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Spatial scales of sensible heat flux variability : representativeness of flux measurements and surface… Schmid, Hans Peter Emil 1988

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S P A T I A L S C A L E S O F S E N S I B L E H E A T F L U X V A R I A B I L I T Y R E P R E S E N T A T I V E N E S S OF F L U X M E A S U R E M E N T S A N D S U R F A C E L A Y E R S T R U C T U R E O V E R S U B U R B A N T E R R A I N By H A N S P E T E R EMIL S C H M I D D i p l . Natw., The Swiss Federal I n s t i t u t e of Technology (ETH). Zurich, Switzerland, 1984 ' A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOHPY i n THE FACULTY OF GRADUATE STUDIES (Geography Department) We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1988 © Hans Peter Emil Schmid, 1988 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 The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) A B S T R A C T The surface character of a suburban area is far from the uniform, smooth and flat planes over which current surface-layer theory is valid and where vertical eddy-fluxes can be assumed to be almost constant horizontally and vertically. The complexity of the surface introduces considerable variability into the atmosphere at small spatial scales. This variability is partly reduced and spatially-averaged by turbulent mixing but s t i l l leaves the concerns about the spatial representativeness of sensible heat flux measurements over a suburban area. The spatial scales of sensible heat flux variability are discussed in terms of the distribution of surface temperature and roughness elements. It is shown that : (1) an eddy-correlation measurement can be considered spatially representative, if its surface zone of influence (.source area) is large enough to include a spatially representative sample of surface temperature and roughness elements. (2) a quantitative measure of spatial representativeness can be estimated by use of the two-dimensional Fourier transform of the surface temperature and roughness element distributions (i.e. by the normalized integrated variance spectrum). (3) the source area of an eddy correlation measurement may be evaluated by a numerical model based on a probability density function plume diffusion model. The source area model developed herein can also be used to estimate the relative influence of specific surface sources or sinks upon an eddy-flux measurement in the surface layer. These concepts are tested in a suburban residential area in Vancouver, B.C., Canada. Remotely sensed surface temperatures and a digitized roughness element inventory are used as data-bases for the Fourier transforms to develop representativeness c r i t e r i a for eddy-flux measurements. A set of sensible heat flux measurements at six sites and the corresponding source area calculations are used to formulate recommendations for the objective evaluation of the spatial representativeness of sensible heat flux measurements over a suburban area. The v a l i d i t y of the suggested evaluation methods is confirmed by the observations. Internal boundary layer growth, estimated by the source area model, compares well with existing work. Some consequences of complex surfaces on the surface layer structure are b r i e f l y discussed. - iv -T A B L E OF C O N T E N T S ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i LIST OF SYMBOLS AND ABBREVIATIONS xiv ACKNOWLEDGEMENTS xx P A R T I : INTRODUCTION 1. Academic and Spatial Context 1 1.1 RATIONALE 1 1.2 STUDY OBJECTIVES 4 1.3 THE STUDY AREA 5 P A R T II : THEORY A N D C O N C E P T S 2. Physical Context 11 3. The Spatial Variability of Sensible Heat Flux 16 3. 1 INTRODUCTION 16 3.2 PRINCIPLES 20 3.3 ANALYSIS 23 3.3.1 Forced Convection 25 3.3.2 Free Convection 30 3.4 SUMMARY 31 4. Spatial Distribution of Surface Temperature and Roughness Elements 33 4. 1 SURFACE TEMPERATURE 33 4.1.1 Infrared Remote Sensing of Surface Temperature 36 - v -4.2 ROUGHNESS ELEMENTS 38 4.2.1 Data Aquisition for the Roughness Inventory 39 5. Spatial Variability: Homogeneity and Representativeness...46 5.1 INTRODUCTION 46 5.2 HOMOGENEITY 48 5.3 REPRESENTATIVENESS 51 5.4 DISCUSSION 55 6. Estimating the Source Area of a Turbulent Flux Measurement over a Patchy Surface 59 6.1 INTRODUCTION 59 6.2 ONE-DIMENSIONAL CROSS-WIND DISCONTINUITY 61 6.3 THE SOURCE AREA IN ONE-DIMENSIONAL INHOMOGENEITY 64 6.4 THE SOURCE AREA IN TWO-DIMENSIONAL PATCHINESS 66 6.5 A RECIPROCAL PLUME MODEL TO ESTIMATE THE SOURCE AREA 73 7. The Source Area Model (SAM) 80 7.1 INTRODUCTION 80 7.2 AN APPLIED DISPERSION MODEL BASED ON METEOROLOGICAL SCALING PARAMETERS 81 7.2.1 The Gryning et.al. (1987) P.D.F. Model (GEA) 81 7.2.1.1 Vertical Dispersion 82 7.2.1.2 Lateral Dispersion 87 7.3 IMPLEMENTATION OF THE DISPERSION SUB-MODEL IN THE SOURCE AREA MODEL 90 7.4 THE FORTRAN-77 CODE OF THE SOURCE AREA MODEL 94 7.5 A STATISTICAL VERSION OF THE NUMERICAL MODEL (mini-SAM) 95 - vi -P A R T III : MEASUREMENTS A N D RESULTS 8. Evaluation of Spatial V a r i a b i l i t y via Spectral Analysis..105 8. 1 INTRODUCTION 105 8.2 SPATIAL VARIANCE OF SURFACE TEMPERATURE 109 8.3 SPATIAL VARIANCE OF ROUGHNESS ELEMENTS 149 8.4 ESTIMATING SPATIAL REPRESENTATIVENESS 165 9. Simultaneous Eddy Correlation Measurements at Two Sites : Five Configurationsons 168 9. 1 INTENTIONS 168 9.2 THE SITES 171 9.3 EQUIPMENT INSTRUMENTATION AND DATA 178 9.4 INSTRUMENT COMPARISON 185 9.5 THE SPATIAL REPRESENTATIVENESS OF EDDY CORRELATION MEASUREMENTS 187 P A R T IV : DISCUSSION A N D C O N C L U S I O N S 10. How Good are the SAM-Estimates ? - Comparison with Existing Work 199 11. The Structure of the Surface Layer over Complex Surfaces 214 12. Summary of Conclusions 221 REFERENCES 224 APPENDIX A : Remotely Sensed Surface Temperatures 231 APPENDIX B : Validation of the P.D.F.-Plume Model 240 APPENDIX C : SAM - Fortran-77 Code 248 APPENDIX D : Data and Results Summary 284 APPENDIX E : S t a t i s t i c a l Indices and Methods 293 - v i i -LIST OF TABLES 2.1 P o s s i b l e framework f o r urban c l imate c l a s s i f i c a t i o n 12 7.1 Standard ized set o f i s o p l e t h dimensions r e l a t i v e to the 0.9 i s o p l e t h 96 7.2 Summary o f mini-SAM polynomial f i t c o e f f i c i e n t s 99 7.3 mini-SAM model v a l i d a t i o n s t a t i s t i c s 103 10.1 Input da ta f o r Cabauw SAM-runs 201 10.2 Computation of exponent of i n t e r n a l boundary l ayer growth 212 A. l T e c h n i c a l d e t a i l s of the remote sens ing f l i g h t s . . . . 2 3 2 B. 1 S t a t i s t i c s f o r CIC/Q model v a l i d a t i o n (with ' P r a i r i e - G r a s s ' data) 241 B.2 S t a t i s t i c s f o r CIC/Q model v a l i d a t i o n (with Hanford-30 data) 245 B.3 S t a t i s t i c s f o r <r -model v a l i d a t i o n (with y Hanford-30 data) 246 - v i i i -LIST OF FIGURES 1.1 Map of the Greater Vancouver Metropolitan Area 6 1.2 The Sunset tower, looking east 8 1.3 View from the top of Sunset tower, looking west 9 1.4 View from the top of Sunset tower, looking east 9 2.1 Idealized arrangement of boundary layer structures over a city 12 3.1 Local Nusselt numbers for crossflow about a circular cylinder. 27 4.1 Area of roughness element inventory 40 4.2 The video-framestore-VZIP configuration 41 4.3 VZIP-image of roughness element inventory, total area 43 4.4 Same as Figure 4.3, close-up of Mainwaring substation 44 5.1 Representativeness vs. sample size ...56 6. 1 Internal boundary layer interfaces with one-dimensional discontinuity 62 6.2 The source area in one-dimensional patchiness 62 6.3 Heat plumes 67 6.4 Estimates of source area dimensions 67 6.5 Schematic cross-section of a P-criterion source area 75 6.6 Integrated effect fraction : a) total integrated effect b) fraction 'P' of the total integrated effect. 77 - ix -7.1 The scaling regions of the atmospheric boundary layer 83 7.2 The source area Is defined by the set of characteristic dimensions of i t s bounding isopleth 92 7.3 Example plot of source area model results 93 7.4 mini-SAM validation scatter-plot : dimension a 100 7.5 Same as Figure 7.4 for dimension b 100 7.6 Same as Figure 7.4 for dimension c 101 7.7 Same as Figure 7.4 for dimension d 101 7.8 Same as Figure 7.4 for dimension x 102 m 7.9 Same as Figure 7.4 for area A 102 0 . 9 8.1 The Frisbee-fiIter 107 8.2 Daytime temperature sub-domains selected for Fourier-transforms 110 8.3 Nighttime temperature sub-domain selected for Fourier-transform I l l 8.4 False colour image of daytime surface temperature in the study area 113 8.5 Dominant spatial scales in suburban city-block system 115 8.6 Same as Figure 8.4; special colour coding to show streets and houses 117 8.7 Perspective view of daytime temperature 'topography' in area (0) 119 8.8 Temperature variance spectrum, daytime, area (0) 120 - x -8.9 Directional distribution of surface temperature variance : daytime, area (0) 121 8.10 Same as Figure 8.8 : daytime, area (1), 128x128 pixel sub-section 123 8.11 Same as Figure 8.10 : whole area 124 8.12 Same as Figure 8.9 : daytime, area (1) 125 8.13 Temperature variance spectrum (radial) : dayt ime, area (1) 125 8. 14 Normalized integrated radial variance spectrum of surface temperature :daytime, area (1) 125 8.15 Sectorial break-up of normalized integrated variance spectrum of surface temperature : daytime, area (1) 128 8.16 Same as Figure 8.8 : daytime, area (2) 131 8.17 Same as Figure 8.9 : daytime, area (2) 132 8.18 Same as Figure 8.13 : daytime, area (2) 132 8.19 Same as Figure 8.14 : daytime, area (2) 132 8.20 Same as Figure 8.15 : daytime, area (2) 133 8.21 Same as Figure 8.8 : daytime, area (3) 136 8.22 Same as Figure 8.9 : daytime, area (3) 137 8.23 Same as Figure 8.13 : daytime, area (3) 137 8.24 Same as Figure 8.14 : daytime, area (3) 137 8.25 Same as Figure 8.15 : daytime, area (3).... 138 8.26 Perspective view of nighttime temperature ' topography' 142 8.27 False colour image of nighttime surface temperature in the study area 143 - xi -8.28 Same as Figure 8.8 : nighttime 144 8.29 Same as Figure 8.9 : nighttime 145 8.30 Same as Figure 8.13 : nighttime 145 8.31 Same as Figure 8.14 : nighttime 145 8.32 Same as Figure 8.15 : nighttime 146 8.33 Area of roughness element inventory showing the sub-domains selected for Fourier transforms 150 8.34 Perspective view of roughness element distribution (128x128 pixel sub-set) 151 8.35 Roughness variance spectrum : area SE 152 8.36 Directional distribution of roughness variance : area SE 153 8.37 Roughness variance spectrum (radial) : area SE 153 8.38 Normalized integrated radial variance spectrum of roughness elements : area SE 153 8.39 Sectorial break-up of normalized integrated variance spectrum of roughness elements : area SE 154 8.40 Same as Figure 8.35 : area SW 156 8.41 Same as Figure 8.36 : area SW 157 8.42 Same as Figure 8.37 : area SW 157 8.43 Same as Figure 8.38 : area SW 157 8.44 Same as Figure 8.39 : area SW 158 8.45 Same as Figure 8.35 : area NE 160 8.46 Same as Figure 8.36 : area NE 161 8.47 Same as Figure 8.37 : area NE 161 8.48 Same as Figure 8.38 : area NE 161 - x i i -8.49 Same as Figure 8.39 : area NE 162 9.1 Measurement sites 172 9.2 Culloden s i t e (looking NE) 173 9.3 Argyle site (looking W) 173 9.4 Waverley s i t e (looking N) 174 9.5 Memorial East s i t e (looking SE) 175 9.6 Memorial East s i t e (looking SW) 175 9.7 Memorial West si t e (looking NE) 176 9.8 Memorial West site (looking W) 176 9.9 Arrangement of instruments on the Sunset tower 181 9. 10 Arrangement of instruments on the mobile tower 181 9.11 Schematic of the Sunset tower and instrumentation 182 9.12 Daily variations of Q at Sunset (dashed) and J H Culloden (solid) 183 9.13 Scatterplot of Q instrument comparison 186 H 9. 14 Scatterplot of Q vs. Q 186 Hmob Hsun 9.15 Scatterplot of vs. 0.9-source area size 189 9.16 Spread of Q„ vs. 0.9-source area size ^ H D I F F (residuals of |Q„ - LOWESS-curve | ) 189 H D I F F 9.17 Same as Figure 9.15, for calibration period 190 9.18 Same as Figure 9.16, for calibration period 190 9.19 Three examples of source areas in the study area 195 10.1 u vs. u (at Cabauw) 201 *22.S » 3 . S 10.2 Cabauw SAM-results (JD 140, 3.5 m) 203 10.3 Cabauw SAM-results (JD 140, 22.5 m) 204 - x i i i -10.4 Cabauw SAM-results (JD 174, 3.5 m) ...205 10.5 Cabauw SAM-results (JD 174, 22.5 m) 206 10.6 Cabauw SAM-results (JD 161, 3.5 m) 207 10.7 Cabauw SAM-results (JD 161, 22.5 m) 208 10.8 Cabauw si t e and surroundings 210 10.9 Exponent of internal boundary layer growth vs. (-L) 212 11.1 Representativeness conditions at Sunset tower 218 A.l Schematic of Falcon-20 aircraft 232 A. 2 MSS scan-head 233 A.3 Plan of ground truth sites at Langara Community College 235 A.4 VZIP-image of temperatures at ground truth sites 236 A. 5 Daytime pixel value-temperature relation 238 B. l Scatterplot of CIC/Q model validation (with 'Prairie Grass' data) 241 B.2 Effects of Dyer (1970) profiles compared to Dyer and Bradley (1982) profiles on p.d.f.-model results 242 B.3 Effects of k = 0.41 compared to k = 0.40 on p.d.f-model results 242 B.4 Scatterplot of CIC/Q model validation (with Hanford-30 data) 245 B.5 Scatterplot of <r-model validation (with y Hanford-30 data) 246 - xiv -L I S T O F S Y M B O L S A N D A B B R E V I A T I O N S Note : o Some symbols are only relevant for one equation and are not l i s t e d here. o The Systeme Internationale (SI) units and their equivalents are used throughout this work, unless otherwise specified. They are indicated in brackets []. a isopleth dimension (Figure 7.2) [m] a i, a 2 constants in non-dimensional profiles (Chapter 7) A - empirical constant (equation (3.5)) - function of the shapefactor s (equation (7.6)) A p source area contained in P-isopleth [m ] b isopleth dimension (Figure 7.2) [m] b f slope of linear functional relationship (Appendix E) b^, b slope of linear regression (Appendix E) B - empirical constant (equation (3.7)) - function of shapefactor s (equation 7.7)) c - isopleth dimension (Figure 7.2) [m] - empirical constant (equation (7.8)) c p specific heat of a i r [J kg _ 1K - 1] c arbitrary constants 1,2.. CIC Cross-wind Integrated Concentration [kg m ] Cov( ' ) covariance d - isopleth dimension (Figure 7.2) [m] - zero displacement height [m] - XV -- coefficient of agreement (equation (E.5)) —2 —1 E water vapour flux density [kg m s ] Efc total effect (equation (6.15)) Exp. peak exposure (Appendix B) [10 kg s m ] E(-) expected value f - frequency - constant in LOWESS (Appendix E) f function of non-dimensional travel time (equation y (7.12)) F - percentage of effective fetch (equation (6.10)) - total irradiance (Appendix A) [W m ] g acceleration of gravity [m s ] GHIS Gryning et.al. (1987) Gr Grashof number (equation (3.6)) h power-spectral density (equation (5.3)) H integrated spectrum power (equation (5.10)) JD Julian Day k von Karman constant K eddy d i f f u s i v i t y [m 2s _ 1] 1 characteristic length scale [m] L - Monin-Obukhov length [m] - length of data^domain (Chapter 5) [m] LAT Local Apparent Time (solar time) m - empirical exponent (equation (3.7)) - sample size (Chapter 5) - xvi -n Nu 0 P p.d.f. P PBL q Q V ^ H D I F F r H R Re Ri RMSE RMSD R2 R2 s SI, S2 - empirical exponent (equation (3.5)) - exponent of internal boundary layer growth (equation (6.4)) - number of data Nusselt number (equation (3.4)) observed variable (Appendix E) empirical constant (equation (7.8)) probability density function - predicted variable (Appendix E) - fraction of total effect contained in P-isopleth (equation (6.16)) Planetary Boundary Layer constant in LOWESS (Appendix E) source strength [kg s 1] _2 sensible heat flux density [W m ] non-dimensional inter-site Q H-difference (equation (9.1)) aerodynamic resistance to heat transfer [s m"1] measure of representativeness (equation (5.7)) Reynolds number Richardson number Root-Mean-Square-Error (equation (E.2)) Root-Mean-Square-Difference (equation (E.2)) residential (2 levels) land-use class (Auer, 1978) coefficient of determination (equation (E.l)) shape factor (equation (7.9)) surface types 1, 2 (Chapter 6) - X V I 1 -SAM Source Area Model t/T non-dimensional travel time (equation (7.12)) y T absolute temperature [K] u wind-speed [m s"1] um roughness wind-speed [m s - 1] U uniform or bulk wind-speed [m s"1] v lateral wind-speed [m s - 1] VZIP Vectrix-Zenith-Image-Processor V A R( ') var i ance w vertical wind-speed [m s - 1] w# convection wind-speed [m s - 1] v source weight per area (equation (7.18)) W weight of a discrete source (Chapter 6) x^ ( i ) distance from leading edge of equilibrium ( i n i t i a l effect) boundary layer (Chapter 6) [m] X L along-wind width of evaporation s t r i p (equation (6.8)) [m] X F F-percent effective fetch (equation (6.10)) [m] x upwind distance of maximum source [m] max X arbitrary variable z height [m] z t height of mixed layer, inversion height [m] z sensor height [m] s Z T roughness length (temperature) [m] Z Q roughness length (wind) [m] z depth of the roughness sublayer [m] - x v i i i -1/T [K - 1] gamma function height of internal boundary layer [m] height of equilibrium boundary layer [m] emissivity a i r temperature, potential temperature [K, or °C] 2 —1 heat conductivity [m s ] wavelength [m] 2 —1 kinematic viscosity [m s ] dummy variable density of dry a i r [kg m ] _3 vapour density [kg m ] - standard deviation - Stefan-Boltzmann constant [W m K ] - aerodynamic surface stress [N m ] - lag (time or space) non-dimensional profile (momentum, heat) normalized concentration [s m ] integrated non-dimensional profile (momentum, heat) wind-direction [rad, or degrees] source weight source weight function (bar) mean of variable (circumflex) denotes the spatial v a r i a b i l i t y of a variable (this is meant simply as a shorthand abbreviation, independent of the s t a t i s t i c a l - x i x -r e p r e s e n t a t i o n ) (•) ( t i l d e ) e s t ima ted va lue o f v a r i a b l e s u b s c r i p t s : ca i ib c a l i b r a t i o n p e r i o d f f o r c e d c o n v e c t i o n f i lm f i l m temperature 1 l amina r c o n v e c t i o n H heat max maximum mob mobi le s i t e s M momentum n n a t u r a l c o n v e c t i o n R r a d i a t i v e temperature sun Sunset s i t e sys s y s t e m a t i c tot t o t a l unsys unsys temat ic - XX -ACKNOWLEDGEMENTS Without the encouragement, patience and shared wisdom of many individuals this thesis could not have evolved from a vague idea to i t s present state. My research supervisor, Dr. Timothy R. Oke, has always been supportive, in more than just academic matters, when help was needed. At the same time he has given me a sense of freedom to work according to my own interests. He provided guidance by putting my research, climatology and science into a greater context, rather than pointing in a particular direction. His sense of humour - and the 'Great Laughter' - always was and always w i l l be appreciated. Similarly, I am indebted to Dr. Douw G. Steyn for his open door and mind, both of which were often used to chase down problems concerning almost anything in the universe. Informal discussions over a glass of beer or through ice-hockey face-masks have helped to shape many ideas for both this thesis and future work. By convincing me to come to fieldschool in Spring of 1985, he just might have changed my l i f e in the very best of ways. The constructive comments from the other two members of my committee, Drs. T. Andrew Black and Gordon A. McBean, were instrumental especially for the f i n a l draft of this thesis. Their help and encouragement Is greatly appreciated. Dr. Keith Knight from U.B.C.'s Statistics Consulting And Research Laboratory has helped to translate some of my ill-formed notions into s t a t i s t i c a l concepts. Janet Whiteside drafted the roughness element inventory and Mark Roseberry upgraded the VZIP system for this work. While working on this thesis, my personal funding was provided by University of B r i t i s h Columbia Graduate Fellowships, Scholarships by the Janggen-Pohn Foundation, Switzerland, as well as Teaching Assistantships from the Geography Department. The research was funded by a grant to Dr. Oke. During the course of the last three and a half years I was greatly influenced by many friends and fellow Graduate Students. I particularly want to thank Helen Cleugh and Sue Grimmond for their support during the 1985/86 fieldseasons and - together with Catherine Souch - for many hours of brainstorming over our morning coffee. Scott Robeson and Matthias Roth were and are invaluable as friends and fellow scientists. I thank my parents and family for their continued support over the years and across so many miles and for sending a l l those 'hints' to remind me where 'Home' is. Lastly, I want to thank Carolyn Porter for being so patient and for being my companion in a l l walks of l i f e . - 1 -PART I : INTRODUCTION 1. Academic and Spatial Context 1.1 RATIONALE In recent years the numerical model l ing o f atmospheric boundary l ayer processes f o r va r ious a p p l i c a t i o n s has r e c e i v e d cons iderab le a t t e n t i o n by researchers i n a l l f i e l d s o f boundary l ayer meteorology. With increased s o p h i s t i c a t i o n and improved s p a t i a l r e s o l u t i o n , such models need to incorpora te s u r f a c e l ayer processes and account f o r t h e i r v a r i a b i l i t y at p r o g r e s s i v e l y f i n e r s p a t i a l s c a l e s . Th is development faces two fundamental d i f f i c u l t i e s : F i r s t l y , most o f the p r i n c i p a l target areas f o r mesoscale models are not c o n s i s t e n t with the h y p o t h e t i c a l , i n f i n i t e , homogeneous and smooth planes f o r which present sur face l ayer s i m i l a r i t y theory i s developed. T h i s i s e s p e c i a l l y t rue f o r urban areas , a g r i c u l t u r a l c r o p - l a n d , orchards and f o r e s t s , i . e . the areas of main human-climate i n t e r a c t i o n , which are h i g h l y v a r i a b l e on a wide v a r i e t y of s c a l e s i n respect to t h e i r s u r f a c e geometry, thermal and moisture regimes. Present s u r f a c e l ayer meteorology does not provide a c o n s i s t e n t t h e o r e t i c a l or conceptual framework f o r t r a n s f e r processes over such heterogeneous t e r r a i n . Secondly, experimental data of energy, mass and momentum f l u x e s over very heterogeneous t e r r a i n are sparse , due to - 2 -methodological problems in the measurement process. Fluxes close to a complex surface are influenced by individual surface features (roughness elements and sources or sinks). This raises the d i f f i c u l t y of interpreting a flux measurement, as well as obtaining flux measurements representative of a large area. The assessment of the spatial v a r i a b i l i t y of surface fluxes is thus greatly impaired by our i n a b i l i t y to identify a specific surface region in the v i c i n i t y of the sensor as the main source area of influence upon a measurement. These d i f f i c u l t i e s apply to any surface turbulent flux of momentum, heat or mass. Interest in the spatial v a r i a b i l i t y of these processes is considered to be j u s t i f i e d in i t s own right. However, i t also arises from specific problems in practical applications, such as the estimation of regional evaporation, obtaining an areal average of roughness windspeed or estimating the total pollutant emission of an area source. The present study focusses on the spatial v a r i a b i l i t y of sensible heat flux. This specific flux has been chosen for a number of reasons, some are fundamental and some are pragmatic in nature. It may be said that the accurate measurement of sensible heat flux is the "easiest" of the various surface turbulent fluxes : sonic anemometer/thermocouple systems are very simple to use and the underlying technology is far advanced compared to instruments for the measurement of other fluxes. In addition, two similar eddy correlation systems for sensible heat flux were readily available to perform simultaneous measurements at two spatially separated sites. - 3 -However, the s e n s i b l e heat f l u x (and i t s s p a t i a l v a r i a b i l i t y ) i s s i g n i f i c a n t a l s o i n more fundamental terms. The thermal regime o f the atmosphere i s a powerful c o n t r o l on i t s t u r b u l e n t s t a t e : i n s t a b l e c o n d i t i o n s , t u rbu lence and m i x i n g a re damped whereas they are g r e a t l y enhanced i n uns t ab l e c o n d i t i o n s by the c o n t r i b u t i o n o f buoyancy to the t u r b u l e n t k i n e t i c energy. D u r i n g the dayt ime and i n the absence o f condensa t ion p rocesses o r c l o u d s , s e n s i b l e heat f l u x i s the main heat input i n t o the atmosphere and thus determines to a grea t ex ten t the thermal regime o f the boundary l a y e r . As a consequence, i t i n f l u e n c e s a l l o t h e r t u r b u l e n t f l u x e s v i a buoyancy and thus feeds back upon i t s e l f i n a s t r o n g l y n o n - l i n e a r manner. The su r f ace s e n s i b l e heat f l u x i s a l s o one o f the p r i n c i p a l c o n t r o l s o f the he igh t o f the mixed l a y e r . S p a t i a l l y r e p r e s e n t a t i v e e s t ima te s o f t h i s f l u x are t h e r e f o r e an e s s e n t i a l input t o any mixed l a y e r model which e v a l u a t e s the volume o f a i r th rough which p o l l u t a n t s , moi s tu re and heat are mixed and d i s p e r s e d . T h i s s tudy seeks to e l u c i d a t e the s p a t i a l s c a l e s o f v a r i a b i l i t y o f s e n s i b l e heat f l u x over a complex s u r f a c e , v i z : a suburban a r e a i n Vancouver, B . C . , Canada. I t a l s o s e t s out to e s t ima te the su r f ace source a r ea which i n f l u e n c e s an eddy c o r r e l a t i o n measurement i n uns t ab l e c o n d i t i o n s . F u r t h e r i t t r i e s to assess the consequences o f a complex su r f ace on the s t r u c t u r e o f the lower p o r t i o n o f the a tmospher ic su r f ace l a y e r . In the f i r s t i n s t ance , the r e s u l t s o f t h i s s tudy are o n l y c h a r a c t e r i s t i c o f the suburban a r e a under c o n s i d e r a t i o n . The method by which they were ob t a ined , however, may be a p p l i e d to a - 4 -v a r i e t y of complex surfaces. Thus, i t s findings o f f e r potential u t i l i t y to both the boundary-layer modelling and measurement communities. The methods used are drawn from f i e l d s such as micrometeorology, a g r i c u l t u r a l meteorology, heat transfer engineering, turbulent d i f f u s i o n modelling, applied s t a t i s t i c s and remote sensing. 