i t y of m o d e l l e d f l u x e s to s u r f a c e t e m p e r a t u r e changes at p o i n t 5 on the West w a l l (IF 1, IP 5) D i f f e r e n c e s (W m~ 2) from a c o n t r o l run w i t h T S =283K. 78 F a c e t Point; T o p 0 . 0 10.1 2 0 . 2 1 2 3 4 5 6 7 8 9 10 0.1 0 . 0 0 . 0 Lo F o c e l Point: Floor L o Focet Point. T o p Surfoce Temperoture Increased By: O 0.2 K A 0.5 K + 1.0 K X 2.0 K «• 3.0 K • 5.0 K F i g u r e 3.7 S e n s i t i v i t y of m o d e l l e d f l u x e s to s u r f a c e t empera ture changes at p o i n t 1 on the West w a l l (IF 1, IP 1 ) . D i f f e r e n c e s (W rn"2) from a c o n t r o l run w i t h T S =283K. 79 Maximum e r r o r s r e s u l t i n g from an o v e r e s t i m a t e of the e x t r a p o l a t e d s u r f a c e temperature of IP 11 on the West w a l l are shown to occur f o r and L * at the top of the o p p o s i t e w a l l and f o r L Q and L * at the canyon top a d j a c e n t to the a f f e c t e d p o i n t . The e r r o r s are l o c a l i z e d and are l e s s than 3.0 W m""2 f or a 5 K i n c r e a s e of the e x t r a p o l a t e d p o i n t . I t would t h e r e f o r e appear tha t the model i s not unduly s e n s i t i v e to the method by which the temperature of the e x t r a p o i n t s i s o b t a i n e d . The t emperature of a s i n g l e p o i n t on the f l o o r (IP 5) has a l s o been v a r i e d in an analogous manner ( F i g u r e 3 . 9 ) . The l a r g e s t d i f f e r e n c e s in L 0 a g a i n are observed d i r e c t l y over the i n c r e a s e d s u r f a c e temperature w i t h a s l i g h t i n d i c a t i o n tha t s m a l l i n c r e a s e s are presen t f or some other p o i n t s on the w a l l s and f l o o r due to m u l t i p l e r e f l e c t i o n s . The major d i f f e r e n c e observed between t h i s t e s t and t h a t f or a p o i n t on a w a l l i s the d i f f e r e n c e i n L 0 observed a c r o s s the canyon t o p . The i n c r e a s e s in L Q at the canyon top over the c o n t r o l run are a p p r o x i m a t e l y 50 to 60 p e r c e n t g r e a t e r for the f l o o r c a s e . A s i g n i f i c a n t i n c r e a s e i s observed at the canyon top even for a 1 degree i n c r e a s e a t a s i n g l e p o i n t on t h e canyon f l o o r . Increase s i n are noted f o r both w a l l s . As a r e s u l t , L * e x h i b i t s a g r e a t e r decrease a c r o s s the t o p . Model r e s u l t s o b t a i n e d from an anomolous temperature l o c a t e d on the f l o o r next to the West w a l l (IP 1) are p r e s e n t e d i n F i g u r e 3 . 1 0 . The l o c a t i o n of maximum d i f f e r e n c e s for L Q and L * on the canyon top s h i f t s w e s t a r d . P o i n t s low on the West w a l l Focet Point: Top 3. 4 5 6 7 8 9 10 + * * ̂  ' O 0.2 K 1 2 3 4 5 6 7 8 9 10 Focel Point: Floor Surfoce Temp«roture Increosed By: A 0.5 K + 1.0 K X 2.0 K «. 2.0 K Figure 3.8 Sensitivity of modelled fluxes to « „ r f a « D ^ r e n c e l (5'S9!? f P ° l n t 1 1 °" t h e W e V S a l T mrxerences (W m from a control run with TS=283K. 81 l i Focet Point: Floor I ; Focet Point: Top Focet Point: T o p ) 2 3 4 5 6 7 8 9 10 - Surfoce Temperature Increased By: O 0.2 K A 0.5 K 4- 1.0 K X 2.0 K « 3 . 0 K *• 5.0 K Figure 3.9 S e n s i t i v i t y of modelled fluxes to surface temperature changes at point 5 on the canyon floor (IF 3, IP 5). Differences (W m 2) from a control run with TS=283K. 82 Focet Point: Top Surfoce Temperature Increased By: O 0.2 K A 0.5 K + 1.0 K X 2.0 K © 3.0 K • 5.0 K F i g u r e 3 .10 S e n s i t i v i t y of m o d e l l e d f l u x e s to s u r f a c e t e m p e r a t u r e changes a t p o i n t 1 on the canyon f l o o r ( IF 3, IP 1 D i f f e r e n c e s (W m~ 2 ) from a c o n t r o l run w i t h T S =283K. 83 and towards the top of the E a s t w a l l e x h i b i t the g r e a t e s t i n c r e a s e in and L * . The s e n s i t i v i t y t e s t s of i n c r e a s i n g a s i n g l e g r i d - p o i n t temperature produce r e s u l t s e a s i l y e x p l a i n a b l e i n terms of the v i e w - f a c t o r s of the g r i d - p o i n t . The most n o t i c e a b l e change i s the i n c r e a s e i n e m i t t a n c e , and hence decrease of net r a d i a t i o n , f o r the p o i n t m o d i f i e d . D i f f e r e n c e s i n the r a d i a t i v e f l u x e s at o t h e r p o i n t s are d i r e c t l y r e l a t e d to the v i e w - f a c t o r of that p o i n t for the a f f e c t e d p o i n t . I n c r e a s e s i n long-wave r a d i a t i o n at a p o i n t r e s u l t in an i n c r e a s e of r e f l e c t e d long-wave for the p o i n t , a l t h o u g h t h i s e f f e c t i s damped somewhat by the h i g h v a l u e s of e m i s s i v i t y of the canyon m a t e r i a l s . Any i n c r e a s e in the temperature of a g r i d - p o i n t for the t e s t s r e s u l t s in an i n c r e a s e i n the f l u x l e a v i n g the canyon top and t h e r e f o r e decreases the net r a d i a t i o n . Except a t the p o i n t of change, d i f f e r e n c e s i n L * on the f l o o r and w a l l s are c o n t r o l l e d by and by L 0 a c r o s s the canyon t o p . C o n s i d e r now the e f f e c t upon the model output i f a g e n e r a l i n c r e a s e or decrease were to a f f e c t a l l t emperature measurements. F i g u r e 3.11 i l l u s t r a t e s the d i f f e r e n c e s from the model run when the s u r f a c e temperature of a l l p o i n t s w i t h i n the canyon i s i n c r e a s e d by an equa l amount. The temperature was i n c r e a s e d by 0 . 2 , 0 . 5 , 1 , 2 , 3 , and 5 K above that of the c o n t r o l r u n . The r e s u l t s show tha t the mode l l ed v a l u e s of L c w i l l i n c r e a s e more or l e s s e q u a l l y f o r a l l p o i n t s w i t h i n the canyon . For a temperature i n c r e a s e of 0.5 K the i n c r e a s e in L Q i s 2 . 5 - 2.6 W m ~ 2 ; w i t h a temperature i n c r e a s e of 1 K for a l l p o i n t s the 84 3 4 5 6 7 8 Focet Point: Floor 21 .9 . „ - < > < , 1 .0 -*i—«r M .0 Lo 21 .1 Focet Point: Top 3 4 5 6 7 8 10 o> 4—t— t_ -* 4--4--4—•—*— r\j _ rn -I 1 1 1 1 1 1 1 1 - ts—©—©— "p 8- s e e i 3 4 5 6 7 8 9 Focet Point: Floor 10 26 2 ' " " o o | U J <>tht£> ~ C o •*(>-_ 16.1 L o 6.0 Focet Point: Top 1 2 3 - 4 5 6 7 8 9 !0o 8 - 8 8 8 8 8 8 8 8 6 o 0.2 K 1 1 1 1 T—=1 1 1 1 I 2 3 4 5 6 7 8 9 10 Focet Point: Floor Surfoce Temperoture Increosed By: 0.5 K + 1.0 K X 2.0 K o 3.0 K • 5.0 K F i g u r e 3.11 te m p e r a t u r e c o n t r o l run S e n s i t i v i t y of m o d e l l e d f l u x e s t o e q u a l s u r f a c e changes a t a l l p o i n t s . D i f f e r e n c e s (W m 2 ) from a w i t h T S=283K. 85 i n c r e a s e i s 5.0-5.2 W m ~ 2 ; an i n c r e a s e of 2 K boos t s L 0 v a l u e s by 10.2-10.3 W m ~ 2 . v a l u e s are i n c r e a s e d , p a r t i c u l a r l y for the lower p o r t i o n s of the w a l l s , due to the e f f e c t of m u l t i p l e r e f l e c t i o n s . L * decreases a c r o s s the canyon t o p , near the tops of the w a l l s and i n the c e n t r e of the f l o o r . 3.5.3 R a d i a t i o n To determine the e f f e c t of v a r i a t i o n s in measured at the canyon top upon mode l l ed v a l u e s of L Q at a l l o ther p o i n t s , a s e n s i t i v i t y t e s t was conducted h o l d i n g the s u r f a c e temperature f i x e d at 283K f or a l l f a c e t s , s e t t i n g the e m i s s i v i t y at the e x p e r i m e n t a l l y determined v a l u e s and i n c r e a s i n g from 363.7 W m - 2 (the c o n t r o l ) to 383.7 W m ~ 2 . As b e f o r e , RSTOP was set at 0.0333. F i g u r e 3.12 p r e s e n t s model output for L Q , L ^ , and L * when i n c r e a s i n g at the canyon top by 1, 2, 5, 10, 15 and 20 W m ~ 2 . Incoming r a d i a t i o n changes in accordance wi th i/>s; the l a r g e s t i n c r e a s e s are seen for the tops of the w a l l s and i n the middle of the canyon f l o o r . The d i f f e r e n c e s in L 0 are m i n o r , even f o r i n c r e a s e s of 15 and 20 W m ~ 2 . A t r e n d towards s l i g h t l y h i g h e r v a l u e s towards the tops of the canyon w a l l s i s n o t e d , m i r r o r i n g the i n c r e a s e i n . The i n c r e a s e over the canyon f l o o r i s smoothed o u t , l i k e l y due to the m u l t i p l e r e f l e c t i o n p r o c e s s . With lower v a l u e s of e m i s s i v i t y g r e a t e r d i f f e r e n c e s may occur due to i n c r e a s e d 86 1 2 3 4 5 6 7 8 Focet Point: Floor 10 Focet Point: Top I 1 1 — T 1 2 3 4 5 6 7 8 9 10 0.4 0.2 0.0 Focet Point: Floor lo Focet Point: Top 1 2 3 4 5 6 7 8 9 10 f -4--4--4—+- —•— -4—- «= =*= =*=#= ft ft ft 1 2 3 4 5 6 7 8 9 10 Focet Point: Floor 8.9 O to Lw Increased By: A 2.0 ' + 5.0 X 10.0 « 15.0 • 20.0 (W m*) F i g u r e 3.12 S e n s i t i v i t y of m o d e l l e d f l u x e s to L i C f D i f f e r e n c e s (W m~ 2) from a c o n t r o l run w i t h T S =283K. 87 r e f l e c t i o n of L ^ ; the e f f e c t of e m i s s i v i t y i s c o n s i d e r e d i n the next s e c t i o n . The changes i n L * are dominated by the changes i n L j and f o l l o w the p a t t e r n s o u t l i n e d above . 3.5.4 E m i s s i v i t y V a r i a t i o n s e x i s t i n the measured v a l u e s of e m i s s i v i t y f o r both the c o n c r e t e b l o c k s used i n the w a l l s and the c o n c r e t e s u r f a c e of the s i t e (see T a b l e 2.3 for the da ta on measured e m i s s i v i t y ) . An average of these v a l u e s i s used i n the mode l . Whi le the s t a n d a r d d e v i a t i o n s of both the c o n c r e t e s u r f a c e and canyon b l o c k i s near 0 .02 , the range of v a l u e s measured i s much h i g h e r . The o u t g o i n g f l u x d e n s i t y of long-wave r a d i a t i o n from a s u r f a c e may be expressed as L Q = e a T s 4 + (1 - e)Li. (3 .5) Note t h a t the e m i s s i v i t y appears twice i n the e q u a t i o n i n an oppos ing f a s h i o n . I f were to e x a c t l y ba lance L Q at a p o i n t the e m i s s i v i t y would have no net e f f e c t (as i n the c o n d i t i o n s for T e s t 2 ) . Under r a d i a t i v e c o o l i n g c o n d i t i o n s , the f i r s t term of (3 .5) w i l l exceed the second so t h a t as the s u r f a c e e m i s s i v i t y decreases the long-wave emit tance w i l l d e c r e a s e . The s e n s i t i v i t y of mode l l ed r a d i a t i o n to v a l u e s of s u r f a c e e m i s s i v i t y was t e s t e d by u s i n g a two se t s of measured input data from a c l e a r , most ly calm evening (Aug. 3 /4; H/W=1.0) . The f i r s t i s from the e a r l y even ing (approx imate ly 0.5 h from s u n s e t ) , when s u r f a c e temperatures are r e l a t i v e l y h i g h and the second i s l a t e at n i g h t (6 hours a f t e r s u n s e t ) . Three e m i s s i v i t i e s were t e s t e d ( a l l canyon f a c e t s are assumed to have equa l v a l u e s of e ) , 0 .96 , the c o n t r o l r u n , 1.00, and 0 .90 . These c o r r e s p o n d to the measured e ± 0.06 which i s the p r o b a b l e e r r o r c a l c u l a t e d f o r e (Appendix D ) . The d i f f e r e n c e s from the c o n t r o l run are p l o t t e d in F i g u r e s 3.13 ( e a r l y evening run) and 3.14 ( l a t e e v e n i n g ) . As e x p e c t e d , L Q i n c r e a s e s wi th e due to the i n c r e a s e d emi t tance of the s u r f a c e . The i n c i d e n t r a d i a t i o n for p o i n t s w i t h i n the canyon a l s o i n c r e a s e s due to the i n c r e a s e of the p o r t i o n of the long-wave i r r a d i a n c e d e r i v e d from other canyon f a c e t s . T h i s i n c r e a s e i s l a r g e s t f or p o i n t s w i t h a low sky v iew- f a c t o r ; near the tops of the w a l l s and a t the m i d - p o i n t of the f l o o r i t i s a minimum and causes l a r g e r L 0 v a l u e s d e s p i t e lower s u r f a c e t e m p e r a t u r e s . L * i s decreased f o r a l l p o i n t s e x c e p t i n g the bottom p o r t i o n s of the w a l l s where a s l i g h t i n c r e a s e i s noted ( F i g u r e 3 . 1 4 ) . E r r o r s i n c o r r e c t l y d e t e r m i n i n g face t . e m i s s i v i t i e s w i l l be l a r g e s t in the e a r l y evening when s u r f a c e t emperatures are h i g h and the f i r s t term of (3.5) dominates . As the canyon c o o l s , assuming c o n s t a n t m e t e o r o l o g i c a l c o n d i t i o n s , the d i f f e r e n c e w i l l l e s s e n and e r r o r s due to e m i s s i v i t y . w i l l d e c r e a s e . Maximum d i f f e r e n c e s of ± 3 . 0 W m~ 2 are observed i n the e a r l y evening f o r L Q and L * a c r o s s the canyon top and L D f or p o i n t s near the tops of the w a l l s . . 89 Focet Point: Top 1 2 3 4 5 6 7 6 9 10 Figure 3.13 S e n s i t i v i t y of modelled fluxes to e, differences (W m - 2) from a control run of e=0.96. C i r c l e s - e=0.90, tr i a n g l e s - e=1.00. Surface temperature d i s t r i b u t i o n used is t y p i c a l for the early evening. Focet Point: Top F i g u r e 3.14 S e n s i t i v i t y of m o d e l l e d f l u x e s t o e, d i f f e r e n c e s ( W m ~ 2 ) from a c o n t r o l run o f e=0.96. C i r c l e s - e=0.90, t r i a n g l e s - €=1.00.. S u r f a c e t e m p e r a t u r e d i s t r i b u t i o n used i s t y p i c a l f o r the l a t e e v e n i n g . 91 3 . 5 . 5 Radiance D i s t r i b u t i o n The A r n f i e l d model p r o v i d e s two s k y - d e r i v e d long-wave r a d i a n c e d i s t r i b u t i o n o p t i o n s (Table 3 . 1 ) , that of Unsworth and M o n t e i t h (1975), Lp = [c + b l n ( u sec/3)] T T " 1 (3 .6) and an i s o t r o p i c d i s t r i b u t i o n = (3.7) P r e v i o u s m o d e l l i n g of the canyon long-wave r e f l e c t i o n c o e f f i c e n t ( A r n f i e l d , 1982) has compared the two d i s t r i b u t i o n s and found the r e s u l t s to be i n s e n s i t i v e to the d i s t r i b u t i o n used . Verseghy (1987) a l s o found i s o t r o p i c r a d i a n c e d i s t r i b u t i o n s to i n c u r r o n l y minor e r r o r s . As a guide to the magnitude of changes i n the r a d i a t i v e f l u x e s which may r e s u l t u s i n g the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n , Tes t 2 was repeated for o p t i c a l water vapour depths of 0.83 and 4.21 ( c o r r e s p o n d i n g to the s a t u r a t i o n vapour p r e s s u r e v a l u e s at 0 and 2 5 ° C r e s p e c t i v e l y ) . The d i f f e r e n c e s from the i s o t r o p i c r e s u l t s are p l o t t e d around the canyon c r o s s - s e c t i o n in F i g u r e 3 . 1 5 . The v a r i a n c e of the r a d i a n c e d i s t r i b u t i o n w i t h z e n i t h ang le i s e v i d e n t i n the r e s u l t s . P o i n t s a t the top of the w a l l s show a l a r g e i n c r e a s e in over the i s o t r o p i c d i s t r i b u t i o n s i n c e they view low z e n i t h a n g l e s of the s k y . At the canyon base , on ly l a r g e z e n i t h ang le s are viewed and a decrease i n L j from the c o n t r o l r e s u l t s . A c r o s s the f l o o r a decrease i n L^ i s observed which i s s t r o n g e s t at m i d - c a n y o n , a g a i n i n accordance w i t h the ^ s of the p o i n t . The Facet Point: Top Focet Point. Top F i g u r e 3.15 S e n s i t i v i t y of m o d e l l e d f l u x e s to r a d i a n c e d i s t r i b u t i o n used . D i f f e r e n c e s (W m~ 2 ) from an i s o t r o p i c d i s t r i b u t i o n u s i n g the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n w i t h u=0.83 cm ( t r i a n g l e s ) and u=4.21 cm ( c i r c l e s ) 93 p a t t e r n of L Q f o l l o w s t h a t of and i s made e v i d e n t by the r e l a t i v e l y low v a l u e of e m i s s i v i t y used i n t h i s t e s t . The average v a l u e of L * a c r o s s the canyon top i s very c l o s e to 0 w i t h the d e c r e a s e near mid-canyon o f f s e t by the i n c r e a s e near ythe edges . A d i f f e r e n c e of a p p r o x i m a t e l y 1.2 W m - 2 from the average L * of T e s t 1 (Tab le 3.6) i s n o t e d . The v a r i a t i o n between the v a l u e s of u employed i n d i c a t e g r e a t e r d i f f e r e n c e s from the i s o t r o p i c c o n t r o l run as u (and t h e r e f o r e s u r f a c e vapour p r e s s u r e ) d e c r e a s e s . Use of the i s o t r o p i c d i s t r i b u t i o n w i l l r e s u l t i n maximum e r r o r s f o r p o i n t s near the tops of the w a l l s under lower s u r f a c e vapour p r e s s u r e . 94 CHAPTER 4. MODEL VALIDATION 4.1 INTRODUCTION T h i s c h a p t e r p r e s e n t s model v a l i d a t i o n u s i n g s e l e c t e d data s e t s . -Examples are drawn from a u t o m a t i c a l l y c o l l e c t e d data chosen to r e p r e s e n t the r e s u l t s o b t a i n e d under a number of d i f f e r e n t c o n d i t i o n s . Parameters which vary i n c l u d e : canyon H / W , m e t e o r o l o g i c a l c o n d i t i o n s , number of model g r i d - p o i n t s , number of samples per g r i d - p o i n t , and the a v e r a g i n g p e r i o d for the model input d a t a . Each set i s composed of s c a t t e r p l o t s of measured versus mode l l ed data f o r each of the long-wave f l u x e s ( L D , L j , L * ) , f o r the Unsworth and M o n t e i t h (UM) (1975) r a d i a n c e d i s t r i b u t i o n u s i n g h o u r l y averages by g r i d - p o i n t to reduce the number of da ta p o i n t s p l o t t e d and enhance the c l a r i t y of the p l o t s . Model performance s t a t i s t i c s are p r e s e n t e d for the average h o u r l y da ta u s i n g the UM r a d i a n c e d i s t r i b u t i o n . The data se t c o l l e c t e d on A ug . 1/2 i s used to compare s c a t t e r p l o t s of both complete and h o u r l y averaged data of each r a d i a n c e d i s t r i b u t i o n . D i s c u s s i o n of the sample s e t s i s o r g a n i z e d a c c o r d i n g to the day of t e s t . A summary of the r e s u l t s i s p r e s e n t e d i n S e c t i o n 4 . 5 . V a l i d a t i o n s e t s c o l l e c t e d but not p r e s e n t e d here are a v a i l a b l e in Appendix F . 95 4.2 MODEL VALIDATION STATISTICS The s t a t i s t i c s u t i l i z e d are adopted from the work of W i l l m o t t (1981,1984) r e g a r d i n g the v a l i d a t i o n of models and e v a l u a t i o n of model per formance . The reader i s r e f e r r e d to these r e f e r e n c e s f o r a f u l l d i s c u s s i o n of the mathemat ica l implementat ion of the s t a t i s t i c s . A b r i e f summary of the e q u a t i o n s used are a v a i l a b l e i n Appendix G . Summary u n i v a r i a t e measures of observed (measured) and p r e d i c t e d (model led) means (0, P r e s p e c t i v e l y i n the t a b l e s ) and s t a n d a r d d e v i a t i o n s ( s D , Sp) are p r o v i d e d as suggested by W i l l m o t t (1981). Measures of e r r o r are p r o v i d e d by the t o t a l r o o t mean square e r r o r (RMSE), the mean a b s o l u t e e r r o r (MAE), and the mean b i a s e r r o r (MBE) ( a c t u a l l y P - 0 ) . F u r t h e r d e c o m p o s i t i o n r e g a r d i n g the type of e r r o r i s a v a i l a b l e from the s y s t e m a t i c root mean square e r r o r (RMSEs) and unsys temat i c e r r o r (RMSEu) and t h e i r r e s p e c t i v e f r a c t i o n s of the t o t a l e r r o r as e x p r e s s e d by MSEs/MSE and MSEu/MSE. I n d i c a t o r s of agreement or c o r r e l a t i o n between the measured and mode l l ed data are p r o v i d e d by the c o e f f i c i e n t of d e t e r m i n a t i o n , r 2 , and the index of agreement, d . The use of the l a t t e r i s advocated by W i l l m o t t (1981) because of the i n s e n s i t i v i t y of r 2 to a d d i t i v e and p r o p o r t i o n a l d i f f e r e n c e s which may e x i s t between the mode l l ed and measured d a t a . The c o e f f i c i e n t s of the l e a s t squares l i n e a r r e g r e s s i o n a ( i n t e r c e p t ) and b ( s lope ) are a l s o p r e s e n t e d to a i d the a n a l y s i s , a l t h o u g h i t w i l l be shown l a t e r the assumption that 96 the measured da ta are f r e e of e r r o r s i s c l e a r l y not v a l i d for some c a s e s . The use of a may be of l i m i t e d u t i l i t y because comparison of measured and mode l l ed L D and L^ i n v o l v e s o n l y a s m a l l range of da ta w i t h magnitudes much g r e a t e r than z e r o . 4.3 MODEL VALIDATION USING DATA COLLECTED IN "POINT' MODE A s m a l l v a l i d a t i o n set was o b t a i n e d u s i n g the CTS i n manual mode to p o s i t i o n the ins t ruments above a canyon g r i d - p o i n t a f t e r which they remained s t a t i o n a r y wh i l e a one- minute average of both model input and v a l i d a t i o n data was c o l l e c t e d . T h i s method of data c o l l e c t i o n i s r e f e r r e d to as ' p o i n t ' mode. The data o b t a i n e d i n t h i s manner are not i n f l u e n c e d by e r r o r s of ins trument response a r i s i n g from t r a v e r s i n g but are more s u s c e p t i b l e to p o s s i b l e e r r o r s due to o b s t r u c t i o n of the sky - view f a c t o r f o r the u n d e r l y i n g s u r f a c e because the ins truments remain f i x e d over a p o i n t on the s u r f a c e . The p o i n t mode a l s o p r o v i d e s i d e n t i c a l a v e r a g i n g p e r i o d s for both the model input and v a l i d a t i o n d a t a . The net e f f e c t s h o u l d be i n c r e a s e d a c c u r a c y of the d a t a . Two p o i n t mode t e s t s were performed p r i o r to the use of the CTS i n automat ic mode to g i v e an i n i t i a l i n d i c a t i o n of model performance and to i d e n t i f y any problems i n the input d a t a . The t e s t s took p l a c e on the n i g h t s of J u l y 19/20 and J u l y 21/22, 1988 u s i n g a canyon H/W of 1.0. V a l i d a t i o n p o i n t s were o b t a i n e d from each of the canyon f a c e t s . 97 The combined t e s t r e s u l t s are p r e s e n t e d i n F i g u r e s 4.1 and 4 . 2 ; the former uses the i s o t r o p i c r a d i a n c e d i s t r i b u t i o n and the l a t t e r the UM r a d i a n c e d i s t r i b u t i o n f o r s k y - d e r i v e d i n c i d e n t long-wave r a d i a t i o n . Each symbol r e p r e s e n t s a v a l i d a t i o n p a i r c o n s i s t i n g of a measured ( a b c i s s a ) , and mode l l ed v a l u e ( o r d i n a t e ) , f or a s i n g l e o c c u r r e n c e ( i e . from one t r a v e r s e ) of one g r i d - p o i n t on a canyon f a c e t . Symbols are a s s i g n e d to the p o i n t s by canyon f a c e t (see c a p t i o n of F i g u r e 4.1) and are c o n s i s t e n t throughout t h i s c h a p t e r . There i s no d i f f e r e n t i a t i o n between p o i n t s on a g i v e n f a c e t on the p l o t s but t h i s i s d i s c u s s e d in the t e x t where a p p r o p r i a t e . The s c a t t e r p l o t of measured v e r s u s mode l l ed L 0 produces e x c e l l e n t agreement between for p o i n t s on the West w a l l and f l o o r . Some s c a t t e r i s observed for p o i n t s on the E a s t w a l l and t o p . Use of the UM r a d i a n c e d i s t r i b u t i o n ( F i g u r e 4.2) has no d i s c e r n a b l e a f f e c t on the s c a t t e r p l o t . I t does however, r e s u l t i n s u b s t a n t i a l improvement of m o d e l l e d L * and , p a r t i c u l a r l y f o r the canyon w a l l s . The p o i n t s showing the g r e a t e s t d i f f e r e n c e when u s i n g the UM d i s t r i b u t i o n are those w i t h h i g h e r \ps, as e x p e c t e d . The p l o t of shows some p o i n t s which l i e above the 1:1 l i n e and which do not appear to be a f f e c t e d by the c h o i c e of r a d i a n c e d i s t r i b u t i o n . These p o i n t s are l o c a t e d on the West w a l l and canyon t o p . Those p o i n t s on the West w a l l are l i k e l y lower on the w a l l where the sky r a d i a n c e d i s t r i b u t i o n has l e s s i n f l u e n c e ; the l a c k of change on the canyon top o c c u r s because the ' m o d e l l e d ' v a l u e of i s L i c t which i s d i r e c t l y measured. T h u s , d i f f e r e n c e s between measured and model led on the canyon 98 July 19/20, 20/21 1988 H:W 1:1 — " 1 1 1 i i_ 390 400 410 420 430 440 Mtosuntd L„ (W m~ 2) -80 -60 -40 -20 0 Measured L* (W m" 2 ) —I 1 1 I L _ 340 360 380 400 420 M«osur»d L, (W m ' 2 ) + West Won ^ East Wall X Floor • Top F i g u r e 4.1 S c a t t e r p l o t s of m o d e l l e d and measured long-wave f l u x e s from the ' p o i n t t e s t s ' . The i s o t r o p i c r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d L ; . 99 July 19/20, 20/21 1988 H:W 1:1 — 1 1 i 1 «00 410 420 430 Meosured L, (W m"*) ' 1 L _ 1_ - 8 0 -60 -40 -20 Measured L« (W m" 2 ) flfnZZ i'r-L ? " t t e r p l o t s of m o d e l l e d and measured long-wave f l u x e s from the ' p o i n t t e s t s ' . The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used f o r mode l l ed L • n o n z e i t n u y / ^ 100 T a b l e 4.1 Model Performance S t a t i s t i c s : P o i n t Mode V a l i d a t i o n S e t . J u l y 19/20, J u l y 21/22 1988. I s o t r o p i c Unsworth and M o n t e i t h S t a t i s t i c L * L D L i L * L 0 L i n , 0 (W m~* P (W m~ 2 35 35 35 35 35 35 ) -36 . .4 418. ,6 382 .2 -36 . ,5 418. ,6 382. ,2 • ) -38 . ,0 419. ,7 381 .6 -37 . .0 419. ,7 382. ,1 s 0 (W m - 2 s p (W m~ * ) 26. ,2 13. ,8 25 . 1 26. ,2 13. .8 25. , 1 ) 25. ,5 13. ,9 24 .5 25. ,3 13. .9 24. ,4 RMSE (w m ~2l 4. .0 4. .0 5. 0 2. .4 4, .0 3. ,5 RMSEs (W 1 . .9 1 . . 1 1 . 2 0. .6 1 . .2 1 . , 1 RMSEu (W m" 2) 3. .5 3. .9 4. 9 2. .3 3. .8 3. ,4 MSEs/MSE 0. .22 0. ,08 0. 06 0. .06 0. .09 0. ,09 MSEu/MSE 0. .78 0. .92 0. 94 0, .94 0, .91 0, .91 MAE (W m" I) 2, .8 2. .9 4. 0 2, .0 2, .9 2, .7 MBE (W m~ 2 ) -1 , .6 1 , .0 -o. 6 -0 , .6 1 , . 1 0, .5 r 2 0, .98 0, .92 0. 96 0, .99 0, .92 0, .98 d 0. .99 0. .98 0. 99 0. .99 0, .98 0, .99 a (W m~ 2) -2 , .9 15. .0 15. 3 -o. .5 17, . 1 14, .6 b 0, .97 0, .97 0. 96 1, .00 0, .96 0, .96 n - number of o b s e r v a t i o n s 0 - Observed mean P - P r e d i c t e d mean s 0 - observed s t a n d a r d d e v i a t i o n Sp - p r e d i c t e d s t a n d a r d d e v i a t i o n RMSE - Root Mean Square E r r o r RMSEs - Root Mean Square E r r o r ( s y s t e m a t i c ) RMSEu - Root Mean Square E r r o r (unsys temat i c ) MAE - Mean A b s o l u t e E r r o r MBE - Mean B i a s E r r o r r 2 - c o e f f i c i e n t of v a r i a t i o n d - index of agreement a - i n t e r c e p t of s imple l i n e a r r e g r e s s i o n b - s l o p e of s imple l i n e a r r e g r e s s i o n 101 top are due to d i f f e r e n c e s between the two i n s t r u m e n t s , the t r a v e r s i n g procedure and the methods of r e c o r d i n g and a v e r a g i n g the model input data compared w i t h the data c o l l e c t e d by the t r a v e r s e d i n s t r u m e n t s . No model e f f e c t s are i n v o l v e d . I t i s a l s o impor tant to r e c a l l tha t on the canyon top L j i s d i r e c t l y measured and L D de termined as a r e s i d u a l , i n c o n t r a s t to the o t h e r canyon f a c e t s . Model performance s t a t i s t i c s f o r both r a d i a n c e d i s t r i b u t i o n s a r e p r e s e n t e d in T a b l e 4 . 1 . The use of the UM r a d i a n c e d i s t r i b u t i o n for L * and r e s u l t s i n a decrease of RMSE by 40% and 50% r e s p e c t i v e l y . The s y s t e m a t i c p o r t i o n of the e r r o r d e c r e a s e s for L * but i n c r e a s e s for when the UM d i s t r i b u t i o n i s u s e d . The index of agreement i s not s i g n i f i c a n t l y a f f e c t e d by the d i s t r i b u t i o n used . J 4.4 MODEL VALIDATION USING AUTOMATICALLY COLLECTED DATA 4 .4 .1 August 1/2 The v a l i d a t i o n set from Aug . 1/2 i n v o l v e d a canyon wi th H/W of 2 . 0 . The p r e v a i l i n g weather was c l o u d l e s s w i t h l i g h t winds a f t e r s u n s e t . The model used 5 g r i d - p o i n t s on the h o r i z o n t a l canyon f a c e t s , the minimum number used by any canyon geometry. V e r t i c a l f a c e t s used 11 g r i d - p o i n t s . Model input data were averaged over 2-minute p e r i o d s and the f l u x e s measured by the 102 traversed instruments were matched to the appropriate model input averaging period. The large number of validation points in Figures 4.3 and 4.4 obscure the d e t a i l s of the plots, however, with the aid of the hourly averages presented in Figures 4.5 and 4.6 a number of observations can be made. Most noticeable i s the d i s t r i b u t i o n of L*. Two d i s t i n c t populations are present for fluxes greater than 50 W m - 2. Of these, points from the East wall l i e consistently below the 1:1 l i n e with increasing differences as L* decreases. Points from the West wall are located above and p a r a l l e l to the 1:1 l i n e . Validation points for the floor are not c l e a r l y v i s i b l e as a separate population and appear to be clustered around the 1:1 l i n e at approximately 30 W m~2. Strong agreement between measured and modelled values i s indicated for points at the canyon top. Second, validation points for L Q from the canyon top and the East wall l i e s i g n i f i c a n t l y above the 1:1 l i n e and in general the scatter of points i s above the 1:1 l i n e as indicated by the positive MBE in Table 4.2. This may be evidence for a s l i g h t over-estimate of the model input surface temperature ( r e c a l l that model vali d a t i o n of s l i g h t temperature increases for canyon facets always resulted in large increases in the emitted long- wave from the canyon top). Differences between L 0 and L* are shown to almost balance in the plots of L^. Very good overal l agreement i s indicated. for points at the canyon top l i e above the 1:1 l i n e but t h i s i s not a function of model performance as outlined in Section 4 . 1 . 103 August 1/2 1988 H:W 2:1 390 400 410 420 430 440 450 Meosured L„ (W m _ J ) -100 -«0 -tO -40 -20 0 Measured L* (W m " 2 ) jooL i i i i . L 100 320 S40 360 MO 400 420 440 Measured L, (W m~ 2 ) + West Wall ^ Eost Woll X Floor a Top F i g u r e 4.3 S c a t t e r p l o t s of mode l l ed and measured long-wave f l u x e s from A u g . 1/2 1988, H/W=2.0 u s i n g the comple te d a t a s e t . The i s o t r o p i c r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d L j . 104 August 1/2 1988 H/W 2.0 390 400 410 420 430 440 450 Measured L 0 (W m" 2 ) —t 1 1 » • - 1 0 0 — 8 0 -60 -40 -20 0 Measured L» (W m - 2 ) 3 4 2 0 3 •O £ 340 o S 320 Soob i • • i L 300 320 340 3*0 3S0 400 420 440 Measured L, (W m" 2 ) + West Woll A East Wall X Floor • T o p F i g u r e 4.4 S c a t t e r p l o t s of m o d e l l e d and measured long-wave f l u x e s from A u g . 1/2 1988, H/W=2.0 u s i n g the complete d a t a s e t . The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d . 105 August 1/2 1988 H:W 2:1 — < 1 • - 8 0 -60 -40 Measured L* (W m" 2 ) u " 1 ' I 300 320 340 360 380 400 420 440 Measurtd L, (W m" 2 ) Wotl A East Wall X Floor • Top F i g u r e 4 .5 S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n data g r i d - p o i n t f o r A u g . 1/2 1988, H / W - 2 . 0 . The i s o t r o p i c r a d i a n c d i s t r i b u t i o n i s used f o r m o d e l l e d 1^ 106 August 1/2 1988 H:W 2:1 i i i • 1 • ' 390 400 410 420 430 440 450 Mtasurad L. (W m - 2 ) — i i i • -100 -80 -60 - 4 0 -20 0 M«asur«d L* (W m" 2 ) 3001* 1 1 i i i i L 300 320 340 360 380 400 420 440 Measured L, (W m" 2 ) + West Won A Eost Woll X Floor o Top Figure 4.6 Scatterplots of hourly averaged v a l i d a t i o n data by grid-point for Aug. 1/2 1988, H/W=2.0. The Unsworth and Monteith (1975) radiance d i s t r i b u t i o n i s used for modelled L^. 107 Use of the UM r a d i a n c e d i s t r i b u t i o n improves model performance f o r the L * and f l u x e s (RMSE reduced by 21 and 17% p e r c e n t r e s p e c t i v e l y ) , p a r t i c u l a r l y for those p o i n t s on the E a s t w a l l . Thus at l e a s t a p o r t i o n of the e r r o r in and L * for p o i n t s on the w a l l i s due to model u n d e r e s t i m a t i o n of those f l u x e s when u s i n g the i s o t r o p i c r a d i a n c e d i s t r i b u t i o n . T a b l e 4.2 shows the r e d u c t i o n of t o t a l e r r o r s and improvements in the index of agreement a c h i e v e d through use of the UM r a d i a n c e d i s t r i b u t i o n . Note tha t the s y s t m a t i c p o r t i o n of the RMSE i n c r e a s e s for L j when the UM d i s t r i b u t i o n i s u s e d . T a b l e 4.2 Model Performance S t a t i s t i c s : August 1/2, • I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . I s o t r o p i c UM S t a t i s t i c L * L Q L i L * L Q L i 0 (W m_^) P (W m 2 ) l U m 2 1 RMSE (W m i) RMSEs (W rn"2) RMSEu (W m" 2) MSEs/MSE MSEu/MSE MAE (W m" 2) MBE (W m~ 2) r 2 d a (W m~ 2) b 75 1 175 30. 4 420. 1 29. 3 421 . 2 25. 9 10. 2 27. 0 10. 1 4. 1 3. 2 1 . 4 1 . 2 3. 9 3. 0 0. 1 1 0. 14 0. 89 0. 86 3. 5 2. 5 1 . 0 1 . 1 0. 98 0. 91 0. 99 0. 98 2. 1 22. 3 1 . 03 0. 95 175 1175 389.8 - 3 0 . 4 391.9 - 2 9 . 5 29.5 25 .9 29.7 26.8 4.5 3.3 2.1 1.1 3 .9 3.1 0.22 0.13 0.78 0.87 3.7 2 .8 2.1 0.8 0.98 0.99 0.99 0.99 2.4 1.7 1.00 1.03 175 1175 420.1 389.8 421.2 391.7 10.2 29.5 10.1 29.3 3. 2 3.7 1 . 2 1 .9 3. 0 3.2 0. 1 4 0.27 0. 86 0.73 2. 5 3.0 1 . 1 1.9 0. 91 0.99 0. 98 0.99 23. 5 6.9 0. 95 0.99 108 S t a t i s t i c s of the h o u r l y averages (Tab le 4.3) y i e l d s reduced RMSE and i n c r e a s e d d for L c , w i th d i s p l a y i n g minor improvements and L * remain ing l a r g e l y unchanged. T a b l e 4.3 Model Performance S t a t i s t i c s : August 1/2, H o u r l y Averaged P o i n t s , I s o t r o p i c and Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . S t a t i s t i c L * I s o t r o p i c Lo L i L * UM L i n - 229 229 229 229 229 229 0 (W m"*) -31 . 3 420.6 389. 3 -31 . 3 420.6 389. 3 P (W m" 2) - 3 0 . 5 421 .9 391 . 4 - 3 0 . 7 421 .9 391 . 2 s 0 (W m - 2 ) 27. 1 10.0 30. 2 27. 1 10.0 30. 2 s p (W m~^) 28. 3 10.4 30. 5 28. 1 10.4 30. 0 RMSE (W m~ 2) 4. 2 2.6 4. 0 3. 3 2.6 3. 1 RMSEs (W m 2 ) 1 . 3 1 .3 2. 1 1 . 0 1 .3 1 . 9 RMSEu (W m 2 ) 4. 0 2.3 3. 4 3. 2 2.2 2. 5 MSEs/MSE 0. 09 0.24 0. 28 0. 09 0.24 0. 37 MSEu/MSE 0. 91 0.76 0. 72 0. 91 0.76 0. 63 MAE (W m" 2) 3. 5 2.0 3. 4 2. 8 2.0 2. 6 MBE (W m~ 2) 0. 8 1 .3 2. 1 0. 6 1.3 1 . 9 r 2 0. 98 0.95 0. 99 0. 99 0.96 0. 99 d 0. 99 0.98 0. 99 0. 99 0.98 0. 99 a (W m~ 2) 1 . 9 - 5 . 8 0. 4 1. 5 - 7 . 9 4. 7 b 1 . 03 1 .02 1. 00 1. 03 1 .02 0. 99 4 .4 .2 August 3/4 The second v a l i d a t i o n set s e l e c t e d for p r e s e n t a t i o n i s from a 1.0 canyon H/W c o l l e c t e d on August 3 /4 . Cont inuous t r a v e r s i n g was m a i n t a i n e d for a 24-hour p e r i o d b e g i n n i n g in the e a r l y morning of August 3. A longer a v e r a g i n g p e r i o d (5-minutes) was used for model input d a t a . Only v a l i d a t i o n data f o l l o w i n g sunset 109 are p r e s e n t e d h e r e . C l o u d l e s s , l i g h t wind c o n d i t i o n s dominated t h r o u g h o u t . The model run uses 10 and 11 g r i d - p o i n t s for h o r i z o n t a l and v e r t i c a l canyon f a c e t s , r e s p e c t i v e l y . Comparison of mode l l ed and measured r e s u l t s ( F i g u r e 4.7) of L Q are s i m i l a r to those of August 1/2. E r r o r s t a t i s t i c s (Table 4.4) are s l i g h t l y g r e a t e r for the UM r a d i a n c e d i s t r i b u t i o n compared w i t h August 1/2. T a b l e 4.4 Model Performance S t a t i s t i c s : August 3 /4 , H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . S t a t i s t i c ( U n i t s ) L * L Q L± n _ 333 333 333 0 (W m *) - 4 5 . 4 435.3 389. 9 P (W m 2 ) - 4 5 . 5 436.5 390. 9 s D (W m-2) 26. 1 12.8 29. 1 s p (W m*^) 26. 2 14.2 28. 1 RMSE (W m" 2) 2. 4 3.3 2. 4 RMSEs (W m~ 2) 0. 1 1 .6 1 . 5 RMSEu (W m~ 2) 2. 4 2.9 1 . 9 MSEs/MSE 0. 003 0.23 0. 37 MSEu/MSE 0. 997 0.77 0. 67 MAE (W m - 2 ) 1 . 8 2.3 1 . 9 MBE (W m~ 2) -o . 1 1 . 1 1 . 0 r 2 0. 99 0.96 0. 99 d 0. 99 0.98 0. 99 a (W m" 2) - 0 . 1 - 3 6 . 9 15. 4 b 1. 00 1 .09 0. 96 D i s t i n c t s u b - p o p u l a t i o n s of L * are aga in n o t e d ; in t h i s case p o i n t s from both w a l l s l i e below the 1:1 l i n e w i t h i n c r e a s i n g e r r o r s f o r s m a l l e r v a l u e s of L * ( p o i n t s h i g h e r on the w a l l ) . The v a l i d a t i o n p o i n t s from the f l o o r are l o c a t e d above the 1:1 l i n e . 110 440 a* =, 420 • 400 E i J80 •D 350 o 340 320 r 320 340 3 6 0 380 400 420 Measured L, (W m"*) 440 + Wesl Wall A East Wall X Floor a T o p F i g u r e 4.7 S c a t t e r p l o t s of h o u r l y averaged v a l i d a t i o n da ta by ? i i ? c ? ° i n J . f o r V4 1 9 8 8 ' H/W=1.0 . The Unsworth and M o n t e i t h U 3 7 5 ; r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d L - . I l l The break in the L* plot for points at the canyon top i s due to the absence of measurements for the period when the CTS was stopped for r e f u e l l i n g the generator. A s l i g h t tendency for the lowest L* values to be under-modelled and/or over-measured is noted. Si g n i f i c a n t improvement in agreement i s achieived for and L* through use of the UM d i s t r i b u t i o n (Table 4.5), p a r t i c u l a r l y for East wall points (the RMSE for L* i s reduced by 44% and by 29% for L^; mean absolute errors show similar improvement). The proportion of systematic error i s reduced for L* but increases for Lt- . The RMSE of L* and L,- is lower than that of August 1/2. Table 4.5 Comparison of RMSE and d S t a t i s t i c s Calculated From Hourly Averaged Data Using Isotropic [IS] and the Unsworth and Monteith (1975) [UM] Radiance D i s t r i b u t i o n s . Units of RMSE are W m~ Aug. 1/2 Aug. 3/4 Aug. 10/11 Aug. 12/13 L* RMSE [IS] 4.2 4.6 10.6 4.7 RMSE [UM] 3.3 2.4 8.2 4.1 d [IS] 0.994 0.992 0.952 0.984 d [UM] 0.996 0.998 0.973 0.989 Li RMSE [IS] 4.0 4.1 8.8 3.8 RMSE [UM] 3.1 2.4 6.7 4.2 d [IS] 0.996 0.995 0.967 0.989 d [UM] 0.997 0.998 0.982 0.987 L o RMSE [IS] 2.6 3.2 2.8 3.1 RMSE [UM] 2.6 3.3 2.9 3.2 d [IS] 0.983 0.985 0.978 0.975 d [UM] 0.984 0.984 0.976 0.973 112 Compared wi th August 1/2, the i n c r e a s e d a v e r a g i n g t imes of the input data do not seem to a d v e r s e l y a f f e c t the v a l i d a t i o n r e s u l t s . The i n c r e a s e d number of g r i d - p o i n t s on the h o r i z o n t a l f a c e t s do not appear to i n c r e a s e model a c c u r a c y as suggested by the s e n s i t i v i t y t e s t s performed on the model u s i n g the e x t e r n a l l y c a l c u l a t e d v i e w - f a c t o r s . 4 . 4 . 3 August 10/11 The v a l i d a t i o n set from the n i g h t of August 10/11 marks the f i r s t t ime that the thermocouples were r e c o n s t r u c t e d and r e - a t t a c h e d to the canyon f l o o r to accomodate the second g r i d s p a c i n g used , in t h i s c a s e , w i th a canyon H/W r a t i o of 0 .67 . There i s i n t e r e s t , t h e r e f o r e , i n how L 0 v a l u e s f o r p o i n t s on the f l o o r compare wi th p r e v i o u s d a t a . The p l o t s of the h o u r l y - averaged v a l i d a t i o n data ( F i g u r e 4.8) i n d i c a t e a g r e a t e r tendency of L 0 for p o i n t s on the f l o o r to l i e below the 1:1 l i n e , i n d i c a t i n g the p o s s i b i l i t y that the new thermocouples o v e r e s t i m a t e the s u r f a c e t e m p e r a t u r e . Ev idence s u p p o r t i n g t h i s a n a l y s i s i s a v a i l a b l e from the c o r r e s p o n d i n g over-measurement of L Q for p o i n t s at the canyon t o p , as expected from the r e s u l t s of the s e n s i t i v i t y t e s t s . P o i n t s at the canyon top c l o s e s t to the West w a l l c o n s i s t e n t l y show the g r e a t e s t e r r o r . T h i s may i n d i c a t e a d d i t i o n a l sources of e r r o r for those p o i n t s . V a l i d a t i o n p o i n t s on the Eas t canyon w a l l a l s o show s u b s t a n t i a l d i f f e r e n c e s i n L Q . A n a l y s i s of i n d i v i d u a l p o i n t s 320 340 360 380 400 Measured L, (W m - 1 ) + West WoH A East Wall X Floor • Top F i g u r e 4 .8 S c a t t e r p l o t s o f h o u r l y averaged v a l i d a t i o n d a t a g r i d - p o i n t f o r A u g . 10/11 1988, H/W=0.67. The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d L 114 r e v e a l s that the e r r o r s i n c r e a s e w i t h he ight up the w a l l . The f i n d i n g s for the West w a l l are s i m i l a r a l t h o u g h the a b s o l u t e magnitude of the d i f f e r e n c e s between the mode l l ed and measured p o i n t s i s l e s s than those on the E a s t w a l l . O v e r a l l the MBE of L Q i s l e s s than tha t c a l c u l a t e d u s i n g the p r e v i o u s two example s e t s , and the RMSE a l s o tend to be s m a l l e r (Tab le 4 . 6 ) . The s c a t t e r p l o t of L * shows r e l a t i v e l y good agreement between mode l l ed and measured p o i n t s at the canyon top and on the f l o o r . However, s e r i o u s d i v e r g e n c e from the 1:1 l i n e i s shown for the canyon w a l l s , which i s not c o r r e c t e d for by the use of the UM r a d i a n c e d i s t r i b u t i o n . D i f f e r e n c e s i n c r e a s e i n v e r s e l y wi th the magnitude of L * . The g r e a t e s t d i f f e r e n c e s are i n d i c a t e d for g r i d - p o i n t s a t the tops of both w a l l s , forming the two d i s t i n c t l i n e s of o u t l i e r s in F i g u r e 4 . 8 . The d e v i a t i o n s of L * d i r e c t l y a f f e c t the p l o t s of L ^ . An e x p l a n a t i o n which accounts f o r the l a r g e d i f f e r e n c e s between mode l l ed and measured v a l u e s of L * , L Q and for the canyon w a l l s i s based i n p a r t upon the o b s e r v a t i o n made d u r i n g data c o l l e c t i o n that these are e r r o r s in ins trument o r i e n t a t i o n w i t h r e s p e c t to the p lane of the w a l l s . These e r r o r s have two o r i g i n s : the f i r s t was a c o n s i s t e n t o v e r - r o t a t i o n of the in s t ruments which o c c u r r e d p r i o r to t r a v e r s i n g the canyon w a l l s . The second i s an observed tendency of the Eas t w a l l to be a n g l e d s l i g h t l y outwards , so t h a t the t r a v e r s e path i s f a r t h e r from the w a l l near the canyon top than at the base of the w a l l . C o n s i s t e n t under-measurement of L 0 and over-measurement of L * w i t h r e s p e c t to the mode l l ed v a l u e f o r the g r i d - p o i n t i s 115 p o s s i b l e for the E a s t w a l l i f the sensors are t i l t e d towards the canyon top ( i e . over r o t a t i o n from the canyon b a s e ) . L Q would be d e c r e a s e d , e s p e c i a l l y near the canyon top and L * would become more p o s i t i v e s i n c e the s i d e of the ins trument f a c i n g outwards from the w a l l would r e c e i v e an i n c r e a s e in r a d i a t i o n from w i t h i n the c a n y o n . A t i l t i n the o p p o s i t e d i r e c t i o n would i n d i c a t e more n e g a t i v e v a l u e s of L * s i n c e a g r e a t e r p r o p o r t i o n of sky would be viewed by the i n s t r u m e n t s . The e f f e c t would be augmented by a w a l l a n g l e d outwards . S i n c e the t r a v e r s e does not c o m p l e t e l y cover the f u l l extent of the t e n t h g r i d - p o i n t a p o r t i o n of the d i f f e r e n c e c o u l d be a t t r i b u t e d to a b i a s of samples i n the average from the lower p a r t of the g r i d - p o i n t where L * would be g r e a t e r , c r e a t i n g f u r t h e r d i f f e r e n c e s . A s i m i l a r e x p l a n a t i o n for the d i f f e r e n c e s of the West w a l l i s u sed . Arguments tha t e r r o r s are due to o v e r - e s t i m a t e d s u r f a c e t empera ture , e s p e c i a l l y for upper p o i n t s on the w a l l , may be d i s c o u n t e d because no l a r g e e r r o r s e x i s t for the canyon t o p . The a d d i t i o n a l sample p o i n t s a f f o r d e d by the i n c r e a s e d g r i d - s p a c i n g on the canyon top and f l o o r do not have a c l e a r a f f e c t on the r e s u l t s , g i v e n the o ther i n f l u e n c e s noted e a r l i e r . I t might be expected that a more a c c u r a t e average would be o b t a i n e d w i t h more samples per g r i d - p o i n t . August 10/11 a l s o marks the f i r s t o c c u r r e n c e of a major v a r i a t i o n i n L ^ c t . An i n c u r s i o n of c l o u d a p p r o x i m a t e l y 5 to 6 hours a f t e r sunset had no n o t i c e a b l e e f f e c t i n the v a l i d a t i o n d a t a . Of concern in t h i s r e g a r d would be the time s c a l e of v a r i a b l e L £ c t w i t h r e g a r d to the model input a v e r a g i n g time (3 116 minutes for t h i s n i g h t ) and the a v e r a g i n g p e r i o d of the t r a v e r s e d data (on the order of 24 seconds f o r w a l l s and 42 seconds for canyon top and f l o o r ) . T a b l e 4.6 Model Performance S t a t i s t i c s : Aug . 10/11 H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . S t a t i s t i c L * L o Lj L n 357 357 357 0 (W n T 2 ) - 3 9 . ,5 410. ,02 3 7 0 . ,58 P (W m~ 2) - 4 4 . ,7 4 1 0 . ,29 3 6 5 . ,62 s 0 (w n T 2 s p (W m 2 ) ) 2 7 . 2 2 . ,4 ,4 10. 9. ,0 .8 2 6 . 22 . ,4 .9 RMSE (W m ~2l 8. ,2 2. ,9 6. ,7 RMSEs (W 7. .4 0. .6 6. ,2 RMSEu (W m" 2) 3. ,6 2. ,8 2. ,6 MSEs/MSE 0. ,81 0, .05 0. .9 MSEu/MSE 0. ,20 0, .95 0. .2 MAE (W m" 1) 6. .3 2. . 1 5. .3 MBE (W m~ 2) - 5 . .2 0. .3 - 5 . .0 r 2 0. .97 0, .92 0. .99 d 0. .97 0, .98 0. .98 a (W m~ 2) - 1 2 . .8 23, .6 46. .5 b 0, .81 0, .94 0, .86 4 .4 .4 August 12/13 S c a t t e r p l o t s of the v a l i d a t i o n data for August 12/13 are p r e s e n t e d i n F i g u r e 4 . 9 . The canyon H/W i s 1 . 3 3 , which uses the same g r i d - p o i n t s p a c i n g as the 0 . 6 7 canyon but o n l y 5 g r i d - p o i n t s . The thermocouples which had been r e c o n s t r u c t e d and r e a t t a c h e d to the canyon f l o o r f o r August 10/11 were used a g a i n . 117 August 12 /13 1988 H:W 1:0.75 1 1 1 1 1 ' • 390 400 410 420 430 440 450 Measured L. (W m"*) -100 Li i i i i i_ -100 -80 -60 -40 -20 0 Measured L * ( W m " ' ) + West Wan A East Wall X Floor a Top F i g u r e 4 .9 S c a t t e r p l o t s o f h o u r l y averaged v a l i d a t i o n d a t a g r i d - p o i n t f o r Aug . 12/13 1988, H/W=1.33. The Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s used f o r m o d e l l e d 1 1 8 Large d i f f e r e n c e s between mode l l ed and measured v a l u e s of L 0 for the top p o i n t s of the Eas t w a l l r e - o c c u r as a r e s u l t . Large v a l u e s of L 0 f or p o i n t s on the canyon top are a l s o e v i d e n t . The RMSE of L c i s i n c r e a s e d s l i g h t l y from the p r e v i o u s day . The remainder of the p o i n t s show r e l a t i v e l y good agreement, w i th the e r r o r s t a t i s t i c s for L * and L^ reduced improved compared to Aug. 10/11. Trends shown i n the s c a t t e r p l o t s of L * are s i m i l a r to those of Aug . 10/11, w i t h g r i d - p o i n t s near the top of the canyon w a l l s showing l a r g e d i f f e r e n c e s . The e f f e c t i s not as pronounced but the range of L * v a l u e s o b t a i n e d i s on ly h a l f of tha t d i s p l a y e d f o r Aug 10/11; RMSE v a l u e s are almost h a l f those for the 0.67 data s e t . Use of the UM r a d i a n c e d i s t r i b u t i o n improves the r e s u l t s , but to a l e s s e r degree than i s o b t a i n e d w i t h Aug. 10/11 (Table 4 . 5 ) . G r i d - p o i n t s from the canyon top and f l o o r are g e n e r a l l y in good agreement. P l o t s of measured v e r s u s mode l l ed L^ i n d i c a t e tha t the mode l l ed v a l u e s of L^ f o r p o i n t s on the canyon top are s l i g h t l y h i g h w i t h r e s p e c t to the t r a v e r s e d v a l u e s . G r i d - p o i n t s on the canyon f l o o r show very good agreement a l t h o u g h the model led v a l u e s are a g a i n s l i g h t l y h i g h w i t h r e s p e c t to measured v a l u e s . P o i n t s from the w a l l s are v i s i b l e as a second c l u s t e r of p o i n t s w i t h a g r e a t e r s l o p e than those from, the f l o o r or w a l l s . Agreement i s improved by use of the UM r a d i a n c e d i s t r i b u t i o n for p o i n t s on the w a l l s , w i t h the p o s i t i v e d i f f e r e n c e s between L 0 model led-measured p o i n t s c a n c e l l i n g the n e g a t i v e d i f f e r e n c e s shown f o r L * . The o c c u r r e n c e of a few p o i n t s , ma in ly on the West 119 w a l l , to p o s i t i o n s above the 1:1 l i n e i s of i n t e r e s t . T h i s r e s u l t s in an o v e r a l l worsening of the s t a t i s t i c a l i n d i c e s for e r r o r and agreement, the on ly case in which t h i s was observed for the examples p r e s e n t e d . Compared wi th the 11/12 of August c a s e , the agreement of i s improved . The data which are s h i f t e d to p o i n t s v i s i b l y above the 1:1 l i n e are the uppermost p o i n t s on both w a l l s and the d i f f e r e n c e s are g r e a t e s t d u r i n g p e r i o d s of i n c r e a s e d and v a r i a b l e c l o u d i n e s s . T h i s i s some ev idence for a breakdown of v a l i d a t i o n d u r i n g c o n d i t i o n s of p a r t i a l c l o u d i n e s s due to d i f f e r e n c e s i n the a v e r a g i n g p e r i o d for model input and v a l i d a t i o n d a t a . The problem i s compounded by the p o s s i b i l i t y of d i f f e r e n t v i e w - f a c t o r s for p a r t i a l l y c l o u d y c o n d i t i o n s between the ins trument measuring L ^ c t and the ins t ruments w i t h i n the canyon when the ins trument o r i e n t a t i o n s are d i f f e r e n t , i e . when the t r a v e r s i n g ins truments are t i l t e d to face the w a l l . T h i s e r r o r shou ld not be a problem when the o r i e n t a t i o n s are the same, and the s c a t t e r p l o t s tend to support t h i s . A n a l y s i s of c l o u d data c o l l e c t e d at Vancouver I n t e r n a t i o n a l A i r p o r t by the AES, and the t r a c e of L j c t w i th t i m e , i n d i c a t e an i n c r e a s e in c l o u d i n e s s to near o v e r c a s t c o n d i t i o n s a f t e r s u n s e t , f o l l o w e d by a p e r i o d of c l e a r i n g 2 to 3 hours from s u n s e t , then 9 to 10 / I0ths c l o u d for the remainder of the n i g h t . 120 T a b l e 4.7 Model Performance S t a t i s t i c s : Aug. 12/13, H o u r l y Averaged P o i n t s , Unsworth and M o n t e i t h Radiance D i s t r i b u t i o n . S t a t i s t i c L * L Q L i n . 280 280 280 0 (W m~t.) - 1 6 . 6 411 .7 395.1 P (W m" 2) - 1 8 . 0 413.8 395.8 s 0 (W rn"2) 19.4 10.0 19.0 s p ^(W m" 2) 19.0 10.2 17.6 RMSE (W m~ 2) 4.1 3.2 4.2 RMSEs (W m 2 ) 1 .5 2.1 1.9 RMSEu (W m~ 2) 3.8 2.4 3.7 MSEs/MSE 0.14 0.4 0.22 MSEu/MSE 0.86 0.6 0.78 MAE (W m" 2) 3 .0 2.3 3.2 MBE (W m 2 ) -1 .3 2.1 0.8 r 2 0.96 0.94 0.96 d 0.99 0.97 0.99 a (W m~ 2) -1 .9 5.4 37.9 b 0.97 0.99 0.91 4 .5 SUMMARY OF VALIDATION RESULTS A number of c o n c l u s i o n s can be drawn from the a n a l y s i s of the v a l i d a t i o n s e t s p r e s e n t e d here and the a d d i t i o n a l data p r e s e n t e d i n Appendix F , v i z : 1. The use of the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n c o n s i s t e n t l y reduces the e r r o r and improves the agreement between mode l l ed and measured L^ and L * for p o i n t s on the canyon w a l l s . 121 2. D i f f e r e n c e s between m o d e l l e d and measured f l u x e s of L 0 and L* a r e shown t o o c c u r on t h e w a l l s , p a r t i c u l a r l y n e a r t h e i r t o p . T h e s e a r e a t t r i b u t e d t o e r r o r s i n t h e p o s i t i o n i n g o f t h e s e n s o r w i t h r e s p e c t t o t h e w a l l . 3. C o n s i s t e n t d i f f e r e n c e s between measured and m o d e l l e d v a l u e s may a r i s e due t o p a r t i a l t r a v e r s e s o f p o i n t s n e a r t h e c o r n e r s of c a n y o n f a c e t s , e s p e c i a l l y t h e c a n y o n t o p . 4. Under v a r i a b l e c l o u d , t h e t r a v e r s e d r a d i o m e t e r s may e x p e r i e n c e d i f f e r e n t v i e w - f a c t o r s f o r c l o u d s t h a n t h e i n s t r u m e n t m e a s u r i n g L £ c t and t h i s c a n l e a d t o d i f f e r e n c e s between m o d e l l e d and m easured r a d i a t i o n v a l u e s . 5. The use of l o n g e r a v e r a g i n g p e r i o d s f o r t h e model i n p u t d a t a t h a n t h a t u s e d t o c o l l e c t t h e t r a v e r s e d d a t a may r e s u l t i n e r r o r s under v a r i a b l e w eather c o n d i t i o n s , but was n o t f o u n d t o be a l a r g e i n f l u e n c e when t h e c o n d i t i o n s were more c o n s t a n t . 6. Agreement between measured and m o d e l l e d f l u x e s d e c r e a s e s somewhat w i t h t i m e due t o a d e g r a d a t i o n i n p e r f o r m a n c e of t h e CTS and wear on t h e t h e r m o c o u p l e s u s e d t o measure s u r f a c e t e m p e r a t u r e . 7. S u r f a c e t e m p e r a t u r e s may have been s l i g h t l y o v e r - e s t i m a t e d by t h e t h e r m o c o u p l e s , however t h e r e s u l t s a r e a c c e p t a b l e c o n s i d e r i n g t h e f i l t e r i n g p r o c e s s n e c e s s i t a t e d by t h e n o i s e . 122 8. The r e s u l t s from canyons which had a low number of g r i d - p o i n t s on some f a c e t s d i d not appear to reduce model a c c u r a c y w i t h the e x t e r n a l l y c a l c u l a t e d v i e w - f a c t o r s used i n the mode l . 9. I n c r e a s i n g the number of samples used to c a l c u l a t e the average measured f l u x e s f o r canyon g r i d - p o i n t s on h o r i z o n t a l canyon s u r f a c e s i n the H/W=0.67 and 1.33 canyons d i d not s i g n i f i c a n t l y enhance the q u a l i t y of the r e s u l t s over those f o r the H/W=1.0 and 2.0 canyons . 10. The o v e r a l l success of the v a l i d a t i o n i s judged to be s a t i s f a c t o r y . The index of agreement, d , i s g e n e r a l l y g r e a t e r than 0.95 and most e r r o r s have been found to be a r e s u l t of measurement p r o b l e m s . The m a j o r i t y of data l i e w i t h i n the c a l c u l a t e d p r o b a b l e e r r o r e s t i m a t e s of Appendix D. 123 CHAPTER 5. MODEL RESULTS USING DIFFERENT ESTIMATES OF SURFACE TEMPERATURE 5.1 INTRODUCTION Chapter 4 conc ludes tha t the A r n f i e l d model can s a t i s f a c t o r i l y p r e d i c t f l u x e s of long-wave r a d i a t i o n under n o c t u r n a l c o n d i t i o n s at p o i n t s w i t h i n s imple canyon s t r u c t u r e s i f the neces sary model input i s a c c u r a t e l y s p e c i f i e d . A very important c o n s i d e r a t i o n for many m o d e l l e r s , however, i s the a v a i l a b i l i t y of a c c u r a t e model input d a t a . The model v a l i d a t i o n was performed u s i n g a f u l l set of input data so t h a t a 1:1 correspondence e x i s t e d between mode l l ed and measured g r i d - p o i n t s for each canyon f a c e t , except the t o p . The input data are d i r e c t l y measured a t each g r i d - p o i n t l o c a t i o n and t h e r e f o r e r e p r e s e n t the most a c c u r a t e and complete input data set a v a i l a b l e . The f e a s i b i l i t y of o b t a i n i n g input of the r e s o l u t i o n and q u a l i t y used for model v a l i d a t i o n for more a p p l i e d uses of the A r n f i e l d model i s l i m i t e d . T h i s chapter i n v e s t i g a t e s the model e r r o r s which may be expected when the p r i m a r y model input of s u r f a c e temperature i s e s t i m a t e d by d i f f e r e n t methods. The necessary model input v a l u e s of e and L £ c t a re r e a s o n a b l y e a s i l y e s t i m a t e d . The e m i s s i v i t y of a wide range of canyon m a t e r i a l s can be o b t a i n e d from t a b l e s , (eg. ASHRAE, 1981). A number of s imple models are a v a i l a b l e for the e s t i m a t i o n of i n c i d e n t long-wave r a d i a t i o n based upon s c r e e n - l e v e l a i r t emperature and measures of h u m i d i t y which may be d e r i v e d from 124 commonly a v a i l a b l e m e t e o r o l o g i c a l measurements. Reviews of these models are g iven by Co le (1976) and B r u t s a e r t (1982) . A d d i t i o n a l c o r r e c t i o n s may be necessary to account for the observed i n c r e a s e in w i t h i n urban areas (Oke and F u g g l e , 1972; E s t o u r n e l et al . , 1983) due to the i n f l u e n c e s of i n c r e a s e d a i r temperature and a e r o s o l s in the urban boundary l a y e r (Oke, 1982) . Of the input r e q u i r e d for the A r n f i e l d model , the s u r f a c e temperatures of canyon f a c e t s are the most d i f f i c u l t to s p e c i f y and are a l s o the dominant c o n t r o l l i n g f a c t o r of model s e n s i t i v i t y when m o d e l l i n g the long-wave r a d i a t i o n at n i g h t . Measurement of s u r f a c e temperature at a l a r g e number of canyon g r i d - p o i n t s i s o f t en not p o s s i b l e . T h i s s e c t i o n p r e s e n t s mode l l ed r a d i a t i o n f l u x d i f f e r e n c e s o b t a i n e d f o r Aug . 