1.2 STUDY OBJECTIVES The objectives of t h i s work can be summarized as follows : 1. To develop a method by which the small scale v a r i a b i l i t y of sensible heat f l u x may be evaluated. 2. To formalize the concept of "representativeness" i n the horizontal s p a t i a l sense and i n the context of sensible heat f l u x measurements. 3. To examine the process by which s p a t i a l inhomogeneities i n the lowest layers of the atmosphere are propagated and diffused v e r t i c a l l y . 4. To develop a model which estimates the r e l a t i v e influence of each surface element on the sensible heat f l u x at any given point i n the surface layer. 5. To discuss the consequences of small scale thermal v a r i a b i l i t y within the surface f a b r i c on the structure of the unstable surface layer of the atmosphere over complex surfaces. 6. To support the hypotheses and th e o r e t i c a l findings of t h i s work by f i e l d observations. - 5 -1.3 THE STUDY AREA The observation programme for this work was conducted in a suburban area of Vancouver, B r i t i s h Columbia, Canada. It has been the target area for a number of process oriented urban climate and micrometeorology studies in the past (e.g. Oke 1979b; Kalanda, 1979; Steyn, 1980; Oke and McCaughey, 1983; Oke and Cleugh, 1987). The present work was completed in parallel with other studies in the same area, which together form a comprehensive suburban surface-layer and coastal/urban boundary-layer research programme (Cleugh, 1988 ; Grimmond, 1988 ; Roth, 1988 ; Steyn and McKendry, 1988). As a result of the abundant "research heritage" in this area, most of the footwork in respect to site selection and surface description and parameterization has already been completed (Kalanda, 1979 ; Steyn, 1980) and can readily be adopted for the present study. A brief summary of the general setting of the study area within the region and the specific characteristics of the main suburban site (the Sunset site) are given in the following. Vancouver is located between the mouth of the Fraser River and Burrard Inlet on the Strait of Georgia (see Figure 1.1). The lower Fraser Valley forms an extensive lowland which i s bounded by the Cascade Range in the south and by the Coast Mountains to the north, both of which reach elevations in the order of 1500 m above sea level. The climatology of Vancouver and the lower Fraser Valley is discussed by Hay and Oke (1976). Weather patterns in winter are typically characterized by the passage of - 6 -Figure 1.1 : Map of the Greater Vancouver Metropolitan. Area - 7 -cyclones and their associated frontal disturbances, resulting in high precipitation and cloud cover, but also r e l a t i v e l y mild temperatures. In contrast, the summers are commonly known for their persistent anticyclonic high-pressure systems. The generally small pressure gradients and the resulting weak synoptic flow are conducive to thermally induced diurnal weather patterns. The most common of these meso-scale systems is the sea-breeze, which is enforced by the effects of mountain/valley-and slope-wind systems (Steyn and Faulkner, 1986). Due to the sea-breeze and the associated advection, the height of the mixed layer is typically confined to only about 500 m (Steyn and Oke, 1982; Steyn and McKendry, 1988). The main micrometeorological site of this study, the Sunset tower, is located in south Vancouver (Figure 1.1) in an area of mainly residential housing (category R2, after Auer, 1978) with occasional schools or commercial neighbourhood centres.The mean building height within a c i r c l e of more than 1 km radius of the tower was evaluated by Steyn (1980) to be 8.5 m. A land-use/roughness element analysis by the same author resulted in a mean roughness length of 0.5 m (using the method of Lettau,1969) and a zero-displacement height of 3.5 m. The tower is located in the compound of the B.C.-Hydro and Power Authority's Mainwaring substation (Figure 1.2). The steel-frame structure has an overall height of 27.5 m. Since the base of the tower is below an escarpment of 5 m, the height above ground needs to be corrected by this amount (see Figure 9.11, in Section 9.3, on instrumentation), so that the maximum Figure 1.2 : The Sunset tower, lo o k i n g east - 9 -F i g u r e 1.3: View from t h e t o p o f Sunset tower , l o o k i n g west F i g u r e 1.4: View from the t o p o f Sunset tower , l o o k i n g e a s t effective height (z-d) is 19 m above the effective zero datum (Steyn, 1980). Figure 1.3 shows the view from the top of the tower in the westerly direction : i t indicates a uniform land-use of residential housing outside the substation compound in this direction. About 100 m to the east of the tower there i s a large school building (height about 20 m, see Figure 1.4). With strong flow from that sector, which is rare in summer and during daytime, this may result in some wake effects. PART II : THEORY AND CONCEPTS 2. Physical Context The present study is directed towards providing a beter understanding of the way in which smal scale surface inhomogeneits influence the turbulent flux of heat in the lowest layers of the atmosphere, particularly over suburban terrain. It focuses on one aspect of a very complex system and is limited to relatively smal horizontal and vertical scales. It is therefore useful and necesary to provide a synopsi  of the physical and conceptual context of this limited aspect, in order to set the stage for the detailed analysis of the spatial scales of sensible heat flux variability. Because of the spatial complexity of the urban terrain it is convenient to consider not one system, but a hierarchy of nested systems and scales, folowing Oke (1984). Being concerned with the interactions betwen the surface and the atmosphere, the upper scale limit of relvance is the planetary boundary layer (PBL). Concepts and problems of boundary layer research have ben reviewed by McBean (Ed., 1979) and more recently by Lenschow (Ed., 1986). Figure 2.1 (from Oke, 1984) shows an idealized arangement of the boundary layer structure over a city. This arangement refers primarily to the daytime structure, where the boundary layer is commonly equated with the convectively driven mixed layer, whose capping inversion forms - 12 -(al "Plume' P8L / Mixed layer UBL L. Ilil , Surface layer " y y-Wau. Rural / Urban Rural Surface layer UBL Roughness layer •<fTT - D - r Y p W " : - A y ^ r S : - ^ r f i < j O r>"ucL Figure 2.1: Idealized arrangement of boundary layer structures over a c i t y (from Oke, 1984). Table 2.1 : Possible framework for urban climate c l a s s i f i c a t i o n (from Oke, 1984). (a) TUftAULCHT BOUNDARY LAYERS Mow C h a r a c t e r i s t i c * 1, Urban canopy l b u l l d l n 9 ) H l o h l y t u r b u l a n t , ! • / • ( (UCD c o n t r o l l e d by rouahnaaa I U M H C I T u r b u l e n t <*»Km l a y e r 11. Urban bounda ry l a y e r (UULt (bl UKAAH M O R P H O L O G Y H i g h l y t u r b u l e n t vekea and p l u M « a , t r a n s i t i o n t o n * T u r b u l a n t , I n c l u d e s awrfac t And n l m d l a y e r s 5 H<i> t y p i c a l l y 10 -2 D - 3 Q ^ t y p i c a l l y 20-40 * D«p«nd * on au r f ace f l u * t * of heat and montn tun . . T y p i c a l l y - day 1 lut, n i g h t 0 .2 >ua L o c a l and n<io Urban u n i t * Urban t t a t u n • Urban c l i m a t e phanocaane. H 01*enal oni*^ W L Sca l e I . B u i I d i n q 3. Canyon S l n o l e l A t l l d l n q , t r c a or garden Urban ( t r e a t and b o r d e r i n g b u i l d i n g * o r t r e a a waka , p l u i M or ahado* Canyon a h e l t e r , c i r c u l a t i o n , ahada, b l o c l Uta te 10at 10» 10* 3On 200* M i c r o ) . B l o c k ( n « I q f i b O u r -C i t y b l o c k , park f a c t o r y coup i t K C l l M t a * of p a r k a , b u i l d i n g c l u s t e r * , cuatulua, a t l n l - b r e e s e e O.Skn Q.SkA 4. L a n d - u a * ten* R e s i d e n t i a l , C (4MM (Cl tk , 1 n i l u t i r l a l a t e . L o c a l c l i e n t at I n c l . w i n d s , c l o u d a o d l -f I c a t i o n Skn Skn L o c a l C i t y Urban a r a a l i t at l a l a n d , urban c i r c u l a t i o n , urban a f f e c t s In g e n e r a l J l l u . 21k« ^ M*SO • D l M f i t l c n t of uni t* i n tho* Lound*ry I m y i t >r i dapLh • of u i & u i i i r u c t u r * * or • of a f fac tad i U o i p h » f t | d l p lan a r a a . M A t t o n a of • o r p h o l o g l c a l b u i l d i n g h a t g h t , O - h w t l d t n g » p a c l n g - 13 -the limit of the PBL (see e.g. Steyn, 1980). The urban surface modifies the lowest layers of the atmosphere, which are (ideally) adjusted to rural surface conditions before their impingement on the city. If the urban area is extensive enough, the growing urban boundary layer w i l l eventually include the entire depth of the PBL. Similarly, a rural boundary layer develops again at the downwind leading edge of the urban/rural transition (see Figure 2.1). Spatial v a r i a b i l i t y within the atmosphere at that scale has received considerable attention in the past by studies focussing on various processes and aspects of climate and does not need further elaboration here (see e.g. Oke (1979a) for a review). Auer (1978) identified several different land-use types within urban areas and the v a r i a b i l i t y of fluxes at the land-use scale has been a focus of part of the US EPA* s Regional Air Pollution Study (RAPS) in St.Louis (e.g. Ching et a i . , 1984). In the present study, spatial v a r i a b i l i t y within the land-use 3 scale is of interest. With length scales up to the order of 10 m and time scales up to one hour, this work looks at v a r i a b i l i t y in the micro-a and micro-|3 scales, according to the Orlansky (1975) classification. The lower portion of Figure 2.1 is a good schematic of the dominant atmospheric and surface components at those scales. The surface layer above the bluff-rough urban or suburban surface needs to be considered in two parts. The upper portion, termed the inertial sublayer, comprises the vertical range of v a l i d i t y of Monin-Obukhov similarity theory (Raupach and Thorn, 1981) and is the result of a double matching process - 14 -of upper-level and surface layer scaling laws (Tennek.es, 1981). In the i n e r t i a l sublayer, vertical profiles of wind, temperature and humidity do not depend on external scale lengths such as the characteristic dimensions of the surface geometry. It is also often referred to as the constant flux layer, because vertical fluxes vary by only about 10 %, both in the horizontal and vertical directions (Dyer and Hicks, 1972). By contrast in the lower portion of the surface layer, the profiles are influenced by individual surface elements and the flow is three dimensional (Raupach and Thorn, 1981). Several names have been coined for this layer, including turbulent wake layer, transition layer or roughness sublayer. The term transition layer refers to the spatial averaging process that applies in this layer : as the spatial atmospheric inhomogeneities, induced by the surface, propagate upwards with eddy-motion, turbulent mixing results in progressive horizontal averaging with increasing height. The spatial inhomogeneity is in transition between the surface, where i t is greatest, and a height in the atmosphere where the horizontal inhomogeneities disappear completely. The upper limit of this transition zone or roughness sublayer is the lower boundary of the i n e r t i a l sublayer (Garratt, 1978a). This feature (Figure 2.1) is examined in the discussion of the present work. Table 2.1 (from Oke, 1984) summarizes the above framework of nested scales and the associated features of urban morphology. The rationale behind this c l a s s i f i c a t i o n of the vertical structure and the relevant horizontal scales is mainly based on phenomena of airflow and turbulent transport and is well suited - 15 -for the present study. The heat, which controls thermal influences on turbulence and diffusion via buoyancy, is received and transformed at the surface. The energy balance associated with this partitioning has been discussed extensively by Oke (1982),Cleugh and Oke (1986) and Oke and Cleugh (1987) with special reference to the study site. Rather than giving a review of current boundary layer theory and surface layer scaling the reader is referred to such texts as Tennekes and Lumley (1972), Nieuwstadt and Van Dop (1981), Pasquill and Smith (1983), and Panofsky and Dutton (1984). These are considered to reflect the "state of the art" of boundary layer theory in the recommendations by Weil (1985) and have been used as works of reference thoughout this thesis. 3. The S p a t i a l V a r i a b i l i t y o f S e n s i b l e Heat F l u x 3.1. INTRODUCTION The principal effect of turbulence in the atmosphere is the mixing of atmospheric constituents and properties. Vertical and horizontal discontinuities and gradients of any kind are drastically reduced by this mixing. Turbulent sensible heat flux is a process which is largely generated at the surface and then transmitted through the overlying atmosphere by turbulent diffusion. The heat carrying eddies are therefore subject to mixing not only in the vertical, but also in the horizontal direction. As a result, surface turbulent fluxes that are induced by a f i e l d of discrete surface patches become spati a l l y averaged with increasing height. Atmospheric measurements of sensible heat flux are thus inherently unable to detect the discrete nature of i t s spatial v a r i a b i l i t y at the surface. An obvious solution to this problem is to assess the spatial v a r i a b i l i t y of surface sensible heat flux (or any turbulent surface layer flux) based on the spatial v a r i a b i l i t y of the surface forcing parameters for this flux. The v a r i a b i l i t y of relevant surface conditions constitutes the maximum possible spatial v a r i a b i l i t y of the flux. The extent to which these spatial differences are reduced with height may then be evaluated i f an estimate for the spatial averaging process by turbulent mixing is available. This spatial averaging process is - 17 -discussed in Chapter 5 and a model for i t s calculation is presented in Chapter 6. In the following, the focus is on the identification of surface parameters that may be used to evaluate the spatial scales of the v a r i a b i l i t y of the sensible heat flux f i e l d . The magnitude of the sensible heat flux w i l l of course change over time due to the temporal v a r i a b i l i t y of the solar energy input into the Earth-Atmosphere system (i.e. diurnal and annual cycles, further modulated by more irregular large-scale weather va r i a b i l i t y ) . This type of temporal v a r i a b i l i t y is s p a t i a l l y coherent over scales at least an order of magnitude larger than the longest spatial dimensions (ca. 5 km) considered in the present study, except in conditions of partial cloud cover and in areas where orography has a significant effect on the mean cloud cover (Hay, 1984). The northern side of Burrard Inlet in Greater Vancouver is subject to strong spatial v a r i a b i l i t y of incoming solar radiation due to orographically induced cloud, but the present study area li e s outside this zone. Therefore, the solar forcing i t s e l f does not introduce substantial spatial v a r i a b i l i t y into the system. Some surface parameters may, however, change temporally as a response to the change in the solar forcing (e.g. temperature or s o i l moisture). Due to differences in surface material the response (e.g. thermal) can vary for different surface elements, so that not only the magnitudes of these surface parameters change, but also their spatial distribution. Obviously, however, the distribution of relevant surface parameters governing sensible heat flux - 18 -v a r i a b i l i t y have to be compiled at a specific time and the analysis based on this distribution is s t r i c t l y speaking only valid for the situation at the time of the data acquisition or for times when the distribution is similar to that situation. To amend this problem, i t is suggested to compare two datasets that are expected to have different surface parameter distributions (e.g. one dataset obtained during the time of maximal convective a c t i v i t y and the other during very stable conditions). Sensible heat flux i s conveniently given in terms of a Monin-Obukhov similarity relation which, in i t s integrated version, takes the form QH = - (9 z -8 Q)-k p c u , / (ln((z-d)/zT)-i/»H(z/L)) . (3.1) This equation is a semi-empirical relation and was developed for heat transfer over ideal surfaces (i.e. for i n f i n i t e , homogeneous planes which contain no large bluff bodies). It is valid only in the in e r t i a l sublayer portion of the surface layer in which turbulent transfer is free of any explici t dependence on external length scales, as a direct consequence of an asymptotic double limit matching process of upper layer similarity and surface layer similarity laws (Tennekes, 1981). Specifically, the lower limit of the i n e r t i a l sublayer, where Z / Z q - » oo, has to be well above the roughness elements. In contrast to ideal conditions, the present work considers the interaction of small-scale heat flux sources or sinks with each other In an environment, where the larger roughness elements (e.g. buildings) are l i k e l y to be of the same spatial scales at which this interaction occurs. Since the distribution of sources and sinks is three-dimensional the diffusion processes between the roughness elements, in the canopy layer, wi l l have to enter our considerations. Clearly, the conditions under which the homogeneous relation (3.1) may be used are not met. Current boundary layer meteorology does not provide a consistent and unequivocal theory for transfer processes in a canopy layer (e.g. Baldocchi and Hutchison, 1987), or even in conditions of horizontal heterogeneity. The scenarios where applications of surface layer theory are of greatest practical interest (i.e. in forests, agricultural crops, urban areas etc.) do not meet the ideal equation conditions in general. Mainly due to the lack of anything better, the relationships developed for homogeneous surfaces have therefore been used, and partly modified with some success, in such situations. This is especially true for modelling purposes (e.g. Taylor (1970), Garratt (1978b), Shuttleworth and Wallace (1985) and others). In these works heterogeneity is limited to one dimensional step changes (Taylor, 1970) or the transfer processes are looked at as averages over larger areas (Garratt, 1978b; Shuttleworth and Wallace, 1985). In the present study the small-scale v a r i a b i l i t y i t s e l f is of interest and concepts such as the f r i c t i o n velocity and especially a roughness length become meaningless when applied to individual surface elements. Therefore (3.1) should not be used to discuss this variability, since i t assumes semi-logarithmic profiles of wind and temperature above each - 20 -surface element with no local advection from neighboring surfaces. The results of empirical or semi-empirical correlations used in hydraulic engineering and adopted by biometeorologists (Monteith, 1973) may be able to provide the most guidance on the subject of small-scale heat flux v a r i a b i l i t y and the relevant controlling variables, since these correlations refer to the local heat flux from a single element, rather than an average over a large area. 3.2 PRINCIPLES The analysis of convective heat transfer problems has been greatly simplified by using non-dimensional groups of the quantities involved. In this way similar transfer situations may be compared : the correlations of the relevant non-dimensional groups provide a qualitative and quantitative expression for the heat transfer regime. For such an analysis the convective heat flux is most conveniently expressed in terms of a temperature difference between the surface and the f l u i d and an aerodynamic resistance r to the transfer : H Q = p c (9 -6 )/ r . (3.2) H K p o z ' H Note that, in an ideal and homogeneous case, (3.1) and (3.2) are compatible through the appropriate expression for r : H r = (l/(k-u ))-(ln((z-d)/z (z/D) (3.3) H * T H The fact that r in (3.2) is a 'black box' and does not assume a - 21 -specific form of profile, such as the semi-logarithmic prof i l e in (3.3), i s the very reason why i t is appropriate for the present purposes. The most important non-dimensional group which contains this r i s the Nusselt number : H Nu = p c l/(ic r ) . (3.4) P H The Nusselt number may be interpreted as the ratio of the conductive thermal resistance to the convective thermal resistance of the f l u i d (Welty et al., 1976) and Is therefore a measure of the efficiency of the convective transfer compared to the conductive transfer of heat in the same medium. Just as the Reynolds number is a convenient way to compare i n e r t i a l and viscous forces associated with geometrically similar bodies immersed in a moving f l u i d , the Nusselt number provides a basis for comparing rates of convective heat loss from similar bodies of different scale, exposed to different wind-speeds and temperature differences (Monteith, 1973). There are two main classes of convective heat transfer according to the forces which drive the flow. In forced convection the transfer is driven by an externally imposed flow such as wind. Here, the turbulence regime which f a c i l i t a t e s the heat flux in the boundary layer is controlled primarily by the geometry of the surface. In this case the dynamical simi l a r i t y of different systems expressed by Re is closely related to the thermal regime expressed by Nu. In air, with a Prandtl number of about 0.71 (independent of temperature), the Nusselt number for - 22 -forced convection can be written, following Monteith (1973), as Nu = A-Ren , (3.5) where values for A and n are tabulated for different types of geometry. Buoyancy induced flows, termed natural or free convection, arise simply because of density variations within the f l u i d (caused by differential heating processes) in a body-force f i e l d , such as gravitation (Jaluria, 1980). In this case the Nusselt number is a function of the Grashof number (and s t r i c t l y also of the Prandtl number). Physically, the Grashof number is the ratio of a buoyancy force times an i n e r t i a l force to the square of a viscous force (Monteith, 1973). It is calculated from Gr = a g l 3 (9 - 9 )/ v2 . (3.6) 0 z ' It follows that in atmospheric free convection Nu can be expressed as Nu = B-Gr™ (3.7) where B and m are tabulated for different geometries. Pure forced or pure free convection are clearly the limiting cases. In non-laboratory conditions both mechanisms w i l l carry some importance and the combined process is commonly known as mixed convection. The c r i t e r i a for forced, free or mixed convection are excellently described by Monteith (1973). As - 23 -indicated above, the driving force for free convection is buoyancy, whereas forced convection is generated by i n e r t i a l forces. The ratio of the strengths of these two limiting regimes is given by the ratio of their respective driving forces which 2 can be written as Gr/Re . This ratio is also known as the Bulk Richardson number in meteorology. Monteith reports rules of thumb regarding the prevalent transfer regime derived from 2 experimental evidence. When Gr/Re > 16, buoyancy forces are much stronger than in e r t i a l forces and the appropriate relation for free convection (3.7) should be used. Forced convection is 2 2 dominant when Gr/Re < 0.1. For intermediate values of Gr/Re , Nu should be calculated both for forced and for free convection and the larger number should be used to estimate the heat flux. 3.3 ANALYSIS The objective of this analysis is to look at the controlling variables for the spatial v a r i a b i l i t y of sensible heat flux and to identify the dominant surface variables. The spatial distributions of these variables may then be used as strong indicators of the spatial v a r i a b i l i t y of sensible heat flux. As a start, the limiting cases of forced and free convection w i l l be discussed. In the following, the spatial v a r i a b i l i t y w i l l be denoted by a circumflex (-) above the symbol for the variable. It may be interpreted as the square root of the spatial variance or the spatial standard deviation of the variable. First, let us consider the heat flux equation i t s e l f . On the - 24 -right-hand-side of (3.2), p and c p may be considered constant in space, they vary only s l i g h t l y with the bulk temperature in the surface layer. Therefore p = 0, c p » 0 and they introduce no v a r i a b i l i t y in Q . On the other hand, the temperature difference H between the surface and some height z is strongly variable in space. It is even conceivable that this mean vertical gradient may change sign over small horizontal scales. When (3.2) is looked at as an analogy to Ohm's law, (9Q~Q ) corresponds to the difference in potential and, i f everything else is held constant, i t controls the flux in a linear fashion. It is possible to simplify this variable for the present purposes, i f z is specified as a height where a l l horizontal gradients vanish. In this Instance 9 is a constant and (0 -G ) s e , so z o z o that the spatial v a r i a b i l i t y of the surface temperature is identified as a strong control of Q independent of the transfer H regime. The assessment of r^ is more problematic. If the temperature difference signifies the magnitude of the potential heat available to the flux, r , or rather 1/r is a measure of the H H efficiency of the f l u i d medium to transfer the sensible heat energy. Its function in the context of sensible heat flux is very similar to the function of a valve in a pipe. Formally in (3.2), f is at least as strong a control on Q as 6 . However, H 6 H O as w i l l be seen, depending on the convection regime, there is a considerable degree of correlation between 9 and f . - 25 -3.3.1 Forced Convection In forced convection, the Nusselt number is a function of the Reynolds number, and the corresponding resistance can be written p e l p e l r p r p Hf . n K A Re K A 1-u (3.8) With the viscosity v, the heat conductivity K , p and c F determined at bulk air temperature (3.8) may be simplified as r = c Hf J (A-l n _ 1-u n) (3.9) and when (3.9) is substituted into (3.2), the heat flux equation becomes Q = c -A (9 -8 ) . l B _ 1 . u B Hf 2 0 2 (3.10) where p-c /c = c . Values for A and n for different geometries r P 1 2 & are given by Monteith (1973) : n Re Flat plates (vert, or horiz.) 0.032 0.8 Cylinders 0.24 0.6 > 2-10 103-5-104 The characteristic length 1 is a measure of the distance over which a boundary layer can develop, i.e. the side length or the diameter of a plate and the diameter of a cylinder. In a suburban environment such length scales relate to buildings (and - 26 -the open spaces between them) and vary typically in the crude range of 10 m to 100 m. With the kinematic viscosity determined at 25 °C, and allowing canopy layer wind-speeds between 1 and 10 m-s 1, the resulting Reynolds number range is 6.5-105 < Re < 6.5-107 . (3.11) Even though the buildings in a suburban environment resemble rectangular boxes more than cylinders, Figure 3.1 is useful to il l u s t r a t e how the Nusselt number can vary on a cylinder as a function of angular distance from the stagnation point. It shows that for high Re, the local Nu may vary up to 50% from the maximum value. This means that even with constant temperatures the v a r i a b i l i t y of the local heat flux extends to 50% of i t s maximum and is controlled entirely by the geometry of the heat source and i t s orientation relative to the flow. A large obstacle, such as a building, also affects the flux on the horizontal surfaces adjacent to i t . In the immediate lee of the building, wind speeds and mixing with the surrounding a i r (and therefore the transport of heat) are greatly reduced due to the sheltering effect of the building and the trapping of a i r in re-circulation c e l l s . This area is known as the cavity zone or bubble (Oke, 1978). In strong flow conditions the cavity length extends to about 1.5 times the height of cubic obstacles in the down-wind direction and increases with the ratio of width to height of the obstacle (Hosker, 1984). In weak wind conditions, however, flow separation at the obstacle edges is less pronounced and the resulting sheltered cavity zone is l i k e l y to 0-Dci;rccl f'Oin stagnation point Figure 3. 1 : Local Nusselt numbers for crossflow about a c i rcu lar cylinder : a) at low Re, b) at high Re (from Welty et al., 1976). - 28 -be much smaller, or even negligible. In addition, as the s t r a t i f i c a t i o n becomes more unstable, a weakened shelter effect is to be expected due to enhanced vertical mixing. After this sheltered bubble zone, in the along-wind direction, the disturbance of the flow by the obstacle results in a turbulent wake zone which is noticeable for a considerable distance downwind. In this wake, although wind-speeds are reduced, the energy removed from the flow is dissipated and turbulence levels are increased. The well developed mixing in this wake region f a c i l i t a t e s the transfer of heat and reduces the effective aerodynamic resistance. In the context of spatial v a r i a b i l i t y i t is interesting to note at this point that because of the well developed mixing in the wake of a large obstacle, the aerodynamic resistance is affected by the flow around the obstacle at a great distance downwind, where a resistance calculated from local conditions alone might be quite different from the effective resistance due to the wake. If the canopy layer contains such large obstacles in more or less regular spacing (e.g. buildings in a suburban area), the flow and mixing conditions are characterized by cumulative wake interaction and generally good mixing, causing what Thorn (1972), and Shuttleworth and Wallace (1985) term a mean canopy airstream. The interesting point here is that in such conditions the aerodynamic resistance seems to vary only at larger spatial scales compared to the surface temperature var i a b i l i t y , due to the 'blurring' effect of the mean canopy flow. Claussen (1987) reports that this blurring is also evident - 29 -on the upwind side of an obstacle or a roughness change due to momentum transport by pressure fluctuations which propagate in a l l directions. This anticipation of a surface change by the flow applies to momentum transfer, where pressure fluctuations have an effect, but only indirectly to scalar fluxes. It seems that in forced convection the aerodynamic resistance r is controlled by the dynamics of the surface canopy layer H system. Parameters which introduce spatial v a r i a b i l i t y include the geometry of the individual surface elements (both shape and spatial dimensions) and the local wind-speed and turbulence regimes (Burns, 1980). Cumulative wake interaction indicates that there i s no continuum of scales in the spatial v a r i a b i l i t y of the wind f i e l d , so that the geometry and the turbulence regimes do not necessarily vary at the same spatial scales. The effect on the heat flux Is that both 9 and r 0 Hf (reflected by the spatial distribution of roughness elements) may assume roles of equal importance on Q . While Q is directly H H proportional to 9q, i t is inversely proportional to r . - 30 -3.3.2 Free Convection The situation is quite different in natural or free convection. Here, the Nusselt number is a function of the Grashof number (see equation (3.7)), so that we can write the corresponding resistance to natural convection : p e l p e l , . V p K p r K B Gr m K B a g l 3 ( 9 - 9 ) u z (3.12) The quantities p, c p, K, V, a and g may be considered constants and w i l l be substituted for by c 3 for simplicity. Equation (3.12) becomes r = c Hn 3' After substitution into (3.2), the heat flux equation for the case of natural convection follows as Q = c± B - l 3 - 1 - ^ -9 ) m + 1 (3.14) Hn 4 0 z where p-cp/c = c^. Values for B and for m are given by Monteith (1973) for the Grashof number ranges to be expected in a suburban environment (i.e. Gr in the order of 10 1 0 - 10 1 2) : B m Horizontal f l a t surfaces 0.13 0.33 Vertical surfaces 0.11 0.33 Considering that B does not change very much, and that m stays - 31 -constant for vertical and horizontal surfaces, Q can be Hn 1 33 considered proportional to (9 -9 ) . Thus, in natural 0 z convection the heat flux is almost completely controlled by the surface temperature and is v i r t u a l l y independent of geometry (B changes + 10%, see above). The conclusion is that in free convection 9 i s the dominant o control over Q , with the geometry having a minor effect. H 3.4 SUMMARY The main parameters introducing spatial v a r i a b i l i t y into the sensible heat flux for the two limiting cases of forced convection and free convection are apparent from equations (3.10) and (3.14) respectively. In free convection the situation is quite clear, with temperature v a r i a b i l i t y being the dominant control, and geometry having only a slight effect. Therefore, the choice of 9 q to represent is obvious in natural or free convection situations. The case becomes more d i f f i c u l t in forced convection. Equation (3.10) indicates that apart from the temperature (which is an important factor even here), the local wind speed and both the type of geometry and the length dimensions involved must be considered. The local wind speed is of course i t s e l f influenced by the geometry of the momentum-extracting roughness elements, which illu s t r a t e s the importance of the surface geometry even more. In the mixed convection situations which are to be expected in the real world, i t would therefore seem that a Q - 32 -representation by 0 q , as is suggested in the free convection case, loses i t s j u s t i f i c a t i o n when the transport process is increasingly characterized by forced convection features (i.e. near neutral thermal s t r a t i f i c a t i o n and windy conditions). It i s concluded that an evaluation of the spatial scales of sensible heat flux v a r i a b i l i t y may be performed adequately by examining the high resolution spatial distribution of surface temperatures and the distribution of dominant roughness elements. Appropriate methods to establish the necessary datasets for the spatial distribution of surface temperature and of roughness elements are developed and discussed in the next chapter. Due to temporal contingency considerations described at the beginning of this chapter, the surface temperature distribution should be evaluated at a time of highly developed convective activity, with a second dataset acquired in very stable conditions, to enable an assessement of the temporally changing temperature distribution on the spatial v a r i a b i l i t y . The overall temporal change of the roughness element distribution is assumed to be negligible. - 33 -4. S p a t i a l D i s t r i b u t i o n of Surface Temperature and Roughness Elements 4.1. SURFACE TEMPERATURE In view of the preceding chapter, a high resolution map of surface temperatures is needed as a base for the evaluation of sensible heat flux spatial v a r i a b i l i t y and in order to obtain an estimate of the dominant spatial scales of this flux. Inevitably this c a l l s for an appropriate definition of the surface and the corresponding surface temperatures, as well as a practical method to measure these surface temperatures at a high spatial resolution. In clas s i c a l micrometeorology over homogeneous planes the surface in respect to turbulent sensible heat flux is usually understood to be the virtual surface at the level (d + z ) above T ground, in the canopy layer. A time-averaged temperature measurement at this level is also a spatial average of a kind, i f the action of wind and turbulence and the resulting micro-advection across various surface elements is considered in Taylor's hypothesis. The micrometeorological definition of the surface and the surface temperature is therefore clearly not suitable for the present problem, where a spatially resolved temperature is wanted. Consider the turbulent sensible heat flux from an individual surface element. It is generated at the top of the thin laminar - 34 -boundary layer, through which the heat is transported by molecular diffusion. The temperature at the top of this laminar boundary layer is often called f i l m temperature (Welty, Wicks and Wilson, 1976) and is the appropriate temperature for equation (3.2), i f r is a resistance involving only the H turbulent heat transfer. With the exception of very smooth surfaces (e.g. ice, glass, polished metal or s t i l l water) this laminar layer i s extremely thin over natural surfaces and common building materials. If i t is assumed that the laminar layer has the same thickness over a l l surface elements, equation (3.2) may be reformulated : Q = p c (0 -0 )/ (r + r ) (4. 1) H P R z ' H Hi' where 0 is the radiation temperature of the surface and r is R HI the resistance to molecular heat transfer across the thin laminar boundary layer. In daytime convective conditions these radiative temperatures are expected to be considerably higher than the film temperatures due to the large temperature gradients across the laminar layer. Assuming that 0 « 0 for R film the limited range of naturally occurring surface temperatures i t follows that also 0 « 0 . I n other words, i f the surface R film temperature distribution is a strong indicator of the spatial v a r i a b i l i t y of sensible heat flux (see Chapter 3), then the radiative surface temperature is a suitable measure for this surface temperature. The same conclusion was drawn by Garratt (1978b) without going into the argument exp l i c i t l y . In his study of transfer - 35 -characteristics from a heterogeneous savannah surface he used infrared radiation temperatures (from both airplane-based and hand-held radiometers) to obtain an overall effective surface temperature. The consistency of heat transfer coefficients, calculated with these surface temperature values, support the v a l i d i t y of this method. In their work on surface energy characteristics in urban and rural areas in and around St. Louis, Dabberdt and Davis (1978) argue in terms of a climatonomic approach to the surface energy balance that the temperature of the material surface is the primary response to the solar forcing and the surface energy fluxes the secondary response. Both Dabberdt and Davis (1978) and Carlson et al. (1981) use radiative surface temperatures (from airplane- and satellite-based radiometers, respectively) in their parameterization schemes for the surface energy fluxes, including the sensible atmospheric heat flux. The apparent reasonableness of the results of both these studies lends support to the use of the radiative surface temperature. In these examples the radiative surface temperature was obtained by remote sensing. Due to the fact that a remote sensor can only 'see' horizontal surfaces, one has to be careful in the use of such data for spatial averaging. The failure to account for the contributions of vertical surfaces to the sensible heat flux could be important especially in built-up areas : the resulting average surface temperature may well be too high during daytime, with the magnitude of the error unknown. This problem is only of l i t t l e concern for the present work, since only the spatial scales of surface temperature v a r i a b i l i t y and - 36 -not the absolute values are of interest. It is concluded that infrared remote sensing of surface temperatures provides a very convenient and physically acceptable method to obtain a high resolution inventory of simultaneous surface temperatures. 4.1.1 Infrared Remote Sensing of Surface Temperature Two sets of remote sensing f l i g h t s over the study area were completed on August 25 and August 26, 1985. The f l i g h t s were commissioned to the Canada Centre for Remote Sensing (CCRS) in Ottawa, Ont.. The instrument used was a Daedalus-1260 multispectral line-scanner (MSS) sensing in the 8-14 um wave-band. Technical details of the fli g h t s , the data format, ground-truth measurements and data calibration are described in Appendix A. The following provides only a brief outline of the f i e l d campaign for contextual purposes and concentrates on the processing of the data. The f l i g h t s were performed using a Falcon-20 twin-jet aircraft travelling at a ground speed of 105 m-s"1 and an average height of 1560 m above ground. The MSS recorded 50 scans/second with 716 pixels/scan. The data were delivered in binary form on computer compatible tape. They were preprocessed by CCRS for instrument calibration, yaw- and roll-correction and orthogonalization (see Appendix A). The times of the fl i g h t s were selected to be centred around the expected maximum development of convective a c t i v i t y at about - 37 -15:30 P. D.T. (14:15 L.A.T.), August 25 and at the expected minimum convective activity, just before sunrise, about 6:20 P.D.T. (5:05 L.A.T.) on August 26, 1985. The weather conditions were ideal for both fl i g h t s , with exceptionally clear skies and calm winds. Ground truth measurements with hand-held radiometers were obtained for various surface types on the grounds of Langara Community College, in the study area (see Appendix A). The data were transferred to a PC based Vectrix-Zenith Image Processing system (VZIP) to view the data and check the f l i g h t coordinates. Comparative distance measurements on a 1:25000 map of the study area provided the average pixel dimensions as 3.26 m (3.28 m) in the N/S direction and 3.21 m (3.23 m) in the E/W direction for the day (night) f l i g h t . The different pixel dimensions for the day and night f l i g h t s result from s l i g h t l y different f l y i n g heights. (Note : there is a 16% overscan on average, see Appendix A). Using the calibration curves from Appendix A, the one-byte integer pixel values were converted to Celsius temperatures as four-byte real values. In this form they were stored on tape, ready for further analysis of the spatial scales of surface temperature on the mainframe computer. - 38 -4.2 ROUGHNESS ELEMENTS In a suburban area the dominant kind of roughness i s that caused by bluff-bodies. A study of the distribution of roughness elements therefore necessarily involves an inventory of the dominant bluff-bodies in the area. To simplify the data structure for such an inventory, the surface elements in the study area were c l a s s i f i e d Into five categories of the most common surface cover types in the mostly residential neighbourhoods of the study area. These classes are : 1. Houses : including a few larger buildings, such as churches, schools, apartment and commercial buildings. 2. Garages : including a l l permanent structures that are too small to be c l a s s i f i e d as houses. 3. Streets : including alleys, parking lots, and other non-vegetated open spaces. 4. Trees : including sizeable shrubs. 5. Grass : including the remaining area of lawns, playing f i e l d s and gardens. These areas may contain small roughness elements such as minor shrubs, fences etc.. The inventory of these roughness elements was performed as a mapping exercise (see below). Since the resulting map already contains the locations and approximate horizontal dimensions of - 39 -these roughness elements, each category was assigned a height to make the mapped information three-dimensional. Residential houses are limited by municipal regulation to heights less than 35 f t (or 10.7 m). Considering the estimate for mean building height of 8.5 m in the same area by Steyn (1980), a rounded average of 10 m was chosen as the assigned height for the category 'Houses'. For 'Garages' the estimate of Steyn (1980) of 3.5 m mean height was adopted. The tree category includes the greatest variation in height. A small number of park trees reach a height of 30-40 m, most boulevard- and garden-trees are not much higher than the houses, whereas shrubs are considerably lower. After examination by eye on site and using photographs, a height of 15 m seemed reasonable as the assigned height for the 'Trees' category. Streets and other open spaces are important on the map because they are areas without large bluff-bodies. To distinguish between the categories 'Streets' and 'Grass', 'Grass' was assigned a height of 0.2 m and 'Streets' of 0.1 m. 4.2.1 Data Aquisition for the Roughness Inventory The area covered by the roughness element inventory is shown in Figure 4.1. The data were compiled in a stepwise process. An airphoto mosaic of the area (scale : 1:2500), dated April, 1987 and available from Vancouver City Hall, was divided into square grid c e l l s of 15 by 15 cm (equivalent to 375 by 375 m), so that the inventory area is made up of a block of 5 "x 4 = 20 grid c e l l s . For each grid c e l l and each of the f i r s t four categories j r r j^£vf r.v:j-;| ~ |- • — ) N . I! ; i~. // .'77—"E -V " K :ll--"|<^[:;|-:lij(''/!l2..~ TR7rr;-k;K(: Tl—i—-«»r»."!«i vi—""•","'rAT i : -1 . . j < j I i i i j ! : ~" rt 11 J. ^ __r^ui>i*L**i_L_:_L.i>Jij^„' 1M (S) : Sunse t Tower ; © : Moun ta in View C e m e t e r y ; © : Memor i a l P a r k ; (R) : K e n s i n g t o n Park © : Gordon Park ; ® : ' H o t - C r o s s e d - B u n s ' ; (J) : L a n g a r a Communi ty C o l l e g e Figure 4.1 : Area of roughness element inventory - 41 -F i g u r e 4 .2 : The v i d e o - f r a m e s t o r e - V Z I P c o n f i g u r a t i o n . G r a p h i c i n f o r m a t i o n i s d i g i t i z e d by the v i d e o camera ( l e f t ) and s t o r e d as a r a s t e r - i m a g e ( f r a m e s t o r e u n i t and m o n i t o r i n the c e n t r e o f the p i c t u r e ) . I t c a n t h e n be sent to the P C - b a s e d VZIP sys t em ( r i g h t ) . - 42 -('Grass' being the residual) the respective elements were handcopied with black ink on to a separate piece of mylar-film, resulting in four arrays of 20 grid cells, or 80 mylar drawings. In the next step each individual mylar copy was graphically digitized by use of a black and white video camera. A framestore-unit made it possible to store the digitized data as binary image files on a PC-hard disk. Because the pixels of the framestore unit are not square and due to a slight lens distortion which skews the image, the square grid cells were mapped into rectangular arrays of 167 x 209 elements. The on-screen editing facility of VZIP was used to 'clean' the images of noise, introduced by the video camera (see Figure 4.2). After the 20 frames for the first four categories were combined to image-arrays of 818 x 836 elements, the greyshades from the framestore scan (ranging from 0 to 255) were transformed into one-byte integer values of ten times the appropriate height. The four classes were then added together ('sandwiched') and all remaining zeros were assigned to the value of the 'Grass' category. Overlaps were handled hierarchically, so that a higher value took precedence over a lower value. The resulting image can be colour coded by VZIP as shown in Figures 4.3 and 4.4. The boundaries of the former grid cells are visible in Figure 4.3 as shifts in N/S direction. These shifts are due to the lens distortion mentioned above. Overall, the alignment and the matching of the different categories is remarkable and can only be attributed to the careful work of the draftsperson (see Acknowledgements). The pixel dimensions were evaluated by comparison with distances on - 43 -t ( l 500m F i g u r e 4 . 