1/2 (H/W=2.0) and Aug. 3/4 (H/W=1.0) when the measured s u r f a c e temperatures were r e p l a c e d by an approximated v a l u e . These are two days w i t h c l o s e t o . i d e a l r a d i a t i v e c o o l i n g c o n d i t i o n s in which geometry e f f e c t s upon s u r f a c e temperature are l i k e l y to be maximized. The schemes used to d e r i v e the s u r f a c e t emperatures are based on sugges t ions made by A r n f i e l d (1976) and W a k e f i e l d (1987) . In the f o l l o w i n g , comparison between the model output u s i n g the approximate s u r f a c e temperatures at a l l g r i d - p o i n t s a g a i n s t tha t u s i n g the measured s u r f a c e temperatures i s made at 5 t imes f o l l o w i n g s u n s e t . These t imes are (except where n o t e d ) : sunset ( s s ) , ss+1 h , ss+2 h , ss+4 h , and ss+8 h . D i f f e r e n c e s between mode l l ed f l u x e s o b t a i n e d u s i n g the a p p r o x i m a t i o n scheme and 125 those model led u s i n g the t rue s u r f a c e temperatures are p l o t t e d around the p e r i m e t e r of s i m p l i f i e d canyon c r o s s - s e c t i o n s . Each p l o t i n c l u d e s a s eparate c r o s s - s e c t i o n for each component of the s u r f a c e long-wave r a d i a t i v e b a l a n c e . I n c i d e n t long-wave at the canyon top i s not i n c l u d e d in the p l o t s because the mode l l ed f l u x i s set equa l to the measured v a l u e of L i C f 5.2 AIR TEMPERATURE When m o d e l l i n g d a i l y mean and i n t e g r a t e d v a l u e s of long-wave f l u x e s , A r n f i e l d (1976) used monthly mean a i r temperature for a r u r a l a i r p o r t s t a t i o n wi th an increment based upon urban l o c a t i o n and the expected heat i s l a n d i n t e n s i t y ( c a l c u l a t e d u s i n g r e l a t i o n s h i p s g i v e n by Oke (1973)) . T h i s temperature r e p l a c e d a l l f ace t temperatures and was a l s o used to e s t imate L i c t . D i r e c t comparison wi th t h i s scheme i s not p o s s i b l e here because on ly n o c t u r n a l long-wave f l u x e s are measured. However, the n o t i o n of u s i n g a i r temperatures to r e p r e s e n t the canyon face t temperatures i s i n v e s t i g a t e d i n two ways: a) Canyon a i r temperature i s measured d i r e c t l y from a thermocouple mounted on the C T S . These temperatures are a v a i l a b l e by g r i d - p o i n t for each f a c e t and can be used to generate an average canyon a i r temperature f o r a complete canyon t r a v e r s e . The a v e r a g i n g p e r i o d for the canyon a i r temperature i s s i g n i f i c a n t l y longer than that of the s u r f a c e temperatures so s u r f a c e temperatures are averaged to match the p e r i o d of the canyon a i r t e m p e r a t u r e . The f i r s t mode l l ed time u s i n g t h i s 126 a p p r o x i m a t i o n i s sunset+0.5 h , because a complete t r a v e r s e i s not always completed p r i o r to t h i s t i m e . b) The s t a n d a r d s c r e e n - l e v e l a i r temperature r e c o r d e d at Vancouver I n t e r n a t i o n a l A i r p o r t by the Atmospher ic Environment S e r v i c e are used to r e p r e s e n t the canyon f a c e t t e m p e r a t u r e s . The r e a d i n g s are taken on the hour and are i n t e r p o l a t e d to the mode l l ed t i m e s . Comparison wi th model output u s i n g measured s u r f a c e temperatures i s made u s i n g s u r f a c e temperatures from the n e a r e s t r e c o r d e d a v e r a g i n g p e r i o d (these match to w i t h i n 5 m i n u t e s ) . 5 .2.1 R e s u l t s : Average Canyon A i r Temperature D i f f e r e n c e s of mode l l ed f l u x e s f o r August 1/2 and August 3/4 are p r e s e n t e d i n F i g u r e s 5.1 and 5.2 r e s p e c t i v e l y . Note - dashed l i n e s are p l o t t e d to h e l p i d e n t i f y between p o s i t i v e and n e g a t i v e d i f f e r e n c e s . Both runs underes t imate the incoming f l u x on a l l canyon f a c e t s . The d i f f e r e n c e s are s m a l l e r near the tops of the w a l l s and towards the middle of the canyon f l o o r where L ^ c t makes up a l a r g e r p o r t i o n of as a r e s u l t of l a r g e r sky-v iew f a c t o r s for those p o i n t s . D i f f e r e n c e s decrease w i th time f o r both w a l l s on Aug . 3/4 as a i r and s u r f a c e temperatures converge ; the lower p o r t i o n s of the w a l l s show an i n c r e a s e wi th time on Aug. 1/2 due to the s t r o n g l y reduced c o o l i n g r a t e s in t h i s canyon geometry. The d i f f e r e n c e s in are s t r o n g l y c o n d i t i o n e d by L D which shows s i m i l a r t empora l t r e n d s . 127 Average Conyon Air Temperature Approximation August 1/2 1988 H/W 2.0 QSunsrf - 0.5 h A Sunset + 1 h + Sunset • 2 h X Sunset I 4 h O Sum* + 8 -23.4 -17.4 - H . 4 Li Differences 2 3 4 5 Focet Point: Floor -8.9 -14.7 -20.6 Li Differences Facet Point: Top F i g u r e 5.1 M o d e l l e d long-wave f l u x d i f f e r e n c e s (W m~2) rtHf.»4„ A u s i n g the average canyon a i r temperature iS plL'i of measured s u r f a c e t e m p e r a t u r e . Data i s from Aug . 1/2 1988? H/W=2.0 Focet Point: Top L o Differences Focet Point: Floor L o Differences Focet Point: Top 1 2 3 4 5 6 7 6 9 10 ° L* Differences Focel Point: Floor I* Differences F i g u r e 5.2 As per F i g . 5.1 f o r Aug . 3/4 1988, H/W=1.0. 129 F l u x e s of L 0 are s t r o n g l y underes t imated when u s i n g the average canyon a i r t empera ture , p a r t i c u l a r l y for p o i n t s near the top of the E a s t w a l l f o l l o w i n g sunset and l a t e r i n the evening at the base of the w a l l s where s u r f a c e temperature i s observed to be h i g h . The e r r o r s decrease wi th time from sunset f o r a l l f a c e t s on Aug . 3 /4 . E r r o r s i n c r e a s e for, the lower p o r t i o n s of the w a l l s w i t h time on Aug . 1/2. D i f f e r e n c e s of L * for p o i n t s on the West w a l l and f l o o r are g e n e r a l l y under 10 W m ~ 2 . The l a r g e s t d i f f e r e n c e s are p o s i t i v e ( i e . L * c a l c u l a t e d u s i n g e s t i m a t e d s u r f a c e temperatures i s l e s s n e g a t i v e than L * c a l c u l a t e d from the measured s u r f a c e temperatures ) and occur due to the l a r g e g r a d i e n t between s u r f a c e and a i r temperature near the top of the Eas t w a l l s h o r t l y a f t e r s u n s e t . These d i f f e r e n c e s are q u i c k l y reduced and become n e g a t i v e towards the end of the n i g h t as h i g h s u r f a c e temperatures in t h i s area c o o l . L * i s o v e r - e s t i m a t e d on the West w a l l , except for p o i n t s near the canyon top l a t e r at n i g h t . ' The canyon f l o o r i s a l s o o v e r e s t i m a t e d , i n c r e a s i n g l y so wi th t i m e , because d i f f e r e n c e s of L Q decrease more s l o w l y than those of . The l a r g e s t d i f f e r e n c e s of L * occur a c r o s s the canyon top because of the l a r g e u n d e r e s t i m a t i o n of L 0 . D i f f e r e n c e s are of the o r d e r 10 - 20 W m ~ 2 , w i t h the maximum l o c a t e d near the canyon m i d - p o i n t ( d i f f e r e n c e s of L * are equa l in magnitude and o p p o s i t e i n s i g n to those of L 0 a c r o s s the canyon top s i n c e does not v a r y ) . 130 5.2 .2 R e s u l t s : A i r p o r t A i r Temperature Use of the a i r p o r t a i r temperature as the f a c e t s u r f a c e temperature for a l l canyon g r i d - p o i n t s y i e l d s an a d d i t i o n a l u n d e r e s t i m a t i o n of L j and L 0 f or a l l canyon f a c e t s by a p p r o x i m a t e l y 5 - 10 W m" 2 ( F i g u r e s 5.3 and 5.4) i n d i c a t i n g a i r p o r t a i r temperatures are c o o l e r than the average canyon a i r t e m p e r a t u r e s . Temporal t r ends are s i m i l a r to those o b t a i n e d u s i n g the average canyon a i r t e m p e r a t u r e . The g r e a t e r changes which occur between the f i r s t two p l o t s of each s e r i e s are a r e s u l t of the use of t emperatures f o r sunset r a t h e r than the h a l f hour a f t e r sunset (as used f o r canyon average t e m p e r a t u r e s ) . C i r c l e s on the p l o t s r e p r e s e n t the d i f f e r e n c e s a t sunse t . O v e r e s t i m a t i o n of L * i n c r e a s e s over tha t shown by the use of average canyon a i r temperature f o r both days; however, the i n c r e a s e a t p o i n t s w i t h i n the canyon i s much s m a l l e r for the H/W=2.0 canyon than f o r the 1.0 c a s e , d i f f e r e n c e s of both and L Q are s m a l l e r for most t i m e s . 5.3 AVERAGE OR MID-POINT SURFACE TEMPERATURE The second a p p r o x i m a t i o n to the measurement of s u r f a c e temperature a t . a l l canyon g r i d - p o i n t s i s based upon the work conducted by W a k e f i e l d (1987) on sampl ing methodolog ies for input da ta needed in models of the A r n f i e l d t y p e . I t assumes an 131 Airport Air Temperature Approximation August 1/2 1988 H/W 2.0 O Sunset A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h « Sunset + 8 h Li Differences Focet Point: Floor l i Differences Focel Point: Top I . Differences Focet Point: Floor Lo Differences Focet Point: Top L* Difference* Facet Point: Floor I . Differences- F i g u r e 5.3 M o d e l l e d long-wave f l u x d i f f e r e n c e s (W m~z) o b t a i n e d u s i n g t h e a i r p o r t a i r t e m p e r a t u r e i n p l a c e of measured s u r f a c e t e m p e r a t u r e . Data i s from Aug. 1/2 1988, H/W=2.0. focet Point: lop L« Differences focet Point: Floor L*> Differences Focet Point: Top \ 2 3 4 5 6 7 6 9 10 « L» Diffetences focet Point: Floor L» Differences Figure 5.4 As per F i g . 5.3 for Aug. 3/4 1988, H/W=1.0. 133 i n f r a r e d rad iometer i s a v a i l a b l e to take measurements of s u r f a c e t e m p e r a t u r e . For homogeneous h o r i z o n t a l canyon s u r f a c e s , a s i n g l e measurement of the c e n t r e of the c o n f i g u r a t i o n was c o n s i d e r e d adequate for the r e p r e s e n t a t i o n of the f a c e t t e m p e r a t u r e . Homogeneous v e r t i c a l canyon f a c e t s a l s o use a s i n g l e temperature measurement of the c e n t r e of the c o n f i g u r a t i o n u s i n g the l a r g e s t ' s p o t ' (area seen by the IR thermometer) p o s s i b l e ( W a k e f i e l d , 1987). D i r e c t comparison of t h i s method i s not p o s s i b l e w i t h the a v a i l a b l e data but two a p p r o x i m a t i o n s may be made. In the f i r s t , the temperature of the m i d - p o i n t of the canyon f a c e t , based upon the s u r f a c e temperature of the c l o s e s t g r i d - p o i n t (or l i n e a r l y i n t e r p o l a t e d between the two n e a r e s t g r i d - p o i n t s ) i s used for a l l p o i n t s on tha t f a c e t . The second uses the average f a c e t t emperature , i . e . the average of a l l p o i n t s on the f a c e t . Averages for the canyon w a l l s i n c l u d e the e x t r a p o l a t e d e l e v e n t h p o i n t and are weighted by g r i d a r e a . 5.3.1 R e s u l t s : Face t M i d - P o i n t Temperature The p l o t s shown in F i g u r e s 5.5 and 5.6 i l l u s t r a t e the d i f f e r e n c e s due to the use of the f a c e t m i d - p o i n t temperature in p l a c e of the t r u e s u r f a c e t e m p e r a t u r e . Trends over t ime for both canyon geometr ies are very s i m i l a r for L * and L 0 ; shows more d i f f e r e n c e s between the two g e o m e t r i e s . 134 Mid-Point Facet Temperature Approximation August 1/2 1988 H/W 2.0 Q Sunset A Sunset + 1 h + Sunset + 2 h X Sunset t « h O Sunset • 8 h Li Differences Focet Point: floor Li Differences Focel Point: Top L« Differences Focel Point: Floor Lo differences Focet Point: Top L* Differences Facet Point: Floor L* Differences Figure 5.5 Modelled long-wave flux differences (W m~z) obtained using the mid-point facet temperature in place of measured surface temperature. Data i s from Aug. 1/2 1988, H/W=2.0. 135 M i d — P o i n t F a c e t T e m p e r a t u r e A p p r o x i m a t i o n A u g u s t 3 / 4 1 9 8 8 H / W 1.0 OSuns«. A Sunset + 1h + Sunset + 2 h X Sunset + 4 h • Sunset + 8 h t i Differences Focel Point: Floor Li Differences Focet Point; Top L* Differences Focet .Point: Floor Differences F i g u r e 5.6 As per F i g . 5 .5 f o r A u g . 3/4 1988, H/W=1.0. 136 On the West w a l l , L j i s i n c r e a s i n g l y o v e r e s t i m a t e d for the upper p o r t i o n s of the w a l l , w i th l a r g e r changes o c c u r r i n g in the H/W=2.0 canyon . The lower h a l f of the w a l l d i s p l a y s i n c r e a s i n g l y n e g a t i v e d i f f e r e n c e s wi th t ime . These changes are d i r e c t l y c o n t r o l l e d by the d i f f e r e n c e s r e c o r d e d for L 0 . The top p o r t i o n s of both canyon w a l l s show i n c r e a s i n g l y p o s i t i v e d i f f e r e n c e s w i t h t ime; those of the H/W=2.0 canyon i n c r e a s e more w i t h t ime , as g r e a t e r changes i n s u r f a c e temperature were observed i n t h i s canyon near the tops of the w a l l s (see Chapter 6 ) . The lower p a r t s of the w a l l d i s p l a y g r e a t e r u n d e r e s t i m a t i o n s w i th time as the reduced c o o l i n g i n c r e a s e s the temperature d i f f e r e n c e between temperature r e c o r d e d at the m i d - p o i n t of the w a l l and that near the base of the w a l l s . The p a t t e r n a c r o s s the f l o o r f o r both runs tends to show g r e a t e r u n d e r e s t i m a t i o n of a t p o i n t s near the w a l l s wi th t i m e , aga in due to reduced c o o l i n g a f f o r d e d by lower 4/s, as f or the base of the canyon w a l l s . The H/W=1 .0 canyon e x h i b i t s o v e r e s t i m a t i o n s near the Eas t w a l l a t t imes near sunset because L 0 i s underes t imated for p o i n t s lower on that w a l l . L c d i f f e r e n c e s are i n i t i a l l y . s l i g h t l y n e g a t i v e f o r the West w a l l and s t r o n g l y so for the top of the E a s t w a l l where h i g h s u r f a c e temperatures o c c u r . The d i f f e r e n c e s for both w a l l s subsequent ly become p o s i t i v e near the tops and n e g a t i v e towards the base , f o l l o w i n g the d i f f e r e n c e between the s u r f a c e temperature at the p o i n t and tha t at m i d - f a c e t . P o i n t s on the f l o o r near the canyon w a l l s a l s o d i s p l a y i n c r e a s e d u n d e r e s t i m a t i o n wi th t i m e . A c r o s s the canyon top i n i t i a l l a r g e 137 n e g a t i v e d i f f e r e n c e s near the E a s t w a l l r e s u l t i n g from u n d e r e s t i m a t e d s u r f a c e temperatures at the top of the E a s t w a l l are reduced wi th time and p o s i t i v e d i f f e r e n c e s near the West w a l l decrease so tha t p o s i t i v e d i f f e r e n c e s e x i s t near each w a l l , f o l l o w i n g the o v e r - e s t i m a t e d L D f or the tops of the w a l l s . The p a t t e r n of d i f f e r e n c e s of L * a c r o s s the West w a l l e s t a b l i s h e d at sunset i s enhanced wi th t ime so tha t p o i n t s near the base of the w a l l are o v e r e s t i m a t e d and p o i n t s above the m i d - p o i n t are underes t imated to a g r e a t e r degree w i t h d i s t a n c e from the face t m i d - p o i n t . Near the top of the E a s t w a l l s u b s t a n t i a l o v e r e s t i m a t i o n of L * o c c u r s due to the p o o r l y mode l l ed v a l u e s of L Q . P o i n t s on the lower p o r t i o n of the w a l l are u n d e r e s t i m a t e d ; s t r o n g l y for the H/W=1.0 canyon and more weakly f o r the 2.0 c a s e . As time a f t e r sunset i n c r e a s e s , the p a t t e r n and magnitude of d i f f e r e n c e s become s i m i l a r to tha t of the West w a l l . A c r o s s the f l o o r there i s a tendency for o v e r e s t i m a t i o n of L * f o r p o i n t s . n e a r the canyon w a l l s , p a r t i c u l a r l y l a t e r at n i g h t , which i s c o n t r o l l e d by the d i f f e r e n c e s in L Q . P o s i t i v e d i f f e r e n c e s of L * are i n i t i a l l y d i s p l a y e d at the canyon top near the E a s t w a l l . These decrease to e s t a b l i s h a p r o f i l e of n e g a t i v e d i f f e r e n c e s f o r p o i n t s c l o s e to w a l l s l a t e r at n i g h t and are aga in c o n t r o l l e d by L D . 138 5.3 .2 R e s u l t s : Average Face t Temperature Use of the average face t temperature in p l a c e of the m i d - p o i n t canyon temperature s h i f t s the l o c a t i o n s where d i f f e r e n c e s s w i t c h from p o s i t i v e to n e g a t i v e and tends to decrease the c a l c u l a t e d d i f f e r e n c e s s l i g h t l y ( F i g u r e s 5.7 and 5 . 8 ) . T h i s o c c u r s because the use of an average s u r f a c e temperature w i l l be a l t e r e d more by the temperature extremes on the f a c e t s , p a r t i c u l a r l y the E a s t w a l l s i n c e a l l p o i n t s are we ighted . O t h e r w i s e , r e l a t i v e l y l i t t l e e f f e c t i s e x e r t e d on the d i s t r i b u t i o n of d i f f e r e n c e s for each of the f l u x e s a c r o s s the canyon f a c e t s at each t i m e . T h i s r e s u l t i s not unexpected because the m i d - p o i n t face t temperatures for the w a l l s are not very d i f f e r e n t from the average s u r f a c e temperature and the d i f f e r e n c e s of s u r f a c e temperatures w i l l t h e r e f o r e be s i m i l a r (see Chapter 6 for p l o t s of s u r f a c e temperature d i s t r i b u t i o n s on canyon f a c e t s ) . The d i f f e r e n c e s on the f l o o r s are b e t t e r mode l l ed u s i n g the average face t temperature l a t e r at n i g h t because the m i d - p o i n t temperature i s g e n e r a l l y the minimum s u r f a c e temperature under calm and. c l e a r c o n d i t i o n s . 1 3 9 Average Facet Temperature Approximation August 1/2 1988 H/W 2.0 O Sunset A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h O Sunset + 8 h L o D i f f e r e n c e s Foce t P o i n t : F loor L . Di f fe rences Foce t Po in t : T o p I 2 3 4 5 L " D i i l e r e n c e s Foce t P o i n t : F l o o r L« Di f fe rences F i g u r e 5.7 M o d e l l e d long-wave f l u x d i f f e r e n c e s (W m"^) o b t a i n e d u s i n g the average f a c e t t emperature i n p l a c e of measured s u r f a c e t e m p e r a t u r e . Data i s from A u g . 1/2 1988, H /W=2.0 .  141 5.4 SUMMARY To summarize the f i n d i n g s of the p l o t s , t a b l e s of the average d i f f e r e n c e s by face t for each f l u x are p r e s e n t e d (Tab le s 5 . 1 - 5 . 6 ) . For a p p l i c a t i o n s w i th an i n t e r e s t i n the f l u x e s l e a v i n g the canyon top ( L * , L 0 ) , the t a b l e s a l l o w comparison of the performance of each a p p r o x i m a t i o n scheme. T a b l e 5.1 Summary of Average D i f f e r e n c e s (W m~ z) by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature A p p r o x i m a t i o n s (Aug. 1/2, H/W=2.0) . A p p r o x i m a t i o n Time From West Eas t Canyon Sunset W a l l W a l l F l o o r (h) A i r p o r t a i r 0 - 2 0 . 0 -20 .0 - 1 8 . 5 temperature 1 - 2 3 . 9 -24 .7 -23.1 2 - 2 2 . 1 - 23 . 5 -22 .0 4 - 2 0 . 9 -22 .4 -21 .2 8 -21.1 - 2 1 . 9 -21 . 1 Average canyon 0.5 . -14 .8 -15 .2 -14.1 a i r temperature 1 - 1 4 . 3 -15 .2 -14 .2 2 - 1 4 . 8 -16 .2 -15 .2 4 -15 .4 - 1 6 . 9 - 1 6 . 1 8 -16 .2 - 1 6 . 9 - 16 . 5 M i d - p o i n t f a c e t 0 - 2 . 9 - 0 . 2 - 0 . 8 temperature 1 -1 .5 0.1 - 0 . 7 2 - 0 . 8 0.3 - 0 . 7 4 0.1 0.3 -1 .1 8 0.4 0.3 -1 .8 Average face t 0 0.3 0.0 1 . 1 temperature 1 0.0 -0.1 -0 .1 2 -0 .1 - 0 . 2 - 0 . 9 4 - 0 . 2 - 0 . 2 - 2 . 0 8 - 0 . 3 - 0 . 3 -3 .1 142 T a b l e 5.2 Summary of Average L 0 D i f f e r e n c e s (W m -^) by Face t I n c u r r e d as a R e s u l t of Us ing Temperature A p p r o x i m a t i o n s (Aug. 1/2, H/W=2.0) . A p p r o x i m a t i o n Time From West E a s t Canyon Canyon Sunset W a l l W a l l F l o o r Top (h) A i r p o r t a i r 0 - 2 5 . 6 - 2 5 . 5 -21 .2 -26.1 temperature 1 -31 .0 - 2 9 . 6 -28 .2 -29 .4 2 - 2 9 . 1 - 2 6 . 8 -28 .0 -26 .4 4 - 2 6 . 9 -24 .4 - 2 9 . 7 - 2 3 . 7 8 -24 .8 - 2 3 . 5 - 3 3 . 5 -22 .3 Average canyon 0.5 - 1 9 . 6 - 1 8 . 8 -16 .0 -19 .0 a i r temperature 1 -19 .4 - 1 7 . 9 -16 .7 -17 .8 2 -20 .2 - 1 8 . 0 -19.1 -17 .6 4 - 2 0 . 3 - 1 7 . 7 - 2 2 . 9 -17.1 8 - 1 8 . 9 - 1 7 . 5 - 27 . 5 -16 .3 M i d - p o i n t f a c e t 0 - 0 . 4 - 4 . 6 - 0 . 7 - 3 . 7 . temperature 1 0.3 - 2 . 3 -1 .2 - 0 . 8 2 0.7 - 1 . 0 -1 .6 0.9 4 0.7 0.5 - 2 . 0 2.8 8 0.9 1 . 1 - 2 . 5 4.1 Average f a c e t 0 - 0 . 2 0.4 0.0 -1 .6 temperature 1 - 0 . 4 -0 .1 0.0 0.1 2 - 0 . 5 - 0 . 3 -0 .1 2.3 4 - 0 . 6 - 0 . 6 -0 .1 2.3 8 - 0 . 7 - 0 . 7 -0 .1 3.4 Of the p r o c e d u r e s t e s t e d , the average f a c e t temperature produces s l i g h t l y s m a l l e r d i f f e r e n c e s at most t imes than does the use of the m i d - p o i n t f a c e t t e m p e r a t u r e . Under the c l e a r and l i g h t wind c o n d i t i o n s t e s t e d , the use of a i r temperature i n p l a c e of the measured s u r f a c e temperatures r e s u l t s i n l a r g e model e r r o r s because s u r f a c e temperatures are c o n s i d e r a b l y warmer, p a r t i c u l a r l y i n the e a r l y e v e n i n g . When m e t e o r o l o g i c a l c o n d i t i o n s are such that a i r and s u r f a c e temperature d i f f e r e n c e s are l e s s , the , f lux d i f f e r e n c e s w i l l be r e d u c e d . The a i r p o r t a i r 143 temperature es t imate produces c o n s i s t e n t l y g r e a t e r d i f f e r e n c e s than the average canyon a i r t emperature , i n d i c a t i n g some s p a t i a l m o d i f i c a t i o n of temperature at the s i t e . T h i s s u p p o r t s the p r a c t i c e of adding a h e a t - i s l a n d increment when u s i n g r u r a l a i r p o r t temperatures ( A r n f i e l d , 1976) . : D i f f e r e n c e s between methods of measurement of the temperatures at the two s i t e s prevent r i g o r o u s comparison between the a i r t e m p e r a t u r e s . T a b l e 5.3 Summary of Average L * D i f f e r e n c e s (W m~ 2) by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature Approx imat ions (Aug. 1/2, H/W=2.0). A p p r o x i m a t i o n Time From West E a s t Canyon Canyon Sunset W a l l W a l l F l o o r Top (h) A i r p o r t a i r 0 5.7 5.4 2.7 26.1 temperature 1 7.1 4.9 5.1 29.4 2 7.0 3.3 6.1 26.4 4 6.0 1 .9 8.6 23.7 8 3.8 1 .6 12.5 22.3 Average canyon 0.5 4.9 3.6 1.9 19.0 a i r temperature 1 5.0 2.7 2.5 17.8 2 5.4 1.8 3.9 17.6 4 4.8 0.8 6.9 17.1 8 2.7 0.6 11.0 16.3 M i d - p o i n t f ace t 0 - 2 . 6 4.3 - 0 . 2 3.7 temperature 1 - 1 . 8 2.4 0.6 0.8 2 - 1 . 5 1.3 0.9 - 1 . 0 4 - 0 . 7 - 0 . 2 0.9 - 2 . 8 8 - 0 . 5 - 0 . 9 0.7 -4 .1 Average f a c e t 0 0.6 - 0 . 4 1 . 1 1.6 temperature 1 0.4 0.0 -0 .1 -0.1 2 0.4 0.2 - 0 . 9 -1.1 4 0.4 0.4 -1 .9 - 2 . 3 8 0.4 0.4 - 3 . 0 -3 .4 144 T a b l e 5.4 Summary of Average L j D i f f e r e n c e s (W m~^) by Face t I n c u r r e d as a R e s u l t of Us ing Temperature A p p r o x i m a t i o n s (Aug. 3 /4 , H/W=1.0) . A p p r o x i m a t i o n Time From West . E a s t Canyon Sunset W a l l W a l l F l o o r (h) A i r p o r t a i r 0 -34 . ,0 - 29 . ,8 -27 . ,8 temperature 1 -28 . ,2 -25 . ,6 -23 . ,4 2 -28 . ,3 -26 , ,5 -23 . ,4 4 -21 . ,2 -20 . ,3 -17 . ,3 8 -14 . ,8 -14 . ,4 -11 . ,5 Average canyon 0.5 -23 . ,6 -20 . ,4 -19 . ,5 a i r temperature 1 -23 . ,5 -20 . ,9 -19 . ,5 2 -17 . .7 -15 . ,8 -14. .7 4 -13 . ,3 -12 . .3 -10. .8 8 - 9 . ,9 - 9 . ,6 - 7 . .5 M i d - p o i n t f ace t 0 - 0 . .5 0. , 1 0. .4 temperature 1 0. .3 0. . 1 0, .6 2 0, . 1 0, . 1 0. .3 4 0. .3 0, . 1 0. .3 8 - 0 . .4 -0 , .4 -0 , .5 Average f a c e t 0 0, . 1 - 0 , . 1 0, .9 temperature 1 0. .0 -0 , . 1 0, . 1 2 -0 , . 1 -0 , . 1 -0 , .2 4 -0 , .2 0, .0 -0 , .6 8 -0 , .2 0, .0 -0 , .8 Some v a r i a t i o n wi th the canyon H/W are n o t e d , a l t h o u g h d i f f e r e n c e s observed may be d u e , i n p a r t to s l i g h t l y d i f f e r e n t m e t e o r o l o g i c a l c o n d i t i o n s on the two d ays . An e x t e n s i o n of the i n v e s t i g a t i o n to o ther canyon H/W r a t i o s has not been a t t e m p t e d . In g e n e r a l , the temporal v a r i a t i o n of d i f f e r e n c e s of a l l f l u x e s i s g r e a t e r in the H/W=1.0 canyon . Canyon a i r t emperatures g i v e lower average L * d i f f e r e n c e s in the H/W=2.0 canyon when compared to the H/W=1.0 c a s e , wh i l e d i f f e r e n c e s u s i n g s u r f a c e temperature a p p r o x i m a t i o n s are l a r g e r i n the H/W=2.0 canyon, p a r t i c u l a r l y 145 f o r t imes j u s t a f t e r s u n s e t . The former e f f e c t i s due to s m a l l e r d i f f e r e n c e s between a i r and canyon s u r f a c e temperatures i n the H/W=2.0 canyon whi l e the l a t t e r may be i s a r e s u l t of sharper v a r i a t i o n s of s u r f a c e temperature a c r o s s canyon f a c e t s for the 2 . 0 canyon at s u n s e t , which are not w e l l r e p r e s e n t e d by the average or m i d - p o i n t f a c e t t e m p e r a t u r e . T a b l e 5 . 5 Summary of Average L 0 D i f f e r e n c e s (W m -^) by Face t I n c u r r e d as a R e s u l t of Us ing Temperature A p p r o x i m a t i o n s (Aug. 3 / 4 , H / W = 1 . 0 ) . A p p r o x i m a t i o n Time From West E a s t Canyon Canyon Sunset W a l l W a l l F l o o r Top (h) A i r p o r t a i r 0 - 4 3 . ,9 - 5 4 . ,2 - 3 8 . ,6 - 4 5 . .6 temperature 1 - 3 7 . ,4 - 4 3 . .6 - 3 4 . . 1 - 3 7 . .7 2 - 3 7 . ,9 - 4 2 . ,4 - 3 6 . ,4 - 3 8 . .0 4 - 2 8 . , 1 - 2 9 . .9 - 2 9 . .6 - 2 8 . .2 8 - 1 8 , ,3 - 1 8 . ,8 - 2 3 . .6 - 1 9 . .4 Average canyon 0 . 5 - 3 0 . ,4 - 3 8 . .3 - 2 6 . . 1 -31 , . 1 a i r temperature 1 - 3 0 . ,8 - 3 7 . .2 - 2 7 . .4 -31 . . 1 2 - 2 2 . .9 - 2 7 . . 5 -21 . .4 - 2 3 . . 1 4 - 1 6 . .9 - 1 8 . .8 - 1 8 . .4 - 1 7 , . 1 8 - 1 1 . ,5 - 1 1 . .9 - 1 6 . .8 - 1 2 . .6 M i d - p o i n t f a c e t 0 - o . ,2 -1 , .6 0 , .6 -1 , .3 temperature 1 0. .2 0, . 5 - o . .2 0. .2 2 0 . .4 0. . 