3 : V Z I P - i m a g e o f roughnes s e lement i n v e n t o r y , t o t a l a r e a . N o n - v e g e t a t e d open space a p p e a r s as b l a c k , lawns as g r e e n , t r e e s as w h i t e , houses as o r a n g e / y e l l o w and g a r a g e s as b l u e . - 44 -- 45 -a 1:25000 map as 1.77 m in the N/S direction and 2.15 m in the E/W direction. The last step of this data aquisition and preparation process involved the transfer of the f i n a l image f i l e to the mainframe computer and the translation of the one-byte integer data into four-byte real values, to prepare them for further analysis of the spatial structure of the roughness element distribution. - 46 -5. S p a t i a l V a r i a b i l i t y : Homogeneity and Representativeness 5.1 INTRODUCTION The turbulent lower part of the boundary layer in which the atmospheric properties and fluxes are closely related to the geometric, thermal, moisture (etc.) characteristics of the surface (Oke, 1978) is known as the atmospheric surface layer. Most surface layer theory therefore refers to some descriptor of these surface characteristics in order to incorporate their effects into the appropriate scaling laws and flux parameterizations. Atmospheric surface layer theory has generally concentrated on the vertical structure of the atmosphere, assuming horizontally uniform surface conditions. S t r i c t l y speaking, this assumption of homogeneity i s never met in any f i e l d situation, since even the most extensive and f l a t grass f i e l d i s s t i l l made up of individual grasses on a matrix of bare s o i l . One might argue, however, that spatial v a r i a b i l i t i e s in surface conditions become irrelevant for turbulent processes when the linear scales of these v a r i a b i l i t i e s approach the size of the smallest eddies in the turbulent atmosphere, i.e. they are related to the Kolmogorov microscale (Monin and Yaglom, 1971). In addition, the common postulate that any turbulence sensor should be placed far above the roughness elements (e.g. z = 100-z ; Tennekes, 1973) m i n 0 ensures that a very vague definition of 'ideal' sites (e.g. Garratt, et al., 1979) is usually sufficient for the homogeneity - 47 -requirements. Over non-ideal sites the surface v a r i a b i l i t y i s reflected in horizontal inhomogeneities in the atmosphere, which may be important at low levels and become increasingly blurred at greater heights (see also Chapter 6). Raupach et al. (1980) and Raupach and Shaw (1982) show that even over inhomogeneous terrain wind profiles converge upon the one-dimensional form, after spatial averaging, so that some homogeneity condition is again valid. Obviously, this spatial averaging process needs to incorporate a large enough domain, to be representative of the surface character. In the case of a very regular pattern of inhomogeneities, such as the experimental configuration in Raupach et al. (1980), the determination of a representative averaging domain seems f a i r l y t r i v i a l . In many f i e l d situations, such as the suburban area in the present study, however, the surface inhomogeneities occur across a variety of scales and the assessement of representativeness and homogeneity becomes d i f f i c u l t and subjective. In the following sections a concept of the terms 'homogeneity* and 'representativeness' is developed. In addition, a mathematical tool is designed to examine the relationship between spatial scale and representativeness quantitatively. Even though two-dimensional inhomogeneity is of concern here, i t is convenient to discuss some principles in one dimension by using the methods of geophysical time series analysis. The results are easily expanded to two dimensions, i f these dimensions are linear and orthogonal. - 48 -5.2 HOMOGENEITY Curiously, the term 'homogeneity' is not commonly defined in standard s t a t i s t i c s texts (e.g. Larsen and Marx, 1981), although homogeneity within a dataset is a fundamental prerequisite for many s t a t i s t i c a l analyses, theorems and laws. In time series analysis the term 'stationarity' is well defined and may be interpreted as homogeneity in time. Priestley (1981) defines stationarity in terms of a random process in time : i f the s t a t i s t i c a l properties of a random process do not change over time (i.e. they are the same at a l l times), the process is called stationary. To express this in a more formal way, suppose some process {XJ is measured at equally spaced points (in time or along a line transect in space). The result is a data series X ^ X 2 > . . . , X r . The process is stationary, i f the mean of { X F C } is independent of the index t and the covariance between X F C and X F C + T depends only on the distance between the points |T|. That is, E ( X T ) = X for a l l t; (5. 1) C 0 V ( X T ) = g(|T|) , (5.2) where g is some function of |x| (Knight, 1987, personal communication). If (5.1) is seen as the expected mean of a block sample of { X T } , centered at t, then (5.1) can obviously only hold for blocks above a certain size. The f i r s t prerequisite for stationarity in time or homogeneity in space therefore relates to the notion of representativeness, which wi l l be discussed - 49 -extensively in the next section. Equations (5.1) and (5.2) describe {Xfc} in the time domain or in the space domain. Alternately, {Xfc} may be characterized by i t s spectral properties. This frequency or wavenumber domain approach decomposes the v a r i a b i l i t y of {Xfc} into frequency components by means of a spectral measure (or a spectral density function, i f no exact periodic components exist, and ignoring the discrete nature of the series for the moment). This spectral density function results from the Fourier transform of the covariance function in (5.2), which may be written as following Priestley (1981). For real data the spectral density function h(f) is a non-negative, symmetric function defined between -0.5 £ f s+0.5, where f is in cycles per space between datapoints,. At. The limiting frequency f = 10.51 corresponds to the Nyquist frequency, the highest frequency which can be unambiguously detected (Kanasewich, 1981). The function h(f) may be determined directly from the data series (after transforming i t so that X = 0) as the square modulus of i t s Fourier transform, also known as the power spectrum. From (5.3) i t follows that h(f)-df is the average (over a l l realizations) of the contributions to the total variance of {Xfc} from components between f and f+df, so that, for real valued {Xt>, h(f) (5.3) - 50 -0 . 5 VAR(Xfc) • <r2 = 2- ^ h ( f ) Af (5.4) o (e.g. Priestley, 1981). Therefore the spectral density function may be viewed as the decomposition of the variance into frequency components and is sometimes called the variance spectrum (e.g. Young and Pielke, 1983). In the case of a discrete data series (and thus a discrete Fourier transform) a l l the integrations above need to be replaced by summations and the lower integration limit in (5.4) needs to be replaced by the Nyquist frequency. Given the preceding, we may consider homogeneity in terms of the spectral distribution of variance. Qualitatively, a data series may be defined to be homogeneous, i f most of i t s variation occurs at relatively high (spatial or temporal) frequencies (the lowest frequency being the size of the entire data domain). Thus the homogeneity of a discrete data series in space or in time is directly reflected in the shape of i t s discrete variance spectrum. If the contributions to the total variance at low frequencies are relatively small, compared to the contributions at higher frequencies, the data series qualifies as homogeneous. Quantitative measures of homogeneity may be designed based on this notion (Knight, 1987, personal communication). The scale corresponding to the frequency at which variance contributions start to become important is related to the term 'representativeness' and w i l l be discussed in the next section. - 51 -In summary i t is concluded that the degree of homogeneity within a data series is a property of the entire series and is dependent on the relative contributions of the (spatial or temporal) frequency components to the total variance. 5.3 REPRESENTATIVENESS The concepts of representativeness and homogeneity are closely interlinked and both are related to the frequency distribution of var i a b i l i t y . However, whereas homogeneity is found to be a property of the entire data domain, representativeness pertains to a block sample out of the domain. It i s associated with the averaging of the data over a certain scale or, in other words, over a portion of the data. The relationship between the two terms can be expressed by reference to the definition of homogeneity given previously : (i) i f a dataset is not completely homogeneous, a completely and unambiguously representative subset does not exist, ( i i ) alternately, i f a representative subset exists in a dataset and the representativeness of the subset depends only on the size, but not on i t s location, the dataset must necessarily be homogeneous at least to the degree of the subset's representativeness. It should be noted that the representativeness of a block sample refers to the size of the sample and not to a specific realization of this sample. It is of course possible that the s t a t i s t i c a l properties of a small sample are coincidentally similar to the entire data domain, but - 52 -in this case the similarity is conditional on the position of the sample. A subset can only be considered truly representative of a dataset, i f there is an in t r i n s i c s i m i l a r i t y and only negligible dependence on the specific configuration. Consider a block sample of m consecutive observations X , X , . . . , X . The mean of this specific sample i s defined as 1 m X = - V X (5.5) ra L, t m t=l Over a l l realizations of such a block of m-observations the expected X is equal to X for a l l observations, ID' E ( X ) = E(x f) = X (5.6) III t t One measure of the representativeness of a block of m observations i s the relative decrease in the variation of X m from the variance of X F C . Thus i t is suggested that the representativeness of m consecutive observations can be defined as VAR(X ) m R(m) = 1 . (5.7) VAR (X T ) Since VAR(X ) is the v a r i a b i l i t y among a l l the different on realizations of the block sample, the quotient on the right of (5.7) must necessarily be smaller than unity and R(m) is seen to vary between zero and unity. It i s again convenient to examine this problem in the - 53 -frequency or wavenumber domain. The frequency components of the total variance (i.e. of VAR(X t)) can be divided into contributions of frequencies that correspond to wavelengths shorter than or equal to the blocklength of the sample and components with longer wavelengths. The expected variance within any one realization of the m consecutive observations may then be interpreted as the high-frequency portion of the variance spectrum and the variance among block sample-means as the low-frequency part, following Young and Pielke (1983), so that E (VAR(X) ) + V A R ( X ) = V A R ( X f ) (5.8) m m t If n i s the total number of observations in the data domain and At the interval that each observation occupies, then L = n-At is the length of the domain. Since f is defined in the previous section as frequency in cycles per interval between observations (At), the wavelength relates to f as X = At/f. Therefore the frequency corresponding to the blocklength of m observations, X = m-At is f = At/A = 1/m and the lowest m m m frequency is 1/n. Frequency f is the delimiter for the m subdivision of the variance density spectrum and, considering equation (5.4), (5,8) may be translated to the frequency domain as 0.5 (f -df) m 2 • ^ h ( f ) Af + 2 • ^ h ( f ) Af = VAR(X t) (5.9) f 1/n m where the discrete nature of the transforms is ignored for - 54 -simplicity and each term corresponds to the respective term in (5.8). (Note that h(0) = 0 since Xfc = 0 after the transformation of the data as described above). The integrated variance spectral distribution function, H(f), is introduced for convenience, following Priestley (1981); f H(f) = 2 • ^ h « ) AC (5.10) 1/n For the present purposes a representation of equation (5.9) as a function of wavelength is more useful than the frequency version. If h(f) = h(At/A) = h (A), (5.9) becomes A 2 • ^ \ ( A ) AA + 2 • ^ \ ( A ) AA = VAR(Xt) (5.11) 2At A +AA m Note that the integration limits have been reversed in this transition from frequency to wavelength presentation. The equivalent to the integrated spectrum then follows as H^U) = 2 • A ^ ' (5.12) 2At so that ^ ( L ) = H(0.5) = VAR(Xfc) (5.13) and - 55 -A m 2 • ^J^U) AA = • (5.14) 2At After substitution into (5.8) and rearranging, i t follows that H.(A ) V A R ( X ) A m m = 1 3 R(m) . (5.15) V A R ( X T ) V A R ( X T ) Thus the normalized wavelength-integrated variance distribution function is identical to the measure of representativeness for a block sample of size m, as introduced in (5.7). In practice the R(m) for a discrete data series of real values are obtained from the square moduli of the series' Fourier transform. R(m) is twice the sum of the components with wavelengths smaller or equal to A , divided by the total m variance. 5.4 DISCUSSION It has been shown that the homogeneity of a dataset and the existence of a representative block sample subset are closely related and dependent on the distribution of variance in the frequency or wavelength domain. A measure of representativeness, R(m) or just R, dependent on the length of a block sample, A , m is defined and identified as the normalized integrated variance spectrum in the wavelength domain. A schematic plot of R(m) versus A is given in Figure 5.1. Apart from showing the shape - 56 -J 1 | | | I I I L sample - leng th 3 m Figure 5.1 : Representativeness vs. sample size (schematic). The shape of the normalized integrated variance spectrum curve determines the homogeneity of the ( in this case hypothetical) data. of the representativeness-sample size relationship, Figure 5. 1 also ill u s t r a t e s the homogeneity of the (hypothetical) data. In section 5.3 i t was indicated that a dataset is homogeneous, i f an i n t r i n s i c a l l y representative subset exists. Since a f i n i t e dataset w i l l always have a f i n i t e variance, a t r i v i a l consequence is that the dataset is always a representative subset of i t s e l f . Thus R becomes unity for A = L. This m c r i t e r i o n alone would qualify a l l datasets as homogeneous. In Section 5.2, however, i t was stated as a requirement for homogeneity that the bulk of the variance contributions occurs at much shorter wavelengths than the length of the data domain. The degree of homogeneity is therefore indicated by the curvature of the R-A relation in Figure 5. 1. If the curvature is convex (negative), the data are homogeneous because the integrated variance increases quickly at small scales and slower at large scales. If there is mostly large scale v a r i a b i l i t y the curve w i l l be concave (positive). A straight line in Figure 5.1 corresponds to a random data structure, as in white noise, with equal v a r i a b i l i t y at a l l scales. If representativeness can be measured by an integration of the variance spectrum in the wavelength domain, i t follows from the above discussion that homogeneity may be considered proportional to the mean derivative of the variance spectrum (i.e. the average slope of the variance spectrum across a l l wavelengths). For spatial v a r i a b i l i t y in two dimensions everything that has been stated above needs to be expanded to consider two indices - 58 -(e.g. t and s) instead of only one. Since the two indices are orthogonal, this expansion is straightforward. The variance spectrum in the wavelength domain can be transformed into polar coordinates and, after integration with respect to angle, can be plotted against radial wavelength (any directionality in the dataset is lost in this way). The resulting representativeness estimates then refer to circular sample blocks of varying radius. In the context of the present work representativeness and homogeneity are of interest in respect to the 'sampling' of surface characteristics and the subsequent spatial averaging process of surface turbulent fluxes by the mixing action of turbulence. Clearly, i t is highly unlikely that these surface characteristics are purely isotropic - especially in a suburban area with preferred street directions. Also, the shape of the 'block sample' which needs to be considered is probably not circular. These points are further discussed in Chapter 9. A suitable definition for a sample involving turbulent diffusion and approximations for i t s dimensions and orientation are presented in Chapters 6 and 7. Together with this present chapter, they provide the basis for an objective assessement of the representativeness of a turbulent flux measurement and for an evaluation of the homogeneity of the area in which the measurement is conducted. - 59 -6. Estimating the Source Area of a Turbulent Flux Measurement over a Patchy Surface 6.1 INTRODUCTION In any measurement procedure i t is of fundamental concern that the quantity measured by the sensor is truly the quantity of interest. In the context of the measurement of surface turbulent fluxes, this is not a t r i v i a l concern, since the flux originates at the surface (and is influenced by i t s characteristics) but the sensor is commonly situated somewhere above, in the atmosphere. Over an inhomogeneous surface, the quantity sensed refers only to those surface elements that influence the measurement. The region of these surface elements may be termed the Source Area. In the following, this notion is used to develop a tool to estimate the dimensions of the source area to which a specific turbulent flux measurement refers. This identification of the contributing surface elements is a f i r s t step to obtain truly representative flux values for a large area. The source area of an eddy correlation measurement of sensible heat flux is defined as the surface area containing the heat sources and/or sinks which influence those a i r parcels which can be carried past the sensor under given external conditions. In other words, i t is that portion of the surface which the instrument 'sees' in an aerodynamical sense. The distances of the upwind, downwind and lateral boundaries of this source area from the sensor location are dependent on the - 60 -characteristics of the flow and on the boundary layer development in the atmospheric layer between the surface and the sensor level. The adjustment of the flow to a new set of surface conditions and the development of the boundary layer after a leading edge has received considerable attention in the past, primarily in the case of a discrete roughness change and a subsequent modification of the Reynolds stress T ( Z ) and wind-speed (u(z) and w(z)) throughout the surface layer. Two of the three equations required for a mathematical solution of this problem are provided by the equations of conservation of mass and horizontal momentum. Various assumptions, such as a res t r i c t i o n to neutral s t r a t i f i c a t i o n , have been used to close the set of equations ( E l l i o t , 1958; Panofsky and Townsend, 1964; Peterson, 1969; Taylor, 1970). Pasquill (1972) reviews and discusses some of these studies. Here, only the general concepts relating to conditions i n the v i c i n i t y of an idealized single cross-wind discontinuity w i l l be reviewed. Subsequently these concepts are adapted to describe the more complex problem of a patchy surface, i.e. two-dimensional inhomogeneity. Finally, an alternative solution is developed in an attempt to solve this problem, following the original proposition by Pasquill (1972) for a surface with patchy roughness. - 61 -6 .2 ONE-DIMENSIONAL CROSS-WIND DISCONTINUITY In the case o f a s i n g l e , d i s c r e t e c r o s s - w i n d d i s c o n t i n u i t y , the degree o f readjustment to the new c o n d i t i o n s can be d e s c r i b e d i n a s t a t i s t i c a l manner and v a r i o u s boundary l a y e r i n t e r f a c e s may be d e f i n e d a c c o r d i n g l y . Here, we d i s t i n g u i s h two such i n t e r f a c e s , f o l l o w i n g Munro and Oke (1975) (see F i g u r e 6 . 1 ) . G i v e n the d i r e c t i o n o f the f l o w i s from s u r f a c e SI a c r o s s the d i s c o n t i n u i t y to su r f ace S2, the f i r s t o f these i n t e r f a c e s marks the b e g i n n i n g o f the m o d i f i c a t i o n s by S2. Below t h i s i n t e r f a c e the f l o w i s not i n e q u i l i b r i u m w i t h SI anymore. T h i s e q u i l i b r i u m may r e f e r to the mechanical t u r b u l e n c e , h u m i d i t y o r thermal s t a t e o f the atmosphere, as the case may be. The he igh t o f t h i s i n t e r f a c e 5(x) i s d e f i n e d as the h e i g h t , up to which the f l o w c o n d i t i o n s d e v i a t e from the S l - e q u i l i b r i u m by more than an a r b i t r a r y f a c t o r ( e . g . , i n Pe t e r son (1969) the i n t e r f a c e f o r Reynolds s t r e s s boundary l a y e r s i s d e f i n e d by p o i n t s a t which the s t r e s s d i f f e r s from the upstream va lue by o n l y 0.1 %). Exper imen ta l s t u d i e s i n bo th l a b o r a t o r y c o n d i t i o n s ( S c h l i c h t i n g , 1968) and a g e o p h y s i c a l s i t u a t i o n (Munro and Oke, 1975) show a growth o f 5(x) p r o p o r t i o n a l to the / 5 power o f x , where x i s the f e t c h o r d i s t a n c e from the l e a d i n g edge (see F i g u r e 6 . 1 ) . The second i n t e r f a c e 5 ' ( x ) , i s the one below which the p r o f i l e i s c o m p l e t e l y ad jus ted and i n e q u i l i b r i u m w i t h S2. Pe t e r son (1969) r e p o r t s evidence tha t the r a t i o o f 5/5* i s g e n e r a l l y about t en . Munro and Oke (1975) f i n d a f a i r agreement w i t h t h i s e s t ima te i n t h e i r measurements over a change from - 62 -z . - o ( x ) S 1 - e q u i 1 i br i um 1 a y e r y ' / ' / ' t r a n s i t i o n zone Wind / ' / / ^ ( E J F - — &'(x) S 2 - e q u i 1 i br i um 1 a y e r SI D S 2 x Figure 6.1 : Internal boundary layer interfaces with one-dimensional discontinuity. The degree of adjustment of the a i r to the downwind surface-type is indicated by a i n i t i a l modification interface (IMIF) and an equilibrium interface (EIF). Figure 6.2 : The source area in one-dimensional patchiness is determined by the distances at which an IMIF and an EIF reach the height of the sensor (z ). s - 63 -tobacco to wheat crops. An intriguingly simple alternative approach was ori g i n a l l y suggested by Miyake, as quoted in Panofsky (1973). In this model, the speed, at which information about changes in surface conditions is propagated upward is assumed proportional to the standard deviation of the vertical velocity fluctuations, so that the interface 5(x) is described by do/dx « <r/u . (6.1) w Hajstrup (1981) assumes that <r can be adequately described by w the relation obtained over homogeneous terrain, so that (6.1) can be written dS/dx « k <f> (z/L)/(ln(z/z ) - tf> (z/L) , (6.2) w 0 M where or /u = d> (z/L) and one of the common empirical w * w relations for <f> (z/L) can be used. In BJajstrup's case he used the Panofsky et.ai. (1977) relation : 0 = (1.5 + 2.9 ( z / L ) 2 ' 3 ) 1 ' 2 . (6.3) w This use of the homogeneous relation in effect assumes an instant equilibrium with the downwind surface. Taylor (1970), who used an identical assumption in his mixing length model, recognizes i t as a major weakness and a source of error of his model, but concludes that in the absence of anything better the homogeneous relations may serve as a f i r s t approximation to the solution. Rao (1975) on the other hand employs Panofsky's w-model with also rather crude assumptions for w. It seems that some sort of assumption for the diffusion characteristics in the - 64 -transition zone are inevitable as a working basis. The canopy layer diffusion discussion between Raupach (1979), Thorn et.al. (1975) and Garratt (1978a, b) support the case in point here. While a diffusion profile within the canopy layer may be accurately parameterized in certain cases, i t s character remains highly dependent on the specific situation, with only l i t t l e value for general application. Rao's (1975) results are interesting, at least qualitatively. He describes the growth of the internal boundary layer as 5 ec x n , (6.4) where n depends only on the Monin-Obukhov length L for a given set of, in his case, upstream conditions. In his calculations n ranges from 0.8 for neutral conditions (confirming E l l i o t t (1958),Peterson (1969) and the results from Munro and Oke (1970)), up to n = 1.5 for a free convection regime. Thus, the internal boundary layer growth appears to be influenced by the s t a b i l i t y regime. 