5 - 0 . .7 0 . . 5 4 0 . .7 .. 1 , .5 -1 . .2 1. . 1 8 0 . . 1 0, .3 - 2 . .0 0 . .5 Average f a c e t 0 - 0 . .2 0, .2 0 , . 1 - 1 . .0 temperature 1 - 0 . .4 - 0 , .4 0, .0 - o , . 1 2 - o , . 5 - 0 , .4 0, .0 0, .2 4 - o . .6 - 0 , . 5 - 0 , . 1 0 , .7 8 - 0 . .4 - 0 , .4 0. .0 0 . .9 146 T a b l e 5.6 Summary of Average L * D i f f e r e n c e s (W rn~ 2) by Face t I n c u r r e d as a R e s u l t of U s i n g Temperature Approx imat ions (Aug. 3 /4 , H/W=1.0) . A p p r o x i m a t i o n Time From West Eas t Canyon Canyon Sunset W a l l Wal l F l o o r Top (h) A i r p o r t a i r 0 9.9 24.5 10.8 45.6 temperature 1 9.2 18.0 10.8 37.7 2 9.5 15.9 •13.0 38.0 4 6.9 9.7 12.3 28.2 8 3.5 4.3 12.2 19.4 Average canyon 0.5 6.8 17.9 6.6 31.1 a i r temperature 1. 7.3 16.3 7.9 31 . 1 2 5.2 11.8 6.7 23. 1 4 3.7 6.5 7.6 17.1 8 1 .5 2.3 9.4 12.6 M i d - p o i n t f a c e t 0 - 0 . 3 1 .6 - 0 . 3 1 .3 temperature 1 0.2 - 0 . 5 • 0.7 -0 .2 2 - 0 . 3 -0 .4 1 .0 - 0 . 5 4 - 0 . 4 -1 .4 1 .5 -1 .1 8 - 0 . 5 - 0 . 7 1 .5 - 0 . 5 Average f a c e t 0 0.3 - 0 . 3 0.9 1 .0 temperature 1 0.4 0.3 0.1 0.1 2 0.5 0.3 - 0 . 2 - 0 . 2 4 0.4 0.4 - 0 . 6 - 0 . 6 8 - 0 . 6 0.5 - 0 . 8 - 0 . 9 A i r t emperature -based a p p r o x i m a t i o n s perform best in the hours soon a f t e r sunset in the H/w=2.0 canyon , but worsen l a t e r compared to L 0 and L j in the 1.0 canyon . S m a l l e r i n i t i a l d i f f e r e n c e s , b e t w e e n a i r and s u r f a c e temperatures in the 2.0 canyon p r o v i d e the b e t t e r i n i t i a l r e s u l t s . These d i f f e r e n c e s i n c r e a s e due to the reduced r a t e of c o o l i n g due to canyon geometry. 147 Differences of L^ are controlled p a r t i a l l y by L Q differences and the 4/s of the point. Where \JJS i s large errors may be reduced because the true value of L^ct was used. Additional tables, (5.7 and 5.8) have been constructed to present the percentage difference from the 'true' modelled values of L * and L Q at the canyon top. Table 5.7 Percentage Difference of Fluxes Leaving the Canyon Top for Various Surface Temperature Approximation Schemes.(Aug. 1/2 1988 H/W=2.0). Approx imation Time From Canyon Top Sunset (h) L o L* Airport a i r 0 -5.9 -24.8 temperatue 1 -6.8 -29.2 2 -6.2 -27.4 4 -5.7 -25.9 8 -5.5 -24.9 Average canyon 0.5 -4.4 -18.3 a i r temperature 1 -4.1 -17.7 2 -4.1 -18.2 4 -4.1 -18.9 8 -4.0 -18.2 Mid-point facet 0 -0.8 -3.5 temperature 1 -0.2 -0.8 2 0.2 1.0 4 0.7 3.1 8 1.0 4.6 Average facet 0 -0.4 -1.5 temperature 1 0.0 0.1 2 0.5 1 . 1 4 0.5 2.5 8 0.8 3.8 These show the percentage errors calculated by dividing the difference obtained by the 'true' value of the flux for averaged 148 v a l u e s of L Q and L * on the canyon t o p . T h i s comparison i s f o r m o d e l l e r s i n t e r e s t e d in the e r r o r s to be expected in f l u x e s e m i t t e d from the canyons . T a b l e 5.8 Percentage D i f f e r e n c e of F l u x e s L e a v i n g the Canyon Top for V a r i o u s S u r f a c e Temperature A p p r o x i m a t i o n Schemes . (Aug. 3/4 1988 H/W=1.0) . A p p r o x i m a t i o n Time From Sunset (h) Canyon Lo Top L * A i r p o r t a i r 0 - 9 . 9 -39 .0 temperature 1 - 8 . 3 -34 .4 2 - 8 . 5 - 3 7 . 7 4 - 6 . 5 -31 .3 8 - 4 . 6 - 2 3 . 6 Average canyon 0.5 - 6 . 8 -27 .3 a i r temperature 1 - 6 . 9 -28 .3 2 - 5 . 2 -23 .0 4 - 3 . 9 - 1 9 . 2 8 - 3 . 0 -15 .4 M i d - p o i n t f a c e t 0 - 0 . 3 -1 .1 temperature 1 0.0 0.2 2 0.1 0.5 4 0.3 1 .2 8 0.1 0.6 Average f a c e t 0 - 0 . 2 - 0 . 9 temperature 1 0.0 -0 .1 2 0.0 0.2 4 0.2 0.7 8 0.2 1 .1 The percentage d i f f e r e n c e s o b t a i n e d when a i r p o r t a i r temperature was used show maximum d i f f e r e n c e s in L Q and L * f o r the hour a f t e r sunset in the H/W=1.0 canyon (Table 5 . 7 ) . T h i s c o i n c i d e s w i t h o b s e r v a t i o n s tha t the maximum heat i s l a n d deve lops s h o r t l y a f t e r sunse t . The maximum d i f f e r e n c e i n the 149 H/W=2.0 canyon i s r e c o r d e d at s u n s e t , w i th a secondary maximum at two hours a f t e r sunse t . Average canyon a i r temperatures reduce the percentage d i f f e r e n c e by 5 - 8%. The d i f f e r e n c e s in the H/W=1.0 canyon u s i n g t h i s a p p r o x i m a t i o n are almost c o n s t a n t w i t h t i m e , which might i n d i c a t e a s imple c o r r e c t i o n f a c t o r c o u l d be s u c c e s s f u l l y employed to improve the s u r f a c e temperature e s t i m a t e s . T h i s f i n d i n g i s not r e f l e c t e d i n the 2.0 canyon however, the d i f f e r e n c e s e x h i b i t a maximum for the hour a f t e r sunset and then reduced d i f f e r e n c e s wi th t i m e , s i m i l a r to those r e c o r d e d for the a i r p o r t a i r temperature a p p r o x i m a t i o n used in the 1.0 canyon . M i d - p o i n t f a c e t temperatures p r o v i d e on ly minor d i f f e r e n c e s i n L 0 . D i f f e r e n c e s are g r e a t e r for L * and show a c o n s i s t e n t i n c r e a s e w i th time for both f l u x e s , due to o v e r e s t i m a t e d s u r f a c e temperature and t h e r e f o r e L Q on the upper p o r t i o n s of the w a l l s . The use of average face t s u r f a c e temperature f u r t h e r improves the percentage d i f f e r e n c e s for nost t i m e s , w i t h s i m i l a r temporal changes and l a r g e r d i f f e r e n c e s r e c o r d e d for L * . I t can be c o n c l u d e d from t h i s a n a l y s i s t h a t : 1. A p p r o x i m a t i o n s to canyon f a c e t temperatures which have as t h e i r b a s i s i n - c a n y o n p o i n t or average s u r f a c e t emperatures w i l l each s i g n i f i c a n t l y outper form a i r t emperature -based e s t i m a t e s . 2. The use of the face t average temperature i s s l i g h t l y s u p e r i o r to the m i d - p o i n t f a c e t temperature in most c a s e s . 150 3. W i t h i n - c a n y o n a i r t e m p e r a t u r e s , averaged over the p e r i o d of a complete t r a v e r s e , y i e l d s u p e r i o r mode l l ed f l u x e s compared to unmodi f i ed a i r p o r t a i r t e m p e r a t u r e s . The r e s u l t s apply to the mos t ly c l e a r and calm c o n d i t i o n s t e s t e d . 4. F u r t h e r c o r r e c t i o n s i n the form of temperature increments due to heat i s l a n d and a i r and s u r f a c e temperature d i f f e r e n c e e f f e c t s c o u l d improve the r e s u l t s when a i r temperatures are used . 5. R e s u l t s c o u l d a l s o be enhanced by the use of a c o r r e c t i o n for the s u r f a c e temperature based upon ^ s for each g r i d - p o i n t and p o s s i b l y the i n i t i a l temperature d i s t r i b u t i o n . 151 C H A P T E R 6. C A N Y O N T E M P E R A T U R E S , R A D I A T I O N , A N D C O O L I N G 6.1 INTRODUCTION F e a t u r e s of the temporal and s p a t i a l v a r i a t i o n of s u r f a c e and a i r temperatures and long-wave r a d i a t i o n w i t h i n the model canyons are presented in t h i s c h a p t e r . C o o l i n g a t v a r i o u s p o i n t s w i t h i n the canyon are compared to show d i f f e r e n t i a l r a t e s dependent upon i n i t i a l s u r f a c e temperature and p o s i t i o n . C o o l i n g of the canyon , r e p r e s e n t e d by the m i d - p o i n t on the canyon f l o o r , i s compared to the c o o l i n g of the open c o n c r e t e s u r f a c e to examine the i n f l u e n c e of canyon geometry. R e s u l t s are compared w i t h those of Oke (1981). The r o l e of a tmospher ic c o n t r o l s on the c o o l i n g of the canyon and open s i t e i s c o n s i d e r e d . 6.2 SURFACE T E M P E R A T U R E D I S T R I B U T I O N S 6.2.1 The D i u r n a l V a r i a t i o n of S u r f a c e H e a t i n g and C o o l i n g in the Model Canyons In a N o r t h - S o u t h a l i g n e d canyon , marked changes in the s u r f a c e t emperature , T s , take p l a c e d u r i n g the p e r i o d of one day . The f o l l o w i n g d e s c r i p t i o n may be assumed to be t y p i c a l of days c h a r a c t e r i z e d by c l e a r , f i n e weather wi th l i g h t winds . The da ta upon which the d e s c r i p t i o n i s based was c o l l e c t e d on August 3 /4 , from a canyon wi th a H/W of 1.0. T h i s canyon was 152 c o n s t r u c t e d p r i o r t o s u n r i s e on-Aug. 3, t h u s , t h e s u r f a c e t e m p e r a t u r e s s t a r t from a p p r o x i m a t e l y e q u a l v a l u e s w i t h no ' h i s t o r y ' of t h e c o o l i n g from t h e p r e v i o u s n i g h t . The s e q u e n c e of e v e n t s i n c a n y o n s w i t h t h e same o r i e n t a t i o n but d i f f e r e n t H/W r a t i o s w i l l be s i m i l a r but t h e p e r i o d o f r e c e i p t of d i r e c t r a d i a t i o n on t h e c a n y o n f a c e t s w i l l d e c r e a s e as H/W i n c r e a s e s . ...Beginning a t s u n r i s e , t h e E a s t c a n y o n w a l l s l o w l y warms as t h e r i s i n g Sun i r r a d i a t e s i t ' s b a c k . As t h e a l t i t u d e o f t h e Sun i n c r e a s e s , t h e i n s i d e t o p o f t h e West w a l l b e g i n s t o r e c e i v e d i r e c t s o l a r r a d i a t i o n ( K ^ ) a n d t h e s u r f a c e t e m p e r a t u r e c l i m b s s h a r p l y f o r t h e d i r e c t l y i r r a d i a t e d p o r t i o n ( F i g u r e 6.1a). The E a s t w a l l ( F i g u r e 6.1b) t h e n warms more q u i c k l y as s h o r t - w a v e r a d i a t i o n i s r e f l e c t e d by t h e West w a l l . F o l l o w i n g e x p o s u r e of t h e e n t i r e West w a l l t o d i r e c t s o l a r r a d i a t i o n i n t h e l a t e m o r n i n g , t h e f l o o r ( F i g u r e 6.1c) n e a r e s t t h e West w a l l b e g i n s t o r e c e i v e K^. A s h a r p peak i n t h e t e m p e r a t u r e d e v e l o p s s i m i l a r t o t h a t d e s c r i b e d f o r t h e t o p of t h e West w a l l e a r l i e r i n t h e m o r n i n g . As t h e e n t i r e f l o o r becomes i r r a d i a t e d , t h e t e m p e r a t u r e d i s t r i b u t i o n becomes l e s s skewed. F i g u r e 6.1c i l l u s t r a t e s t h e t e m p o r a l s p a c i n g o f maximum s u r f a c e t e m p e r a t u r e f o r p o i n t s 1 and 5 on t h e c a n y o n f l o o r , where p o i n t 1 i s n e a r e s t t h e West w a l l and 5 i s n e a r m i d - c a n y o n . The s u r f a c e t e m p e r a t u r e of t h e open s i t e has been p l o t t e d f o r c o m p a r i s o n . C o o l i n g of t h e West w a l l b e g i n s p r i o r t o s o l a r noon a s t h e a n g l e o f i n c i d e n c e of d i r e c t s o l a r d e c r e a s e s . The i n i t i a l c o o l i n g i s f o l l o w e d by a p e r i o d o f r e l a t i v e l y c o n s t a n t o r 153 (a) West Wall (b) (c) o o 4> 3 "5 » O L E Q> o o 3 30 25 20 •t i ns 1 K 10 - * ! 1 1 1000 2000 Time (PDT) East Wall T J 6 0 0 1000 2000 Time (PDT) Canyon Floor 0600 45 o 40 3 "5 k_ ID 35 CL E V 1— 30 <D O a 25 3 v> 20 -><-'•-. « 1 1 - 1 \ M S — i ; Open - / 1 ) 1 I 1 v> V \ 1 1 1 1 1 1 1 1000 2000 Time (PDT) 0600 F i g u r e 6.1 D i u r n a l v a r i a t i o n of s e l e c t e d s u r f a c e t e m p e r a t u r e s i n a canyon w i t h H/W=1.0, Aug. 3/4. 154 s l i g h t l y i n c r e a s i n g s u r f a c e temperatures which are m a i n t a i n e d through most of the a f t e r n o o n , r e s u l t i n g in d i s t i n c t s h o u l d e r s on the p l o t s i n F i g u r e 6 . 1 a . These are due p r i m a r i l y to r a d i a t i o n r e f l e c t e d from the E a s t w a l l w i t h some a d d i t i o n a l e f f e c t from heat conducted through the w a l l . S i m i l a r shapes of p l o t s for were observed by Nunez (1975) and Nunez and Oke (1976) . A f t e r s o l a r noon, the E a s t w a l l becomes the r e c i p i e n t of and t h e r e f o r e e x h i b i t s the h i g h e s t t e m p e r a t u r e s . The r a t e of temperature i n c r e a s e i s not as great as t h a t observed on the West w a l l s i n c e s u b s t a n t i a l warming has a l r e a d y taken p l a c e by r e f l e c t i o n from the o p p o s i t e w a l l and some component of t r a n s m i s s i o n of heat from the o u t s i d e of the w a l l . E v e n t u a l l y , shadows are c a s t by the West w a l l which p r o g r e s s a c r o s s the f l o o r towards , and l a t e r up, the Eas t w a l l . P o i n t s on the canyon f l o o r and E a s t w a l l e x p e r i e n c e r a p i d c o o l i n g once i n shadow and a g a i n c r e a t e a h i g h l y skewed s p a t i a l temperature d i s t r i b u t i o n . As sunset approaches , o n l y the top of the E a s t w a l l and the back of the West w a l l r e c e i v e d i r e c t r a d i a t i o n . The maximum s u r f a c e temperature near the top of the E a s t w a l l i s r ecorded w i t h i n 90 minutes be fore sunset and exceeds t h a t of the open s u r f a c e . T r a n s m i t t a n c e of heat through the West w a l l p r o b a b l y o f f s e t s some of the c o o l i n g on the inner f a c e t . The presence of a l a r g e hangar to the northwest of the canyon s i t e cas t shadows over the canyon p r i o r to l o c a l sunset and a shor t p e r i o d of d i r e c t i r r a d i a n c e on the E a s t w a l l i s l o s t , so tha t c o o l i n g i s s l i g h t l y enhanced. 155 In the e a r l y e v e n i n g , the West w a l l d i s p l a y s l o c a l l y c o o l e r temperatures near the top and bottom o f . t h e w a l l , w i th those the base a t t r i b u t e d to the p r o x i m i t y of the r e l a t i v e l y c o o l canyon f l o o r at t h i s t i m e . The temperature of the West w a l l exceeds the temperature of the lower h a l f of the E a s t , due to h e a t i n g from b e h i n d . The top of the Eas t w a l l undergoes r a p i d c o o l i n g and a ' c r o s s - o v e r ' of the o ther p l o t t e d p o i n t s i s observed between one and two hours f o l l o w i n g sunset a f t e r which the top p o i n t d i s p l a y s the minimum T s . A s i m i l a r c r o s s - o v e r of the top p o i n t from the West w a l l i s observed a p p r o x i m a t e l y one hour p r i o r to sunse t . A c r o s s - o v e r of p o i n t s 1 and 5 on the canyon f l o o r o c c u r s i n the hour be fore s u n s e t . The c r o s s - o v e r p o i n t s s i g n a l a r e v e r s a l of the l o c a t i o n of maximum and minimum temperature l o c a t i o n upon the canyon f a c e t s and i n d i c a t e t h a t the c o o l i n g i s s t r o n g l y governed by p o s i t i o n i n the canyon , and t h e r e f o r e \ps. C o n t i n u a l c o o l i n g i s observed to the end of the measurement p e r i o d wi th the minimum temperature for a l l f a c e t s r e c o r d e d j u s t p r i o r to s u n r i s e . Rates of c o o l i n g over the f a c e t s converge as the i n i t i a l temperature d i s t r i b u t i o n of the f a c e t i s r e v e r s e d . P o i n t s on the canyon f l o o r and West w a l l d i s p l a y a p p r o x i m a t e l y equa l r a t e s of c o o l i n g four hours from sunse t ; the r a t e s f o r the East w a l l are dependent upon p o s i t i o n , i n d i c a t i n g that the i n i t i a l temperature d i s t r i b u t i o n s t i l l a f f e c t s the c o o l i n g r a t e of the p o i n t s w i t h i n the canyon . 156 6.2 .2 Temperature D i s t r i b u t i o n s on Canyon F a c e t s S u r f a c e temperature d i s t r i b u t i o n s f o r a mid-canyon l e n g t h c r o s s - s e c t i o n are p r e s e n t e d for three d i f f e r e n t H/W r a t i o s ; 2 . 0 , 1.0 and 0 .41 . The m e t e o r o l o g i c a l c o n d i t i o n s of the three days were g e n e r a l l y c l e a r and c a l m , however emphasis i s p l a c e d upon d i f f e r e n c e s in the shape of the p r o f i l e s r a t h e r than a b s o l u t e magni tudes , s i n c e d i r e c t comparison r e q u i r e s c o n s i s t e n t , equa l c o n d i t i o n s for the f u l l h e a t i n g and c o o l i n g p e r i o d . Canyon c r o s s - s e c t i o n diagrams are used to presen t the s p a t i a l and temporal changes of the d i s t r i b u t i o n of s u r f a c e temperatures a c r o s s the canyon f a c e t s . The f a c e t temperature d i s t r i b u t i o n s for a canyon wi th a H/W of 2.0 are shown in F i g u r e 6 . 2 a . The d i s t r i b u t i o n s are markedly d i f f e r e n t at sunse t ; the West w a l l i s c h a r a c t e r i z e d by an r e l a t i v e l y s t r a i g h t temperature p r o f i l e whi le the E a s t w a l l e x h i b i t s h i g h v a l u e s of T s near the top of the w a l l r e s u l t i n g from the i r r a d i a t i o n p r i o r to s u n s e t . The West w a l l shows a smal l i n c r e a s e in temperature for p o i n t s 7-9 which may i n d i c a t e r e c e i p t of r e f l e c t e d r a d i a t i o n from the d i r e c t l y i r r a d i a t e d top p o r t i o n of the E a s t w a l l . The lowest t h r e e p o i n t s are s l i g h t l y warmer than the r e s t . The temperature d i f f e r e n c e from top to base over the West w a l l i n c r e a s e s w i t h time to j u s t over 4 ° C 8 hours from s u n s e t . The E a s t w a l l e x h i b i t s a peak T s at the top of the w a l l at sunset w i t h the lower p o r t i o n of the w a l l d i s p l a y i n g a r e l a t i v e l y s t r a i g h t p r o f i l e because i t has been shaded for a s u b s t a n t i a l p e r i o d of t i m e . By 4 hours a f t e r 157 S u r f o e e T e m p e r a t u r e F o c e t P o i n t : F l o o r S u r f a c e T e m p e r a t i Sorfoc. Twr««r<*ur« Fac»t Pomt: Floor Surlace T«mp«.otir. A Suraef + 1 h + Sunset + 2 h X Sunset + 4 h « Sunset F i g u r e 6.2 S p a t i a l and t empora l v a r i a t i o n of s u r f a c e t e m p e r a t u r e d i s t r i b u t i o n s i n ( top) a canyon w i t h H/W=2.0, (midd le ) H/W=1.0 canyon , (bottom) 0.41 canyon . 158 s u n s e t , the temperature p r o f i l e of the Eas t w a l l resembles tha t of the West w a l l , w i t h s u r f a c e temperatures warmer at the base of the w a l l . From sunset onwards, the minimum temperature on the f l o o r o c c u r s i n the c e n t r e of the canyon . S u r f a c e temperatures on the West s i d e are c o n s i s t e n t l y s l i g h t l y h i g h e r than those on the E a s t s i d e . Two reasons may account f o r t h i s : a) the base of the West w a l l i s s l i g h t l y warmer than tha t of the E a s t w a l l at s u n s e t , which may reduce the r a t e of r a d i a t i v e c o o l i n g due to the warmer s u r r o u n d i n g s , and b) c o n d u c t i o n of heat through the f l o o r from areas to the West of the canyon . The d i s t r i b u t i o n s of T s for the f a c e t s of an H/W=1.0 canyon are p r e s e n t e d in F i g u r e 6 .2b . At s u n s e t , the maximum temperature d i f f e r e n c e on the Eas t canyon w a l l i s a p p r o x i m a t e l y 6 ° C . In c o n t r a s t to the H/W=2.0 canyon, the temperature p r o f i l e decreases s t e a d i l y towards the base of the w a l l , r e f l e c t i n g the l a t e r t imes at which the w a l l was shaded in the more open geometry. The p o s i t i o n of the maximum T s at sunset i s one g r i d - p o i n t lower than the H/W=2.0 c a n y o n . The West w a l l d i s p l a y s a very s t r a i g h t i n i t i a l p r o f i l e w i t h no i n d i c a t i o n of the warmer temperatures near the top as in the H/W=2.0 c a s e . The temperature d i f f e r e n c e between the base and top of the w a l l i n c r e a s e s w i t h time up to four hours a f t e r sunset a f t e r which i t remains r e l a t i v e l y c o n s t a n t at a p p r o x i m a t e l y 3 ° C . The temperature d i s t r i b u t i o n a c r o s s the canyon f l o o r at sunset i s marked by a s l i g h t d i p near the West w a l l and a p l a t e a u of s l i g h t l y h i g h e r temperatures from the canyon m i d - 159 p o i n t e x t e n d i n g to the o p p o s i t e w a l l . The d i p i s a s s o c i a t e d w i t h the c o o l i n g of t h i s s e c t i o n of the canyon which has been o c c u r r i n g s i n c e i t was shaded s h o r t l y a f t e r s o l a r noon. The h i g h e r temperatures towards the e a s t e r n s i d e of the f l o o r are a r e s u l t of the l a t e r t ime at which they were shaded, a l t h o u g h t h e i r c o o l i n g may a l s o have been reduced by r e f l e c t i o n and r a d i a t i v e warming from the warm Eas t w a l l through t h e • a f t e r n o o n and e a r l y e v e n i n g . Four hours from s u n s e t , the temperature d i s t r i b u t i o n a c r o s s the canyon f l o o r e x h i b i t s a shal low concave curve wi th the p o i n t s neares t the w a l l s h a v i n g the h i g h e s t t e m p e r a t u r e s . By 8 hours a f t e r sunset t h i s curve has f u r t h e r deve loped to show a minimum near the canyon c e n t r e . The t h i r d case i l l u s t r a t e d i s the s m a l l e s t H/W b u i l t , 0.41 ( F i g u r e 6 . 2 c ) . For t h i s H/W the w a l l h e i g h t was reduced by two b l o c k s r e s u l t i n g i n the l o s s of four thermocouples from each w a l l . The data p r e s e n t e d are from Aug. 22/23 and the s u r f a c e temperatures are not a v a i l a b l e at s u n s e t . The w a l l s aga in d e v e l o p a sha l low S-shaped temperature p r o f i l e . The peak of T s near the top of the Eas t w a l l so apparent i n the H/W=1.0 and 2.0 cases near the top of the Eas t w a l l i n the hour f o l l o w i n g sunset i s much l e s s e v i d e n t , i n i t ' s p l a c e i s a broad i n c r e a s e over p o i n t s 3 -5 . Compared w i t h the H/W=1.0 and 2.0 canyons , the l o c a t i o n of the maximum T s appears to be f u r t h e r from the top of the w a l l in the e a r l y e v e n i n g . The canyon f l o o r d i s p l a y s s u b s t a n t i a l l y warmer temperatures on the Eas t s i d e of the canyon i n the e a r l y e v e n i n g , r e s u l t i n g from the r e l a t i v e l y l a t e t ime at which they were shadowed by the West w a l l and p o s s i b l y due to 160 some r e f l e c t i o n of s o l a r r a d i a t i o n a n d / o r r a d i a t i v e warming from the Eas t canyon w a l l . Warmer temperatures for p o i n t s neares t the w a l l s on each s i d e of the canyon are c l e a r l y v i s i b l e , the i n c r e a s e i n s u r f a c e temperature i s a p p r o x i m a t e l y 1.5 °C over t h a t at m i d - c a n y o n . Comparison of a b s o l u t e temperatures i s d i f f i c u l t because of v a r y i n g c o n d i t i o n s under which the canyons heated and c o o l e d . Of i n t e r e s t i s whether there e x i s t s a l i m i t in terms of canyon H/W at which reduced c o o l i n g r a t e s a f f o r d e d by l a r g e r canyon H/W i s overcome by lower a b s o l u t e temperatures due to reduced s o l a r input d u r i n g the day . The data a v a i l a b l e cannot be used to determine t h i s and t h i s might b e t t e r be t e s t e d under-more c o n t r o l l e d c o n d i t i o n s . 6 .2 .3 C o o l i n g of Canyon F a c e t s F o l l o w i n g Sunset The p r e v i o u s s e c t i o n d e s c r i b e d the temperature d i s t r i b u t i o n s on canyon f a c e t s at g iven t i m e s . H e r e , the c o o l i n g of v a r i o u s p o i n t s upon the f a c e t s i s examined. The i n f o r m a t i o n i s an expans ion of the r e s u l t s d i s c u s s e d i n 6.2.1 f o r t imes a f t e r s u n s e t . For w a l l s , three p o i n t s per face t are p l o t t e d , ( e x c e p t for the H/W=0.41 case when o n l y the_upper - and lowermost p o i n t s are p r e s e n t e d ) , r e p r e s e n t i n g the lower , mid , and uppermost p o r t i o n s of the w a l l (model p o i n t s 1, 5, and 10 r e s p e c t i v e l y ) . These p o i n t s r epresen t a range of sky-v iew f a c t o r s from a minimum at the base of the w a l l t o a maximum at the t o p . The canyon f l o o r i s r e p r e s e n t e d by two p o i n t s which a l s o cover 161 almost the f u l l range of \ / / s ; p o i n t 1 near the base of the West w a l l , and p o i n t 5 near the midpoint of the canyon . P l o t s f o r three H/W r a t i o s are p r e s e n t e d : 1.0, 2 . 0 , and 0.41 u s i n g the f i l t e r e d s u r f a c e temperatures for the same days as p r e s e n t e d in the p r e v i o u s s e c t i o n . The f i l t e r e d s u r f a c e temperatures have been used for p l o t t i n g . The p l o t s c l e a r l y i n d i c a t e the importance of p o s i t i o n (and t h e r e f o r e \ps) upon the c o o l i n g . P o i n t s deep w i t h i n the canyon e x h i b i t a more l i n e a r c o o l i n g than more exposed p o i n t s . The c o o l i n g of p o i n t 10 on the w a l l s over tha t of the lower p o i n t s i s most c l e a r l y e v i d e n t f o r the l a r g e s t H/W r a t i o , 2.0 ( F i g u r e 6 . 3 ) . The topmost p o i n t on the w a l l s may even c o o l at a r a t e exceed ing that of the open s u r f a c e because of the h i g h e r s u r f a c e temperatures j u s t p r i o r to sunset due to the more f a v o u r a b l e ang le of s o l a r i r r a d i a t i o n . C o o l i n g c o n t i n u e s throughout the n i g h t up to s u n r i s e . The c o o l i n g r a t e s over each face t become more equa l a p p r o x i m a t e l y 4 hours a f t e r s u n s e t . T h i s i s w e l l i l l u s t r a t e d by the p l o t s of West w a l l c o o l i n g in F i g u r e s 6.4 and 6.5 f o r the H/W=1.0 and 0.41 canyons . The l a r g e s t r e d u c t i o n in the c o o l i n g r a t e o c c u r s f o r p o i n t s near the top of the E a s t w a l l due to t h e i r i n i t i a l l y h i g h s u r f a c e t e m p e r a t u r e . There i s no i n d i c a t i o n of a l e v e l l i n g - o u t of the c o o l i n g r a t e s which might have suggested the approach of canyon temperatures to an e q u i l i b r i u m v a l u e . The r e v e r s a l of the temperature p r o f i l e on the Eas t w a l l i s seen to be completed by four hours a f t e r sunse t . 162 West Wall 1 2 3 4 5 6 7 Time From Sunset (h) East Wall 1 2 3 * 5 6 7 Time From Sunset (h) Canyon Floor 0 1 2 3 4 5 6 7 8 Time From Sunset (h) Figure 6.3 Cooling of selected points on canyon facets; H/W=2.0 canyon, August 1/2. (a) West wall, (b) East wall, (c) Floor. 163 28 S 24 CL E £ 22 West Wall PI 1 PI s P» 10 \ ^ » ^ ^ - J 1 1 • —1_J 1 _1 .1 2 3 4 5 6 7 Time From Sunset (h) East Wall 1 2 3 4 5 6 7 Time From Sunset (h) Canyon Floor 2 3 4 5 6 7 Time From Sunset (h) F i g u r e 6.4 C o o l i n g of s e l e c t e d p o i n t s on canyon f a c e t s : canyon H/W=1.0, August 3 /4 . (a) West w a l l , (b) E a s t w a l l , (c) F l o o r . 164 West Wall 2 2 .. (J 21 o © w 3 2 0 O e> 19 o. E V 1— 18 €> o o 17 u 3 to 16 1 2 3 4 5 6 7 8 Time From Sunset (h) East Wall . 2 3 o 2 2 3 21 O a> a. 2 0 E 19 o 18 o o 17 i_ 3 IO 16 \ % *\ \ « i H ! 1 1 I 1 * s _ V 1 1 1 1 2 3 4 5 6 7 8 9 Time From Sunset (h) Canyon Floor 1 2 3 4 5 6 7 8 Time From Sunset (h) F i g u r e 6.5 C o o l i n g of s e l e c t e d p o i n t s on canyon f a c e t s ; canyon H/W=0.41, August 2 2 / 2 3 . (a) West w a l l , (b) E a s t w a l l , (c) F l o o r . 165 For l a r g e r H/W, the m i d - p o i n t of the f l o o r i s at a l l t imes c o o l e r than p o i n t s towards the w a l l s . C o o l i n g r a t e s on the canyon f l o o r are more equa l w i t h i n the H/W=2.0 canyon than i n the more open canyons . 6.3 AIR TEMPERATURES 6.3.1 Average Canyon A i r Temperatures Average canyon a i r t e m p e r a t u r e s , c a l c u l a t e d over the p e r i o d of one complete t r a v e r s e are compared wi th the r e c o r d e d a i r t emperatures at Vancouver I n t e r n a t i o n a l A i r p o r t ( F i g u r e s 6 .6 , 6 . 7 ) . The canyon a i r temperatures are c o n s i s t e n t l y warmer than those r e c o r d e d at the a i r p o r t , except f o r b r i e f p e r i o d s on Aug . 