6.3 THE SOURCE AREA IN ONE-DIMENSIONAL INH0M0GENEITY With this concept of a growing internal boundary layer, the source area for a point at height z g can be described in terms of an i n i t i a l modification interface (IMIF) and an equilibrium boundary layer interface (EIF). In the one-dimensional case the source area is limited upstream by a region which is too far away to have a considerable effect on the flow at the reference point (see Figure 6.2) - 65 -The upwind leading edge of this region is marked as -x , e denoting a distance from the point of reference of |x | in the e upwind direction. This location is defined by 5*(|x |) = z (6.5) e s which states that the equilibrium boundary layer interface with i t s leading edge at x = -x , reaches a height of z at x = 0 e s (i.e. after a distance Ix I). Thus, being in equilibrium with the surface downstream from -x , the profile below 5' is not affected by the surface effects of the upstream area. Since 5' reaches a height z at x = 0, the sensor is not affected by the s surface upstream of -x . e The downstream boundary of the source area i s defined in a similar fashion. It is postulated that the surface downstream from that boundary has no effect on the conditions at (x = 0, z = z g ) . The corresponding relation is therefore 6(1x1) = z (6.6) i s where the boundary i s located at x = -x . This of course means that the i n i t i a l modification interface, with i t s leading edge at x = -x , reaches the height z at x = 0, after a distance i s I x J . Thus, the surface conditions downstream from x = -x are not f e l t up to a height z at x = 0. s In summary, the source region of a reference point at height z in one-dimensional patchiness is defined by an upstream s equilibrium boundary layer interface, intersecting at the point of reference with an i n i t i a l modification boundary layer interface (see Figure 6.2). - 66 -6.4 THE SOURCE AREA IN TWO-DIMENSIONAL PATCHINESS Taking the preceding example a step further, a situation arises where a series of one-dimensional discontinuities are encountered in the source area. In this case, things become rapidly very complex, seeing that several different internal boundary layers are superimposed upon each other. Each interface w i l l be affected by a s l i g h t l y different local s t a b i l i t y regime and therefore w i l l have a different growth rate (see equation (6.4) and Rao, 1975), making i t necessary to use a bulk s t a b i l i t y parameter as an average. The internal boundary layer approach becomes even more unsuitable i f the problem is extended to two-dimensional patchiness, as in a more typical real-world surface. Clearly, the concept of internal boundary layers cannot be continued to two-dimensional discontinuities because the extent of the source area is not only controlled by vertical diffusion and the mean wind, but also by lateral diffusion (i.e. in the y-direction). A more suitable procedure for this case was f i r s t suggested by Pasquill (1972). In his approach the developing zone of influence downwind from a surface element is treated as a plume. Thermal plumes, consisting of temperature anomaly zones in the boundary layer, have been identified by Holmes (1970) from a series of airborne temperature transects over a patchy area in southern Alberta, Canada (see Figure 6.3). Both 'hot plumes' and 'cold plumes' have been observed to start at the leading edge of - 67 -b > a -> • * < W I N D ' Dimensions in km Height Stability G cnerally smooth G cnerally rough Zo - 3 c m Zo - 1 m a b C a !> C unstable 0-2 0-8 0-1 Q-l 0-4 0-1 50 m neutral 0-6 2-8 0-4 : 0-3 l'S ' 0-2 stable 5 40 6 2-9 28 3-4 unstable 0-3 1-5 0-2' 0-3 0-9 0 1 100 m neutral 1-4 7 1 0-7 4-4 0-6 stable 13 (270) 21 10 (130) 14 unstable OS' 3-2 0-J 0 5 2-1 0-3 200 m neutral 3 25 3 1-6 17 2 unstable 1-S 9 1-2 11 6 0-7 400 m neutral 10 130 12 6 90 8 Figure 6.4 : Estimates of source area dimensions (Pasquil l , 1972) are dependent on the sensor height, atmospheric s t a b i l i t y and the surface roughness. - 68 -a surface patch and are noticeable for a considerable distance downwind (the height exaggeration in Figure 6.3 is somewhat deceiving : the distance from A to D is about 40 km). In Pasquill's approach the patchy surface is thus regarded as am array of elementary 'sources' from which a property is emitted. Pasquill termed this source area the 'effective fetch'. This use of the term 'fetch' is not to be confused with the traditional meaning of the word in meteorology, where i t denotes a distance rather than an area. In the following, both 'source area' and 'effective fetch' w i l l be used as.synonyms. To become familiar with this alternative concept of elementary sources and their plumes, rather than boundary layer interfaces, i t is convenient to take a step back and again look at the one-dimensional problem, before proceeding to two-dimensional patchiness. Gash (1986) estimates the effect of a limited fetch for an evaporation measurement by considering an upwind continuum of elementary line sources, each occupying an infinitesimal s t r i p of width Sx. For a sensor mounted at a height z above the zero-plane displacement, water vapour s diffusing from a distance x will on average be sensed with concentration p (x.z ). Applying a solution by Calder V s (Gash,1986) to the basic diffusion equation for an i n f i n i t e cross-wind line source (Q) of passive particles in conditions of neutral s t r a t i f i c a t i o n , and assuming a uniform windfield, gives Q -Uz /ku^x p (x.z ) = e 3 (6.7) V s . k u^x where U is the uniform wind-speed, assumed to apply over the - 69 -whole neutrally buoyant boundary layer. If E = Q/Sx and letting 5x approach zero, (6.7) may be integrated to give the concentration caused by a f i n i t e s t r i p of width X l > Gash (1986) then considers the vertical concentration gradient and obtains, after differentiation with respect to z and integration with s respect to x, an equation for the vertical concentration gradient at z as caused by the diffusion from a s t r i p of depth s x : L ^ v ^ Z s ^ ^ -Uz /ku x „, s * L . (6.8) = - e 3z k u z s * s As X l approaches infinity, (6.8) takes the form of the flux gradient equation for evaporation in homogeneous and neutral conditions and unlimited fetch, dp (z ) E V 3 = (6.9) dz ku z s * s He then considers humidity measured at z with an i n f i n i t e s upwind fetch and a gradient given by (6.9). A percentage of that gradient (and therefore of the flux) w i l l be the result of diffusion from within a distance x^ , from the point of measurement. He terms that distance the 'F percent effective fetch*. It is given by the ratio of (6.8) and (6.9), which, on taking logarithms and rearranging, gives Uz s x =- l/ln(F/100) . (6.10) ku. Thus, with a suitable choice of U/u#, an estimate of the - 70 -effective fetch for a measurement at height z under neutral s conditions is given. Gash continues the argument to estimate the fractional error of an evaporation measurement due to a step-change in evaporation rate at xp. For the present study two points are of importance. Gash (1986) presents the effective fetch in terms of a percentage, realizing that the total effect sensed by the instrument is due to a l l elementary sources upwind, but that the elementary source weight decreases with distance, as represented by the concentration in (6.7). The second point i s one noted also by Gash (1986) : the diffusion calculations over a surface inhomogeneity (and therefore also an energy input inhomogeneity) are not f u l l y internally consistent. Although different fluxes from the two surface types are considered for the evaluation of the concentration distribution, the s t a b i l i t y and flow conditions are assumed to be horizontally uniform. Thus, in effect, the energy balance constraint i s violated. Again, as mentioned in respect of the assumptions of Taylor (1970) and Rao (1975), homogeneous horizontal flow and s t r a t i f i c a t i o n are assumed, even though the energy input into the system at the surface is not. Atmospheric scientists are familiar with assumptions of this kind. A famous example is the set of Boussinesq approximations for the Navier-Stokes equations, where the effects of buoyancy and compressibility are accounted for by neglecting compressibility everywhere except in connection with gravity (e.g. Nieuwstadt and van Dop, 1981). As in the present case, only the most direct effects of the phenomenon in question are - 71 -considered, and a f i r s t order approximation is obtained. Keeping these limitations for the simple one-dimensional case in mind, we proceed to two-dimensional patchiness and the original work by Pasquill (1972). Pasquill's analysis for the effective fetch or source area is developed for a 'momentum plume', where the surface elements appear as individual sinks rather than sources. Regarding a particular elementary area in isolation, the momentum d e f i c i t which this 'sink' w i l l produce at a given height w i l l rise to a maximum at a certain distance downwind and then f a l l off continuously as distance is further increased. The functional form is precisely that contained in the theoretical ground-level concentration distribution from an elevated source, i f the familiar reciprocal relation for the distribution from ground and elevated sources is adopted (Smith,1957) and, i f for convenience the Gaussian form of vertical distribution is used as a working approximation (Pasquill, 1972). Pasquill uses the well known form of the Gaussian plume model for his computations, assuming complete reflection on the ground : *(x,y,0) u 1 = exp Q 710* o * y z where <r « x q and or /or = constant (independent of distance). z y z With the shape of the isopleth given by (6.11), a cr i t e r i o n for the source area in terms of the particle-source analogy is 2cr exp -2<r (6.11) - 72 -defined : the area bounded by the % /2 isopleth is chosen, max i.e. the area in which the concentration from an elementary unit point source is greater than one half of the maximum concentration from such a source, as sensed at the same height. Differentiation of (6.11) with respect to x (and with y = 0) gives the expression for Y '• max X /Q max 2 <r / ( T T Z e u <r ) z s y (6.12) so that the x /2 isopleth is given by max 1 = exp 1 -2ar r 2 •> z s • exp • 2<r 2 z (6.13) (note that since both or and or are functions of x, (6. 13) is y z two-dimensional and symmetric about the x-axis). Pasquill (1972) tabulates the dimensions of this region relative to the receptor location for a set of height, roughness and s t a b i l i t y conditions (see Figure 6.4). Pasquill's source area estimates are the f i r s t attempt to cope with patchy var i a b i l i t y . In view of recent developments in diffusion theory, there are some points in his approach which may be updated in a revised version of the same concept. - 73 -6.5 A RECIPROCAL PLUME MODEL TO ESTIMATE THE SOURCE AREA As mentioned, Pasquill (1972) assumes Gaussian diffusion in both the horizontal and the vertical directions for his source area computations. This assumption allows him to use Smith's (1957) reciprocal theorem, which states that the concentration at ground level downwind of a point- or line-source at height z s is identical to the concentration at the same x and y but at height z , due to an exactly similar source at ground level s (Smith, 1957). Smith's proof of this theorem r e l i e s heavily on the assumption that the distribution of velocity fluctuations in any direction is symmetrical, which indeed i t is assumed to be in the Gaussian model. In his review paper on updating applied diffusion models Weil (1985) states that, while Gaussian diffusion i s acceptable in the horizontal, real plumes are clearly not Gaussian in the vertical. The implication i s that, with skewed vertical diffusion, the symmetry assumption in Smith (1957) is invalid and the concept of reciprocity, as employed by Pasquill (1972), should be modified or replaced. It w i l l be shown in the following that a s l i g h t l y different concept of reciprocity is valid in the context of the present problem, regardless of the functional form of diffusion characteristics. In particular, i t w i l l not be necessary to assume that an elevated source at height z generates the same s effect pattern on the ground as a ground source w i l l produce at height z . s As in the one dimensional case, the by now familiar - 74 -assumption of a horizontally homogeneous windfield and an average atmospheric s t r a t i f i c a t i o n has to be applied. Hunt and Simpson (1982) mention that surface changes of a limited amplitude have only very weak dynamical effects and the zone of influence is the same for a l l such changes. In the present study i t is assumed that horizontal windshear is negligible over the spatial and temporal scales of concern, and that Monin-Obukhov scaling is applicable as a f i r s t approximation. Consider a sensor of humidity, temperature, wind-speed or their fluctuations at x = 0, y = 0 and at a height z in the s surface layer (see Figure 6.5). An arbitrary elementary source on the ground and in a sector upwind from the sensor w i l l create a time averaged concentration distribution whose maximum Is located a certain distance directly downwind from this source. If this source is moved a given distance in the x or y direction, the concentration distribution w i l l s h ift in a complementary fashion, due to the horizontally homogeneous flow. Accordingly, the sensor at z w i l l experience higher or lower s concentrations, depending on the x/y-location of the source. Let the source location which causes the maximum concentration at the sensor be termed the maximum source location. Obviously, the effect at the sensor will decrease, when the source is moved away from the maximum source location in any direction. The total effect that the sensor experiences is determined by the weighted contributions of a l l sources upstream, and the resulting two-dimensional source weight distribution function fi(x,y) has i t s maximum at the maximum source location. mox h e i g h t wind P-SOURCE A R E A Figure 6.5 : Schematic c r o s s - s e c t i o n of a P - c r i t e r i o n source area. It i s def ined as the reg ion i n which any point source causes a minimum r e l a t i v e e f f e c t l eve l of xp at the sensor l o c a t i o n . The corresponding source weight d i s t r i b u t i o n fi(x,y) i s equivalent to the p l a n - p r o j e c t i o n of the z - e f f e c t l eve l d i s t r i b u t i o n of a v i r t u a l source beneath the s sensor (and with a v i r t u a l wind i n the reverse d i r e c t i o n ) . - 76 -Consider furthermore an arbitrary c r i t e r i o n ' P' to define a sensed concentration or effect level x^, as the minimum sensed effect level, for a source to belong to the P-source area. Again, i f the source is moved away from the maximum source location in any direction, the sensed effect level w i l l eventually decline to the x\p level. The geometric location of a l l point sources, whose effect level at the sensor location equals x^, forms a closed curve, which is the boundary of the P-source area. This curve is in effect a tracing (on the ground) of the r isopleth (on an imaginary plane at level z ) derived P s from the source at the maximum source location. Therefore, the source area may be obtained by a simple geometric translation of the Xp isopleth (see Figure 6.5). Similarly, the source weight distribution function (flcx.y)), with u = Q(x,y) is a geometric translation of the concentration or effect level distribution function at level z . Mathematically, the P-source area outline s and the xp isopleth (or the source weight distribution function and the concentration distribution at z , respectively) are s identical except for a change of axes. In practice, Q(x,y) may be obtained by simply reversing the wind direction, placing a virtual source at the ground underneath the sensor, and projecting the virtual effect level distribution at z down onto s the ground (see Figure 6.5). This concept is similar to the reciprocal plume of Pasquill (1972), but i t is independent of Smith's (1957) reciprocity theorem, however, and is not constrained by any particular form of vertical or horizontal diffusion. - 77 -F i g u r e 6.6 : I n t eg ra t ed e f f e c t f r a c t i o n : a) t o t a l i n t e g r a t e d e f f e c t b) f r a c t i o n ' P ' o f the t o t a l i n t e g r a t e d e f f e c t - 78 -Having established the source weight distribution and the outline for a P-source area, a suitable form for the c r i t e r i o n ' P' which defines the effect level isopleth is needed. An obvious suggestion is that 'P' is the fraction of the total effect which is contributed by the P-source area. As mentioned above, the total effect (E t) at the sensor location is the weighted sum of a l l upwind sources : 1 For point sources the summation becomes a double integral of the weight distribution function £2(x,y), and considering the effect of each source relative to i t s source strength, the drop out : + 00 +00 E t = J* J* 0(x,y) dx dy (6. 15) -oo 0 so that E is the volume under the Q(x,y)-function (see Figure 6.6a). The horizontal area bounded by each w = wp isopleth (i.e. wp corresponding to xp before the translation) represents that fraction 'P' of which is contained underneath the Q(x,y) surface portion bounded by the w p-isopleth (Figure 6.6b) : + 00 +00 P = J\fQ(x,y) dx dy / J" J" Q(X,y) dx dy . (6.16) u=w -oo o P It follows that ' P* is the portion of the total integrated effect which is contributed by the P-criterion source area, bounded by the weight distribution function isopleth w = wp. The P-source area is therefore defined in a similar fashion - 79 -to the F-percentage effective fetch of Gash (1986) and may/ be seen as an extension of i t to two-dimensional inhomogeneity. Depending on the kind of plume model chosen to describe the source diffusion characteristics, however, the source area calculations are not limited to neutral conditions. The present work is aimed at unstable thermal s t r a t i f i c a t i o n and the source area model presented in the next chapter i s applicable only in those conditions. - 80 -7. The Source Area Model (SAM) 7.1 INTRODUCTION The concepts of the source area model (SAM) are based on the diffusion characteristics of a ground-level point source. A short-range plume model forms the core of the source area model. The choice of plume diffusion model is therefore of prime importance, since i t contains a l l the physics and boundary layer parameterization schemes that w i l l eventually be reflected in the source area results. In the following the diffusion model selected for use is described, with special focus on adaptions for the purposes at hand and the computer implementation. It is then tested with data from independent passive tracer releases at ground level. These test results are presented in Appendix B, together with the FORTRAN-77 code of the source area model. Since the source area i t s e l f is not directly measurable, this is the only possible validation of the physical processes determining the source area model. - 81 -7.2 AN APPLIED DISPERSION MODEL BASED ON METEOROLOGICAL SCALING PARAMETERS In his recommendations for applied plume dispersion models for ground level sources, Weil (1985) mentions three classes of models, into which recent developments of convective boundary layer scaling are incorporated. These include the classic Gaussian model, the probability density function (p.d.f.) approach and 'impingement' models, such as the one by Venkatram (1980). A l l three classes use Gaussian concentration profiles in the lateral direction, but only the Gaussian model also assumes a symmetric distribution in the vertical. Observations of real plumes make clear, however, that the assumption of homogeneous turbulence and Gaussian diffusion in the vertical direction does not hold in the presence of a surface boundary (Pasquill and Smith, 1983). Therefore the Gaussian approach has to be rejected especially for the case of ground-level releases. 'Impingement' models have been designed primarily to cope with the special features of buoyant plumes (Weil, 1985) and are not suitable for the present study. This leaves the p.d.f. model class as the most appropriate and up-to-date applied plume modelling approach for present purposes. 7.2.1 The Gryning et ai. (1987) P.D.F. Model (GHIS) The p.d.f. model selected for this work was presented by Gryning et al. (1987) (hereinafter referred to as GHIS). While i t is easy to use on a routine basis, i t incorporates recent - 82 -developments in the scaling of boundary layer turbulence and s t a b i l i t y parameters. This section summarizes the findings of GHIS and follows the format of that paper closely. GHIS divide the boundary layer into a number of scaling regimes depending on the relative height within the boundary layer and the atmospheric s t a b i l i t y conditions (see Figure 7.1). The present application is concerned only with unstable daytime conditions and is limited to heights less than a few tens of metres. From Figure 7.1 this means that surface layer scaling applies in most daytime situations. In addition, GHIS point out, based on Holtslag's (1984) study on the 'Prairie Grass' data, that surface layer predictions compare well with free convection layer results at small distances. As mentioned, spread in the lateral direction may safely be assumed to be Gaussian and axial diffusion becomes Important only In conditions where u/w, s 1.2 (Willis and Deardorff, 1976) (i.e. at windspeeds below about 1 m-s-1, as a rule-of-thumb for most situations). It follows that the plume model may be separated into a cross-wind integrated concentration (CIC) model of the vertical spread to which a Gaussian component is applied to account for the lateral distribution. 