2/3 and 11/12 ( F i g u r e 6 . 7 ) . I n g e n e r a l , the canyon a i r temperature f o l l o w s that observed at the a i r p o r t . D i f f e r e n c e s between the two f l u c t u a t e w i th t ime; .of course the once h o u r l y measurements from the a i r p o r t do not r e s o l v e the s m a l l e r temporal f l u c t u a t i o n s e x h i b i t e d by the canyon t e m p e r a t u r e . Some of these f l u c t u a t i o n s may a r i s e from t r a v e r s e s which were s topped for a p e r i o d b e f o r e be ing comple ted . C o n s i d e r i n g the e f f e c t s of wind d i r e c t i o n i t seems t h a t for winds i n the a long-canyon d i r e c t i o n the d i f f e r e n c e between the canyon and a i r p o r t temperatures i s r e d u c e d . The two H/W=2.0 canyon cases i l l u s t r a t e t h i s t r e n d . For most c a s e s , a l a r g e a i r temperature d i f f e r e n c e o c c u r s s h o r t l y a f t e r sunset (but t h i s i s o f t e n r e d u c e d , p a r t i c u l a r l y when the wind s h i f t s from w e s t e r l y 166 T 1 * — t * — i ^ p 0 2 4 6 8 10 Time From Sunset M 1*—I 1 1 1 1 1 1 1 r 0 2 <S 6 8 10 Time From Sunset (h) -1 1 1 1 1 1 1 1 i 1 r 0 2 A 6 8 10 Time From Sunset Ch) • Canyon Air Temperature A Airport Air Temperature * Wind Direction F i g u r e 6.6 Average canyon a i r t emperature and a i r t e m p e r a t u r e s r e c o r d e d a t Vancouver I n t e r n a t i o n a l A i r p o r t f o r (a) A u g . 1/2, (b) A u g . 3 / 4 , (c) Aug . 2 2 / 2 3 . 167 0 2 A 6 8 10 Time f r o m Sunset (h) Time F r o m Sunset (h) F i g u r e 6.7 Average canyon a i r t empera ture and a i r t emperatures r e c o r d e d a t Vancouver I n t e r n a t i o n a l A i r p o r t f o r (a) Auq 2/3 (b) A u g . 10/11 , (c) Aug . 11/12. Symbols as per F i g u r e 6 6 168 at sunset to a more n o r t h e r l y d i r e c t i o n ) . The l a r g e s t temperature d i f f e r e n c e observed i s over 3 °C in the H/W=2.0 canyon on August 2/3 ( F i g . 6 . 7 a ) . When the wind was e a s t e r l y , and the canyon geometry p r o v i d e d maximum s h e l t e r . For most of the 2.0 data the wind i s n o r t h e r l y , and the winds tend to be c h a n n e l l e d through the canyon so the s h e l t e r i n g e f f e c t of the l a r g e H/W i s l o s t . Other l a r g e c a n y o n - a i r p o r t temperature d i f f e r e n c e s are recorded in the H/W=1.0 canyon on Aug . 3/4 ( F i g u r e 6.6b) and the H/W=0.41 canyon on Aug . 22/23 ( F i g u r e 6. 6c ) . The c o n t r o l of geometry upon c o o l i n g can t h e r e f o r e have two types of e f f e c t s upon the a i r t emperature : l a r g e r aspec t r a t i o s reduce s u r f a c e c o o l i n g and , depending upon wind d i r e c t i o n , may p r o v i d e g r e a t e r s h e l t e r . 6 .3 .2 S p a t i a l D i s t r i b u t i o n of Canyon A i r Temperatures Canyon a i r temperatures at p o i n t s 40-50mm above the s u r f a c e of the canyon f a c e t s are p r e s e n t e d for the same canyons as the s u r f a c e temperature d i s t r i b u t i o n s ( F i g u r e 6 . 8 ) . The temperatures have been c o r r e c t e d for c o o l i n g or warming which o c c u r r e d d u r i n g the t r a v e r s e c i r c u t . R e c a l l a l s o that the t r a v e r s e cannot i n c l u d e p o i n t s in the c o r n e r s of the canyon and there may be some b i a s due to incomplete t r a v e r s e s of g r i d - p o i n t s near the ends of each canyon f a c e t . A i r temperatures for p o i n t s a c r o s s the canyon top are i n c l u d e d . 169 Focot Point: Top 1 2 3 4 5 I ,1 ' ' Air Ttmptrott** Focot Point: Floor Air Tomperotu-4 Facet Point: Top 1 2 3 4 5 6 7 6 3 10 I 1 1 1 I i ' ' i I ' . ' ' „ 1 ' . 1 —H i i i i i—i—i—r—i i—i—i—i—r— . ' .5 19 9 22.3 I 2 3 4 5 6 7 8 9 10 24.0 21.6 19.2 *• TimptrottM Focot Point: rioor Air Tomorqluro Foot Point: Top O Sunset A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h « Sunset + 8 h F i g u r e 6.8 S p a t i a l and t empora l v a r i a t i o n s of a i r temperature above canyon f a c e t s in ( top) a canyon w i t h H/W=2.0, (middle) H/W=1.0 canyon , ' and (bottom) 0.41 c a n y o n . 170 A i r temperatures in each of the canyons are c o n s i s t e n t l y lower than the c o r r e s p o n d i n g s u r f a c e t e m p e r a t u r e s . In g e n e r a l , s p a t i a l v a r i a t i o n s are minor over most f a c e t s . The H/W=2.0 and 1.0 canyons show a s m a l l i n c r e a s e of a i r temperature towards the base of the w a l l s . The l a r g e i n c r e a s e of s u r f a c e temperature noted for the top of the E a s t w a l l near sunset i n a l l canyons i s shown on ly as a s m a l l i n c r e a s e of a i r temperature on the E a s t s i d e of the canyon t o p . In each case a i r temperatures over the E a s t w a l l are s l i g h t l y warmer at a l l t imes than over the West w a l l . The on ly n o t a b l e s p a t i a l d i s t r i b u t i o n s a c r o s s the canyon f l o o r i s an i n c r e a s e towards both w a l l s for the sunset+8 h l i n e in the 0.41 H/W canyon and p o s s i b l y i n the H/W=2.0 canyon . Temperatures over the f l o o r are warmer than those at the canyon top wi th s m a l l d i f f e r e n c e s in the H/W=0.41 canyon ( 0 . 2 - 0 . 6 ° C ) and l a r g e r d i f f e r e n c e s i n the 1.0 and 2.0 canyons (0.2 - 1 . 9 ° C ) . The d i f f e r e n c e s remain r e l a t i v e l y c o n s t a n t w i t h time in the H/W=2.0 and 0.41 canyon, however the d i f f e r e n c e s are much s m a l l e r for t imes l a t e r at n i g h t in the H/W=1.0 canyon . T h i s may be a s s o c i a t e d w i t h an i n c r e a s e d wind speed observed i n the e a r l y morning of Aug . 4. A i r t emperatures a c r o s s the canyon top are s l i g h t l y warmer near the E a s t w a l l i n each of the canyons for most of the p l o t t e d t i m e s , which agrees wi th the o b s e r v a t i o n of a warmer E a s t w a l l . 171 6.4 Long-Wave R a d i a t i o n F i g u r e s 6 .9-6.11 presen t the f l u x e s of L * , L^, , and L 0 m o d e l l e d for p o i n t s around the same H/W=2.0, 1.0 and 0.41 canyon c r o s s - s e c t i o n s used to present a i r and s u r f a c e t e m p e r a t u r e s . The mode l l ed va lues use the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n . Use of mode l l ed v a l u e s a l l o w s a l l p o i n t s around the c r o s s - s e c t i o n to be shown at a g i v e n t i m e . A separate c r o s s - s e c t i o n i s used to present each f l u x . L ^ a c r o s s the canyon top i s not p r e s e n t e d as i t i s cons tant for a l l p o i n t s at each t i m e . I n c i d e n t long-wave on the canyon f l o o r e x h i b i t s s i m i l a r s p a t i a l v a r i a t i o n s in a l l three canyons wi th h i g h e r v a l u e s near the base of the w a l l s , e s p e c i a l l y l a t e r at n i g h t . E a r l y i n the even ing there i s a s m a l l tendency f o r L^ to be l a r g e r for p o i n t s on the f l o o r c l o s e to the West w a l l because these have the g r e a t e s t v i e w - f a c t o r for the top of the warm E a s t w a l l . L a r g e r v a r i a t i o n s a c r o s s the f l o o r occur wi th more open geometr i e s ; the H/W=2.0 canyon shows l i t t l e v a r i a t i o n due to the s m a l l i / / s and the almost i s o t h e r m a l lower p o r t i o n s of the w a l l s . The canyon w a l l s for the H/W=1.0 and 2.0 cases d i s p l a y an i n c r e a s e of L^ towards the base of the w a l l s ; the i n c r e a s e i s c l o s e to l i n e a r f o r the H/W=1.0 canyon but i n c r e a s e s more r a p i d l y near the top of the H/W=2.0 canyon . The shape of the p r o f i l e s of L^ on the canyon w a l l s i s m a i n t a i n e d throughout the n i g h t i n each of the cases p r e s e n t e d . The H/W=0.41 canyon d i s p l a y s an i r r e g u l a r p a t t e r n of L^ on the w a l l s w i t h a decrease noted for the m i d - p o r t i o n of each 172 August 1/2 1988 H / W 2.0 F i g u r e 6.9 S p a t i a l and t e m p o r a l v a r i a t i o n s of m o d e l l e d r a d i a t i v e f l u x e s i n an H/W=2.0 c a n y o n . Top - Li. m i d d l e - bot tom - L * . 1 173 August 3/4 1988 H / W 1.0 Facet Point: Top Facet Point: Top O Sunset A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h «> Sunset + 8 h F i g u r e 6.10 S p a t i a l and t e m p o r a l v a r i a t i o n s of m o d e l l e d r a d i a t i v e f l u x e s i n an H/W=1.0 c a n y o n . Top - L ; , m i d d l e - L 0 , bot tom - L * . 174 August 22/23 1988 H/W 0.41 Focet Point: top Facet Point: top O Sunset A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h O Sunset + 8 h F i g u r e 6.11 S p a t i a l and t e m p o r a l v a r i a t i o n s of mode l l ed r a d i a t i v e f l u x e s i n an H/W=0.41 c a n y o n . Top - L ^ , m i d d l e bottom - L * . 175 w a l l . T h i s f e a t u r e i s present for each of the t imes p l o t t e d and may be a r e s u l t of n u m e r i c a l a p p r o x i m a t i o n s i n the model r o u t i n e s , as the L j d e r i v e d from the measurements of L * and L D made by the t r a v e r s e d r a d i o m e t e r s e x h i b i t s smoother changes a c r o s s the w a l l s i n a manner s i m i l a r to the o ther canyons . The p l o t s of L Q for the canyon w a l l s and f l o o r are i d e n t i c a l i n shape to those d e s c r i b e d f o r the canyon s u r f a c e t empera ture . L 0 on the canyon top shows an i n c r e a s e towards the East canyon w a l l a t , and s h o r t l y a f t e r sunset i n a l l three cases because of the very warm s u r f a c e temperatures t h e r e . L a t e r at n i g h t , L 0 i s s l i g h t l y h i g h e r towards the middle of the canyon because the tops of the w a l l s are the c o o l e s t p o r t i o n of the canyon . The net r a d i a t i o n a c r o s s the canyon f l o o r shows h i g h e r v a l u e s towards the w a l l s i n the H/W=1.0 and 0.41 canyons wi th the g r e a t e s t d i f f e r e n c e s in the e a r l y e v e n i n g . The H/W=2.0 canyon i s the r e v e r s e , w i t h lower v a l u e s at a l l t imes near the canyon w a l l s . T h i s i n d i c a t e s g r e a t e r r a d i a t i v e c o o l i n g near the canyon w a l l s . However, the s u r f a c e temperature d i s t r i b u t i o n s and p r o f i l e s of L 0 show an i n c r e a s e towards the w a l l s which becomes s t r o n g e r w i t h time which would support reduced c o o l i n g at these p o i n t s . Measured data are not a v a i l a b l e for comparison w i t h these p o i n t s to c o n f i r m t h i s f e a t u r e . One p o s s i b i l i t y i s that the s u r f a c e temperature i s anomolously warm on the f l o o r near the w a l l s due to c o n d u c t i o n from o u t s i d e the canyon . T h i s might i n c r e a s e L Q for the p o i n t and r e s u l t in a net decrease i n L * . A second p o s s i b i l i t y l i e s in n u m e r i c a l e r r o r s w i t h i n the model 176 w h i c h a r e g r e a t e r f o r c o r n e r g r i d - p o i n t s ( A r n f i e l d , p e r s . comm. 1989). The i r r e g u l a r i t i e s n o t e d f o r L^ i n t h e 0.41 c a n y o n a f f e c t t h e p r o f i l e s o f L* on t h e w a l l s . On t h e cany o n t o p L* i s r e d u c e d f o r p o i n t s n e a r t h e E a s t w a l l i n t h e e a r l y e v e n i n g and c l o s e t o b o t h w a l l s l a t e r on. S i m i l a r r e s u l t s f o r t h e c a n y o n t o p a r e shown i n t h e H/W=1.0 and 2.0 c a n y o n s . Of t h e t h r e e c a n y o n s , L* i s s m a l l e s t f o r t h e H/W=1.0 c a n y o n i n t h e f i r s t two h o u r s a f t e r s u n s e t . L a t e r , t h e l o w e s t v a l u e s a r e r e c o r d e d i n t h e H/W=2.0 c a n y o n and L* i n c r e a s e s as H/W d e c r e a s e s . T h i s i s an i n d i c a t o r o f t h e r e d u c e d c o o l i n g w h i c h o c c u r s i n c a n y o n s w i t h g r e a t e r H/W e a r l y i n t h e e v e n i n g . 6.5 SUMMARY OF RESULTS: TEMPORAL AND SPATIAL VARIATION OF IN-CANYON TEMPERATURES AND RADIATION The p r e v i o u s s e c t i o n s have o u t l i n e d t e m p o r a l and s p a t i a l v a r i a t i o n s o f t h e i n - c a n y o n q u a n t i t i e s of long-wave r a d i a t i o n , s u r f a c e and a i r t e m p e r a t u r e s . A.summary o f r e s u l t s i s a s f o l l o w s : 1. D u r i n g t h e day, s u r f a c e t e m p e r a t u r e s a r e s t r o n g l y a f f e c t e d by K^. T h i s r e s u l t s i n a s p a t i a l l y skewed t e m p e r a t u r e d i s t r i b u t i o n a c r o s s c a n y o n f a c e t s w h i c h a r e s u n l i t , s h a d e d o r p a r t i a l l y s h a d e d . The t i m e of maximum T s v a r i e s a c r o s s t h e f a c e t as K; r e a c h e s . a maximum f o r a p o i n t . 177 2 . Maximum s u r f a c e t e m p e r a t u r e s on t h e t o p p o r t i o n of t h e E a s t w a l l a r e o b s e r v e d t o be h i g h e r t h a n t h a t of t h e open s i t e . Maximum T s v a l u e s i n c r e a s e f o r p o i n t s h i g h e r on t h e w a l l s and t o w a r d s t h e m i d d l e o f t h e c a n y o n b e c a u s e of a l o n g e r e x p o s u r e t i m e f o r K^. 3. T e m p e r a t u r e on t h e West w a l l d i s p l a y s a d i s t i n c t s h o u l d e r f r o m t h e l a t e m o r n i n g t o e a r l y e v e n i n g due p r i m a r i l y t o r e f l e c t e d r a d i a t i o n from t h e o p p o s i t e w a l l w i t h a s m a l l component due t o c o n d u c t i o n o f h e a t t h r o u g h t h e w a l l . T h i s f e a t u r e i s not as w e l l d e f i n e d f o r t h e E a s t w a l l i n t h e m o r n i n g . 4. S p a t i a l d i s t r i b u t i o n s o f t e m p e r a t u r e a c r o s s c a n y o n f a c e t s a f t e r s u n s e t a r e c o n d i t i o n e d by t h e i r i n i t i a l ( s u n s e t ) t e m p e r a t u r e and t h e i r p o s i t i o n i n t h e c a n y o n . The r o l e o f \£s i n s h a p i n g t h e t e m p e r a t u r e p r o f i l e s r e l a t e d d e v e l o p s t h r o u g h t h e n i g h t as d i f f e r e n t i a l c o o l i n g t a k e s p l a c e . 5. V e r y warm s u r f a c e t e m p e r a t u r e s n e a r t h e t o p of t h e E a s t w a l l a r e o b s e r v e d w e l l i n t o t h e n i g h t . The p o s i t i o n o f maximum T s on t h e E a s t w a l l a t s u n s e t a p p e a r s t o be low e r on t h e c a n y o n w a l l as t h e H/W d e c r e a s e s . 6. C o o l i n g r a t e s a r e i n i t i a l l y h i g h , p a r t i c u l a r l y f o r p o i n t s w i t h h i g h s u r f a c e t e m p e r a t u r e s a t s u n s e t . The r a t e s t e n d t o become more e q u a l w i t h t i m e . C o o l i n g i s n e a r l y l i n e a r f o r p o i n t s 178 deep w i t h i n t h e c a n y o n . T h e r e i s no r e d u c t i o n o f t h e c o o l i n g r a t e t o i n d i c a t e t h e a p p r o a c h o f an e q u i l i b r i u m t e m p e r a t u r e . 7. A i r t e m p e r a t u r e s a f t e r s u n s e t a r e a l w a y s c o o l e r t h a n s u r f a c e t e m p e r a t u r e s . O n l y m i n o r s p a t i a l v a r i a t i o n s a c r o s s i n d i v i d u a l f a c e t s a r e n o t e d a l t h o u g h t h i s may be due i n p a r t t o t h e t r a v e r s e l e n g t h s w h i c h do n o t i n c l u d e t e m p e r a t u r e s f o r p o i n t s i n t h e c a n y o n c o r n e r s , where more change may be p r e s e n t . 8. A i r t e m p e r a t u r e s a r e warmer o v e r t h e E a s t t h a n t h e West w a l l and warmer o v e r t h e f l o o r t h a n t h e t o p . S m a l l e r v e r t i c a l d i f f e r e n c e s a r e o b s e r v e d i n t h e more open c a n y o n s . No c o n s i s t e n t v a r i a t i o n s w i t h t i m e were o b s e r v e d ; wind s p e e d and d i r e c t i o n may r e d u c e o r i n c r e a s e d i f f e r e n c e s . 9. Wind d i r e c t i o n i s shown t o be an i m p o r t a n t c o n t r o l l i n g f a c t o r i n t h e d e v e l o p m e n t of c a n y o n "heat i s l a n d s ' , and i s c a p a b l e of o v e r c o m i n g geometry i n f l u e n c e s . 10. Long-wave r a d i a t i o n a t p o i n t s w i t h i n a c a n y o n i s d i r e c t l y r e l a t e d t o t h e T s and \[/s o f t h e p o i n t measured. 11. Net long-wave a t t h e c a n y o n t o p i s t h e l o w e s t i n t h e 2.0 c a n y o n l a t e a t n i g h t , b e c a u s e of t h e r e d u c e d r a t e o f c o o l i n g t h r o u g h t h e n i g h t . L* d e c r e a s e s t o w a r d s t h e canyon w a l l s i n t h e H/W=2.0 c a n y o n , w h i c h i s not seen e l s e w h e r e . 179 6.6 CANYON VERSUS OPEN SITE SURFACE COOLING C o n t r o l of s u r f a c e c o o l i n g by s u r f a c e geometry has been c i t e d p r e v i o u s l y (Oke, 1981) as a p o s s i b l e cause of the s u r f a c e urban heat i s l a n d . R e s u l t s are p r e s e n t e d here comparing the c o o l i n g of the open s i t e w i th that of the canyon . Canyon c o o l i n g i s r e p r e s e n t e d by a p o i n t or p o i n t s a t canyon m i d - w i d t h on the f l o o r . Use of the canyon f l o o r m i d - p o i n t f a c i l i t a t e s comparison wi th o ther p u b l i s h e d r e s u l t s ; o ther measures of c o o l i n g of the canyon c o u l d a l s o be made, f or example a i r temperature or n e t - r a d i a t i o n . Examples from each of the H/W r a t i o s t e s t e d are p r e s e n t e d i n S e c t i o n 6 . 6 . 1 . The b a s i s for s e l e c t i o n was based upon homogeneity of c o n d i t i o n s through the n i g h t and the achievement of the d e s i r e d s i m u l a t i o n c o n d i t i o n s (see C h a p t e r s 1 and 2 ) . The p r e s e n t a t i o n format i n c l u d e s a p l o t of the decrease of s u r f a c e temperature w i t h time of the open s i t e and the canyon . For H/W r a t i o s of 2.0 and 1.33 the m i d - p o i n t temperature on the canyon f l o o r i s used . For o ther canyons the average of the two p o i n t s c l o s e s t to the middle are used . The c o o l i n g curves are o v e r l a i n by a p l o t of the open s i t e net long-wave r a d i a t i o n , L * 0 . The time s e r i e s of L j c t and wind speed (u) are p l o t t e d above the c o o l i n g d iagram, u s i n g the same time s c a l e to permit easy - in ter - compar i son of p l o t s . T a b l e 6.1 p r e s e n t s average m e t e o r o l o g i c a l s t a t i s t i c s r e c o r d e d at Vancouver I n t e r n a t i o n a l A i r p o r t f or the p e r i o d 0600 - 1800 (PDT) for each day shown. 180 T a b l e 6.1 D a i l y Average M e t e o r o l o g i c a l C o n d i t i o n s 0600-1800 (PDT) Augus t , 1988. Date T x a max T d u C a 1 C O 2 T s T s max Q* O* ^ max ( ° C ) (°c) ( ° C ) (km/h) (/10) ( /10) ( ° C ) ( ° C ) (W m" 2) 1 18.3 21 .6 11.7 8.7 2.2 2.2 31 .8 41.8 239 420 3 21.6 24.4 12.6 4.0 0.8 0.2 37.5 45.2 284 410 3 8 17.8 20.9 11.4 7.5 9.8 9.4 24.3 29. 1 78 227 10 18.4 20.5 13.6 12.1 5.5 5.1 11 18.5 21 .2 13.8 9.6 3.8 1 .8 30.5 40. 1 228 415 12 17.8 21.0 12.0 6.6 0.5 0.5 30.7 40.4 230 405 14 17.6 20.8 12.3 6.8 8.9 6.9 20 16.8 19.5 10.5 10.2 7.3 6.4 22 18.7 21 .9 13.7 10.4 0.0 0.0 1. Ca - C l o u d amount 2. Co - C l o u d o p a c i t y 3. A v e r a g i n g p e r i o d 0710-1800 for T s and Q* 181 Where a v a i l a b l e , net r a d i a t i o n and s u r f a c e temperature data from the open s i t e are a l s o i n c l u d e d . These, data are of use when d e f i n i n g the c o n d i t i o n s d u r i n g the p e r i o d of canyon h e a t i n g and f o r comparing the s i m i l a r i t y of c o n d i t i o n s between two d a y s . 6 .6 .1 Canyon and Open S u r f a c e C o o l i n g : S u r f a c e Geometry C o n t r o l s F i g u r e s 6 .12-6 .17 present r e s u l t s for each of the H/W t e s t e d , a long w i t h an a d d i t i o n a l data set f o r the H/W=1.0 canyon comparing the v a r i a b i l i t y of c o o l i n g under l e s s than i d e a l c o n d i t i o n s . P l o t s are p r e s e n t e d in order of d e c r e a s i n g canyon H/W. A reduced r a t e of c o o l i n g w i t h l a r g e r canyon H/W i s immediate ly a p p a r e n t . The d i f f e r e n c e between canyon and open s u r f a c e temperatures i n c r e a s e s w i th time from sunse t , w i th the g r e a t e s t r a t e of change near sunset i n a l l c a s e s . The d i f f e r e n c e of the s lope of the two c u r v e s i s reduced wi th time and w i t h more open geometr ies ( F i g u r e 6 . 1 7 ) . The t o t a l temperature decrease of the open s i t e at 8 hours from sunset i s a p p r o x i m a t e l y 8-9 °C for the data c o l l e c t e d under x i d e a l ' c o n d i t i o n s . Of the H/W r a t i o s t e s t e d , the most p o o r l y r e p r e s e n t e d in terms of x i d e a l ' m e t e o r o l o g i c a l c o n d i t i o n s i s the 1.33 H/W r a t i o ( F i g u r e 6.13) and the 0.41 canyon ( F i g u r e 6 . 1 7 ) , which a l t h o u g h c l e a r and calm at n i g h t , were preceded by l e s s than i d e a l c o n d i t i o n s in the day (see T a b l e 6 . 1 ) . 182 - -I 1 1 1 1 1 1 1 1 r~ 0 1 2 3 4 5 6 7 8 9 Time From Sunset (h) F i g u r e 6.12 Bot tom: Compar i son of canyon ( s o l i d l i n e ) and open s i t e ( a l t e r n a t i n g l o n g and s h o r t dashed l i n e ) c o o l i n g o v e r l a i d w i t h L*p ( s h o r t dashed l i n e ) . L ^ c t . (top) and wind speed ( u ) , ( m i d d l e ; are a l s o p r e s e n t e d . Data from Aug. 1/2, H/W=2.0 . 183 F i g u r e 6.13 As per F i g u r e 6 . 1 2 . Data from Aug . 14/15, H/W-1.. 33. 184 F i g u r e 6.14 H/W=1.0. As per F i g u r e 6 .12 . Data from Aug . 3 / 4 , canyon 185 F i g u r e 6.15 As per F i g u r e 6 . 1 2 . Data from Aug . 8 / 9 , canyon H/W=1.0. 186 £ — i 1 1 1 1 1 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 E.O 7.0 8.0 9.0 eg 1 £ 8H o 9_ tn o _ J o o „ m Time From Sunset (h) — i 1 1 1 1 1 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 BO 9.0 Time From Sunset (h) V) c to £ o V ) O « o « Q <D l_ "5 v. <D CL £ <D E Time From Sunset (h) F i g u r e 6.16 As per F i g u r e 6 . 1 2 . Data from Aug. 11/12, canyon H/W=0.67. 1 8 7 Time From Sunset (h) F i g u r e 6.17 As p e r F i g u r e 6.12. D a t a from Aug. 20/21, c a n y o n 188 In a l l c a s e s , the c o o l i n g r a t e s , both i n and o u t s i d e the canyon , at the end of the n i g h t are r e l a t i v e l y l a r g e . The c o n c r e t e base of the s i t e , and the u n d e r l y i n g m a t e r i a l , thought to be sand , appears to draw upon a l a r g e s t o r e of heat and i t ' s thermal a d m i t t a n c e , u, i s p r o b a b l y h i g h , (range 150 -2 ,370 J m~ 2 s ~ 1 / / 2 K - 1 w i th t y p i c a l v a l u e s of 1300 J m~ 2 s " 1 / 2 K ~ 1 Oke, 1981). The l i n e a r i t y of c o o l i n g i n s i d e the canyon agrees f a v o u r a b l y wi th both the s c a l e model and c i t y data p r e s e n t e d i n Oke (1.981 )' ( F i g u r e 6 . 1 8 ) . However, the r e l a t i v e l y l a r g e r a t e s of c o o l i n g measured at the open s i t e are i n g r e a t e r c o n t r a s t w i t h wooden x r u r a l ' model r e s u l t s . R e s u l t s from an open c o n c r e t e s l a b used by Oke (1981) show b e t t e r agreement, w i th a much more l i n e a r c o o l i n g than the wooden model ( F i g . 6.18) and a much reduced d i f f e r e n c e between the r a t e s of c o o l i n g measured a t the s t a r t of the c o o l i n g p e r i o d and those measured at the end. Of the example c i t y - r u r a l c o o l i n g r a t e s p r e s e n t e d , that of U p p s a l a shows the best agreement w i t h r e s p e c t to the r a t e of c o o l i n g l a t e at n i g h t . I n i t i a l c o o l i n g r a t e s for both canyon and open s i t e s appear to be l e s s than those of Oke (1981) . I t i s p o s s i b l e tha t the shadows c a s t by the nearby hangar a c r o s s the canyon p r i o r to the l o c a l time of sunset are of some importance to the low i n i t i a l r a t e s of c o o l i n g . T h i s may cause c o o l i n g to take p l a c e at a g r e a t e r r a t e than would n o r m a l l y be the case be fore l o c a l sunse t . The g r e a t e r r a t e s of c o o l i n g observed l a t e at n i g h t are p r o b a b l y a f u n c t i o n of the thermal p r o p e r t i e s of the c o n c r e t e canyon f l o o r . The wooden models of Oke (1981) tend to 'use up' t h e i r f i n i t e heat s t o r e and approach 1 8 9 12 - 14 - O Montreal •> — -O Vancouw ' «v o Upptalt 10 12 TIME (h) Figure 6.18 Cooling of urban and r u r a l surfaces observed from (a) scale models, (b) the c i t i e s of Montreal (H/W=3.29), Vancouver (H/W=1.50) and Uppsala (H/W=0.76). From Oke (1981). F i e l d observations made by Oke and Maxwell (1975) (Montreal and Vancouver) and Hogstrom et a l . (1978) (Uppsala). 190 Figure 6.19 Temporal changes of heat islan d i n t e n s i t y generated using (a) a scale model and (b) observed i n t e n s i t i e s from Montreal, Vancouver and Uppsala. From Oke (1981) . 191 an e q u i l i b r i u m t e m p e r a t u r e . T h i s e f f e c t i s reduced i n the c i t y - r u r a l examples p r e s e n t e d and f u r t h e r wi th the c o n c r e t e m a t e r i a l used h e r e , i n d i c a t i n g the thermal admit tance of the c o n c r e t e s u r f a c e used i n t h i s study may exceed tha t of both the wooden models and urban areas p r e s e n t e d i n Oke (1981) . Attempts to model the c o o l i n g u s i n g the r a d i a t i v e c o o l i n g model of Groen (1947) and a s imple s u r f a c e energy ba lance model (Lyons , Oke, and S t e y n , p e r s . comm.) model based upon the f o r c e - r e s t o r e method proposed by B l a c k a d a r (1976), Bhumralkar (1975) and D e a r d o r f f (1978) were l a r g e l y u n s u c c e s s f u l . Imprec i se knowledge of thermal p r o p e r t i e s of the m a t e r i a l and of boundary c o n d i t i o n s necessary for the energy ba lance model reduce the e f f e c t i v e n e s s of model led r e s u l t s . The Groen f o r m u l a t i o n a l s o r e q u i r e s tha t both L ^ c t and the r a t e of decrease of L * wi th temperature change remain c o n s t a n t through the n i g h t . N e i t h e r of which were observed to be the c a s e . The Groen approach may a l s o r e q u i r e an i n i t i a l t ime o ther than sunset for the c a l c u l a t i o n of some p a r a m e t e r s , Kondo and Haginoya (1989) suggest t=0 i s a p p r o x i m a t e l y 30 min p r i o r to s u n s e t . A g a i n , the shadows c a s t by the hangar over the s i t e may change t h i s t i m e . The composi te p l o t ( F i g u r e 6.20) of the temperature d i f f e r e n c e between the canyon and open s i t e shows the d i s t i n c t e f f e c t s of the v a r i o u s canyon geometr ies l a t e r at n i g h t . V a r i a t i o n s i n the m e t e o r o l o g i c a l c o n d i t i o n s on some days r e s u l t in the merging of the three i n t e r m e d i a t e H/W g e o m e t r i e s , e a r l y i n the c o o l i n g p e r i o d . Comparison of the temperature d i f f e r e n c e s measured from-the model canyon w i t h those for s c a l e and r e a l 192 Time From Sunset (h) F i g u r e 6.20 T e m p o r a l d e v e l o p m e n t o f t h e t e m p e r a t u r e d i f f e r e n c e between t h e m i d - p o i n t o f t h e c a n y o n f l o o r and t h e open c o n c r e t e T e m p e r a t u r e d i f f e r e n c e s have been s e t t o z e r o a t s u n s e t D a t a a r e f r o m d a y s w i t h m o s t l y c l e a r c o n d i t i o n s ; H/W=2.0 Aug. 1/2 1.33 Aug. 14/15, 1.0 Aug. 3/4, 0.67 Aug. 11/12, 0.41 Aug. 20/21. 193 canyons in Oke (1981) ( F i g u r e 6.19) do not show the c h a r a c t e r i s t i c e a r l y i n c r e a s e to a maximum and then a l i n e a r decrease of temperature d i f f e r e n c e w i t h t ime . The d i f f e r e n c e s in F i g u r e 6.20 i n c r e a s e t h r o u g h o u t , b u t . a t a d e c r e a s i n g r a t e w i th t i m e . In the data here the r a t e of temperature decrease in the urban canyon never exceeds that of the open s i t e . T h e r e f o r e , the o b s e r v a t i o n tha t the t ime of maximum i n t e n s i t y o c c u r s l a t e r as the H/W i n c r e a s e s (Oke, 1981) cannot be c o n f i r m e d h e r e , but the r a t e of i n c r e a s e in the temperature d i f f e r e n c e s f o r a l l t imes i n c r e a s e s w i th H/W, i n agreement w i t h those of Oke (1981) . I t s h o u l d be noted that the heat i s l a n d i n t e n s i t i e s measured in the t h r e e c i t i e s p r e s e n t e d in F i g u r e 6.19 are d e r i v e d from a i r temperatures r a t h e r than s u r f a c e temperatures as used in Oke (1981) and the presen t work. The o v e r a l l e f f e c t of geometry upon the development of a canyon heat i s l a n d i s a l s o shown i n F i g u r e 6.21 where the maximum temperature d i f f e r e n c e i s p l o t t e d v e r s u s \j/s f or the m i d - p o i n t of the canyon f l o o r . The data are drawn from days which were most ly c l e a r wi th l i g h t winds . O v e r a l l , a g e n e r a l l i n e a r t r e n d i s noted f o r the H/W t e s t e d , s u p p o r t i n g the e a r l i e r r e s u l t s of Oke (1981) . The r e g r e s s i o n r e l a t i o n o b t a i n e d i s Tmax = 6 » 6 0 " 7 « 3 0 ^s ( 6 « 1 ) w i t h an r 2 of 0.94 and a s tandard e r r o r of T m = v of 0 . 3 6 ° C . 194 C a n y o n H / W 2 .0 1.33 1.0 0 . 6 7 0.41 i • o 5 ATmax = 6 . 6 0 - 7 . 3 0 ^ - O p e n  4 O 2 r = 0 . 9 4 ly o n  - 3 • 8 \ o \ D CJ X n 2 o \ . E * I— <i 1 i i 0.3 0.4 0 .5 0 .6 0.7 0.8 S k y V i e w - F a c t o r F i g u r e 6.21 S u r f a c e t e m p e r a t u r e d i f f e r e n c e s between t h e c a n y o n a n d open s i t e s a t e i g h t h o u r s a f t e r s u n s e t f o r t h e f i v e H/W t e s t e d . 195 6 .6 .2 Canyon and Open Surface C o o l i n g : Atmospher ic C o n t r o l s The c o o l i n g of a s u r f a c e i s s t r o n g l y c o n t r o l l e d by the wind speed and r a d i a t i o n , the former a i d i n g in the c o n v e c t i o n of heat from s u r f a c e s as t u r b u l e n t s e n s i b l e heat and the l a t t e r governs the r a t e of r a d i a t i v e heat t r a n s f e r . The e f f e c t s of L ^ c t and u upon the s u r f a c e c o o l i n g i s e v i d e n t i n s e v e r a l of the p l o t s . An i n c r e a s e in u a p p r o x i m a t e l y 2 hours a f t e r sunset on Aug. 1/2 ( F i g u r e 6.12) i s c o r r e l a t e d w i t h an i n c r e a s e d r a t e of c o o l i n g of the open s u r f a c e . A drop in windspeed on the even ing o f , A u g . 11/12 appears to be r e l a t e d to a reduced r a t e of s u r f a c e c o o l i n g at t h i s t ime a l t h o u g h the e f f e c t s are not l a r g e . The d i r e c t i o n of i n f l u e n c e of changes in u does not f o l l o w t h a t commonly observed for r u r a l areas at n i g h t i n which an i n c r e a s e i n u i s a s s o c i a t e d wi th enhanced t r a n s f e r of s e n s i b l e heat towards the s u r f a c e and a r e d u c t i o n of the c o o l i n g r a t e . The c l a s s i c case i s dependent upon the f o r m a t i o n of an i n v e r s i o n l a y e r above the s u r f a c e , which has been observed not to occur w i t h i n urban canyons (Nakamura and Oke, 1988). The e f f e c t of wind under these c o n d i t i o n s would thus be to enhance the c o o l i n g of the s u r f a c e by c o n v e c t i v e t r a n s f e r of heat away from the s u r f a c e i n the form of t u r b u l e n t s e n s i b l e h e a t . P o s i t i v e f l u x e s of t u r b u l e n t s e n s i b l e heat have been observed over urban areas (Yap and Oke, 1974; C h i n g et a l . , 1983; Cleugh and Oke, 1986). The dependence of s u r f a c e c o o l i n g on L ^ c t i s marked. F l u c t u a t i o n s of L ^ c t a f f e c t i n g the s u r f a c e temperature are shown i n F i g u r e s 6.1,3, 6.14 and 6 .15 . The e f f e c t i n c r e a s e s w i th the ^ s 196 of the s u r f a c e , so t h a t maximum e f f e c t s are e x p e r i e n c e d by the open s i t e . The data from August 8/9 ( F i g u r e 6.15) i l l u s t r a t e the much reduced r a t e of c o o l i n g under c l o u d y c o n d i t i o n s , a l t h o u g h the daytime heat input to the canyon was a l s o reduced by c l o u d . D i f f e r e n c e s between the canyon and the open c o n c r e t e are a l s o reduced compared w i t h Aug . 3/4 ( F i g u r e 6 . 1 4 ) . -The even ing of August 14/15 ( F i g u r e 6.13) i n d i c a t e s warming of the open s i t e as L i c t i n c r e a s e s . T h i s o b s e r v a t i o n i s s u p r i s i n g but not u n i q u e . Temperatures w i t h i n the canyon were a l s o observed to i n c r e a s e at these t i m e s . The net long-wave r a d i a t i o n of the s u r f a c e remains n e g a t i v e so tha t r a d i a t i v e warming i s not i n d i c a t e d . E x p l a n a t i o n for the warming may be r e l a t e d to the method of mount ing . I f convergence of heat were to occur s l i g h t l y below the s u r f a c e due to a reduced r a t e of heat l o s s from the s u r f a c e wh i l e heat c o n t i n u e s to be conducted upwards through the c o n c r e t e f o l l o w i n g the e s t a b l i s h e d s u b s u r f a c e temperature g r a d i e n t (the thermocouple j u n c t i o n s were s l i g h t l y b u r i e d ) a r i s e i n temperature might o c c u r . The p e r i o d of i n c r e a s e i s s h o r t - l i v e d a f t e r which temperatures a g a i n d e c r e a s e , s u g g e s t i n g the r e - e s t a b l i s h m e n t of a new s u b - s u r f a c e temperature g r a d i e n t . P l o t s of L ^ c t w i th time on very c l e a r n i g h t s ( F i g u r e s 6 .12 , 6 .14, 6 .16 , 6.17) i n d i c a t e a s l i g h t decrease of the order of 10 - 20 W m~ 2 over the n i g h t , the shape of the decrease v a r i e s from n e a r - l i n e a r ( F i g u r e s 6 . 1 6 , 1 7 ) , to v a r i a b l e ( F i g u r e 6.14) to g r e a t e r r a t e s of decrease w i t h t ime ( F i g u r e 6 . 1 2 ) . These o b s e r v a t i o n s , '• i f t y p i c a l of c l e a r n i g h t s , are of importance 197 b e c a u s e s e v e r a l m o d e ls of r a d i a t i v e c o o l i n g assume L ^ c t t o be c o n s t a n t . The r e d u c t i o n of L j c t i s e x p e c t e d b e c a u s e a i r t e m p e r a t u r e d e c r e a s e s i n t h e l o w e r a t m o s p h e r e and t h i s i s t h e l a y e r w h i c h c o n t r i b u t e s most t o t h e i n c o m i n g long-wave f l u x . Net long-wave o v e r t h e open s u r f a c e i s a l w a y s a t a minimum a t s u n s e t under c l e a r c o n d i t i o n s . I t t h e n i n c r e a s e s f o r t h e f i r s t p a r t o f t h e n i g h t a t c l o s e t o a l i n e a r r a t e a f t e r w h i c h t h e r a t e s l o w s w i t h t i m e . 6.7 SUMMARY OF RESULTS: CANYON AND OPEN SURFACE COOLING 1. I n c r e a s e d c a n y o n H/W r e d u c e s t h e r a t e of c o o l i n g o b s e r v e d a t t h e m i d - p o i n t o f t h e ca n y o n f l o o r under i d e a l r a d i a t i v e c o o l i n g c o n d i t i o n s . Under l e s s t h a n i d e a l c o n d i t i o n s , t h e d i f f e r e n c e s a r e much l e s s . 2. The g r e a t e s t d i f f e r e n c e i n c o o l i n g r a t e s between c a n y o n s i s o b s e r v e d n e a r s u n s e t . The r a t e s become more e q u a l w i t h t i m e and a r e a p p r o x i m a t e l y l i n e a r a t . t h e end of t h e n i g h t . 3. C o o l i n g o f t h e c a n y o n s and t h e open c o n c r e t e a t t h i s s i t e d i f f e r from r e s u l t s f o r o t h e r s c a l e models and c i t i e s , and w i t h t h a t p r e d i c t e d by m o d e l s . A number o f f a c t o r s may a c c o u n t f o r t h e s e d i f f e r e n c e s i n c l u d i n g t h e use of s u r f a c e as oppo s e d t o a i r t e m p e r a t u r e s , t h e t h e r m a l p r o p e r t i e s o f t h e ca n y o n f l o o r , 198 shadowing o f t h e s i t e p r i o r t o l o c a l s u n s e t , and d i f f i c u l t i e s i n o b t a i n i n g a c c u r a t e model i n p u t d a t a . 4. C o m p o s i t e p l o t s of t h e i n c r e a s e i n t e m p e r a t u r e d i f f e r e n c e between t h e canyon and open s i t e shows t h e c a n y o n 'heat i s l a n d ' grows most r a p i d l y f o l l o w i n g s u n s e t , but n e v e r s t o p s i n c r e a s i n g o v e r t h e measurement p e r i o d . The p l o t o f maximum t e m p e r a t u r e d i f f e r e n c e v e r s u s i//s a g r e e s w e l l w i t h e a r l i e r r e s u l t s , w i t h a g e n e r a l l y l i n e a r d e c r e a s e o v e r t h e r a n g e o f H/W t e s t e d . 5. I n c r e a s e d wind s p e e d a t n i g h t a p p e a r s t o enhance c o o l i n g f o r b o t h t h e open s i t e and c a n y o n s u g g e s t i n g t h e a b s e n c e of i n v e r s i o n c o n d i t i o n s o v e r t h e s i t e . 6. I n c r e a s e s of L ^ c t have a r a p i d and marked e f f e c t upon s u r f a c e t e m p e r a t u r e and c o o l i n g r a t e s , c a u s i n g i n c r e a s e s i n s u r f a c e t e m p e r a t u r e f o r p o i n t s w i t h l a r g e \ps i n s e v e r a l c a s e s . 7. L ^ c t d e c r e a s e s w i t h t i m e on v e r y c l e a r n i g h t s , however, t h e n a t u r e o f t h i s d e c r e a s e i s n o t c o n s i s t e n t . L* shows minimum v a l u e s n e a r s u n s e t a f t e r w h i c h i t i n c r e a s e s a t a r a t e w h i c h d e c r e a s e s w i t h t i m e . 199 CHAPTER 7. CONCLUSIONS 7.1 ACHIEVEMENT OF THE RESEARCH OBJECTIVES In Chapter 1, three main o b j e c t i v e s were o u t l i n e d : (1) the v a l i d a t i o n of the A r n f i e l d model (1976, 1982) f o r n o c t u r n a l long-wave r a d i a t i v e f l u x e s , (2) the i n v e s t i g a t i o n of model e r r o r s which r e s u l t from u s i n g l e s s than the f u l l model input data s e t , and (3) the s tudy of the e f f e c t s of urban s u r f a c e geometry upon the c o o l i n g of urban a r e a s . For d e t a i l s of the r e s u l t s , r e f e r to the i n d i v i d u a l c h a p t e r summaries. The f i r s t o b j e c t i v e has been met, s u b j e c t to the l i m i t a t i o n s of the methodology used to measure the independent long-wave f l u x e s and the no i se i n c u r r e d i n the s u r f a c e temperature measurements. Notab le f i n d i n g s i n c l u d e : the i n c r e a s e in model a c c u r a c y for low numbers of g r i d - p o i n t s when v i e w - f a c t o r s are r e - c a l c u l a t e d u s i n g the N u s s e l t Sphere method, a s l i g h t but c o n s i s t e n t i n c r e a s e i n a c c u r a c y of L^ and L * when the Unsworth and M o n t e i t h (1975) r a d i a n c e d i s t r i b u t i o n i s s p e c i f i e d , and very good r e s u l t s o v e r a l l when m o d e l l i n g f l u x e s at the canyon t o p , which form the lower boundary c o n d i t i o n s for l a r g e r s c a l e a tmospher ic p r o c e s s e s . Four a p p r o x i m a t i o n s to the f u l l model input f o r s u r f a c e temperature were made. These were based upon: (1) a i r temperature measurements made w i t h i n and (2) o u t s i d e the canyon, and on (3) average and (4) f a c e t m i d - p o i n t t e m p e r a t u r e s . 200 Temporal and s p a t i a l v a r i a t i o n s w i t h i n the canyon were p r e s e n t e d . I t was c o n c l u d e d tha t s u r f a c e t emperature -based e s t imates were s u p e r i o r to those based upon a i r t emperatures ; the g r e a t e s t d i f f e r e n c e s were o b t a i n e d when u s i n g unmodi f i ed s c r e e n - l e v e l a i r temperature e s t i m a t e s taken i n d e p e n d e n t l y at Vancouver I n t e r n a t i o n a l A i r p o r t . The use of such d a t a , commonly a v a i l a b l e in c l i m a t e a r c h i v e s , f or t h i s s c a l e of urban m o d e l l i n g i s not a d v o c a t e d . A s u b s t a n t i a l data base of canyon and open s i t e c o o l i n g u s i n g d i f f e r e n t canyon H/W v a l u e s and under v a r y i n g weather c o n d i t i o n s was g a t h e r e d . S p a t i a l and temporal v a r i a t i o n s of r a d i a t i v e f l u x e s and a i r and s u r f a c e temperatures were p r e s e n t e d . Sur face geometry has a c l e a r i n f l u e n c e under c o n d i t i o n s when r a d i a t i v e c o o l i n g dominates w i th the g r e a t e s t d i f f e r e n c e i n c o o l i n g r a t e s between canyon H/W observed near s u n s e t . When c o n d i t i o n s d e t e r i o r a t e , s u r f a c e geometry e f f e c t s are r e d u c e d . Comparison wi th p r e v i o u s r e s u l t s and models shows d i f f e r e n c e s which may be due to the use of s u r f a c e as opposed to a i r t e m p e r a t u r e s , thermal p r o p e r t i e s of the m a t e r i a l s used , and l o c a l s i t e f a c t o r s . 7.2 RECOMMENDATIONS AND FUTURE RESEARCH T h i s r e s e a r c h has v a l i d a t e d the urban canyon r a d i a t i o n model of A r n f i e l d (1976, 1982) to the s c a l e of i n d i v i d u a l g r i d - p o i n t s u s i n g a unique methodology. The success of the v a l i d a t i o n of 201 t h i s model a l s o suppor t s o ther models which are based upon s i m i l a r p r i n c i p l e s . The methodology employed i s recommended as an a l t e r n a t i v e to o b t a i n i n g measurements in r e a l c i t i e s , which are c o m p l i c a t e d by l o g i s t i c a l d i f f i c u l t i e s , in cases where such measurements can be made without undue s c a l i n g r e s t r i c t i o n s . S c a l i n g r e s t r i c t i o n s need to be f u r t h e r i n v e s t i g a t e d i n o r d e r for more d i r e c t comparisons to be made wi th f i e l d o b s e r v a t i o n s and for t h i s approach to be used to s tudy t u r b u l e n t f l u x e s . The v a l i d a t i o n of the f u l l energy ba lance w i t h i n urban canyons s h o u l d be made. The a c c u r a c y of some n u m e r i c a l methods of c a l c u l a t i n g v iew- f a c t o r s shou l d be examined. I f e r r o r s can be r e d u c e d , lower numbers of g r i d - p o i n t s can be used to ach i eve a g i v e n a c c u r a c y , w i th consequent r e d u c t i o n s i n c o m p u t a t i o n a l t ime r e q u i r e d f o r models . The development of a p p r o x i m a t i o n s to model input shou ld be made with the goa l of a c h i e v i n g r e a s o n a b l y a c c u r a t e model r e s u l t s w i th c u r r e n t l y a v a i l a b l e data s o u r c e s . I t has been shown that a i r p o r t a i r t e m p e r a t u r e s , which are o f t e n the data a v a i l a b l e i n c l i m a t e a r c h i v e s , d i d not r e p r e s e n t canyon s u r f a c e temperatures w e l l . C o r r e c t i o n of these temperatures based upon known heat i s l a n d i n t e n s i t i e s and the temporal and s p a t i a l v a r i a t i o n s of s u r f a c e temperature in urban canyons c o u l d be attempted i n model form, be fore p a s s i n g the r e s u l t s to models such as tha t of A r n f i e l d (1976, 1982). C o o l i n g models ( B r u n t , 1941; G r o e n , 1947; L y o n s , Oke and Steyn p e r s . comm.) d i d not agree w e l l wi th measured r e s u l t s f o r 202 both open and canyon s i t e s a l t h o u g h not a l l the model input data was a v a i l a b l e . F u r t h e r work on the a c c u r a c y and the assumptions of these models would be h e l p f u l . The model canyons undergo s i g n i f i c a n t c o o l i n g p r i o r to s u n s e t , f u r t h e r r e s e a r c h shou ld ensure measurements are a v a i l a b l e from t h i s p e r i o d of t i m e . The r e l a t i o n between canyon H/W, s u r f a c e t e m p e r a t u r e s , and p e n e t r a t i o n of i n t o the canyon to c o o l i n g c o u l d be f u r t h e r i n v e s t i g a t e d , perhaps under more c o n t r o l l e d c o n d i t i o n s to determine whether a p a r t i c u l a r H/W e x i s t s a f t e r which lower a b s o l u t e s u r f a c e temperatures tha t are due to i n c r e a s e d shad ing w i t h i n the canyon are not o f f s e t by reduced c o o l i n g a t n i g h t . 203 REFERENCES A i d a , M. 1982. 'Urban Albedo as a F u n c t i o n of the Urban S t r u c t u r e - A Model E x p e r i m e n t ' , Boundary-Layer Met. 23, 405-413. A i d a , M. and Gotoh , K. 1982. 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SPECIFICATION OF SENSOR TRAVERSING SPEED AND v DELAY INTERVAL A.1 INTRODUCTION The s e n s o r t r a v e r s i n g s p e e d and t h e d e l a y i n t e r v a l were d e t e r m i n e d from an a n a l y s i s o f s e n s o r r e s p o n s e t o a n t i c i p a t e d c h a n g e s i n t h e o u t g o i n g and n e t long-wave f l u x d e n s i t i e s a c r o s s c a n y o n f a c e t s . The a p p r o a c h i s b a s e d upon t h e r e s p o n s e o f a t e m p e r a t u r e s e n s o r t o c h a n g e s i n e n v i r o n m e n t a l t e m p e r a t u r e as d e s c r i b e d i n F r i t s c h e n and Gay ( 1 9 7 9 ) . A p r e r e q u i s i t e f o r t h e a n a l y s i s i s an e s t i m a t e of t h e t i m e c o n s t a n t f o r t h e m i n i a t u r e net r a d i o m e t e r s u s e d . No v a l u e i s a v a i l a b l e f o r t h e e x a c t model u s e d i n t h i s work, but M o n t e i t h (1972) l i s t s t h e r e s p o n s e t i m e of a p r e v i o u s S w i s s t e c o m i n i a t u r e n e t r a d i o m e t e r (Model S1) as 98 p e r c e n t e q u i l i b r a t e d t o a s t e p change i n 25 s. U s i n g t h e s e d a t a and t = (1 - e _ t / T ) ( A .1) where t i s t h e e l a p s e d t i m e and t h e r i g h t hand s i d e i s t h e a d j u s t m e n t c o m p l e t e d t o a s t e p c h a nge, a t i m e c o n s t a n t r , o f 6.3905 s was c a l c u l a t e d . The a p p r o a c h of F r i t s c h e n and Gay (1979) assumes a s t a t i o n a r y i n s t r u m e n t r e s p o n d i n g t o a t e m p o r a l change i n t h e e n v i r o n m e n t a l t e m p e r a t u r e . H e r e , t h e d e t e r m i n a t i o n of s e n s o r t r a v e r s i n g s p e e d assumes s p a t i a l c h a n g e s i n r a d i a t i o n t o be much g r e a t e r t h a n 212 temporal changes for any p a r t i c u l a r p o i n t ; i e . the change i n L Q or L * a c r o s s a canyon f a c e t i s much l a r g e r than any change induced by h e a t i n g or c o o l i n g i n the t ime i t takes to t r a v e r s e the f a c e t . The response of the ins trument i s t h e r e f o r e a f u n c t i o n of d i s t a n c e , but i s e a s i l y c o n v e r t e d to t ime when d i v i d e d by the t r a v e r s i n g speed. The e q u a t i o n govern ing sensor response i s dQ/dt = - 1 / T (Q - Q A ) (A .2 ) where dQ/dt i s the change of measured r a d i a t i o n w i t h time (here t ime i s the time neces sary to t r a v e r s e a f a c e t and r e p r e s e n t s s p a t i a l r a t h e r than temporal changes) and Q A i s the a c t u a l or t r u e r a d i a t i o n . I f changes i n Q A over the f a c e t s are known, the sensor response may be c a l c u l a t e d f o r the t r a v e r s e s a c r o s s the canyon f a c e t s . To represen t Q A model led d i s t r i b u t i o n s of L Q and L * were generated u s i n g the A r n f i e l d model and canyon temperature data c o l l e c t e d in O c t o b e r , 1987 f o r a H/W=1.0 canyon (see F i g u r e A . 1 ) . To a f i r s t a p p r o x i m a t i o n , . t h e net and outgo ing r a d i a t i o n d i s t r i b u t i o n on the w a l l s f o r most t imes a f t e r sunset may be r e p r e s e n t e d by a ramp change, and tha t of the top and f l o o r by a p o r t i o n of a p e r i o d i c or s i n u s o i d a l change ( i f not r e p r e s e n t e d by a ramp change ) . 213 October 10/11 1987 H/W 1.0 O Surest A Sunset + 1 h + Sunset + 2 h X Sunset + 4 h O Sunset + 8 h Focel Point: Top F i g u r e A . 1 M o d e l l e d d i s t r i b u t i o n s of L * (bottom) over canyon f a c e t s . O c t . 10, ( t o p ) , Lo (middle ) and 1987. 214 A. 2 CANYON WALLS 1 W i t h a c o n s t a n t o r ramp change o f L 0 or L*, t h e change of t h e r a d i a t i o n s e n s e d by t h e r a d i o m e t e r o v e r t i m e ( i e . d i s t a n c e / s p e e d ) may be w r i t t e n dQ/dt = - 1/T (Q - B t ) (A.3) where Q r e p r e s e n t s t h e r a d i a t i o n m e asured by t h e s e n s o r , B i s th e r a t e o f change of t h e t r u e r a d i a t i o n Q A w i t h t i m e t , and T i s t h e i n s t r u m e n t t i m e c o n s t a n t . I t i s assumed t h a t Q i s e q u a l t o Q A a t t h e s t a r t of t h e t r a v e r s e . The g e n e r a l s o l u t i o n o f (A. 3.) i s Q = B ( t - T) + C e _ t / T (A.4) and i f Q = Q A a t t h e s t a r t of t h e t r a v e r s e t h e n C = BT and Q - Q A = -BT(1 - e _ t / T ) . (A.5) When t >> r , ( i e . t h e t r a v e r s i n g t i m e i s much g r e a t e r t h a n t h e i n s t r u m e n t t i m e c o n s t a n t ) , (A.5) may be r e d u c e d t o Q - Q A = - B T . (A.6) G i v e n t h e change of L 0 o r L* o v e r t h e w a l l and t h e s p e e d o f t r a v e r s e , t h e s l o p e , B, may be c a l c u l a t e d . S p e c i f y i n g t h e l e n g t h 215 of t r a v e r s e and ins trument time c o n s t a n t , v a r i o u s speeds or t imes of t r a v e r s e may be t e s t e d f o r the d i f f e r e n c e between Q and Q A . Two such t e s t s are p r e s e n t e d i n T a b l e A.1 and cases from each t e s t are i l l u s t r a t e d in F i g u r e A . 2 . T a b l e A.1 True and Measured R a d i a t i o n For a Sensor T r a v e r s e d A c r o s s a Canyon W a l l . Change i n F l u x D e n s i t y of R a d i a t i o n Over a W a l l : (a) 15 W m ~ 2 , (b) 30 W m ~ 2 T r a v e r s e Time (s) Speed (m s~I) X1 0 ~ 2 Slope B (W m" 2 s 1 ) (W m z ) (a) 300 0.335 0.05 0.32 240 0.419 0.0625 0.40 180 0.558 0.0833 0.53 120 0.838 0.125 0.80 60 1 .675 0.25 1 .60 (b) 300 240 180 120 60 0.335 0.419 0.558 0.838 1 .675 0.10 0.125 0. 167 0.25 0.50 0.64 0.80 1 .07 1 .60 3.20 With g r e a t e r changes of r a d i a t i o n over the f a c e t , and wi th f a s t e r t r a v e r s i n g t i m e s , i t i s apparent that the d i f f e r e n c e between the t r u e and measured r a d i a t i o n grows. The t r a v e r s i n g t imes t e s t e d were chosen somewhat a r b i t r a r i l y but are c o n s t r a i n e d by a lower l i m i t governed by the maximum speed of the t r a v e r s i n g u n i t and the i n c r e a s e in v i b r a t i o n s of the sensors at those speeds , and an upper l i m i t r e s t r i c t e d by the t o t a l amount of time needed to complete an e n t i r e t r a v e r s e . To 216 SENSOR RESPONSE TO A RAMP CHANGE 0 5 10 15 20 25 30 Time From Start ot Traverse (s) F i g u r e A . 2 Sensor response to a ramp change i n L Q or L * over a w a l l . T o t a l change over the w a l l i s 30 Wm" 2 . S o l i d l i n e s - a c t u a l r a d i a t i o n , dashed l i n e - measured r a d i a t i o n . T r a v e r s e t imes a r e : A - 60 s, B - 180 s, C - 300 s. 217 keep t h e d i f f e r e n c e (Q - Q A) below 1 W m~"2 f o r t h e c h a n g e s t e s t e d , a s p e e d of o v e r t h r e e min m~1 i s n e c e s s a r y . A.3 CANYON TOP AND FLOOR F o r t h e s i n u s o i d a l change t h e g o v e r n i n g e q u a t i o n , ( A . 2 ) , may be . r e w r i t t e n a s dQ/dt + Q/T = ( Q . / i O s i n w t (A.7) where CJ r e p r e s e n t s t h e a n g u l a r f r e q u e n c y of o s c i l l a t i o n CJ = 2 7 T / P e r i o d (A.8) and t h e a m p l i t u d e of t h e o s c i l l a t i o n s . The s o l u t i o n of t h e f i r s t o r d e r l i n e a r d i f f e r e n t i a l e q u a t i o n w i t h t h e c o n s t a n t s o l v e d f o r Q = Q A = 0 = t as b e f o r e , y i e l d s Q = Q 1 w T [ 1 + ( c j T ) 2 ] " 1 / 2 e " t / T + Q, [ 1 + ( C J T ) 2 ] - 1 / 2 sin(uJt - a r c t a n w r ) . (A.9) As i l l u s t r a t e d i n F i g u r e A.3, t h e measured s i g n a l w i l l be b o t h a t t e n u a t e d and l a g g e d b e h i n d t h e t r u e s i g n a l . The a t t e n u a t i o n f a c t o r a = [1 + ( C J T ) 2 ] - 1 / 2 (A.10) 218 SENSOR RESPONSE TO A SINUSOIDAL CHANGE Time From Start of Traverse (s) F i g u r e A . 3 Sensor re sponse to S o l i d l i n e s - a c t u a l r a d i a t i o n r a d i a t i o n . T r a v e r s e t imes a r e : f o r a 1 m canyon w i d t h . a s i n u s o i d a l change i n L Q or L * . , dashed l i n e s - measured A - 60 s, B - 180 s, C - 300 s 219 and t h e l a g L = arctancjT (A. 1 1 ) when t >> T , and t h e l a g t i m e = L/OJ. The a t t e n u a t i o n , l a g and l a g t i m e c a l c u l a t e d a t s e v e r a l t r a v e r s i n g s p e e d s a r e p r e s e n t e d f o r r a d i a t i o n d a t a o f a g i v e n a m p l i t u d e i n T a b l e A.2. T a b l e A.2 A t t e n u a t i o n and Lag Time o f Canyon F l o o r / T o p D a t a f o r V a r i o u s T r a v e r s e Times A m p l i t u d e : 10 W m~2 O n e - h a l f p e r i o d r e p r e s e n t s t h e d i s t r i b u t i o n T r a v e r s e Time P e r i o d A t t e n u a t i o n F a c t o r Lag Lag Time (s ) ( s ) ( s ) 300 600 0.998 0.067 6.381 240 480 0.997 0.083 6.375 180 360 0.994 0.111 6.364 120 240 0.986 0.166 6.332 60 120 0.948 0.323 6.167 F o r t r a v e r s e t i m e s of 3 m i n u t e s o r g r e a t e r p e r metre t h e amount o f a t t e n u a t i o n i s n e g l i g i b l e and t h e l a g t i m e i s o n l y v e r y s l o w l y c h a n g i n g . F o r t r a v e r s e s of 2 m i n u t e s o r l e s s p e r m e t r e , t h e a t t e n u a t i o n becomes more p r o n o u n c e d and l a g t i m e s d e c r e a s e . G i v e n a l a g t i m e o f 6.4 s and a speed o f t r a v e r s e o f 0.55X10~ 2 m s ~ 1 , a measurement made a t a p a r t i c u l a r p o i n t would be l a g g e d by a p p r o x i m a t e l y 35.6 mm i n d i s t a n c e . 220 A.4 DELAY INTERVAL The d e t e r m i n a t i o n o f s e n s o r t r a v e r s i n g s p e e d b a s e d upon s e n s o r r e s p o n s e t o t h e d i s t r i b u t i o n of r a d i a t i o n a c r o s s t h e c a n y o n f a c e t s assumes Q A = Q a t t h e s t a r t o f a t r a v e r s e a c r o s s e a c h f a c e t . To a c h i e v e t h i s , upon c o m p l e t i o n of a t r a v e r s e and r o t a t i o n o f t h e s e n s o r s a t e a c h c o r n e r , t h e i n s t r u m e n t s must r e m a i n s t a t i o n a r y f o r a p e r i o d o f t i m e i n o r d e r t o e q u i l i b r a t e t o t h e r a d i a t i v e b a l a n c e of t h e new f a c e t . The d i f f e r e n c e i n r a d i a t i v e b a l a n c e between the two f a c e t s i s e q u i v a l e n t t o a s t e p c h a n g e U s i n g t h e t i m e c o n s t a n t of t h e r a d i o m e t e r s , t h e t i m e needed t o a d j u s t t o a g i v e n f r a c t i o n o f t h e new l e v e l o f r a d i a t i o n may be d e t e r m i n e d from ( A . 