7.2.1.1 Vertical Dispersion Nieuwstadt and van Ulden (1978) show that the vertical dispersion from a ground level source in the surface layer is adequately described by K-models, using Monin-Obukhov simil a r i t y theory to account for the effects of s t a b i l i t y on the vertical - 83 -UNSTABLE N E U T R A L S T A B L E 01MENSI0NLESS STABILITY h/'L Figure 7.1 : The scaling regions of the atmospheric boundary layer, shown as function of the dimensionless height z /h and the s t a b i l i t y parameter h/L, where h is the height of the boundary layer, (from GHIS). - 84 -structure of turbulence. It is commonly assumed that the d i f f u s i v i t y of passive particles, K, may be approximated by the d i f f u s i v i t y of heat, K , i.e. : H K = K = k u z/<p (z/L) . (7.1) H * H It is interesting to note that the present study assumes the reverse analogy to describe a heat plume based on a model designed for a particle or gas plume, so that the sequence of assumptions has performed a f u l l c i r c l e . D i f f u s i v i t y profiles as in (7.1) together with the familiar diabatic log-linear wind profile have often been used for numerical solutions of the crosswind integrated diffusion equation as given by van Ulden (1978) : u-d(cic)/dx = a(K - a ( c i o / a z ) / a z (7.2) H With the boundary conditions given by Pasquill and Smith (1983, p. 96) as CIC 0 as x,z oo CIC oo at x = z = 0 K -a(CIC ) /az -+ 0 as z 0, x > 0 H and when the profiles of wind and d i f f u s i v i t y are replaced by simple power laws, an approximate analytical solution becomes possible and is expressed by van Ulden (1978) as CIC/Q = (A/zu)-exp j - (B-z/z)sJ- (7.3) with - 85 -oo z = z-CIC(z) dz CIC(z) dz (7.4) o 0 as the weighted mean height of the particles that have travelled a distance x and being the mean horizontal velocity of the particles. The exponent 's* in (7.3) is termed the shape factor and is closely related to the exponents of the power law profiles of wind-speed and d i f f u s i v i t y (van Ulden, 1978). The x-dependency of CIC/Q in (7.3) is contained in this x-dependent exponent as well as in z. B and A follow from a substitution of (7.4) into (7.3) (Nieuwstadt and van Ulden, 1978) as where I* is the gamma function. CIC/Q is thus obtained from the mean plume height z, the shape parameter s, the mean transport velocity u and the functions A and B. GHIS summarize equations for the calculation or approximation of these quantities in their appendix. A relation between travel distance, roughness length, Monin-Obukhov length and the mean plume height is approximated by van Ulden (1978) and may be written for a ground source and unstable conditions : (7.5) A = s - r(2/s ) / [ r ( l/s)] s (7.6) B = r(2/s)/ r ( i / s ) s (7.7) - 86 -x = (z/k 2)-[ln(cz/z ) - i/r (cz/L)]-[l - pa z/(4L)]~ 1 / 2. (7.8) 0 M 1 The coefficients p and c depend on s but are rather insensitive to variations of s. van Ulden (1978) suggests a value of p = 1.55 for practical applications and Gryning et al. (1983) propose c = 0.4 for unstable conditions. After f i t t i n g the power law of wind speed to the diabatic form of the log-linear pro f i l e and the power law of d i f f u s i v i t y to equation (7.1), with perfect correspondence at the height z = c-z, the shape factor s i s determined by Gryning et al. (1983) as 1 - a c-z/(2L) (1 - a c-z / L ) ~ x / * s = 1 + - — (7.9) 1 - a c-z/L ln(c-z/z ) - </• (c-z/L) l o M for unstable conditions. The constants a and a are related to the choice of the 1 2 non-dimensional profile functions for heat and momentum <p and <p . In the present application the functions suggested by Dyer M and Bradley (1982) are used with a,^ = 14 and a g = 28. Sensitivity tests conducted here show that differences between this and the Dyer (1974) version of a i = a g = 16 are minimal (see Appendix B). Finally, the mean horizontal plume velocity u i s approximated by van Ulden (1978) for L < 0 u = u/k-[ln(c-z/z o) - 0 M(c-z/D] . (7.10) In practice, equations (7.8),(7.9) and (7.10) are used as a - 87 -t r i p l e t in an i n i t i a l value iteration to obtain z, s and u for a given set of x, z and L. With these values, A in (7.6) and B in o (7.7) are known and CIC/Q may be obtained from (7.3). The shape factor, s, varies with s t a b i l i t y and downwind distance. In most real situations this variation seems to range between values of 1 and 2 at short range and unstable conditions (Gryning et al., 1983). When s = 2, (7.3) becomes Gaussian. Therefore the Gaussian model is a special case of a p.d.f. model. In the present application s was prevented from exceeding a value of 2 (by exponential damping- for s > 1.9) to avoid numerical i n s t a b i l i t i e s in the model. Pasquill and Smith (1983) present an equivalent approximation for the x-s relation taking a s l i g h t l y different route. 7.2.1.2 Lateral Dispersion Lateral dispersion is caused by turbulence and horizontal wind-shear. In unstable conditions with high turbulence intensities and over a short range only, the latter process becomes insignificant (Pasquill and Smith, 1983). Consequently, the cross-wind spread of the plume, <r , may be related to y turbulence via the standard deviation of lateral wind fluctuations, cr^ , following Taylor (1921). From experimental observations i t is concluded by GHIS that the lateral profile of a plume closely resembles a Gaussian distribution. Once the appropriate CIC is known, the time-averaged concentration at any point may therefore be - 88 -obtained from %<x,y,z) = CIC(x.z) V2ir ar exp 2<r (7.11) where x(x,y,z) is the normalized concentration (i.e. conc./Q). Taylor's formula may be written as ar = cr t-f (t/T ) y • y y (7.12) following GHIS, where t is the travel time (= x/u), and T is y the Lagrangian time scale for lateral dispersion, so that f is a function of the non-dimensional travel time t/T . In a review y of various suggestions for the form of the f function, Irwin (1983) found the best results with a version by Draxler (1976) : f = l/[l+(t/2T ) 1 / 2] y y (7.13) The choice of T obviously has a significant effect on the resulting f and introduces considerable uncertainty into the cr y y estimates. For very short distances and travel times f y approaches unity and a linear model in place of (7.12) appears to be a reasonable simplification : cr * ar • x/u y v (7.14) This linear approximation was also proposed by Pasquill and Smith (1983) for short distances from the source. Comparisons of the results given by this model with the data from 35 unstable runs from the Hanford-30 data series (see Appendix B) showed - 89 -good agreement. Since the concept of a Lagrangian time scale as a scaling parameter becomes questionable over highly complex surfaces (and close to the roughness elements) this linear model which does not rely on estimates of a T has been adopted for y the present work. <r^  is not usually available on a routine basis. It may be estimated from the standard fluctuations of wind direction or , <P following Panofsky and Button (1984), as <r « or -u (7.15) V If) If <r^  cannot be obtained, an alternative parameterization via the mixing height, z^ is proposed by GHIS. For ground sources : (o-/u) 2 * 0.35-V * Z ->2/3 i k-L (7.16) In summary, the p.d.f.-plume model by GHIS (for unstable, surface layer scaling conditions and ground-level sources) provides a concentration estimate for any point within close range of the source. Appendix B gives the results of an independent comparison of the model with tracer diffusion observations . The required input information for the model includes u , L, z and either or , or or z . * o v <p 1 - 90 -7.3 IMPLEMENTATION OF THE DISPERSION SUB-MODEL IN THE SOURCE AREA MODEL Chapter 6 shows how a reciprocal plume model may be used to estimate the weighted source area of a sensor at a level z . s This reciprocal plume is identical to the virtual plume from a virtual source on the ground beneath the sensor, dispersed.by a virtual wind in the opposite direction to the real wind. The source area for a given significance level i s defined as the area bounded by a certain isopleth of this virtual plume. In order not to confuse the real plume with the virtual plume, the 'concentration' of the virtual plume was termed 'effect level' or source weight, <*>, where each value is scaled by the maximum along the centerline. The source weight, w, of a point (-x,-y,0) in respect to a sensor at point (0,0,z ) is identical to the concentration, %' , s at a point (+x,+y,z ), scaled by the maximum concentration on s the centerline (i.e. on the line where y = 0; z = z ), due to a s source at a point (0,0,0) : o>(-x,-y,0) = %'(x,y,z ) / y (z ) . (7.17) s ' max s In the following, an axis reversal is assumed, so that the negative sign for the x and y position and the primes of % m a y be lef t out. The concentration f i e l d of a continuous point source on the ground is defined for any level in the surface layer by equations (7.3), (7.6)-(7.11) and (7.14). Consequently, the - 91 -normalized concentration %(x,y,z) may be determined f o r any point (x,y, z ) , once the s i t e conditions Z q and d and the meteorological s c a l i n g parameters L, u and cr are known. In practice numerical i n s t a b i l i t i e s and overflow conditions l i m i t the distance i n the x-direction f o r which x i s numerically defined to regions with x £ x , and x i s determined using a m i n m i n b i s e c t i o n search routine. Subsequently, % and i t s x-location m a x are obtained by a s i m i l a r 'regula f a l s i ' search along the centerline. In order to calculate the location of a given is o p l e t h the GHIS-model equations are solved numerically f o r any one member of the t r i p l e t (%,x,y) when the other two elements are given. Because of the l a t e r a l symmetry of d i f f u s i o n only p o s i t i v e y have to be considered. Nested combinations of forward and backward stepping, bisec t i o n and secant algorithms are used f o r t h i s purpose. With these numerical tools the effect l e v e l f i e l d may be integrated portion-wise, along isopleths of decreasing f r a c t i o n s of x • The integration i s performed by a double Simpson's m a x routine. The integral over the whole effect f i e l d ( i . e . down to the zero effect level) i s approximated by a simple second order extrapolation, i n which the curvature of the la s t three nodes i s preserved. The integrated bands are then translated into f r a c t i o n s of the t o t a l integrated effect and serve as nodes f o r a cubic spline interpolation to map rounded values of integrated effect f r a c t i o n to an appropriate effect level isopleth. For each rounded integrated effect f r a c t i o n a set of c h a r a c t e r i s t i c F i g u r e 7 . 2 : The s o u r c e a r e a i s d e f i n e d by the set o f c h a r a c t e r i s t i c d i m e n s i o n s o f i t s b o u n d i n g i s o p l e t h . a : d i s t a n c e to downwind edge o f the i s o p l e t h b: d i s t a n c e from the down-wind edge to the p o i n t where t h e i s o p l e t h i s w i d e s t c : d i s t a n c e f rom the p o i n t where t h e i s o p l e t h i s w i d e s t t o the upwind edge d : maximum w i d t h o f the i s o p l e t h a+b+c: maximum f e t c h a : minimum f e t c h In a d d i t i o n , the a x i a l d i s t a n c e t o t h e maximum s o u r c e V o c a t i o n i s g i v e n by the s o u r c e a rea m o d e l . SOURCE AREA MODEL - RESULTS e SAM Calculations for Jul.Day. 212 at 12:30 L A X at SUNSET - 1 Wind at sensor level (sp./dir.) : 3.4 / 270.0 ; Zi : n/e ; SV : 0.560 Stability (Zs/L) : -0.075 ; U» : 0.300 ; Sensor height : 22.5 All data are in SI units Figure 7.3 : Example plot of source area model results - 94 -source area dimensions is determined from these isopleths. These dimensions include (see Figure 7.2), a : the distance from the sensor to the downwind centerline limit, b : the distance from the downwind centerline limit to the distance with the maximum width, c : the distance from that point to the upwind centerline limit and d : the maximum half-width of the isopleth. The area bounded by each of these isopleths i s evaluated using a Simpson's integration algorithm. This area, the characteristic dimensions for each rounded integral effect fraction and the x-distance of the maximum source location in respect to the sensor, form the principal results for a given source area model run. As an option, these results may be presented graphically, in a three dimensional plot, such as in Figure 7.3. 7.4 THE FORTRAN-77 CODE OF THE SOURCE AREA MODEL The FORTRAN-77 code of the source area model consists of a set of 23 nested subroutines and functions, adding up to some 2100 lines (including comments). The model is independent of any library routines (other than the intr i n s i c FORTRAN functions) and may be run on a mainframe computer or on a PC. On the UBC MTS-G mainframe computer (accelerated Amdahl 5850) each run takes about 7 seconds of CPU-time (ca. 20 seconds with DISSPLA plot). The time per run on a IBM-PC/XT averages about 12 minutes. Depending on the main program which c a l l s the shell - 95 -subroutine, the model may be run in a loop with imported input variables, or interactively, where the user is prompted for the model input. The entire code is presented in Appendix C. 7.5 A STATISTICAL VERSION OF THE NUMERICAL MODEL (mini-SAM) As mentioned above, a SAM run is quite time consuming and may be too tedious in situations where only a rough estimate is necessary. The results of 715 runs on a PC and on the UBC mainframe computer, with a wide range of input values, have been used to approximate the chararacteristic dimensions of the isopleths by a polynomial f i t . The ratio of the dimensions of the various integrated effect fraction isopleths in a given run remains f a i r l y constant over a very wide range of input values. The whole set of isopleths may therefore be approximated, i f the characteristic dimensions of one are known. The errors of these approximations normally do not exceed a few percent. Table 7.1 shows the standardized set of isopleth dimensions in relation to the 0.9-isopleth together with the expected standard errors. Each area between two sequential isopleths represents an equal portion (i.e. 10 %) of the total integrated effect, although their sizes vary by more than an order of magnitude. Therefore, the relative source weight of an elementary area depends on i t s position relative to the set of isopleths. The source weight per area, w, may then be written in terms of the Table 7. 1 : Standardized set of isopleth dimensions relative tp the 0.9 isopleth. The source weight per area, w , is given p relative to w (see equation (7.20)). int .eff Ap +/ - % a +/ - 'A b +/• 1 4 C +/ - % d +/• „ 1 4 V * o . i 0. 10 0.0126 7.3 2.1082 2.0 0.0787 4 3 0.1068 9 0 0.1357 3 1 1 .00 0.20 0.0311 6.4 1.7706 1 .4 0.1275 4 6 0. 1728 8 .6 0.2053 2 5 0.68 0.30 O.0571 5.6 1.5719 1.2 0.1817 4 5 0.2372 8 5 0.2700 2 0 0.48 0.40 O.0931 5.0 1.4347 1 .0 0.2420 4 6 0.3062 7 8 0.3359 1 7 0.35 0.50 0.1421 4.6 1.3314 0.9 0.3107 4 2 0.3819 7 5 0.4060 1 4 0.26 0.60 0.2107 4 . 3 1.2472 0.8 0.3955 4 3 0.4651 7 1 0.4849 1 3 0. 18 0.70 0.3147 3.7 1.1716 0.7 O.5016 4 3 0.5711 6 0 0.5821 1 1 0. 12 0.80 0.5047 2.5 1.0942 0.6 0.6663 4 3 0.7207 5 1 0.7242 0 8 0.066 0.90 1 .0 1 .0 1 .0 1.0 1 .0 0.025 - 97 -source weight distribution function Q(x,y) (and normalized by the total effect E ) as x+Ax y+Ay / v = J J £l(x,y) dx dy /(Ax - A y E t ) . (7.18) x y / For the average source weight per area between isopleths P and (P-0.1), wp, follows v = 0. 1 / [A - A ] , (7. 19) p ' L p p-o.r so that the wp may be normalized with the average source weight per area for the smallest isopleth, WQ , as A o.i w/v m (7.20) ^ °-1 [A - A ] 1 P P - 0 . 1 J Ap is the area bounded by the P-isopleth. This ratio, wp/wQ x» is a measure for the importance of a unit area of a given surface inhomogeneity depending on i t s position relative to the sensor. For example, a disturbance somewhere between the 0.9-isopleth and the 0.8 isopleth is of l i t t l e consequence to the measurement, compared to a similar" disturbance within the 0. 1-isopleth, and w^/WQ ^ indicates how much less important i t is. The standardized set of source weights per area, relative to the 0.1 isopleth is also contained in Table 7.1. The 0.9-isopleth characteristic dimensions have been approximated in terms of z , z /z , u . L and o* by f i t t i n g a s s 0 * v set of orthogonal polynomials to the model results, using the - 98 -Free-format Triangular Regression Package from the UBC Computing Centre (Le, 1983). After a transformation of variables the general form of the polynomial for dimension X (X = a, b, c, d, x , A) is given 1 i max by : In (Xt) = n n l 2 E a • ( l n z ) k + V a •(ln(z /s )) k+ ^ U k s u 12k s 0 k=0 k=0 n n E a i 3 k . ( l n u . ) k + E a U k . ( l n - L ) 1 k=0 k=0 n E a • (ln <r ) u l B k v k=0 ; (i = 1,2 6). (7.21) In this way each characteristic dimension is approximated by one linear equation of five independent variables. The f i r s t two variables are constant for any given site, which reduces the number of variables for a specific s i t e to three meteorological controls. Table 7.2 summarizes the regression constants a (as in 6 i j k equation (7.21)) and Table 7.3 contains the summary s t a t i s t i c s of the polynomial f i t . Figures 7.4 - 7.9 show the scatterplots of the comparison between source area model runs and the polynomial approximations. Though the number of terms necessary to approximate a l l the dimensions i s large, the accessibility of source area estimates is increased considerably by this s t a t i s t i c a l version of the ln tai i i i ln (b) 1 2 1 In lc) i i * ln Id) 1 4 1 In (xm) In (A) variable 00 2. 6652154 -22.573248 -21.003838 - 1.2944537 -20. 997688 -45. 205787 constant 11 - S. 3899568 0.83391462 4.9746758 1.3945269 - 1. 4336980 13. 830045 I n t : ) • '? 1 . SB07017 - 2.23S5605 - 0.10884310 0. 86714119 - 6. 2979254 l n ' d > • 13 - 0. 19333715-. 0.51065259 - 0. 10049848 1. 3733871 ln'<z I « 14 — - - 0.43476245E -01 — - - 0. 10B24610 l n ' d ) • 21 0. 32944544 17.244166 13.980797 0. 14192713 16. 326991 23.235270 In (z /z ) • o 22 - 0. 257286a3E-01 - 4.9954973 - 4.0627402 - 0.64599710E-02 - 4. 7530946 - 6. 7472282 In* (z /z ) » o 23 0. 17321196E-02 0.61049839 0.497B679S 0. 5B4269B5 0. B243109B In'(z /z ) • o 24 — - - 0.26358443E -01 - 0.21545915E -01 - 0. 25351517E -01 - 0. 35580032E -01 In" (z /z ) a O 31 0. 12238973 0.18670063 0.23067548 - 0.97617692 0. 10103001 - 0. 63979019 In tu_) 32 - 0. 320B1366E-01 0.25348158E -01 - 0.360962S7E -01 0.38747429E- 01 0. 12667125 - 0. 69395441E -01 In 1 t u > > 33 - 0. 79245576E-01 - 0.10104497 - 0.26010320 — - - 0. 39B53203 In' (uJ 34 — - - 0.74067156E -01 — - - 0. 10672003 In 4 <uj 41 1. 7965164 2. 1733248 1.778B597 0.82672876 1. 9027159 3. B322710 ln <-L) 42 - 0. 25708377 - 0.27445103 - 0.72653457E -01 - 0.582B7161E- 01 - 0. 25730721 - 0. 4904 2522 In*(-L) 43 0. 120770B6E-01 0.11555328E -01 - 0.21603852E -01 • 0. 116B5460E -01 0. 20778019E -01 I n 1 t - L ) 44 — - 0.17S39108E -02 — - — - In ' t -L) 51 0. 78914032E-01 0.19311151 0.15864185 0.99485475 0. 1962273B 1. 2280201 ln (cr ) 32 0. 53996396E-01 0.11940725 0.9109B499E -01 0. 20817542 0. 13440652 I n 2 ( a J 33 - 0. 32373654E-01 - 0.69132516E -01 - 0.59892080E -01 - 0. 74461739E -01 - 0. B6402650E -01 In' (a ) V 54 — - — — - 0. 37152093E -01 — - I n 4(CT V> Table 7.2 : Summary of mini-SAM polynomial f i t coefficients. Multiply coefficient with variable and add vert ica l ly , to obtain the characterist ic dimensions for the 0.9-source area (see eq. (7.21)). - 100 -100 - 40 30 40 50 60 70 SAM Isoplech-dimension ' a ' (m) 00 Figure 7 .4 : mini-SAM validation scatter-plot dimension a 3000 F 2500 2000 1500 1000 100 1000 1500 2000 SAM Isopleth-dimension ' b ' (m) 2500 Figure 7 .5 : Same as Figure 7 .4 for dimension b - 101 -2000 1500 - 1000 500 400 000 1200 SAM Isopleth-dimenston ' c ' (m) 1600 2000 Figure 7.6 : Same as Figure 7.4 for dimension c 3000 h 2500 H .2000 .1500 h ;1000 -500 0 1000 2000 3000 4000 SAM Isopleth-dimension ' d ' (m) Figure 7.7 : Same as Figure 7.4 for dimension d - 102 -300 -- 103 -0 Table 7.3 : mini-SAM model val idation s ta t i s t i c s a b c d xm A 0 . 9 n 715 • 715 715 715 715 715 R 2 • 96 • 92 • 94 • 93 .96 • 93 d • 99 .98 • 99 .98 • 99 • 98 0 26.08 7h].k 701.3 316. k 83.0 817.6 P 26.02 737.6 697-8 312.7 82.8 795.3 RMSE„ „• tot 3.03 120. 8k. 8 93.7 8.3*1 420.9 RMSE 0.18 2k. 6 15.8 50. k 1.08 162.4 sys RMSE 3.02 118. 83-3 79.0 8.27 388.3 unsys un i t s m m m m m 10-5 m. - 104 -numerical source area model (mini-SAM). A programmable pocket calculator may be used to compute the dimension estimates from equations (7.21). A SHARP PC-1403 pocket computer is able to perform this calculation in approximately 30 seconds. Together with Table 7.1, estimates of the source area dimensions at different significance levels and the relative importance of a specific inhomogeneity within the source area may be evaluated quickly. - 105 -PART III : MEASUREMENTS AND RESULTS 8. E v a l u a t i o n o f S p a t i a l V a r i a b i l i t y v i a S p e c t r a l A n a l y s i s 8.1 INTRODUCTION A method to quantify the representativeness of a measurement and the homogeneity of a dataset was presented in Chapter 5. It involves the computation of a two-dimensional Fourier transform of a discrete dataset. The variance spectrum is obtained by taking the squared modulus of the transform and a quantitative measure of representativeness is available from the integrated variance distribution curve in the radial wavelength domain. The relevance of remotely sensed surface temperatures during daytime and at night and the construction of an inventory of roughness elements was argued in Chapter 3 and the data acquisition was described in Chapter 4. The discrete Fourier transform (DFT) package used in this application (Moore, 1984) is most efficient when datasets with x- and y-dimensions of integral powers of 2 are used. For this reason, the DFT was performed on quadratic sub-arrays with dimensions up to 512 x 512 data-points. The two-dimensional spectra were obtained in a similar way to that described in' Steyn and Ayotte (1985). The data were f i r s t averaged, and the average subtracted from each point, in order to remove the large magnitude spike at zero wavenumber which - 106 -represents the mean data value. This array was then multiplied by a circular f i l t e r of identical size, having a value of unity over most of the domain and a cosine taper at the fringes. This taper gradually reduces the values to zero in a cosine fashion across an annulus with a width of 10% of i t s outer radius. Values at a l l greater r a d i i w i l l be zero. This f i l t e r takes a shape similar to a f l y i n g disk and was therefore named the "Frisbee-fliter" (see Figure 8.1). It is applied to avoid high frequency noise introduced by the edge discontinuities of the dataset (Justice, 1981). The tapered data array could be plotted in a perspective view using a DISSPLA plotting routine (see below). After transforming the data into wavenumber domain, the returned complex array was separated into i t s real and imaginary portions from which the frequency components of the variance could be obtained as the square moduli. The visualization of the two-dimensional variance spectrum by a perspective view plot can be used to assess the directionality or degree of isotropy of the data and illu s t r a t e s the dominant spatial scales within the dataset. Dominant scales ( i f any exist) are expected to show up as peaks in the variance spectrum (see following sections). Finally, the variance spectrum array (which is given in Cartesian co-ordinates for the wavenumber domain) can be transformed into the wavelength domain and plane polar co-ordinates as described in Chapter 5. Both the spectrum and the wavelength dependent integrated variance distribution may be plotted for individual sectors or integrated over a l l CIRCULAR "FRISBEE"-FILTER o Plot of circular filter applied to the dataset to avoid high frequency noise Figure 8.1 : The Frisbee-f iIter - 108 -sectors. It was suggested in Chapter 5 that this integrated or cumulative relative variance distribution is equivalent to a measure for the representativeness of a circular block-sample with a diameter equal to the wavelength of the distribution curve. The spectral power components are subject to two kinds of uncertainty relating to the magnitudes and the wavenumbers of the individual components. The latter type of uncertainty i s due to the discrete nature of the data, so that the spectral values are given only at the natural wavenumbers (2/N, 3/N N/N; N = number of data points in one dimension). The power of any one component (e.g. at wavenumber i/N) must be interpreted as the power of a wavenumber band (i-i/2)/N £ k > (i+1/2), where k is the wavenumber, rather than of one specific wavenumber (Priestley, 1981; pp. 429ff). If the spectral components are plotted against wavelength (A), this 'spectral smearing' appears to be more dominant at longer wavelengths because the given wavelength resolution decreases as The spectral distribution of power (or variance) within a wavenumber band is unknown. The usual procedure to handle the question of the amplitude uncertainty in spectral analysis involves the composition of an average spectrum from several independent realizations, so that the problem can be treated s t a t i s t i c a l l y . In the present context, however, single realizations are treated as individual cases and thus a s t a t i s t i c a l treatment of the