1 ) . Changes between f a c e t s a r e assumed t o be g r e a t e r t h a n t e m p o r a l c h a n g e s o c c u r r i n g d u r i n g t h e d e l a y t i m e . The CTS has a b u i l t i n d e l a y of 27 s w h i c h i s i n v o k e d a f t e r e a c h i n s t r u m e n t r o t a t i o n . T h i s p r o v i d e s an a d j u s t m e n t o f 0.985. The l a r g e s t c h a n g e s i n r a d i a t i o n o c c u r d u r i n g i n s t r u m e n t r o t a t i o n t o and from t h e c a n y o n t o p , w i t h much s m a l l e r c h a n g e s o c c u r r i n g between t h e w a l l s and f l o o r . T a b l e A.3 p r e s e n t s t h e a d j u s t m e n t t o a s t e p change c o m p l e t e d w i t h i n c r e a s i n g t i m e u s i n g (A.1) f o r a t i m e c o n s t a n t of 6.3905 s. 2 2 1 T a b l e A.3 A d j u s t m e n t C o m p l e t e d t o a S t e p Change i n R a d i a t i o n U s i n g a M i n i a t u r e Net R a d i o m e t e r . E l a p s e d Time (s ) A d j u s t m e n t C o m p l e t e d 5 0.543 10 0.791 15 0.904 20 0.956 2 5 * 0.980 27* 0.985 30 0.991 35 0.996 * D e l a y b u i l t i n t o CTS A.5 CONCLUSIONS T h i s a n a l y s i s s e r v e s o n l y as a g u i d e t o c h o o s i n g an a p p r o p r i a t e t r a v e r s i n g s p e e d and d e l a y i n t e r v a l and does not p r o v i d e any form of c o r r e c t i o n f a c t o r . I t i s b a s e d on s i m p l e a p p r o x i m a t i o n s t o t h e a c t u a l r a d i a t i o n d i s t r i b u t i o n s w i l l o c c u r a c r o s s c a n y o n f a c e t s and i s a compromise between f a c t o r s . B a s e d on t h e r e s u l t s of t h e p r e c e e d i n g c a l c u l a t i o n s , a t r a v e r s i n g s p e e d o f a p p r o x i m a t e l y 5.5 cm s ~ 1 (3 min m~ 1) was c h o s e n t o b e s t p r o v i d e a s p e e d w h i c h would a l l o w c o m p l e t e c a n y o n t r a v e r s e s i n a s h o r t t i m e , w h i l e not i n c u r r i n g l a r g e l a g s o r a t t e n u a t i o n s of th e m easured s i g n a l s . E r r o r s i n c u r r e d by t h e c h o s e n s p e e d and a s s u m p t i o n s of r a d i a t i o n d i s t r i b u t i o n s a r e d i s c u s s e d i n A p p e n d i x 222 APPENDIX B. DATA PROCEDURES B.1 DESCRIPTION OF RECORDED DATA The d a t a r e c o r d e d by t h e CR21X m i c r o l o g g e r may be d i v i d e d i n t o two c a t e g o r i e s : t r a v e r s e d and s t a t i o n a r y . T r a v e r s e d d a t a i n c l u d e s t h e r a d i a t i o n and t e m p e r a t u r e measurements made by t h e t r a v e r s i n g s e n s o r s : L*, L c , T c , and T a . Th e s e d a t a a r e r e c o r d e d as samples a t a f i x e d t i m e i n t e r v a l ( e i t h e r 1 o r 2 s ) . A l s o 1 r e c o r d e d i s a number r e p r e s e n t i n g t h e s t a t e of t h e t r a v e r s e ; 1 r e p r e s e n t s measurements t a k e n d u r i n g a d e l a y p e r i o d , 0 r e p r e s e n t s t h o s e made d u r i n g a t r a v e r s e . S t a t i o n a r y d a t a i n c l u d e : s u r f a c e t e m p e r a t u r e , L i c t , w ind d i r e c t i o n , and L * 0 . The s t a t i o n a r y d a t a a r e sampled a t f i x e d i n t e r v a l s (15 s d u r i n g t r a v e r s i n g ) and a v e r a g e s a r e r e c o r d e d a t i n t e r v a l s of 1 - 3 min d u r i n g t r a v e r s i n g o p e r a t i o n s . B.2 ASSIGNMENT OF TRAVERSED DATA TO GRID-POINTS T r a v e r s e / d e l a y i n f o r m a t i o n p r o v i d e d by t h e r e c o r d e d 0 o r 1, t o g e t h e r w i t h e s t i m a t e s o f t h e s p e e d of t r a v e r s i n g f o r e a c h f a c e t a l l o w e a c h sample t o be e q u a t e d t o a d i s t a n c e f r o m t h e s t a r t o f t h e t r a v e r s e o f e a c h f a c e t . T h e n c e , u s i n g t h e s t a r t and end l o c a t i o n of t h e t r a v e r s e i n r e l a t i o n t o t h e f a c e t d i m e n s i o n s , and g r i d - p o i n t d i m e n s i o n s , e a c h sample may be a s s i g n e d t o a g r i d - p o i n t . 223 Once a f a c e t number and p o i n t number have been a s s i g n e d t o e a c h sample, t h e y may be c o n d e n s e d i n t o a v e r a g e s o v e r any number of g r i d - s q u a r e s . T h i s assumes t h a t t h e a v e r a g e o f samples a c r o s s t h e g r i d - q u a r e i s a good a p p r o x i m a t i o n of t h e v a l u e a t t h e g r i d - p o i n t , w h i c h i s t h e l o c a t i o n o f t h e m o d e l l e d v a l u e . B.3 SIGNAL FILTERING THE SURFACE TEMPERATURE F i g u r e B.1a p r e s e n t s s u r f a c e t e m p e r a t u r e s measured on t h e e v e n i n g o f Aug 1/2. When compared w i t h F i g u r e B.1b f o r t h e e v e n i n g o f A u g u s t 25/26 i t i s c l e a r t h a t t h e r e i s a n o i s e component i n t h e f o r m e r d a t a above t h a t w h i c h m i g h t n o r m a l l y be e x p e c t e d . R e s u l t s i n d i c a t e t h e measurement of t h e d e l a y v o l t a g e and t h e use o f an A/C a d a p t e r t o power t h e 21X a r e t h e p r i n c i p a l c a u s e s . The n o i s e i s most o b v i o u s i n t h e s u r f a c e t e m p e r a t u r e d a t a , but can be f o u n d i n t h e t r a v e r s e d a t a a s w e l l . I t seems l i k e l y t h a t t h e n o i s e i s s i m i l a r t o AC n o i s e ( i e . 60 H z ) , u n f o r t u n a t e l y t h i s c a n n o t be a s c e r t a i n e d b e c a u s e t h e r e c o r d e d s i g n a l i s sampled a t a much s l o w e r r a t e ( u s u a l l y 15 s e c o n d i n t e r v a l s ) and a v e r a g e d o v e r a 1, 2 o r 3 m i n u t e p e r i o d . S i n c e t h e n o i s e i s o u t s i d e t h e N y q u i s t c r i t i c a l f r e q u e n c y ( i n t h i s c a s e 1/6 min.) i t becomes f o l d e d o r a l i a s e d i n t o t h e f r e q u e n c y r a n g e - f c < f < f c . G i v e n t h e s e n s i t i v i t y of t h e A r n f i e l d model t o even s m a l l c h a n g e s i n t h e i n p u t s u r f a c e t e m p e r a t u r e ( C h a p t e r 3) i t i s d e s i r a b l e t o r e d u c e t h e e f f e c t s of n o i s e i n t h e c o o l i n g c u r v e s . 224 IS - • 1 1 ' ' ' • 1 ' 0 1 2 S « S t 7 6 T i m * F r o m S u n s v t ( h ) F i g u r e B .1a S u r f a c e t e m p e r a t u r e s , even ing of Aug . 1/2 . 2* 23 • Floor H> S ? 22 • 3 • E • 20 ^ ^ j . WoB 8 IP 5 u o T 1» 3 </> • 18 IP 17 •J ' 0 2 4 6 8 T l m « F r o m S u n * « l ( h ) F i g u r e B . l b S u r f a c e t e m p e r a t u r e s , e v e n i n g of A u g . 2 5 / 2 6 . 225 I t was d e c i d e d n o t t o m o d i f y t h e t r a v e r s e d a t a as t h e l e v e l o f n o i s e i s n o t as g r e a t and some d e g r e e of s m o o t h i n g i s a c h i e v e d when the t r a v e r s e d a t a i s r e d u c e d t o v a l u e s f o r e a c h g r i d - p o i n t . B.3.1 Methods Used • O n e of a number o f p o s s i b l e a l t e r n a t i v e s t o smooth t h e t e m p e r a t u r e d a t a i s t o c o n s t r u c t a l o w - p a s s f i l t e r . The method, s t a t e d s i m p l y , r e q u i r e s t h e d e f i n i t i o n of a s e t of f i l t e r c o e f f i c e n t s , f o l l o w e d by t h e t r a n s f o r m a t i o n o f t h e d a t a t o t h e f r e q u e n c y domain, where t h e y a r e m u l t i p l i e d by t h e f i l t e r c o e f f i c i e n t s and r e - t r a n s f o r m e d t o t h e t i m e - d o m a i n . The c r i t i c a l o p e r a t i o n i s t h e d e f i n i t i o n o f t h e f i l t e r c o e f f i c i e n t s . To d e f i n e t h e c o e f f i c i e n t s a s t o p / p a s s f r e q u e n c y i s c h o s e n t o f i l t e r o n l y t h e f r e q u e n c i e s w h i c h d e f i n e t h e n o i s e . A p l o t o f t h e power s p e c t r a l d e n s i t y of t h e d e t r e n d e d t e m p e r a t u r e d a t a c a n be u s e d t o i n v e s t i g a t e t h e f r e q u e n c i e s a t w h i c h n o i s e e x i s t s . From t h e s a m p l i n g theorem, two p o i n t s / c y c l e a r e t h e minimum needed t o d e f i n e a s i n e wave of t h e N y q u i s t c r i t i c a l f r e q u e n c y (0.5 x s a m p l i n g i n t e r v a l ) . To e n s u r e th e n o i s e d e f i n e d by c o n s e c u t i v e samples i s w e l l d e f i n e d ( t h e h i g h e s t f r e q u e n c y v i s i b l e n o i s e ) t h e i n p u t d a t a were r e s a m p l e d u s i n g l i n e a r i n t e r p o l a t i o n between t r u e d a t a p o i n t s . The t o t a l number o f p o i n t s i s now 2xN-1 (no e x t r a p o l a t i o n ) . R e s a m p l i n g a l s o has t h e a d v a n t a g e of a l l o w i n g more f r e q u e n c i e s t o be d e f i n e d i n t h e power s p e c t r u m when u s i n g F o u r i e r T r a n s f o r m methods of c o m p u t i n g t h e power s p e c t r a l d e n s i t y . 226 A p l o t of t h e power s p e c t r a l d e n s i t y o b t a i n e d u s i n g t h e Maximum E n t r o p y Method ( P r e s s e t a l . 1986) o f t h e r e s c a l e d and d e t r e n d e d t e m p e r a t u r e d a t a from A u g u s t 1/2 (West W a l l IP 1-5) i s shown i n F i g u r e B.2. The f r e q u e n c y s c a l e i s b a s e d upon the d a t a b e i n g sampled one t i m e u n i t a p a r t . The l a r g e peak e v i d e n t n e a r z e r o i s p r o b a b l y a r e s u l t of t h e d e t r e n d i n g o f t h e d a t a (Hamming, 1983); a c c u r a t e d e t r e n d i n g o f t h e d a t a was made d i f f c u l t by t r u e v a r i a t i o n s i n t h e c o o l i n g c u r v e ; up t o a t h i r d o r d e r p o l y n o m i a l has been s u b t r a c e d f r o m t h e d a t a . The t i m e s e r i e s of t h e f i r s t p o i n t on t h e West w a l l had t h e most l i n e a r d e c r e a s e of t e m p e r a t u r e and was t h e r e f o r e d e t r e n d e d most s u c c e s s f u l l y ( F i g u r e B . 3 ) . A s e p a r a t e p l o t o f i t ' s power s p e c t r u m i s shown i n F i g u r e B.4. The h i g h f r e q u e n c y n o i s e d e f i n e d by c o n s e c u t i v e o r i g i n a l p o i n t s i s e v i d e n t a t a f r e q u e n c y of 0.25. D e f i n i t e p e a k s a r e a l s o p r e s e n t a t f r e q u e n c i e s o f 0.062, .094, and 0.117, w i t h t h e peak a t 0.094 c o n t a i n i n g a l m o s t t w i c e t h e power o f t h e o t h e r p e a k s . The n o i s e d e f i n e d by t h i s peak c a n a l s o be seen i n t h e t i m e s e r i e s p l o t . Note t h a t b e c a u s e t h e d a t a a r e sampled a t d i f f e r e n t r a t e s a n d / o r a v e r a g e d o v e r d i f f e r e n t l e n g t h s o f t i m e , t h e power s p e c t r a i n v o l v e d i f f e r e n t a l i a s i n g of t h e n o i s e and s h i f t t h e l o c a t i o n o f a p p a r e n t n o i s e p e a k s . W i t h l o n g e r a v e r a g i n g p e r i o d s , t h e h i g h f r e q u e n c y n o i s e i s r e d u c e d . F i l t e r c o e f f i c i e n t s may now be d e f i n e d f o r a g i v e n s t o p / p a s s f r e q u e n c y e s t i m a t e d from th e power s p e c t r u m . F i g u r e B.5 p l o t s t h e t r a n s f e r f u n c t i o n v a l u e s f o r p o s i t i v e f r e q u e n c i e s of a low- 227 0.20 h F i g u r e B.2 Power s p e c t r u m o f r e s c a l e d and d e t r e n d e d d a t a f o r p o i n t s 1 t o 5 on t h e West w a l l on Aug. 1/2. 228 o o D " o OJ CL E OJ O O D if) ~o OJ TJ C OJ ~a> Q - 0 . 1 - 0 . 3 100 200 3 0 0 4 0 0 T i m e 500 F i g u r e B.3 D e t r e n d e d s u r f a c e t e m p e r a t u r e o f p o i n t 1 on t h e West w a l l ; Aug. 1/2. Time i s i n u n i t s o f sa m p l e s f r o m t h e s t a r t o f r e c o r d i n g . 229 Frequency F i g u r e B.4 Power spec trum of r e s c a l e d and d e t r e n d e d s u r f a c e t emperature shown i n F i g u r e B.3. 230 LOW PASS FILTER SF=0.03 Frequency F i g u r e B.5 T r a n s f e r f u n c t i o n o f a l o w - p a s s f i l t e r w i t h a s t o p f r e q u e n c y ( S F ) o f 0.03 f o r d i f f e r e n t t r u n c a t i o n s o f t h e F o u r i e r s e r i e s . 231 p a s s f i l t e r w h i c h p a s s e s f r e q u e n c i e s up t o 0.03. The c u r v e s a r e o b t a i n e d f r o m t h e F o u r i e r s e r i e s f o r H ( f ) = 1 , 0 < | f | < 0.03 (B. 1) H ( f ) = 0, 0.03 < | f | <0.5 (B.2) and assumes t h a t t h e t r a n s f e r f u n c t i o n i s s y m m e t r i c a b o u t f=0. The s e r i e s , t r u n c a t e d t o k terms i s k H ( f ) = 2Fs + I (2/rrk) s i n 2 7 r k f s cos27rkf ( s i n (jrk/n)/vrk/n) 2 (B.3) n=1 where t h e s q u a r e d s i n e t e r m i s a s m o o t h i n g t e r m (Hamming, 1983). F i g u r e B.5 shows t h a t a f t e r a p p r o x i m a t e l y 40 t e r m s , t h e t r a n s f e r f u n c t i o n i s v e r y c l o s e t o t h e i d e a l c u r v e (1 up t o t h e s t o p f r e q u e n c y 0 a f t e r ) . The i d e a l c u r v e may be u s e d but t h e i m p u l s e r e s p o n s e c a n , and i n t h i s c a s e d o e s , have r i n g i n g a t f r e q u e n c i e s c o r r e s p o n d i n g t o t h e s h a r p edges o f t h e f i l t e r ( P r e s s e t a l . 1986). F i g u r e B.6 i l l u s t r a t e s t h e o r i g i n a l and f i l t e r e d d a t a u s i n g 40 terms t o a p p r o x i m a t e t h e t r a n s f e r f u n c t i o n f o r s t o p f r e q u e n c i e s o f 0.2, 0.075, and 0.03. A s i m p l e l i n e a r d e t r e n d i n g and r e s c a l i n g o f t h e o r i g i n a l d a t a was p e r f o r m e d . C l e a r l y , r e m o v a l o f o n l y t h e h i g h e r f r e q u e n c i e s does n o t a d e q u a t e l y smooth t h e f l u c t u a t i o n s . T h i s i s n o t u n e x p e c t e d s i n c e t h e h i g h f r e q u e n c y n o i s e a l i a s e d i n t o t h e d a t a when sampled w i l l have components even a t low f r e q u e n c i e s . The d a nger o f s e t t i n g t h e 232 c u t o f f f r e q u e n c y a t t o o low a f r e q u e n c y i s t h e l o s s of some t r u e i n f o r m a t i o n ; w h i l e 0.03 may seem low i n t h i s r e g a r d , i t does n o t a p p e a r u n r e a s o n a b l e i n v i s u a l t e r m s . The d a t a need not be r e s c a l e d t o use t h e f i l t e r , however, n o t e t h a t t h e s t o p f r e q u e n c y i s two t i m e s t h a t u s e d p r e v i o u s l y when u s i n g t h e a s s u m p t i o n of d a t a sampled one t i m e u n i t a p a r t i n b o t h t h e o r i g i n a l and r e s c a l e d d a t a . A s e c o n d method of d e t e r m i n i n g a f i l t e r f o r n o i s y d a t a i s t h a t of o p t i m a l o r Wiener f i l t e r i n g . F o l l o w i n g t h e p r o c e d u r e s o u t l i n e d i n P r e s s e t a l . 1986, t h e power s p e c t r u m o f t h e d a t a i s p l o t t e d u s i n g a l o g s c a l e and an e x t r a p o l a t i o n o f t h e s i g n a l and n o i s e p o r t i o n s of t h e p l o t a r e made ( F i g u r e B . 7 ) . The f i l t e r f u n c t i o n i s d e f i n e d a s F ( f ) = | S ( f ) | 2 / ( | S ( f ) | 2 + | N ( f ) | 2 (B.4) where | S ( f ) | 2 and | N ( f ) | 2 a r e d e r i v e d from | S ( f ) | 2 + | N ( f ) | 2 P c ( f ) = | C ( f ) | 2 0<f<fc (B.5) w i t h Pc t h e power s p e c t r a l d e n s i t y of t h e measured s i g n a l C. S ( f ) and N ( f ) a r e t h e s i g n a l and n o i s e f r e q u e n c i e s r e s p e c t i v e l y . The a p p r o a c h i s n o t s e n s i t i v e t o t h e method u s e d t o o b t a i n t h e power s p e c t r u m ( P r e s s e t a l . 1986); i t o n l y r e q u i r e s t h e g e n e r a l shape of t h e n o i s e and s i g n a l r e g i o n o f t h e p l o t . E x c e p t f o r a few s m a l l p e a k s , F i g u r e B.7 e x h i b i t s a g e n e r a l l y f l a t t a i l o f n o i s e . T h i s method assumes t h a t t h e s i g n a l and n o i s e a r e 233 F i g u r e B.6 F i l t e r e d s u r f a c e t e m p e r a t u r e on Aug. 1/2 u s i n g v a r i o u s l o w - p a s s f i l t e r s . P l u s s i g n s - m e a s u r e d s u r f a c e t e m p e r a t u r e , s h o r t d a s h e d l i n e - SF = 0.2, l o n g d a s h e d l i n e - SF = 0.075, s o l i d l i n e - SF = 0.03. Time h a s u n i t s o f samples from t h e s t a r t . 234 io 7 F 10 6 - 105 . Frequency F i g u r e B.7 Power s p e c t r a l d e n s i t y o f s u r f a c e t e m p e r a t u r e ( p o i n t 1 on t h e West w a l l ) f o r Aug. 1/2. The s t r a i g h t h o r i z o n t a l l i n e r e p r e s e n t s t h e a v e r a g e ' t a i l * o f n o i s e . The s i g n a l + n o i s e c u r v e h a s been e x t r a p o l a t e d . 235 u n c o r r e l a t e d , an a s s u m p t i o n p a r t i a l l y v i o l a t e d h e r e , as e v i d e n c e d by a p o r t i o n o f t h e s i g n a l p e a k i n g a t t h e h i g h e r f r e q u e n c i e s . However, due t o a l i a s i n g , t h e c h a r a c t e r o f t h e n o i s e i s made more ' w h i t e ' as i t i s f o l d e d i n t o t h e s p e c t r u m (Hamming, 1983). C o n t i n u i n g , we c a n d e f i n e t h e f i l t e r f u n c t i o n ( F i g u r e B . 8 ) . Note t h a t i t ' s shape b e a r s marked r e s e m b l a n c e t o t h e low p a s s f i l t e r f u n c t i o n c o n s t r u c t e d e a r l i e r , d i f f e r i n g o n l y i n t h e s t o p f r e q u e n c y . G i v e n t h i s s i m i l a r i t y , a l o w - p a s s f i l t e r was c o n s t r u c t e d t o a p p r o x i m a t e t h e o p t i m a l f i l t e r . A s t o p f r e q u e n c y of 0.015 was s e l e c t e d b a s e d upon t h e p l o t o f t h e o p t i m a l f i l t e r f u n c t i o n . D a t a from o t h e r d a y s y i e l d e d s i m i l a r r e s u l t s . U s i n g 105 t e r m s , t h e t r a n s f e r f u n c t i o n c l o s e l y a p p r o x i m a t e s t h e o p t i m a l f i l t e r . U s i n g t h e l o w - p a s s f i l t e r a s a r e p r e s e n t a t i o n o f t h e o p t i m a l f i l t e r , t h e o r i g i n a l d a t a f r o m F i g u r e B.6 a r e f i l t e r e d and t h e r e s u l t s a r e shown i n F i g u r e B.9. B.4 EXTRAPOLATION OF ADDITIONAL SURFACE TEMPERATURES ON CANYON WALLS The a d d i t i o n of an e x t r a g r i d - p o i n t n e a r t h e t o p o f t h e w a l l s i n t h e m o d i f i e d A r n f i e l d model ( C h a p t e r 3) r e q u i r e s two a d d i t i o n a l s u r f a c e t e m p e r a t u r e s . M e a s u r e d d a t a a r e n o t a v a i l a b l e f o r t h e s e g r i d - p o i n t s , t h e r e f o r e t h e t e m p e r a t u r e must be e s t i m a t e d by e m p i r i c a l o r t h e o r e t i c a l means. In t h e a b s e n c e o f an a n a l y t i c a l f u n c t i o n t o d e s c r i b e t h e t e m p e r a t u r e of t h e 236 F i g u r e B.8 F i l t e r f u n c t i o n d e r i v e d from an o p t i m a l f i l t e r ( s o l i d l i n e ) , compared t o a l o w - p a s s f i l t e r ( d a s h e d l i n e ) u s i n g 105 terms and a s t o p f r e q u e n c y o f 0.015. 237 24.0 F i g u r e B . 9 F i l t e r e d s u r f a c e temperature u s i n g an ' o p t i m a l ' low- pass f i l t e r . T o p : A u g . 1/2. Bottom: The f i r s t 75 measurement i n t e r v a l s of A u g . 1/2 f o r comparison w i t h F i g . B . 6 . 238 a d d i t i o n a l g r i d - p o i n t s , i n t e r p o l a t i o n / e x t r a p o l a t i o n methods may be u s e d . The p r o b l e m i n t h i s c a s e i s one o f e x t r a p o l a t i o n , and i s made more d i f f i c u l t by t h e d i f f e r e n c e s o f t h e s t r u c t u r e and p o s i t i o n o f t h e c a p p i n g b l o c k s compared t o t h e r e m a i n d e r of t h e c a n y o n w a l l s . R e c a l l t h a t t h e c a p p i n g b l o c k i s a s o l i d s l a b p l a c e d a c r o s s t h e - u p p e r row o f c o n c r e t e b l o c k s . I t i s p r o b a b l e t h a t , due t o t h e v a r i a t i o n i n t h e t h e r m a l p r o p e r t i e s and p o s i t i o n o f t h e c a p p i n g b l o c k s , t h e e x t r a p o l a t i o n o f t h e t e m p e r a t u r e d i s t r i b u t i o n c u r v e o v e r t h e w a l l t o t h e c a p p i n g b l o c k w i l l n o t be smooth. A low o r d e r e x t r a p o l a t i o n of t h e t e m p e r a t u r e o v e r t h e t o p p o r t i o n o f t h e w a l l has been s e l e c t e d t o e s t i m a t e t h e s e t e m p e r a t u r e s . P r o f i l e s o f t h e s u r f a c e t e m p e r a t u r e d i s t r i b u t i o n up and down t h e c a n y o n w a l l s ( C h a p t e r 5) i l l u s t r a t e a s i g n i f i c a n t a l t e r a t i o n i n shape between t h e w a l l s and o v e r t i m e . In p a r t i c u l a r , t h e d i s t r i b u t i o n o f t e m p e r a t u r e up t h e E a s t w a l l a f t e r s u n s e t shows a l a r g e peak n e a r t h e t o p o f t h e w a l l . In s u c h c a s e s t h e t e m p e r a t u r e t o be e x t r a p o l a t e d i s r e l a t e d t o o n l y one o r two g r i d - p o i n t s l o w e r t h a n i t s e l f and a low o r d e r o f e x t r a p o l a t i o n i s n e c e s s a r y . L a t e r a t n i g h t t h e t e m p e r a t u r e d i s t r i b u t i o n i s much smoother and a h i g h e r o r d e r of e x t r a p o l a t i o n c a n be s p e c i f i e d . P o l y n o m i a l e x t r a p o l a t i o n assumes t h e t e m p e r a t u r e s o f t h e known p o i n t s a r e c o r r e c t so t h a t i n d a t a where n o i s e was o r i g i n a l l y p r e s e n t t h e f i l t e r e d d a t a a r e u s e d i n t h e e x t r a p o l a t i o n p r o c e d u r e . The s e n s i t i v i t y of t h e e x t r a p o l a t i o n t o t h e o r d e r u s e d f o r a s e l e c t i o n of d a t a i s p r e s e n t e d i n C h a p t e r 239 3. W h i l e t h e p r o b a b i l i t y o f e r r o r s i s h i g h , t h e p o i n t s t o be e x t r a p o l a t e d a r e o n l y u s e d t o p r e v e n t e x c e s s e r r o r s i n t h e m o d e l l e d r a d i a t i o n f o r t h e t e n t h g r i d - p o i n t on e a c h w a l l and a r e n o t u s e d as a p o i n t i n t h e v a l i d a t i o n o f t h e measured and m o d e l l e d d a t a . The c l o s e s t p o i n t s a f f e c t e d by e x t r a p o l a t i o n e r r o r s ( c o r n e r p o i n t s on t h e c a n y o n t o p ) a r e a l s o u s u a l l y o m i t t e d ( d e p e n d i n g on l e n g t h of t r a v e r s e ) . No e f f e c t on n e i g h b o u r i n g p o i n t s on t h e w a l l i s a c h i e v e d i n t h e model ( s e e s e n s i t i v i t y t e s t s , C h a p t e r 3 ) . The s m a l l a r e a o f t h e e x t r a g r i d - p o i n t a l s o r e d u c e s t h e e r r o r s w h i c h i t may c a u s e i n o t h e r m o d e l l e d p o i n t s , r e l a t i v e t o t h e e r r o r of a n o r m a l g r i d - p o i n t . 240 APPENDIX C . CALIBRATION OF THE BARNES PRT-4A INFRARED THERMOMETER A c a l i b r a t i o n c u r v e f o r an i n f r a r e d (IR) thermometer may be o b t a i n e d by m e a s u r i n g t h e i n s t r u m e n t o u t p u t f o r v a r i o u s known t e m p e r a t u r e s of a b l a c k b o d y s u r f a c e . The c a l i b r a t i o n c u r v e s h o u l d c o v e r t h e range o f t e m p e r a t u r e s e x p e c t e d t o be e n c o u n t e r e d i n t h e f i e l d . The c a l i b r a t i o n c u r v e was c r e a t e d and c o n f i r m e d two methods. In t h e f i r s t , a s t i r r e d , c o n s t a n t t e m p e r a t u r e water b a t h was u s e d as a t e s t s u r f a c e (Huband, 1985). The e m i s s i v i t y o f water i t s e l f i s h i g h (0.972 D a v i e s e t a l . 1971; 0.993 B u e t t n e r and K e r n , 1965) and when combined w i t h t h e p o l i s h e d m e t a l s u r f a c e s of t h e b a t h p r o d u c e s an e f f e c t i v e e m i s s i v t y of t h e w a t e r - b a t h s y s t e m o f n e a r u n i t y ( F u c h s and T a n n e r , 1966). A p r o f i l e o f s i x t h e r m o c o u p l e s was p l a c e d i n t h e b a t h t o o b t a i n a v e r t i c a l t r a n s e c t o f t e m p e r a t u r e was a c r o s s t h e w a t e r / a i r b o u n d a r y so t h a t s u r f a c e t e m p e r a t u r e c o u l d be d e t e r m i n e d by i n t e r p o l a t i o n . The IR thermometer was mounted o v e r t h e b a t h and v i e w e d t h e water s u r f a c e t h o u g h an a p e r t u r e c u t i n t h e s t y r o f o a m i n s u l a t i n g c o v e r o f t h e b a t h . To o b t a i n p o i n t s a t l o w e r t e m p e r a t u r e s a low t e m p e r a t u r e water b a t h f i l l e d w i t h a w a t e r - g l y c o l m i x t u r e was u s e d . A s e c o n d method, d e r i v e d from F u c h s and T a n n e r ( 1 9 6 6 ) , was u s e d t o c o n f i r m t h e c u r v e o b t a i n e d from t h e water b a t h . T h r e e t h e r m o c o u p l e s (30 awg) were cemented i n s h a l l o w g r o o v e s c u t i n a p l y w o o d p l a t e . - w h i c h was p a i n t e d f l a t b l a c k . A s m a l l cone w i t h a 241 p o l i s h e d aluminum i n n e r s u r f a c e was c o n s t r u c t e d and t h e IR thermometer was u s e d t o view t h e s u r f a c e t h o u g h t h e apex o f t h e c o n e . The cone i n c r e a s e s t h e e f f e c t i v e s u r f a c e e m i s s i v i t y so t h a t t h e s u r f a c e under t h e cone behaves a p p r o x i m a t e l y as a b l a c k b o d y when t h e t e m p e r a t u r e o f t h e cone e q u a l s t h a t o f t h e s u r f a c e . S u r f a c e t e m p e r a t u r e measurements made w i t h t h e t h e r m o c o u p l e s may t h e n be compared t o t h e i n s t r u m e n t o u t p u t . F i g u r e A3.1 i l l u s t r a t e s t h e r e s u l t s f r o m t h e two p r o c e d u r e s . The d a t a from t h e f i r s t method (open s q u a r e s ) a r e f i t by a c u r v e u s i n g a s e c o n d o r d e r p o l y n o m i a l ( s o l i d ) and t h e d e s t - f i t s t r a i g h t l i n e ( d a s h e d ) . The d i f f e r e n c e between t h e two l i n e s i s m i n o r o v e r most of t h e range of s u r f a c e t e m p e r a t u r e s e n c o u n t e r e d i n t h e f i e l d . The p o i n t s from t h e s e c o n d method a r e a l s o i n c l u d e d f o r c o m p a r i s o n ( c r o s s e s ) . Use o f t h e p o l y n o m i a l r e l a t i o n i s recommended f o r v e r y h i g h o r low s u r f a c e t e m p e r a t u r e s . 242 T. = -13.318 • 1.2747 (rnV) - 0.004 (nM>) 10 20 30 40 Instrument Output (mV) F i g u r e C . 1 C a l i b r a t i o n c u r v e f o r B a r n e s M o d e l PRT-4A I n f r a r e d Thermometer. 243 APPENDIX D. ERRORS D.1 INTRODUCTION T h i s a p p e n d i x d e s c r i b e s t h e e r r o r s i n t h e measured v a r i a b l e s u s e d f o r model i n p u t and model v a l i d a t i o n . The bounds of e r r o r p l a c e d upon t h o s e v a r i a b l e s u s e d i n model i n p u t c a n be u s e d i n c o n j u n c t i o n w i t h t h e s e n s i t i v i t y a n a l y s i s of C h a p t e r 3 t o d e t e r m i n e e f f e c t s on m o d e l l e d r a d i a t i o n . The i n s t r u m e n t a t i o n e r r o r s have been c o l l e c t e d f r o m many s o u r c e s : o r i g i n a l c a l i b r a t i o n c e r t i f i c a t e s , m a n u f a c t u r e r ' s s p e c i f i c a t i o n s and v a r i o u s t e x t s . F o r some i n s t r u m e n t a t i o n , t h e r e i s d o c u m e n t a t i o n t o l i s t t h e component e r r o r s w h i c h t o g e t h e r c o m p r i s e a t o t a l e r r o r i n t h e measurement. The a b s o l u t e component e r r o r s may be summed o r combined u s i n g a r o o t mean s q u a r e (RMS) a p p r o a c h . The e r r o r s of f u n c t i o n s w h i c h c o n s i s t s of s e v e r a l v a r i a b l e s have been c a l c u l a t e d u s i n g p r o b a b l e e r r o r a n a l y s i s a s d e s c r i b e d i n F r i t s c h e n and Gay ( 1 9 7 9 ) . F o r a f u n c t i o n Y c o n s i s t i n g of v a r i a b l e s x 1 f x 2 ... x n , t h e g e n e r a l e q u a t i o n f o r t h e a b s o l u t e e r r o r o f Y i s EY = Ex. 6Y/6x] + E x 2 SY/6x 2 + ... +' E x n • 6 Y / 6 x n (D. 1 ) where t h e Ex^ a r e t h e a b s o l u t e e r r o r s o f e a c h o f t h e v a r i a b l e s . A b s o l u t e e r r o r p r o v i d e s a 'worst c a s e ' e r r o r e s t i m a t e where a l l 244 e r r o r s a c t i n t h e same d i r e c t i o n t o p r o v i d e a maximum e r r o r . The p r o b a b l e e r r o r of Y assumes t h e e r r o r s o f t h e x^ a r e n o r m a l l y d i s t r i b u t e d and t h a t some c o m p e n s a t i o n o f t h e i n d i v i d u a l e r r o r s i s l i k e l y t o o c c u r . The f o r m u l a f o r t h e p r o b a b l e e r r o r of Y i s PeY = [ ( P e x , 6 Y / 6 x r ) 2 + ... + ( P e x n 8 Y / 6 x n ) 2 ] 1 / 2 . (D.2) The d i f f e r e n t i a l s o f Y w i t h r e s p e c t t o e a c h o f t h e x^ a r e s u b s t i t u t e d i n t o (2) and t y p i c a l v a l u e s o f t h e v a r i a b l e s a r e use d t o g e n e r a t e PeY. The v a l u e s o f Pex^ a r e t h e e r r o r s i n t h e v a r i a b l e s , t y p i c a l l y t h e RMS e r r o r , and a r e o b t a i n e d from component i n s t r u m e n t e r r o r s , o r e s t i m a t e d by o t h e r means. D.2 SURFACE TEMPERATURE ERRORS E r r o r s t o be c o n s i d e r e d when making a s u r f a c e t e m p e r a t u r e measurement u s i n g a 21X m i c r o l o g g e r w i t h a b u i l t - i n r e f e r e n c e t e m p e r a t u r e i n c l u d e : t h e r e f e r e n c e t e m p e r a t u r e , t h e t h e r m o c o u p l e o u t p u t , t h e v o l t a g e measurement u s e d t o e s t i m a t e t h e t e m p e r a t u r e , t h e r e f e r e n c e and o u t p u t l i n e a r i z a t i o n , method of a t t a c h i n g t h e t h e r m o c o u p l e t o t h e s u r f a c e and n o i s e ( C a m p b e l l S c i e n t i f i c , 1984). T a b l e D.1 summarizes t h e component e r r o r s o f a s u r f a c e t e m p e r a t u r e measurement t a k e n d u r i n g t h e day under h i g h s o l a r r a d i a t i o n (Case 1), and under t y p i c a l n i g h t t i m e c o n d i t i o n s (Case 2 ) . 