amplitude uncertainty is meaningless. Consequently, the interpretation of - 109 -individual spikes in any given spectrum can be valid only for the specific sample from which the spectrum was obtained. The discussion of the spectra in the following and the inferred correspondence of individual spikes or groups of spikes with geometric patterns in the data should be interpreted with the above in mind. 8.2 SPATIAL VARIANCE OF SURFACE TEMPERATURE Figure 8.2 shows the square blocks within the daytime surface temperature data domain that were selected for the Fourier transformations. The area used from the nighttime temperatures is shown in Figure 8.3. The dominant street layout in the study area is perpendicular In the E/W and N/S directions, except for a small area in the north-eastern corner of the data domain, NE of Kingsway, where the pattern is approximately diagonal to the cardinal directions. Area (0) (Figure 8.2) was e x p l i c i t l y chosen to l i e in this region to compare the effect of a change in street directions to other regions. Area (1) contains few large 'inhomogeneities' (i.e. larger than block size non-residential areas), with the exception of Kensington Park and Tecumseh Park. Area (2), on the other hand, is characterized by large 'inhomogeneities', namely Memorial Park and Mountain View Cemetery. Area (3) is a high density residential neighbourhood with no major ' inhomogeneities' . The square block of the night data (Figure 8.3) is almost equivalent to area (1) of the 11 I • ' Li © : Sunset Tower; © : Mounta in View Cemetery ; © : Memor ia l P a r k ; © : K e n s i n g t o n Park © : Gordon Park ; © : ' H o t - C r o s s e d - B u n s 1 ; © : L a n g a r a Community C o l l e g e Figure 8 -2 : Daytime temperature sub-domains selected for Fourier-transforms. Area (0): 128x128, areas (1), (2): 512x512, area (3): 256x256 pixels. Areas (1) and (2) have a part ia l overlap. i i:-;vvv (D : Sunset Tower ; © : Mounta in View C e m e t e r y ; © : Memor ia l P a r k ; (R) : K e n s i n g t o n Park © : Gordon Park ; © : 1 H o t - C r o s s e d - B u n s 1 ; © : L a n g a r a Community C o l l e g e Figure 8-3 : Nighttime temperature sub-domain selected for Fourier-transform - 112 -daytime temperatures ( i t is merely shifted to the north a l i t t l e bit, due to a different flight-path at night). The surface temperature data are best viewed as false colour images on VZIP. Figure 8.4 shows the radiative temperature distribution of the central section of the daytime data domain. The major landmarks in the area are clearly recognizable : Kingsway cuts diagonally through the north eastern corner and the f i r s t major north/south street from the right is Victoria Street. The false colour coding maps the coldest radiation temperatures (colder than the a i r temperature which was almost O 30 C) as blue. Most street surfaces are considerably warmer, as is expected due to their low albedo values. They show up as red, corresponding to radiation temperatures around 40 °C. Dry grass areas of lawns, parks and cemeteries (Mountain View Cemetery is shown near the lef t edge of Figure 8.4) appear with colours of red, green and yellow corresponding to radiation temperatures o between 40 and 50 C. Irrigated grass is of course considerably cooler. The two large grey rectangles in the upper right centre of Figure 8.4 are irrigated soccer fi e l d s in Kensington Park. The hottest surface temperatures correspond to roofs, such as the prominent feature of the "Hot Crossed Buns" (apartment building roofs) in the top right quadrant of Figure 8.4, the roofs of commercial buildings along Victoria Street and the dotted rows of residential house roofs. These surfaces are coded as white, corresponding to temperatures above 50 °C. Sensor o saturation is reached at about 60 C. The very few pixels that show saturation can be traced to hot a i r outlets of a i r F i g u r e 8.4 : F a l s e c o l o u r image o f d a y t i m e s u r f a c e t e m p e r a t u r e i n the s t u d y a r e a - 114 -conditioners or other commercial activity. Clearly Figure 8.4 contains enough spatial information to be used as the base for a detailed c i t y map. In other words the distribution of radiative surface temperatures reflects both the regularities and the breaks in regularities, relating to the dominant objects and surface types found in a suburban area. The spatial scales of the regularities seem to be governed by the street spacing, which in turn determines the spacing of house-rows, rows of boulevard trees and alleys. Further, in more or less regular intervals the streets are wider than usual because they serve as major avenues or thoroughfares (e.g. Victoria Street on the right and Fraser Street near the l e f t edge of Figure 8.4 or, perpendicular to these, 41 Avenue just north of Memorial Park and the "Hot Crossed Buns"). Property lot-sizes on the other hand determine the house spacing along the individual streets and avenues. These findings are not surprising and might seem t r i v i a l . They become important though, i f the analysis of the spatial temperature distribution is generalized, so that the results and conclusions become independent of the specific temperature configuration at the time of the remote sensing f l i g h t (see Section 8.4). The argument here is that the dominant spatial scales of surface temperature variation are reflected in the (more easily accessible) distribution of permanent structures in the same area and are therefore similar at a l l times. (The causal relationship is obviously the other way around, but that is unimportant at the moment). This hypothesis w i l l be examined - 115 -1 l a rge c i t y - b l o c k or ' s u p e r - b l o c k ' 850 m 2 a l o n g - r o w - s t r e e t spac ing ( a l l e y spac ing) 100 m 3 a c r o s s - r o w - s t r e e t spac ing 210 m 4 house-row ( s t r e e t / a l l e y ) spac ing 50 m 5 i n te r -house spac ing (a long row) 11-15 m Figure 8.5 : Dominant spatial scales in suburban ci ty-block system - 116 -by comparing the daytime temperature distribution (Figure 8.4) with an entirely different night-time configuration (see below). In summary, i t appears that the spatial distribution of surface temperature in Figure 8.4 has a dominant component of a highly regular pattern and a minor component of breaks in this regularity caused by parks, school grounds etc. It is expected that the wavelength of the spacing of these regular features w i l l show up as peaks in the variance spectrum. The linear dimensions of these regular spacings were estimated from Figure 8.4 and from a topographic map of the area. They are summarized schematically in Figure 8.5. The spacing of major streets or avenues, which are typically wider than normal streets, is approximately 850 m, and the area bounded by two pairs of them could be termed a ' superblock'. With no park or school interrupting the pattern such a superblock contains 32 blocks, arranged in a four by eight array, as shown in Figure 8.5. The spacing of the streets along the house-rows (i.e. the shorter block length) is about 100 m on average, which makes the street-to-alley spacing about 50 m. With two rows of houses per block this gives a 50 m house-row spacing. The street spacing in the direction of the longer block dimension amounts to about 210 m. Street widths vary considerably, which explains the apparent inconsistency of 8 short blocks adding up to 850 m. The number of houses per row in a block varies typically between 10 and 15 with an average spacing of 11 to 15 m. Oke (personal communication) measured an inter-house spacing of about 23 m in the same area as a weighted mean of along-row and across-row house spacing. A similar estimate, given the preceeding spacings F i g u r e 8 . 6 : Same as F i g u r e 8 . 4 ; s p e c i a l c o l o u r c o d i n g to show s t r e e t s and houses - 118 -of rows, and houses along rows, is found here. These values are order of magnitude estimates and claim no high accuracy. It is expected, however, that the 50 m, 100 m or 200 m scales may be identified as peaks in the variance spectra. Figure 8.6 illust r a t e s the dominant way in which these scales are reflected in the surface temperatures. In this image selective false colour coding shows only small bands of radiation temperature, corresponding to the common street temperature and the roof temperatures. Figure 8.7 shows a perspective view of the surface temperature ' topography' of the square area (0) with a side length of about 0.4 km, after i t has been multiplied with the "Frisbee-fliter". The two main street axes (and accordingly the house-row axes) are NNE/SSW and WNW/ESE. The "ridge" and "canyon" sequence created by this street pattern corresponds to "waves" propagating perpendicularly to the street-axis direction, i.e. in WNW/ESE and NNE/SSW directions respectively. The corresponding two-dimensional variance spectrum of area (0) is shown in Figure 8.8. The bi-birectionality of the data is clearly visible, with two branches of spectral ridges, a major sharper one in the SSW direction and a minor but wider one in the ESE direction. Because of symmetry relative to the origin, only half the spectrum is shown. After transforming the spectrum into polar co-ordinates, the angular variance distribution relative to the total was computed. It is displayed as a rose diagram with 16 sectors in Figure 8.9. Because of the discrete width of the sectors the two SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temp, variance spectrum in wavenumber domain Figure 8.8 : Temperature variance spectrum, daytime, area (0) SPATIAL VARIANCE ANALYSIS OF S U R F A C E T E M P E R A T U R E N w \ I CO I Total Variance = 110.79 Rose Diagram of Angular Variance-% Distribution F i K u r e 8 - 9 : Directional distribution of surface temperature variance: daytime, area (0) - 122 -main axes do not appear to be orthogonal and the NW/SE arm in Figure 8.8 seems to be distributed on two sectors in Figure 8.9. The domain of this particular transform (with a sidelength of only 400 m) is too small to compute a meaningful integrated spectral variance distribution, the importance of the street-pattern in determining the anisotropy and directionality of this daytime surface temperature distribution i s clearly indicated. In area (1) (see Figure 8.2) streets and avenues run in the E/W and N/S directions respectively. The transform of a small subsection of this area (400 m diameter) illu s t r a t e s this change in directionality, from that of area (0) very nicely (see Figure 8.10). When the domain for a transform is larger, the discrete spectrum becomes very dense, spikes and peaks become sharper, and a grid-plot such as Figure 8.10 is incapable of showing a l l the detail : only the highest peaks are distinguishable. Figure 8. 11 shows the equivalent to Figure 8.10 for the whole domain of area (1). The impressive structure of the two variance branches is lost in this plot due to the lack of resolution in the plotting grid. The rose diagram for the whole of area (1) (Figure 8.12), however, illustrates that this structure is s t i l l conserved. Apart from the two cardinal axes, corresponding to the street directions in this case, the area would qualify as isotropic, but as expected, the influence of the street grid on the anisotropy is very strong. Note that the total variances of areas (0) and (1) compare quite well, i f one considers that (1) is sixteen times larger than (0) (the standard deviations d i f f e r SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temp, variance spectrum in wavenumber domain ELgure 8.10 : Same as Figure 8.8 : daytime, area (1), 128x128 pixel sub-section SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temp, variance spectrum in wavenumber domain Figure 8.11 : Same as Figure 8.10 : whole area - 125 -SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE W I r po.o IS. i 1 E 13.0 30.0 Total Variance = 129.43 Rose Diagram of Angular Var iance-% Distribution F 1 gure 8.12 : Same as Figure 8.9 : daytime, area (1) I" •0 0 2 io-» r o v . U f i e l A X (in km) Figure 8.13 : Temperature variance spectrum (radial) : daytime, area (1) Figure 8.14 : Normalized integrated radial variance spectrum of surface temperature :daytime, area (1) - 126 -by only 0.85 °C. ). In polar co-ordinates the variance spectrum may be plotted against radial wavelength by integrating over a l l angles. The radial variance spectrum of the daytime surface temperature in area (1) is given in Figure 8.13. The spectral variance density has been divided by the wavelength to account for the proper visual weighting of the spectrum in a logarithmic plot. The values have been scaled by the maximum occurring variance density, since absolute magnitudes are unimportant here. Due to a limit in the number of spikes that can be drawn, the smallest ones had to be omitted. Note that this plot is not a real integral with respect to angle. It i s merely a "sweep" 360° around the origin, so that a l l the spikes are aligned on one linear, radial axis. In contrast, the integrated variance spectrum plot (below) involves true integrations. The interesting features in Figure 8. 13 are the three high peaks. The dominant one corresponds to a wavelength of about 25 m and the two minor ones to around 20 m and 50 m respectively. Following from the discussion of Figure 8.5, the 50 m peak can easily be identified as the scale of the house row and street/alley spacing. Figure 8.4 showed that both street surfaces and house roofs are relatively warm. Therefore i t is suspected that the regular 25 m spacing of house rows and streets or alleys is reflected in the cluster of high peaks at the 25 m wavelength in Figure 8.13. The origin of the minor peak at about 20 m is less clear. It might correspond to the inter-house spacing along the rows. 20 m is rather long for - 127 -that, but could be attributed to 'spectral smearing' (see above). Figure 8.14 shows the integrated radial variance spectrum for area (1). In normalized form, the integrated spectral variance is equivalent to the measure for representativeness, R, of a specific wavelength, as described in Chapter 5. Here, R increases very quickly at short wavelengths and levels off at about 200 m, where R is already well above 0.9. This result means that a circular block sample of 200 m diameter i s , on average, more than 90 % representative of the temperature distribution in area (1). In addition, the very strong curvature away from the diagonal across the plot indicates a high degree of homogeneity (compare with Chapter 5 and Figure 5.1) : the contribution to the. total variance made by the longer wavelengths is minimal, compared to that by the shorter wavelength components. Further implications of these findings are discussed below, after examining more examples of daytime and nighttime temperature distributions and the roughness element distribution. The integrated radial variance spectrum is broken up into 16 component sectors of width 22.5° (only 8 sectors are relevant, for symmetry reasons) and is displayed in Figure 8. 15a-h. The influence of each sector on the anglewise integrated variance (Figure 8.14) is equivalent to the relative contribution of that sector to the total variance, as in Figure 8. 12. It is apparent from Figure 8.15a-h that the variation among sectors is very small. In general i t is observed that the R-values of the - 128 -DrrCCRATtO RJ.DUL V*RIAMCE-SPtCTRUU Of SURftkCT TCUPCIUTURC IMTICRATCD RASUL VARUNCr-SPECTBUU Or SURT1CE m i P E R A T U R t , w r t C R m o •uouLv.RUMCE-sPr.cTRUw or s U R r * c r TEMPERATURE < MTICIUITD RADIAL VARIAHCE-SPECTRUU or SURPACE T W E R A ™ . . 5 o.« c) Stcter .» - 20Z.S*tt t.2S* (SS*/NNE) o.o o.2 o.4 o.a o.o i.o i.a 1.4 j.a Vaw«l«n0i't X f in JkmJ d) Stelor . • - rdO.O**M.25* (S/H) 0.0 0.2 0.4 0 A OA 1.0 1.2 1.4 i.fl *awl*n9iK X f in kmj Figure 8.15 : Sectorial break-up of normalized integrated variance spectrum of surface temperature : daytime, area (1) - 129 -1NTECRATED RADIAL VARIANCE-SPECTRUM Of SURrACE TEUPERATURE [NTECRATED RADIAL VARIANCE-SPECTRUM O f SURFACE TEMPERATURE INTECRATEO RADIAL VARIANCE-SPECTRUM O r SURrACE TEMPERATURE INTEGRATED RADIAL VARIANCE-SPECTRUM Or SURTACI TEMPERATURE (Figure 8.15, continued) - 130 -dominant sectors rise quickest, whereas minor sectors contain some large scale variance components, but they are rel a t i v e l y insignificant for the spectrum as a whole. A possible reason for this effect is the more or less irregular distribution of inhomogeneities such as small parks which may contribute to the variance in any direction, according to their position relative to each other. In area (2) the N/S components of both the 25 m and the 50 m wavelengths are much more distinct than in area (1) (see Figure 8.16). This is to be expected since the comparison of areas (1) and (2) i n Figure 8.2 shows that the streeet/alley- and street/ house-row-spacing in N/S direction i s much more dominant in area (2) than in area (1). Figure 8.2 shows that area (2) contains a series of large scale inhomogeneities in NW/SE direction, namely Mountain View Cemetery, John Oliver School and Memorial Park. In fact, area (2) was sp e c i f i c a l l y chosen because of these features. Although i t cannot easily be determined from Figure 8.16, the large peaks at very low wavenumber l i e in the NW/SE sector. This conclusion is confirmed by the sectorial break-up of the integrated variance spectrum (see below). Curiously, the double peak near wavenumber 40 km"1, corresponding to 20 m and 24 m wavelengths, already observed in area (1) is also present in the E/W-direction in Figure 8.16, but is not identifiable in the N/S direction. The shape of the rose diagram for area (2) (Figure 8.17) is as expected and compares closely with the rose plot of area (1). SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temp, variance spectrum in wavenumber domain Figure 8.16 : Same as Figure 8.8 : daytime, area (2) - 132 S P A T I A L V A R I A N C E A N A L Y S I S O F S U R F A C E T E M P E R A T U R E N poc / ' o w i i r^£% 30.0 I5.01—^<q S S C J ' ' 1 E 1 IS.0 30.0 \ ***. 9 * ***>-s T o t a l V a r i a n c e = M 9 . 5 6 R o s e D i a g r a m o f A n g u l a r V a r i a n c e - / ! D i s t r i b u t i o n F i g u r e 8.17 : Same as F i g u r e 8.9 : dayt ime, a r e a (2) 10-* IO'1 10° 0.0 0.2 0.4 O.fl 0.8 1.0 1.2 1.4 I.a ravUngtn X (in km) TawUngth \ (in km) F i g u r e 8.18 : Same as F i g u r e 8.13 : dayt ime, a r ea (2) F i g u r e 8.19 : Same as F i g u r e 8.14 : dayt ime, a rea (2) - 133 -INTEGRATED RADIAL VARIANCE - SPECTRUM Or SURFACE TEMPERATURE INTEGRATED RADIAL VARIANCE-SPECTRUM Or SURrACE TEMPERATURE : o.« a) 0.0 0.2 0.4 0.4 0.8 1.0 1.2 1.4 1.0 favtwngth X (in km) INTEGRA TED RADIAL VARIANCE-SPECTRUM Or SURfACt TEMPERATURE I N T E G R A T E D R A D I A L V A R I A N C E - S P E C T R U U or S U R F A C E T E U P E R A T U R E d ) Sielor .• - 160.0'tt 1.25* (S/N) o.o o.z 0.4 0 6 a.a 10 1.2 1.4 1.6 favtltnglh X (in km) Figure 8.20 : Same as Figure 8.15 : daytime, area (2) - 134 0-0 0.2 0.4 0.0 0.0 1.0 1.2 1.4 l.fl 0.0 0.2 0.4 0.6 0 6 10 1.2 1.4 1.0 r*v*ltnQtn k (\n km) JTavWtnftA X (in km) (Figure 8.20, continued) - 135 -Also the radial variance spectrum (Figure 8.18) and the R-curve (Figure 8.19) compare quite well with their equivalents from area (1). A l l three peaks or peak clusters identified in the radial variance spectrum for area (1) are again present and even more dominant (Figure 8.18). Representativeness (Figure 8.19) reaches a value of about 0.9 at 0.2 km wavelength and levels off towards longer wavelengths, as seen before. The two large steps at approximately 750 m and 1200 m may be attributed to the large scale inhomogeneities in this area. Figures 8.20a-h support the general impression gained from the equivalent for area (1). The long wavelength contributions in the non-street directions are stronger than in area (1). This is especially true for the SE/NW and the SSE/NNW sectors (Figure 8.20e,f), which confirms the interpretation that the series of peaks in the NW/SE direction in area (2) is responsible for large long-wave peaks in that sector (and, as i t turns out, the adjacent sector). While the dominant street/house row/alley sequence occurs in the E/W direction in area (1), the street block arrangement is turned by 90 degrees in area (3), and the dominant sequence occurs in the N/S direction (Figure 8.2). The very distinct peak at wavenumber 40 km-1 in the N/S direction (Figure 8.21) and the N/S bias of the rose diagram (Figure 8.22) are clear proofs of that observation. Note that the domain of area (3) is only a quarter of the size of areas (1) and (2). The regular street-block structure in this area (3) is almost undisturbed by large scale inhomogeneities (Figure 8.2) and accordingly, the low wavenumber peaks in Figure 8.21 are comparatively small. SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temp, variance spectrum in wavenumber domain Figure 8.21 : Same as Figure 8.8 : daytime, area (3) - 137 -SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE 1 poc o / \ 30.0 i5.o ^ l?.0 30.0 s T o t a l Variance = 1 5 4 . 1 4 R o s e D i a g r a m o f Angular Var iance-% D i s t r i b u t i o n Figure 8.22 : Same as Figure 8.9 : daytime, area (3) 10* 1 1 0 " 10° 0.0 0.1 0.2 0.3 0.4 0.S 0.6 o.? o a Watttlrngth X {in km) Vawtmgtn X (ir\ tmj Figure 8.23 : Same as Figure 8.13 : daytime, area (3) Figure 8.24 : Same as Figure 8.14 : daytime, area (3) - 138 -IKTEC RATED RADIAL VARIANCE-ST ECl RUU Or SURfACC TEMPERATURE 0.2 0.3 0.4 0.5 O.fl 0.7 WavtlingtS A (in km) IH TEC RATED RADIAL VAJIUHCE-SPCCTnUU Of SURFACE TEUPERATURE rNTTGRATED RADIAL VARIANCE-SPECTRUM Of SURrACE TEUPERATURE INTEGRA TED RADIAL VARIANCE-SPECTHUU Of 5URTACC TEMPERATURE c) SieUr - Z0t.i*tlt.2S* (SST/NNE) 0.2 0.3 0.4 0.5 0.0 0.7 *a\ttlt*\gth A (in km) 0.0 0.1 0.2 0.3 0.4 0.4 O.a ITavtltngth A (in km) Figure 8.25 : Same as Figure 8.15 : daytime, area (3) - 139 0.0 0.1 0.2 0.3 0.4 0.S 0.0 O.T 0.0 0.0 0.1 0.2 0.3 0.4 O.J 0 0 0.7 0 » r t v . l n i f f A A f o , km) f a i . a l «n ; IA A f in km) INTEGRATED IU0UL VARIANCE-SPECTRUM Or SUREACE TEMPERATURE INTEGRA TCP RADIAL VARIANCE-SPECTRUM Of SURf ACE TEMPERATURE 0.3 0.4 0.5 0.0 rovUnfltx X (tn km) (Figure 8.25, continued) - 140 -Figures 8.23 and 8.24 are a familiar sight by now. The simil a r i t y with both area (1) and area (2) is striking especially for the R-curve (note that the maximum wavelength is only half of the ones for areas (1) and (2)). The peaks at 20, 25 and 50 m observed in areas (1) and (2) are also present here, however, they are supplemented by two additional peaks at about 15 m and 35 m. Since area (3) is smaller than areas (1) and (2), small inhomogeneities or irregularities in the spacing structure of streets and houses can cause higher peaks in the variance spectrum than for the larger areas. The nature of these peaks is uncertain. It might be traceable to a shift of the houses towards the streets or towards the alleys (note that 15 + 35 = 50) or might reflect the spacing between garages, trees etc. , but this is not of prime importance here. Also the break-up of the integrated variance into the various sectors compares well with the previous results (see Figure 8.25). The large long wavelength contributions in the non-cardinal sectors are apparent rather than significant since these sectors contribute l i t t l e to the total (compare with the rose diagram in Figure 8.22). This strong consistency between the spectral variance distributions between different areas, and even different domain sizes, indicates an overall homogeneity to the spatial temperature distribution in the data domain. It also allows a claim for v a l i d i t y of the results for more than one particular configuration (since the temperature configurations of areas (1), (2) and (3) are different, but the results are very similar). The results for the daytime case can be further - 141 -generalized when they are compared with the results for the nighttime temperature distribution. Figure 8.26 shows a sub-section of the nighttime surface temperature distribution in a perspective view after i t has been multiplied by the Frisbee-filter. Compared to the daytime case, the temperature v a r i a b i l i t y i s much smaller (compare with Figure 8.7). The street surfaces are by far the warmest surface-type and roof-tops are coldest, due to their large sky view factor, which f a c i l i t a t e s radiative cooling, as il l u s t r a t e d by the VZIP false colour image in Figure 8.27 (see also Roth, Oke and Emery, 1988). The variance spectrum of the nighttime temperature distribution (Figure 8.28) reflects this diminished v a r i a b i l i t y : the sharp peaks are considerably smaller than the highest ones for areas (1) and (2) of the daytime cases (the automatic scaling of the spectrum power axis in this plot exaggerates the peaks in the nighttime case). Whereas in the daytime configurations the 40 km-1 wavenumbers (= 25 m spacing) were the most important components, this street/house/alley spacing appears to be almost negligible in the nighttime case. In the daytime both street surfaces and house roofs are relatively warm and represent "ridges" in the temperature "topography", with "troughs" of cooler, often irrigated, vegetation in between (see Figure 8.4) so that the street/house/alley spacing corresponds to the distance between "wavecrests". At night, the situation is different : streets are warmest, vegetation is intermediate and house-roofs are coldest SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE Plot of surface temperature data in space domain Elgure 8.26 : Perspective view of nighttime temperature 'topography* s a t ! * -HWIMI i f l i tHHi l *• MtfL 4Mf M « • " • ' • K M « M 2f2 *t M l / *M *». CO 1Mb. 4 I Figure 8.27 : False c o l o u r image of nighttime s u r f a c e temperature i n the study area SPATIAL VARIANCE ANALYSIS OF S U R F A C E TEMPERATURE Plot of surf. temp, variance spectrum in wavenumber domain Figure 8.28 : Same as Figure 8.8 : nighttime - 145 -SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE N . - - - o " --*" d rt ' 9 30.0 l i - o - ^ S> ~~*P.O 30.0 \ q s T o t a l V a r i a n c e = 2 1 . 