245 T a b l e D.1 E r r o r Summary: S u r f a c e T e m p e r a t u r e E r r o r D e s c r i p t i o n Amount o f E r r o r (°C) C a s e 1 Case 2 P e r c e n t 1 o f T o t a l E r r o r 2 R e f e r e n c e T e m p e r a t u r e 0.6 0.7 28 39 T h e r m o c o u p l e O u t p u t - ANSI S t a n d a r d 1.0 1.0 - 1% S l o p e E r r o r 0.2 0.04 9 2 V o l t a g e Measurement 0.06 0.07 3 4 R e f e r e n c e L i n e a r i z a t i o n 0.001 0.001 <1 <1 O u t p u t L i n e a r i z a t i o n 0.001 0.001 <1 <1 Mount i n g 0.75 0.5 36 28 N o i s e 0.5 0.5 24 28 Sum RMSE 2.91 I 2.11 2 1 .47 \ 1 . 10 2 2.77 1 .81 1 .41 1 .00 U s i n g ANSI s t a n d a r d e r r o r A s s u m i n g 1% s l o p e e r r o r R e f e r e n c e t e m p e r a t u r e e r r o r s c o n s i s t o f e r r o r s i n t h e t h e r m i s t o r u s e d t o measure t h e r e f e r e n c e t e m p e r a t u r e w i t h i n t h e 21X (±0.2 °C i n t h e range 0 - 4 0 °C) and d i f f e r e n c e s i n t e m p e r a t u r e a l o n g t h e t e r m i n a l s t r i p on t o p o f t h e 21X ( g e n e r a l l y l e s s t h a n 0.3 °C when t h e i n s u l a t i n g t e r m i n a l s t r i p c o v e r i s u s e d ) . The f i r s t e r r o r i s e s s e n t i a l l y f i x e d w i t h r e s p e c t t o t h e d a t a c o l l e c t e d however t h e s e c o n d e r r o r w i l l become s m a l l e r o r l a r g e r d e p e n d i n g upon how c l o s e l y t h e m u l t i p l e x e r t e r m i n a l s t r i p t e m p e r a t u r e matches t h e 21X when t h e 21X r e f e r e n c e t e m p e r a t u r e i s u s e d . E r r o r s a c r o s s t h e 21X t e r m i n a l s t r i p c o v e r a r e f u r t h e r r e d u c e d by p l a c i n g t h e e n t i r e 246 d a t a l o g g e r w i t h i n an i n s u l a t i n g box, w h i c h m i n i m i z e s t e m p e r a t u r e g r a d i e n t s a r o u n d t h e d a t a l o g g e r . T h e r m o c o u p l e o u t p u t e r r o r s a r e e n t e r e d as e i t h e r t h e ANSI s t a n d a r d w h i c h i s t h e m a n u f a c t u r e r s s t a n d a r d f o r t y p e - T t h e r m o c o u p l e w i r e o r as t h e p r o d u c t of t h e s l o p e e r r o r o f t h e Seebeck c o e f f i c i e n t w i t h t h e d i f f e r e n c e i n t e m p e r a t u r e between t h e s u r f a c e t e m p e r a t u r e and measurement t e m p e r a t u r e . When e x p r e s s e d i n t h e l a t t e r form, e r r o r s w i l l become s m a l l a s t h e s u r f a c e t e m p e r a t u r e and r e f e r e n c e t e m p e r a t u r e t e n d t o c o n v e r g e . V o l t a g e meaurement i s g e n e r a l l y 0.05% of t h e f u l l s c a l e range u s e d t o measure t h e t h e r m o c o u p l e v o l t a g e . In a l l c a s e s t h e 5 mV r a n g e was u s e d , w h i c h i n t h e e n v i r o n m e n t a l r a n g e o f t e m p e r a t u r e s c o n v e r t s t o a t e m p e r a t u r e e r r o r o f 0.6 - 0.7°C. R e f e r e n c e and o u t p u t l i n e a r i z a t i o n e r r o r s o c c u r a s a r e s u l t o f a p p r o x i m a t i o n s t o t h e c o n v e r s i o n o f v o l t a g e t o t e m p e r a t u r e , and a r e n e g l i g i b l e compared t o t h e o t h e r e r r o r s . M o u n t i n g e r r o r s i n c l u d e c o n d u c t i o n e r r o r s , r a d i a t i o n e r r o r s e t c . w h i c h r e s u l t f r o m t h e p r e s e n c e of a t h e r m o c o u p l e on t h e measurement s u r f a c e . E a c h o f t h e s e component e r r o r s c a n be e s t i m a t e d from f o r m u l a s ( e g . F r i t s c h e n and Gay, 1979) when p r e c i s e i n f o r m a t i o n i s a v a i l a b l e on t h e h e a t t r a n s f e r p r o p e r t i e s o f t h e s u r f a c e , t h e r m o c o u p l e and s u r r o u n d i n g s . H e r e , an e s t i m a t e o f t h e e r r o r has been made u s i n g t h e r a n g e o f t e m p e r a t u r e d i f f e r e n c e s about t h e 1:1 l i n e o f t h e c o m p a r i s o n of s u r f a c e t e m p e r a t u r e a c c u r a c y shown i n C h a p t e r 2. E r r o r s a r e l i k e l y t o be h i g h e r i n t h e d a y t i m e due t o l a r g e r r a d i a t i o n e r r o r s . 247 The f i n a l e r r o r component l i s t e d i s n o i s e . P l o t s o f u n f i l t e r e d t e m p e r a t u r e i n d i c a t e d f l u c t u a t i o n s a b o u t t h e mean of up t o 0.5 °C. W h i l e t h e e r r o r s i n t h e f i l t e r e d d a t a a r e t h o u g h t t o be much l e s s , t h i s t e r m has been r e t a i n e d as a c a u t i o n a r y m easure. U s i n g t h e 1% s l o p e e r r o r i n p l a c e o f t h e ANSI e r r o r , i t i s c l e a r t h a t t h e e r r o r i n t e m p e r a t u r e measurements i s made up p r i m a r i l y o f r e f e r e n c e t e m p e r a t u r e e r r o r s , m o u n t i n g e r r o r s , and n o i s e . I t i s l i k e l y t h a t e r r o r s i n t h e r e f e r e n c e t e m p e r a t u r e and n o i s e a r e o f t e n l e s s t h a n i s p r e s e n t e d i n T a b l e D.1, t h e r e b y i m p r o v i n g t h e e r r o r e s t i m a t e . T o t a l e r r o r e s t i m a t e s f o r n i g h t t i m e t e m p e r a t u r e s r a n g e from 2.77 °C ( a b s o l u t e sums w i t h ANSI s t a n d a r d e r r o r ) t o 1.00 °C (RMSE w i t h a 1% s l o p e e r r o r ) . D.3 EMISSIVITY ERRORS S u r f a c e e m i s s i v i t y was e s t i m a t e d u s i n g t h e g o v e r n i n g e q u a t i o n l i s t e d i n T a b l e D.2b. T s and T r , e a c h c o n s i s t of a s i n g l e measurement w h i l e T^ was o b t a i n e d by s a m p l i n g T^ a t s e v e r a l z e n i t h a n g l e s and m u l t i p l y i n g t h e v i e w - f a c t o r f o r t h e p o r t i o n of sky v i e w e d w i t h t h e v a l u e of T^-. Component e r r o r s a r e shown i n T a b l e D.2a. 248 T a b l e D.2a E r r o r Summary: E m i s s i v i t y ( T s , T k , T r ) E r r o r D e s c r i p t i o n Amount Of E r r o r I n f r a r e d Thermometer 0 . 5 ° C 1 V o l t a g e Measurement 0 . 03 °C 2 Sum 0 . 53 U C RMSE 0 . 50 ° c R e f e r e n c e : B a r n e s E n g i n e e r i n g Co. R e f e r e n c e : C a m p b e l l S c i e n t i f i c (1984) S u b s t i t u t i n g t y p i c a l v a l u e s of T r , T s and T k (293.66, 294.16, and 281.16 K r e s p e c t i v e l y ) i n t o t h e e q u a t i o n f o r Pee ( T a b l e D.2b) a p r o b a b l e e r r o r of 0.06 i n t h e s u r f a c e e m i s s i v i t y i s o b t a i n e d when t h e e r r o r of T r , T s , and T k a r e s e t t o 0.5 K e a c h . T a b l e D.2b P r o b a b l e E r r o r A n a l y s i s : E m i s s i v i t y G o v e r n i n g E q u a t i o n : e = ( T r 4 - T k 4 ) / ( T s 4 - T k 4 ) P r o b a b l e E r r o r of e: Pee = [ ( P e T r 6 e / 6 T r ) 2 + ( P e T s 6 e / 5 T s ) 2 + ( P e T k 6 e / 8 T k ) 2 ] 0 • 5 P a r t i a l E r r o r s : 6 e / 5 T r = 4 T r 3 / ( T S 4 - T k 4 ) 6 e / 6 T s = -4 T s 3 ( T r 4 - T k 4 ) / ( T S 4 - T k 4 ) 6 e / 5 T k = 4 T k 3 ( T r 4 - T S 4 ) / ( T S 4 - T k 4 ) Because T k i s t a k e n a t d i s c r e t e z e n i t h a l and a z i m u t h a l a n g l e s i t i s an a p p r o x i m a t i o n t o t h e t r u e h e m i s p h e r i c a l a v e r a g e and t h e e r r o r s w i l l be- g r e a t e r t h a n f o r T r and T s . A n a l y s i s of t h e 249 p a r t i a l e r r o r s r e v e a l s t h a t t h e change i n e m i s s i v i t y w i t h r e s p e c t t o i s minor (±0.003 f o r t y p i c a l v a l u e s u s e d ) ; t h u s t h e e s t i m a t e o f p r o b a b l e e r r o r r e m a i n s s a t i s f a c t o r y f o r g r e a t e r e r r o r s i n T^.. D.4 RADIATION ERRORS D.4.1 L i c t E r r o r s The i n s t r u m e n t u s e d t o measure L ^ c t was an E p p l e y P r e c i s i o n I n f r a r e d R a d i o m e t e r ( p y r g e o m e t e r ) , Model PIR. T a b l e D.3a p r o v i d e s a summary of e r r o r s of t h i s i n s t r u m e n t and t h e r e c o r d i n g s y s t e m u s e d (21X M i c r o l o g g e r ) . T a b l e D.3a E r r o r Summary: L ^ c t E r r o r D e s c r i p t i o n Amount of E r r o r C a l i b r a t i o n 2 % 1 T e m p e r a t u r e Dependence L i n e a r i t y C o s i n e R e s p o n s e R e c o r d e r E r r o r s 2 % (-20°C - 40°C) 2 1 % (0 - 700 W m~ 2) 2 i n s i g n i f i c a n t f o r d i f f u s e s o u r c e s ; <5 % from n o r m a l i z a t i o n 2 0.05 % of FSR Sum " 5.05 % ( d i f f u s e s o u r c e ) RMSE 3.00 % R e f e r e n c e : L a t i m e r ( 1 9 7 2 ) . R e f e r e n c e : E p p l e y I n s t r u m e n t M a n u a l . 250 The p r o b a b l e e r r o r a n a l y s i s r e c o g n i z e s t h e g o v e r n i n g - e q u a t i o n i s s i m p l y t h e i n s t r u m e n t o u t p u t m u l t i p l i e d by t h e c a l i b r a t i o n i c o e f f i c i e n t ( T a b l e D.3b). T a b l e D.3b P r o b a b l e E r r o r A n a l y s i s : L ^ c t G o v e r n i n g E q u a t i o n : L ĉ^ = C ( I O ) ; C = C a l i b r a t i o n C o e f f i c i e n t IO = I n s t r u m e n t O u t p u t P r o b a b l e E r r o r of L ^ c t : P e L i c t = [ ( P e C 6 L i c t / 8 C ) 2 + ( P e l O 8 L i c t / « I O ) 2 ] 0 • 5 P a r t i a l E r r o r s : 6 L i c t / 6 C = IO 6 L i c t / 5 I O = C The component e r r o r o f t h e i n s t r u m e n t c a l i b r a t i o n i s t r e a t e d s e p a r a t e l y from t h e o t h e r e r r o r s . F o r t h e i n s t r u m e n t u s e d t h e c a l i b r a t i o n c o e f f i c i e n t i s 215.98 W m~ 2 mV - 1, t h e e r r o r i n C i s 2%, and t h e e r r o r i n t h e i n s t r u m e n t o u t p u t i s 2.24% (RMSE). T h e r e f o r e an i n s t r u m e n t o u t p u t o f 1.6 mV ( t y p i c a l n i g h t t i m e v a l u e ) y i e l d s a p r o b a b l e e r r o r e s t i m a t e of 10.37 W m - 2. D.4.2 L * 0 E r r o r s Net r a d i a t i o n o v e r t h e open s i t e was measured u s i n g a m i n i a t u r e n e t r a d i o m e t e r . The i n s t r u m e n t and r e c o r d i n g e r r o r s a r e l i s t e d i n T a b l e D.4. 251 T a b l e D.4 E r r o r Summary: L * D E r r o r D e s c r i p t i o n • Amount o f E r r o r C a l i b r a t i o n 2.5 % T e m p e r a t u r e Dependency 0.012 % B a l a n c e 1 . 0 % R e c o r d i n g E r r o r s 0.05 % o f FSR Sum 3.562 % RMSE 2.693 % R e f e r e n c e : O r i g i n a l M a n u f a c t u r e r ' s S p e c i f i c a t i o n s The p r o b a b l e e r r o r a n a l y s i s i s s i m i l a r t o t h a t f o r L i c t . The c a l i b r a t i o n c o n s t a n t f o r L * 0 was 164.93 W m~2 mV _ 1, t h e e r r o r of c a l i b r a t i o n i s 2.5% and t h e r e m a i n d e r o f t h e i n s t r u m e n t e r r o r i s 1.0% (RMSE). F o r an i n s t r u m e n t o u t p u t o f -0.1 mV, t h e p r o b a b l e e r r o r of L * D i s 0.4 W m~2; f o r an o u t p u t of -0.8 mV t h e e r r o r i s a p p r o x i m a t e l y 3.6 W m~2. D.4.3 L* E r r o r s E r r o r s i n t h e r a d i a t i v e f l u x e s measured by t h e t r a v e r s e d i n s t r u m e n t s have two a d d i t i o n a l e r r o r s t o t h o s e l i s t e d i n T a b l e D.4. T h e s e r e s u l t from t h e t r a v e r s i n g p r o c e d u r e and a r e composed of e r r o r s r e s u l t i n g f r o m t h e f i n i t e i n i t i a l d e l a y l e n g t h , and t h e e r r o r s of i n s t r u m e n t r e s p o n s e t o c h a n g e s o f r a d i a t i v e f l u x e s o v e r c a n y o n s u r f a c e s . The e r r o r t o t h e i n i t a l d e l a y has been c a l c u l a t e d a s 1.5 %, w h i c h i s t h e a d j u s t m e n t r e m a i n i n g t o a s t e p 252 change a f t e r a d e l a y l e n g t h of 27 s e c o n d s ( T a b l e A 1 . 3 ) . The d e s i g n e r r o r f o r a change i n L* o f 30 W m~2 o v e r a 1 m c a n y o n f a c e t t r a v e r s e d i n 180 s e c o n d s i s a p p r o x i m a t e l y 3 p e r c e n t . A w o r s t c a s e e r r o r o f 10 % has been i d e n t i f i e d by u s i n g t h e o b s e r v e d c h a n g e s i n L* between a d j a c e n t p o i n t s on t h e c a n y o n w a l l s . The e r r o r summary of L* i s p r e s e n t e d i n T a b l e D.5. T a b l e D.5 E r r o r Summary L* ( T r a v e r s e d ) E r r o r D e s c r i p t i o n Amount of E r r o r C a l i b r a t i o n 2.5 % T e m p e r a t u r e Dependency 0.012 % B a l a n c e 1 . 0 % R e c o r d i n g E r r o r s 0.05 % of FSR I n i t i a l D e l a y 1 . 5 % R e s p o n s e t o Change i n L D 3 % ( d e s i g n ) 10 % ( w o r s t c a s e ) Sum 8.06% ( d e s i g n ) , 15.06% ( w o r s t c a s e ) RMSE 4.30% 10.46% A p r o b a b l e e r r o r a n a l y s i s o f ,L* y i e l d s e r r o r s r a n g i n g from 0.7 - 1.7 W m~2 f o r an i n s t r u m e n t o u t p u t o f -0.1 mV, and between 5.7 - 13.9 W m~2 f o r an o u t p u t of -0.8 mV u s i n g t h e w o r s t c a s e RMS e r r o r f o r t h e i n s t r u m e n t o u t p u t ( N o t e : T a b l e D.5 g i v e s t h e t o t a l e r r o r s i n c l u d i n g t h e c a l i b r a t i o n c o m p o n e n t ) . 253 D.4.4 L c E r r o r s Use of a b l a c k b o d y c a v i t y w i t h a m i n a t u r e n e t r a d i o m e t e r t o measure L 0 i n c u r s a d d i t i o n a l e r r o r s due t o t h e need f o r a t e m p e r a t u r e measurement. The component e r r o r s a r e l i s t e d i n T a b l e D.6a. T a b l e D.6a E r r o r Summary L Q ( T r a v e r s e d ) E r r o r D e s c r i p t i o n Amount o f E r r o r C a l i b r a t i o n 2.5 % T e m p e r a t u r e Dependency 0.012 % B a l a n c e 1 . 0 % R e c o r d i n g E r r o r s 0.05 % Of FSR I n i t i a l D e l a y 1 . 5 % R esponse t o Change i n L Q 3 % ( d e s i g n ) 10 % ( w o r s t c a s e ) R e f e r e n c e T e m p e r a t u r e 0.4 °C T h e r m o c o u p l e O u t p u t ANSI S t a n d a r d 1% S l o p e E r r o r 1 .0 °C 0.02 °C V o l t a g e Measurement 0.07 °C R e f e r e n c e L i n e a r i z a t i o n 0.001 °C O u t p u t L i n e a r i z a t i o n 0.001 °C Sum ( r a d i a t i o n e r r o r s ) 8 . 0 6 % d e s i g n 15.06 % w o r s t RMSE ( r a d i a t i o n e r r o r s ) 4.30 % 10.46 % c a s e Sum ( t h e r m o c o u p l e e r r o r s ) 0.49 °C 1% s l o p e 1.47 °C ANSI RMSE ( t h e r m o c o u p l e e r r o r s ) 0.41 °C e r r o r 1.08 °C 254 The t h e r m o c o u p l e e r r o r s a r e b a s e d upon a c a v i t y t e m p e r a t u r e o f 20.0 °C, a r e f e r e n c e t e m p e r a t u r e o f 18.0 °C and a t e m p e r a t u r e g r a d i e n t a c r o s s t h e t e r m i n a l s t r i p o f 0.2 °C. A p r o b a b l e e r r o r a n a l y s i s i s p r e s e n t e d i n T a b l e D.6b. In g e n e r a l , t h e i n s t r u m e n t o u t p u t i s s m a l l a t n i g h t d u r i n g t r a v e r s i n g ( e x c e p t a t t h e c a n y o n t o p ) so t h a t t h e p r o b a b l e e r r o r i s d o m i n a t e d by t h e T c e r r o r s . P r o b a b l e e r r o r s r a n g e from w o r s t c a s e e s t i m a t e s o f o v e r 10 W m - 2 a t t h e c a n y o n t o p t o l e s s t h a n 3 W m~2 f o r d e s i g n r a d i a t i o n e r r o r s and t h e r m o c o u p l e e r r o r s u s i n g a 1% s l o p e e r r o r ( T a b l e D . 6 c ) . T a b l e D.6b P r o b a b l e E r r o r A n a l y s i s : L 0 G o v e r n i n g E q u a t i o n : L 0 = C (IO) + a T c 4 P r o b a b l e E r r o r of L c : P e L D = [ ( P e C 6 L 0 / 6 C ) 2 + ( P e l O 6 L o / 6 I 0 ) 2 + ( P e T c 6 L 0 / 6 T C ) 2 + (Pea 6 L 0 / 6 a ) 2 ] 0 * 5 P a r t i a l E r r o r s : 6 L 0 / 6 C = IO 6 L Q / 6 I 0 = C 5 L 0 / 6 T C = 4 a T c 3 6 L Q / 5 a - 0 255 T a b l e D.6c T y p i c a l P r o b a b l e E r r o r s : L Q T c 10 10 L 0 measured P e l O PeTc P e L n (°C) (W m 2.) (mV) (W m - 2) (mV) (°C) (W m 2 ) 289.6 -80.4 -0.516 318.1 -0.018 0.41 4.1 1 -0.052 1.08 10.3 2 289.8 1.9 -0.013 401.9 0.0004 0.41 2.3 0.0013 1.08 5.9 290.2 25.8 0.166 427.7 0.0058 0.41 2.5 0.0168 1.08 6.6 P e l O c a l c u l a t e d u s i n g RMSE d e s i g n (3.5%) P e l O c a l c u l a t e d u s i n g RMSE w o r s t c a s e (10.46%) 256 A P P E N D I X E . M O D E L I N P U T I n p u t t o t h e model f o r a t y p i c a l n o c t u r n a l r u n i s l i s t e d i n T a b l e E . 1 . P a r a m e t e r s l i s t e d f u l l y d e f i n e n o c t u r n a l r a d i a t i v e c o n d i t i o n s . Model v a l u e s needed f o r d a y t i m e r u n s , e g . s t a n d a r d i r r a d i a n c e o f s h o r t - w a v e and d i f f u s e r a d i a t i o n , s o l a r a z i m u t h and z e n i t h a n g l e a r e not l i s t e d . In a d d i t i o n t o t h e l i s t e d i n p u t d a t a , m o d i f i c a t i o n s t o t h e model r e q u i r e t h a t t h e c a n y o n l o c a t i o n ( f a c e t number and p o i n t ) and v a l u e o f t h e measured o b s e r v a t i o n s o f L c and L * be e n t e r e d . 257 T a b l e E . I Model I n p u t f o r N o c t u r n a l Model Runs. M o d e l Name D e f i n i t i o n Comments H W L THETA E T LD v a r i o u s ID IT IF IP JRAD IRADL P1L P2L P3L RSTOP IVFIN IVFOT ICHEK Canyon h e i g h t Canyon w i d t h Canyon h a l f - l e n g t h Canyon o r i e n t a t i o n E m i s s i v i t y T e m p e r a t u r e L i c t V i e w - f a c t o r s D a t e Time (PDT) F a c e t Number P o i n t Number D e f i n e s r a d i a t i o n b u dget components t o be c a l c u l a t e d I d e n t i f i e s t h e r a d - i a n c e d i s t r i b u t i o n f o r s ky d e r i v e d l o n g - wave R a d i a n c e d i s t r i b u t i o n p a r a m e t e r 1 R a d i a n c e d i s t r i b u t i o n p a r a m e t e r 2 R a d i a n c e d i s t r i b u t i o n p a r a m e t e r 3; Z e n i t h a l o p t i c a l water p a t h (u) c o n v e r g e n c e c r i t e r i a f o r c a n y o n m u l t i p l e r e f l e c t i o n r o u t i n e s s o u r c e . o f v i e w - f a c t o r s f i l e t o s a v e view f a c t o r s s e t s c h e c k t o a v o i d re- c a l c u l a t i n g v a r i a b l e s a r b i t r a r y u n i t s ?« n n n d e g r e e s from n o r t h measured; a r r a y measured; a r r a y m easured; a t p l a n e o f cany o n t o p a r r a y s f o r p a i r s of can y o n f a c e t s t i m e w r i t t e n from d a t a l o g g e r . End of the a v e r a g i n g p e r i o d u s e d . From 1 t o 4 : 1 W a l l A, 2 W a l l B, 3 F l o o r , 4 Canyon Top From 1 t o 10 f o r e a c h f a c e t a r r a y o p t i o n s : i s o t r o p i c , U n sworth and M o n t e i t h (UM) (1975) I n t e r c e p t c o f t h e UM f o r m u l a S l o p e f r o m UM f o r m u l a (cm), o p t i o n a l ; f o r use w i t h long-wave r a d . d i s t . o f U n s w o r t h and M o n t e i t h (1975) i n u n i t s o f Wm - 2 i n t e r n a l c a l c u l a t i o n o r r e a d f r o m e x t e r n a l f i l e 258 APPENDIX F. ADDITIONAL VALIDATION DATA SETS F.1 AUGUST 2/3 The c o m p l e t e v a l i d a t i o n d a t a s e t c o l l e c t e d on A u g u s t 2/3, 1988, from a can y o n w i t h a H/W o f 2.0. T a b l e F.1 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 2/3, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and U n s w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . S t a t i s t i c L* I s o t r o p i c L 0 L i L* UM L o L i n 1 1 24 1 1 24 1 1 24 1 1 75 1 175 1 1 75 0 (W m t.) -32. 6 429.3 396. 7 -30. 4 420. 1 389. 8 P (W m 2 ) -30. 9 430.8 400. 0 -29. 5 421 .2 391 . 7 s 0 (W m 2) 26. 9 11.9 31 . 6 25. 9 10.2 29. 5 s p (W m~ z) 28. 2 11.8 31 . 6 26. 8 10.1 29. 3 RMSE (W m~2) 4. 9 4.0 5. 6 3. 3 3.2 3. 7 RMSEs (W m" 2) 2. 0 1 .7 3. 3 1 . 1 1 .2 1 . 9 RMSEu (W m 2 ) 4. 5 3.6 4. 5 3. 1 3.0 3. 2 MSEs/MSE 0. 17 0.18 0. 35 0. 1 1 0.14 0. 26 MSEu/MSE 0. 83 0.82 0. 65 0. 89 0.86 0. 74 MAE (W m - 2) 4. 3 3.1 4. 7 2. 8 2.5 3. 0 MBE (W m 2 ) 1 . 8 1 .5 3. 3 0. 8 1 . 1 1 . 9 r 2 0. 98 0.91 0. 98 0. 99 0.91 0. 99 d 0. 99 0.97 0. 99 0. 99 0.98 0. 99 a (W m~ 2) 3. 0 24.9 7. 2 1 . 7 23.5 6. 9 b 1 . 04 0.95 0. 99 1 . 03 0.95 0. 99 259 F.2 AUGUST 8/9 The c o m p l e t e v a l i d a t i o n d a t a s e t c o l l e c t on A u g u s t 8/9, 1988, from a c a n y o n w i t h a H/W o f 1.0. T a b l e F.2 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 8/9, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and U n s w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . S t a t i s t i c I s o t r o p i c UM L* Lo L i L* L Q L i 64 1064 1064 1064 1064 1064 14.6 415. 9 401 .3 -14.6 415.9 401 .3 14.1 417.2 403. 1 -14.3 417.2 402.9 11.0 6.2 11.6 11.0 6.2 11.6 8.2 5.5 9.1 9.4 5.4 9.5 3.6 1 .9 4.0 3.0 2.0 4.1 3.0 1.6 3.4 1.9 1.6 3.1 2.0 1.0 2.2 2.3 1 .2 2.7 0.69 0.71 0.72 0.40 0.64 0.57 0.31 0.29 0.28 - 0.60 0.36 0.43 3.2 1 .6 3.2 2.5 1.7 3.3 0.5 1 .3 1 .8 0.4 1 .3 1 .7 0.94 0.97 0.94 0.94 0.95 0.92 0.96 0.98 0.96 0.98 0.97 0.96 -3.5 55.8 99.6 -2.2 63.7 89.7 0.73 0.87 0.76 0.83 0.85 0.78 n 0 P (W n T 2 ) (W m 2) (W m"2) (W m~ 2) RMSE (W m 2 ) RMSEs (W m 2 ) RMSEu (W m 2 ) MSEs/MSE MSEu/MSE MAE MBE r 2 d (W m 2 ) (W m 2 ) a b (W m 2 ) 260 F.3 AUGUST 11/12 The c o m p l e t e v a l i d a t i o n d a t a s e t c o l l e c t on A u g u s t 11/12, 1988, from a can y o n w i t h a H/W o f 0.67. T a b l e F.3 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 11/12, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and U n s w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . I s o t r o p i c UM S t a t i s t i c L* L 6 L i L* L 0 L i n 0 (W m~ 2) P (W m_2) s 0 (W m_2) s p (W m z ) RMSE (W m~ 2) RMSEs (W m_ 2) RMSEu (W m 2 ) MSEs/MSE MSEu/MSE MAE (W m~ 2) MBE (W m 2 ) r 2 d a (W m~ 2) b 975 975 -42.7 411.3 -47.6 412.9 27.0 13.0 23.0 13.2 9.1 3.5 6.8 1.6 6.0 3.1 0.56 0.21 0.44 0.79 6.4 2.4 -4.9 1.6 0.93 0.95 0.97 0.98 -12.3 7.4 0.83 0.99 975 975 368.5 -42.7 365.3 -47.2 27.4 27.0 24.2 24.1 6.9 6.9 4.9 5.5 4.9 4.2 0.50 0.64 0.50 0.36 5.1 5.2 -3.2 -4.5 0.96 0.97 0.98 0.98 46.3 -9.5 0.87 0.88 975 975 411.3 368.5 412.9 365.7 13.0 27.4 13.2 25.1 3.6 5.1 1.7 3.8 3.2 3.4 0.22 0.56 0.78 0.44 2.5 4.0 1.6 -2.8 0.94 0.98 0.98 0.99 8.9 31.4 0.98 0.91 261 F.4 AUGUST 14/15 The c o m p l e t e v a l i d a t i o n d a t a set. c o l l e c t on A u g u s t 14/15, 1988, from a c a n y o n w i t h a H/W o f 1.33. T a b l e F.4 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 14/15, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and U n s w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . S t a t i s t i c L * I s o t r o p i c L * UM n 0 P (W m~ 2) (W m 2 ) (W m 2 ) (W m 2 ) RMSE (W n T 2 ) RMSEs (W m~ 2) RMSEu (W m~ 2) MSEs/MSE MSEu/MSE MAE MBE r 2 d (W n T 2 ) (W m~ 2) a b (W rn"2) 960 960 960 960 960 960 -25. 1 408. 7 383. 5 -25. 1 408. 7 383. 5 -28. 2 410. 9 382. 7 -28. 1 410. 9 382. 8 23. 7 9. 5 25. 7 23. 7 9. 5 25. 7 22. 1 8. 7 24. 2 22. 3 8. 7 24. 0 6. 3 3. 6 4. 4 5. 0 3. 7 3. 6 3. 7 2. 5 2. 0 3. 4 2. 5 2. 0 5. 1 2. 6 3. 9 3. 6 2. 7 3. 0 0. 34 0. 48 0. 21 0. 46 0. 46 0. 31 0. 66 0. 52 0. 79 0. 54 0. 54 0. 69 4. 8 2. 7 3. 6 3. 8 2. 8 3. 0 -3. 0 2. 2 -o. 8 -2. 9 2. 2 -0. 7 0. 95 0. 91 0. 97 0. 97 0. 90 0. 98 0. 98 0. 96 0. 99 0. 99 0. 96 0. 99 -5. 4 •54. 0 26. 8 -4. 8 56. 3 27. 3 0. 91 0. 87 o. 93 0. 93 0. 87 0. 93 262 F.5 AUGUST 22/23 The c o m p l e t e v a l i d a t i o n d a t a s e t c o l l e c t on A u g u s t 22/23, 1988, from a c a n y o n w i t h a H/W o f 0.41. T a b l e F.5 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 22/23, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and U n s w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . S t a t i s t i c I s o t r o p i c UM L* L 0 L i L * L o L i 865 865 865 865 865 865 -48.8 413.1 364.3 -48.8 413. 1 364.3 -51.8 412.3 360.4 -51 .5 412.3 360.8 22.5 8.7 22.6 22.5 8.7 22.6 18.3 8.6 19.1 20.4 8.5 21.1 7.2 3.1 7.0 4.6 3.1 5.2 5.7 1 . 1 5.5 3.6 1 . 1 3.9 4.4 2.9 4.2 2.9 2.9 3.4 0.63 0.13 0.62 0.61 0.13 0.56 0.37 0.87 0.38 0.39 0.87 0.44 5.2 2.2 5.0 3.5 2.3 4.2 -3.0 -0.8 -3.8 -2.7 -0.8 -3.5 0.94 0.89 0.95 0.98 0.88 0.98 0.97 0.97 0.97 0.99 0.97 0.99 -13.4 29.0 60.5 -7.8 30.3 24.7 0.79 0.93 .0.82 0.90 0.93 0.92 n 0 P (W m~ 2) (W m~ 2) (W m~ 2) (W m 2 ) RMSE (W m~ 2) RMSEs (W n T 2 ) RMSEu (W m~ 2) MSEs/MSE MSEu/MSE MAE MBE r 2 d (W m 2 ) (W m 2 ) a b (W m" 2) 263 F.6 AUGUST 23/24 The c o m p l e t e v a l i d a t i o n d a t a s e t c o l l e c t on A u g u s t 23/24, 1988, from a c a n y o n w i t h a H/W of 0.67. T a b l e F.6 Model P e r f o r m a n c e S t a t i s t i c s : A u g u s t 23/24, I n d i v i d u a l V a l i d a t i o n P o i n t s , I s o t r o p i c and Uns w o r t h and M o n t e i t h (1975) R a d i a n c e D i s t r i b u t i o n . S t a t i s t i c I s o t r o p i c UM L* L o L i L * L o L i 956 956 956 956 956 956 0 (W m t) --40.8 420.4 379.6 -40.8 420.4 379.6 P (W m 2 ) -44.3 419.7 375.4 -43.9 419.7 375.8 s 0 (W n T 2 ) 24.8 9.4 25.7 24.8 9.4 25.7 s p (W m l ) 19.9 9.1 21.6 20.8 9.1 22.5 RMSE (W m~ 2) 8.5 3.2 7.7 6. 1 3.2 5.8 RMSEs (W m" 2) 6.6 1 .0 6.2 5.3 1 .0 5.1 RMSEu (W m" 2) 5.4 3.0 4.5 3.2 3.0 2.7 MSEs/MSE 0.60 0.10 0.65 0.75 0.10 0.77 MSEu/MSE 0.40 0.90 0.35 0.25 0.90 0.23 MAE (W m" 2) 6.6 2.3 5.8 4.9 2.4 4.8 MBE (W n T 2 ) -3.5 -0.6 -4.2 -3.2 -0.6 -3.8 r 2 0.93 0.89 0.9 0.98 0.99 0.99 d 0.96 0.97 0.9 0.98 0.97 0.99 a (W m~ 2) -12.8 34.6 63.7 -10.1 35.9 46. 1 b 0.77 0.92 .0.8 0.83 0.91 0.87 264 APPENDIX G . STATISTICAL INDICES OF MODEL PERFORMANCE G.1 SUMMARY UNIVARIATE STATISTICS Observed 0, and p r e d i c t e d P, means and s t a n d a r d d e v i a t i o n s , _ n 0 = ( Z 0 ^ n ~ 1 (G.1) i = 1 _ n P = ( I P i ) n " 1 (G.2) i = 1 s o [n (I O i 2 ) - (2 O i ) 2 n _ 1 ( n - 1 ) - 1 ] 0 - 5 (G.3) i=1 i=1 s D = [n ( I P i 2 ) - (Z P i ) 2 n _ 1 ( n - 1 ) _ 1 ] ° - 5 (G.4) * i=1 i=1 G.2 COEFFICIENTS OF LEAST-SQUARES LINEAR REGRESSION The i n t e r c e p t a , and s lope b . n n n n (Z P A M I O,- 2 ) - (Z O i M Z Oi P i ) i = 1 i=1 i = 1 i = 1 a = (G.5) n n n (Z O i 2 ) - (Z O i ) 2 i=1 i=1 n n n n (Z O i P i ) - (Z O i M Z P i ) i=1 i=1 i=1 b = (G.6) n (Z O i 2 ) - (Z O i ) 2 i=1 i=1 265 G.3 MEASURES OF ERROR The root mean square error, RMSE, systematic and unsystematic portions of the RMSE (RMSES and RMSEU respectively); mean absolute error, MAE, and mean bias error, MBE. RMSE = [n~ 1 I (Pi - O i ) 2 ] 0 ' 5 (G.7) i = 1 RMSE S= [n~ 1 Z (Pi - O i ) 2 ] 0 ' 5 (G.8) i = 1 RMSEu = [n~ 1 Z (Pi - P i ) 2 ] 0 - 5 (G .9) i = 1 where Pi = a + bOi, (G.10) and a and b are the c o e f f i c i e n t s of simple linear regression. 11 MAE = n" 1 Z |Pi - Oi| (G.11) n Z i = 1 n MBE = n 1 Z (Pi - Oi) (G.12) i = 1 266 4 INDICATORS OF CORRELATION The coefficient of determination, r , and index of agreement, n n n [n (Z O i P i ) - (Z O i M Z P i ) ] 2 i=1 i=1 i=1 r 2 = _ _ (G.13) [n (Z O i 2 ) - (Z O,) 2 ] [n (Z P i 2 ) - (Z P i ) 2 ] i = 1 i = 1 i = 1 i = 1 Z ( P i - O i ) 2 i = 1 d = (G.14) where £ [ |Pil + | O i N 2 i=1 P i = P i - 0 (G.15) O i = O i - 0. (G.16)

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