0 5 R o s e D i a g r a m o f A n g u l a r V a r i a n c e - % D i s t r i b u t i o n Figure 8.29 : Same as Figure 8.9 : nighttime 10-' I 0 - 1 10° 0.0 0.2 Oi 0 « 0.8 1.0 1.2 l.« 1.0 V a v . L n o l n X fin km) ' ov tUnff lA X f i n km) Figure 8.30 : Same as Figure 8.13 : nighttime Figure 8.31 : Same as Figure 8.14 : nighttime - 146 -IWTECRATED RADIAL VARIAHCE-SPECIRUM or SURrACE TEMPERATURE a) 00 0.2 0.4 0.0 0.0 t.O 1.2 |.4 fivtltnfflh \ (in km) INTEGRATED RADIAL VARIANCE-spEcrnuw or swrxci TEMPERATURE 00 OS t.O 1.2 **av«Unf(/t k (in km) INTEGRATED RADIAL VARIANCE-SPECTRUM Of SURrACC TEUPERATURC c) S*<el»r .• - !0t.fm.2S' (SSf/NNE) 0.0 0.8 0.4 0.0 0.8 1.0 1.2 1.4 rawing** A (in km) IHTECRATE0 RACIAL VARIANCE-SPECTRUM Of SURFACE TEMPERATURE Figure 8.32 : Same as Figure 8.15 nighttime - 147 IHTCCRATXO RADIAL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE 0.0 0.2 0.4 0.0 0.6 1.0 1.2 1.4 t.8 Wavttnotn X (in km) INTEGRATED RADUL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE 1.0, , 0.0 0.2 0.4 O.a 0.8 10 1.2 1.4 16 WaviitnftK A (in km) INTEGRATED RADIAL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE g) Stet»r - 112.5* kt t .23* (ESt/WHW) INTEGRATED RADIAL VARIANCE-SPECTRUM OF SURFACE TEMPERATURE oo o.2 o.4 o.a o.a 1.0 1.2 1.4 i.a rawtingth * (in km) J'av*Itna(A X (in km) (Figure 8.32, continued) - 148 -(Figure 8.27). As a result the "wavecrest" to "wavecrest" distance corresponds to the street/alley spacing and is twice as long as the street/house/alley spacing. Accordingly the 20 km 1 wavenumber peaks (= 50 m wavelength) are dominant at night, as illustrated in Figure 8.28. The peak in the centre, at very low wavenumbers seems to be similar to the one in Figure 8.11 for area (1) of the day case, but i t is much lower. The nighttime rose diagram (Figure 8.29) is almost identical to Figure 8.12 : the street directions dominate. Note that the total variance has decreased by more than a factor of 6 between the daytime and the nighttime temperature distributions. In correspondence with the perspective view of the variance spectrum, the radial plot in Figure 8.30 shows a dominant cluster of peaks at about 50 m wavelength, but not at 25 m. The relat i v e l y greater importance of larger scales is also apparent in the integrated spectrum (Figure 8.31) : an R of 0.9 is only reached at 0.4 km. However, this effect should be viewed in relation to the magnitude of the total variance. If the temperature distribution is generally f l a t , the error resulting from an insufficiently representative sample i s s t i l l expected to be very small. More important here is to note that a circular block sample with 200 m diameter (which was shown to have an R-value of well over 0.9 during the day, when temperature differences are large) is s t i l l about 80 % representative even with the much decreased total variance at night. The break-up of the total variance into sectors (Figure 8.32a-h) shows similar characteristics to the daytime cases. The - 149 -large scale contributions to the variance are most apparent in the non-street axis directions. In summary the spectral distribution of surface temperature variance seems to be consistent for several different daytime configurations and the few differences when compared to the nighttime case, are of minor importance. The spatial structure is in general relatively homogeneous and the dominant spatial scales can easily be correlated with the spacing of permanent structures. 8.3 SPATIAL VARIANCE OF ROUGHNESS ELEMENTS The area covered by the roughness element inventory (see Chapter 4) is shown in Figure 8.33. Since the pixel dimensions of the dig i t i z a t i o n have a non-unity ratio, the three 512 x 512 pixel blocks used for the Fourier transform are not quadratic (in space). They are indicated in Figure 8.33 by the dashed lines and labelled according to their position within the inventory area. The heights of the surface cover elements in a VZIP image are assigned colours, was already presented in Figures 4.3 and 4.4. A small section of the area is shown in a perspective view plot in Figure 8.34. The distinct street-canyon morphology is clearly v i s i b l e and over-emphasized by the approximately twofold height exaggeration in this plot. It is therefore no surprise that the house-row spacing of 50 m forms the strongest component in the variance spectrum shown in Figure 8.35. In fact, the (D : Sunset Tower; (G) : Mountain View C e m e t e r y ; (0) : Memor ia l P a r k ; (R) : K e n s i n g t o n Park © : Gordon Park ; © : ' H o t - C r o s s e d - B u n s 1 ; 0 : L a n g a r a Community C o l l e g e Figure 8.33 : Area of roughness element inventory showing the sub-domains selected for Fourier transforms. The three sub-domains have dimensions of 512x512 pixels. They a l l overlap each other partially. SPATIAL VARIANCE ANALYSIS OF R O U G H N E S S ELEMENTS Figure 8.34 : Perspective view of roughness element d is tr ibut ion (128x128 pixel sub-set). The high spikes correspond to trees, the larger blocks are buildings and the low spikes garages. Grass and streets are not distinguishable. SPATIAL VARIANCE ANALYSIS OF R O U G H N E S S ELEMENTS Plot of rough, elem. variance spectrum in wavenumber domain Figure 8.35 : Roughness variance spectrum : area SE - 153 -SPATIAL VARIANCE ANALYSIS OF ROUGHNESS ELEMENTS w 15.0 30.0 1 E T o U l V a r i a n c e = 6 2 . 2 0 R o s e D i a g r a m of A n g u l a r V a r i a n c e - % D i s l r i b u l i o n Figure 8.36 : Directional d is tr ibut ion of roughness variance : area SE 0.4 o.s rav«Unft \ X (in tm) Figure 8.37 : Roughness variance spectrum (radial) :' area SE Figure 8.38 : Normalized integrated radial variance spectrum of roughness elements : area SE - 154 -INTEGRATED RAOUL VARIANCE-SPECTRUM Of ROUGHNESS ELEMENTS INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHHESS ELEMENTS 0.4 O.fl Wavlmgth k (in km) IWTECRATED RADIAL VARIANCE-SPECTRUM Or ROUGHNESS ELEMENTS INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHKnSS ELEMENTS V a v W r n t f l A A (in km) Figure 8.39 : Sec t o r i a l break-up of normalized integrated variance spectrum of roughness elements : area SE - 155 -p-rac RATED RAOIAL vARiANCE-sprcnunj or ROUGHNESS ELCUEHTS »"av(mgth K (\n km) INTEGRATED RAOIAL VARIANCE-SPECTRUU Or ROUCHNESS EUUCNT5 0.4 o.a Ifavtlmgth \ (in km) ( F i g u r e 8.39, con t inued) SPATIAL VARIANCE ANALYSIS OF R O U G H N E S S ELEMENTS Plot of rough, elem. variance spectrum in wavenumber domain igure 8.40 : Same as Figure 8.35 : area SW - 157 -SPATIAL VARIANCE ANALYSIS GF ROUGHNESS ELEMENTS N --27 o s Total Voriance = 62.75 Rose Diagram of Angular Var iance - / ! Distribution Figure 8.41 : Same as Figure 8.36 : area SW Figure 8.42 : Same as Figure 8.37 : area SW ifowUnglh \ (in ttn) Figure 8.43 : Same as Figure 8.38 : area SW - 158 -0 0 0.2 0.4 O.A OA 0.0 0.2 0.4 0.0 o.a Tavcttngth X (in km) ITawlmgth \ (in km) INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUGHNESS ELEMENTS INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHNESS ELEMENTS 0.4 o.a f<t\nt*ngth X (in km) favtUngtK X (in km) Figure 8.44 : Same as Figure 8.39 area SW - 159 -INTEGRATED RADIAL VARIANCE-SPECTRUM Or ROUCHNESS ELEMENTS IHTEC RATED RADIAL VARIANCE-SPECTRUM Or ROUGHNESS ELEMENTS e) 0.2 0.4 0.6 O.S WnwtmgtK X f i n km) INTEGRATED RADIAL VARIANCE-SPECTRUM Or ROUCKKESS ELEMENTS r*vltngth X (in km) INTEGRATED RADIAL VARIANCE-SPECTRUM or ROUCHNESS ELEMENTS h) Sictor _ 90.0*iJ I Si* (£/*) oo o.2 o.4 o.a ' « v f J « n « t A A (in km) (Figure 8.44, continued) SPATIAL VARIANCE ANALYSIS OF R O U G H N E S S ELEMENTS Eigure 8.45 : Same as Figure 8.35 : area NE - 161 -SPATIAL VARIANCE ANALYSIS OF ROUGHNESS ELEMENTS r . - --o-«•» •J •' 9 / „ jai -w i — r ^ ~ ^ 30.0 li^ O—• X^J ^ C T j i E v** >,.o 30.q o c Total Variance = 57.97 Rose Diagram of Angular Variance-?! Distribution Figure 8.46 : Same as Figure 8.36 : area NE Figure 8.47 : Same as Figure 8.37 : area NE Figure 8.48 : Same as Figure 8.38 : area NE - 162 -WTECRATEQ RADIAL VARIAMCE-SPECTRUV or ROUCKHESS ELCUEHTS IKTECRATED RADIAL VARIANCE-SPECTRUM Or ROUCKNESS ELCUEHTS a) s.eiw . • - i47.s'in.ts' (rsr/ri/t) o.< o.e Jawtmgtti X (in km) 0.2 0.4 0.0 ' a v a l t n g t v X (In km) INTEGRATED RADIAL VARIANCE-SPECTRUM Or ROUCHHCSS ELEMENTS ; INTEGRATED RADIAL VARIANCE-SPECTRUM or ROUCHNESS ELEMENTS d) Sictor - 130.0* tit.25' (S/N) 0.4 0.0 0.0 r*vii*tigt\ \ (in km) Figure 8.49 : Same as Figure 8.39 area NE - 163 -(HTEGRATED RADIAL VARIANCE-SPECTRUM OF ROUGHNESS ELEMENTS INTEGRATED RADIAL VARIANCE-SPECTRUM OF ROUCHNESS ELEMENTS 0.* 0.0 Wav*l*ngt\ \ (in km) [NTECRATED RADIAL VARIANCE-SPECTRUM OF ROUGHNESS ELEMENTS g) Stetor - ttZ.S'tlt.ZS' (ESC/WHW) INTEGRATED RADIAL VARIANCE-SPECTRUU Of ROUGHNESS ELEMENTS WavltnQtn A (in km) r*v*l»netn X (in km) (Figure 8.49, continued) - 164 -results of the spatial variance analysis of roughness elements are so similar to the results of both the daytime and the nighttime surface temperature analyses, that the roughness results need to be discussed only briefly. The various plots in Figures 8.35 to 8.49 are perfectly analogous to those for the surface temperature analysis. They mostly speak for themselves and are included here for completeness. Summarizing i t may be mentioned that the results for the three sub-sections of the roughness inventory domain, labelled NE, SE and SW, are very similar, both to each other and to their analogous in the surface temperature analysis. The street axes are the dominant factors determining the anisotropy of the area and the house-row spacing (or the street/alley spacing) is the dominant spatial scale. This spatial scale might be the only significant difference to the spatial structure of the daytime surface temperature distribution, where the street/house/alley spacing (about half the length of the house-row spacing) is superior, due to the effect described in Section 8.2. The strong curvature of the integrated variance spectrum curve (see e.g. Figure 8.38) confirms the homogeneity of the area found for the surface temperatures. Similarly, the representativeness measure, R, reaches a value well over 0.9 at wavelengths of only 0.2 km (see e.g. Figure 8.38). - 165 -8.4 ESTIMATING SPATIAL REPRESENTATIVENESS The preceding discussion of results from the spatial variance analysis of surface temperature and roughness elements indicates clearly that the dominant spatial scales of variance are governed by the spacing of permanent structures or surface types. In particular the consistency of the integrated radial variance curves for both temperature and roughness suggests a generalized description of the spatial structure of meteorological surface parameters in terms of the distribution of permanent structures and surface cover patterns. It was stated in Chapter 3 that the spatial v a r i a b i l i t y of sensible heat flux and the representativeness of measurements of this flux may be examined by analysing the spatial distribution of surface temperature and the dominant roughness elements. Therefore, given the finding that the spatial distributions of surface temperature and roughness elements are similar, i t follows that the common scale of representativeness for surface temperatures and roughness elements is also the scale of representativeness for a sensible heat flux measurement. Choosing an arbitrary R-value of 0.9 as a c r i t e r i o n for a s u f f i c i e n t l y representative block sample, i t also follows from the preceding analyses that a circular block sample with a diameter of 200 m w i l l meet this crit e r i o n in most cases in the present study area. 200 m happens to be four times the length of the single most dominant spatial scale identified : the average spacing between house-rows across the streets or alleys and the - 166 -spacing of the street and alley pairs. Alternately, i t is two times the spacing of the street-pair forming the long sides of a block. Due to the anisotropy of the area (bias of variance towards the street axes), i t is important that a block sample involves an area across the house-rows, since i t is in that direction that the variance is concentrated (as reflected in the rose diagrams of the previous sections). This precaution is of course t r i v i a l in the case of a circular block sample, since the dimension is 200 m in a l l directions. It is not t r i v i a l for a non-circular (e.g. e l l i p t i c a l ) sample. The source area considerations in Chapter 6 show that the shape of the "sample" of a turbulent flux measurement is only rarely circular. To be conservative, i t may therefore be recommended that the smallest axial dimension of a block sample should be at least four times the house-row spacing, in order to meet the representativeness cr i t e r i o n of R -£ 0.9. This scale of representativeness (four times the house-row spacing) is valid, on average, in suburban Vancouver, B.C., where the measurements that are the basis for this conclusion were taken. It may easily be extended, however, to other c i t i e s with suburban areas of similar character (e.g. the land-use class R2 of Auer (1978)). It is very d i f f i c u l t to speculate at this point, whether these findings may be used for non North American c i t i e s or for complex surfaces in general. Further work in this area is potentially very interesting for a wide research community. The measure of representativeness, R, was formulated in Chapter 5 to minimize the dependence of this representativeness - 167 -on a specific realization of the sample. Some contingency upon the configuration of the sample remains, however. For example, i f the sample is located over a large-scale inhomogeneity (e.g. a large park), the result i s probably not representative of the whole area. If i t is possible to monitor the location of the sample, such unrepresentative cases can easily be identified. The combination of the source area model presented in Chapters 6 and 7 and the representativeness estimate developed here serves as an objective criter i o n to accept or reject a particular sensible heat flux measurement as being spat i a l l y representative. In the next chapter practical examples are used to show how the size of the source area may affect the spatial representativeness of a turbulent flux measurement. - 168 -9. Simultaneous Eddy Correlation Measurements at Two Sit e s : Five Configurations 9. 1 INTENTIONS Consider two eddy correlation measurements of sensible heat flux at two different locations in the same area and at approximately the same height in the surface layer. Source area calculations for the two sensors, following the method described in Chapter 6, are expected to be similar as far as their size and orientation relative to the respective sensor are concerned. However, due to the different location of the two sensors, the actual location of the two source areas on the ground w i l l be different. Therefore, i f the surface conditions in the area have considerable spatial v a r i a b i l i t y , i t is possible that the two measurements wi l l have different magnitudes. It is suggested in Chapters 5 and 8 that the spatial representativeness of measurements depends on the size of their source areas : the larger the two source areas are, the higher the spatial representativeness of the measurements and therefore the smaller the difference between the two measurements, i.e. the smaller the site uncertainty. If the source areas are small, and the measurements are not representative, the deviation between the two measurements will not necessarily be large, but i t w i l l be very variable from case to case. The two measurements - 169 -might, by chance, be very similar, depending on the type of surface elements that the two source areas include. Over a large number of realizations i t is therefore expected that the spread in the differences between the two measurements is wider when the source area is small and that i t becomes smaller as the source area increases. It follows that the s i t e uncertainty of the measurement is inversely dependent on the size of the source area. A large number of eddy correlation measurements of sensible heat flux density (Q ) at paired sites were conducted in the H summer of 1986 to examine the above hypothesis. These measurements and the resulting relationship between the range of the observed flux differences and the source area size (using the 0.9-level source areas as computed by the source area model described in Chapter 7) are discussed in the following sections. In Chapter 8 a spatial scale of representativeness was determined for the source area of sensible heat flux measurements in the study area, based "on the spatial distribution of surface characteristics. This scale and the concept of representativeness are developed in terms of discrete block samples, where the elements within the sample a l l have the same non-zero weight and those outside have none. The source area of a time averaged turbulent flux measurement, however, is not a block sample in the above sense. Its source weight distribution function reaches a maximum at a single point and then f a l l s off asymptotically on a l l sides, as described in Chapters 6 and 7 (see e.g. Figure 7.3). In - 170 -accordance with this notion, the source area i t s e l f i s defined in terms of the fraction of the total integrated effect i t contains : the source area for each integrated effect fraction level contains the source areas of a l l lower levels (see Chapter 6). Obviously, i t Is d i f f i c u l t to decide which level of effect fraction i s relevant for estimates of representativeness. Each measurement "samples" (in the course of an averaging time period) over a very large area and the source area for every measurement would Inevitably be considered representative, i f only a high enough integrated effect level is chosen. (Note that the source area for an integrated effect level of unity i s always i n f i n i t e l y large in theory. ) The source weight function demonstrates, however, that points far away from the maximum source location have practically no influence on the measurement (Chapter 6). If a dependence of the s i t e uncertainty of the measurements at the 0.9-level source area size can be demonstrated, such a relationship may help to decide how large the 0.9-level source area needs to be for the site uncertainty of the measurements to become negligible (compared to the instrument uncertainty). It is anticipated that this c r i t i c a l 0.9-level source area w i l l have dimensions that are considerably larger than the scale of representativeness as determined in Chapter 8, because i t includes a large portion with almost no weight, for the reasons described above. It is suggested that the appropriate level of the source area can be found by matching the smallest axial dimension of the various source area levels with the scale of - 171 -representativeness and at the same time meeting the size requirements of the 0.9-level of this source area. In the following sections the measurement sites, the instrumentation and the data processing involved w i l l be outlined. A f i e l d comparison of the two eddy correlation systems is used to determine the instrument uncertainty, which in turn helps to evaluate the site-related portion of the differences between the measured Q values at the two sensors. After the H 0.9-level source area dimensions have been computed using the meteorological conditions of each hourly Q -observation above 10 H 2 W/m , a method is suggested to evaluate the contingency of the site uncertainty on the source area size. 9.2 THE SITES During late July, a l l of August and early September, 1986, simultaneous sensible heat flux density measurements were taken using two similar eddy-correlation systems (see below). One set was held fixed at the Sunset meteorological tower, while the other set was mounted on a mobile tower. During the measurement period this mobile tower was operated at five different locations (see Figure 9.1). The Sunset tower si t e served as the main si t e and a l l meteorological data necessary for the SAM-runs were collected at that site (see below). The Sunset tower site was described in Section 1.3, here a brief description of the five mobile sites - 172 -S : Sunset Tower 1 : Culloden Site 2 : Argyle Site 3 : Waver ley Site k : Memorial East Site 5 : Memorial West Site SCALE 500 m 1 km Figure 9.1 : Measurement s i tes F i g u r e 9 . 3 : A r g y l e s i t e ( l o o k i n g W) F i g u r e 9-4 : W a v e r l e y s i t e ( l o o k i n g N) F i g u r e 9 . 7 : M e m o r i a l West s i t e ( l o o k i n g NE) F i g u r e 9 . 8 : M e m o r i a l West s i t e ( l o o k i n g W) - 177 -is given. With the mean daytime winds expected to be governed by the westerly sea-breeze (Hay and Oke, 1973; Steyn and Faulkner, 1986), sites (1) Culloden, (2) Argyle and (3) Waverley were chosen to be spread out in the N/S direction (see Figures 9.1 and 9.2-9.4). Culloden and Argyle are both separated from the Sunset tower by a distance in the order of 1 km, whereas Waverley is only about 400 m distant. Sites (4) Memorial East and (5) Memorial West were selected to l i e just downwind and upwind respectively of Memorial Park during sea-breeze conditions, in order to examine the influence of this major inhomogeneity on the sensible heat flux. These two sites (Figures 9.5-9.8) are separated by approximately 500 m, and l i e 1 km (West) and 780 m (East) away from Sunset. With authorization from Vancouver City, Engineering Dept., i t was possible to park the mobile tower-trailer in any regular car parking space and operate i t for a period of up to three or four days. A pick-up camper-unit provided security for the data-loggers and instruments and served as a shelter for the operator at night. Residents' reactions to the odd-looking "spying device" ranged from interest, inquisitiveness and even admiration through total indifference to open suspicion and mistrust. The most notable comment came from a resident of Culloden Street, who complained that the tower "attracts mosquitoes and bugs" (sic !). In general, however, there were no d i f f i c u l t i e s with such potential problems as residents' claims for parking space or vandalism. A more detailed description of each site i s given in Appendix D. - 178 -9.3 EQUIPMENT INSTRUMENTATION AND DATA The mobile tower was the key piece of equipment for this study. Because of the limited space available at the side of residential streets and along parks without impeding normal t r a f f i c , i t was necessary to operate the mobile tower without s t a b i l i z i n g guys (see Figures 9.2 to 9.8, above). The model used is a pneumatic system of nine telescopic sections by Hilomast Ltd. (UK). It is extendable to a height of 28 m, driven by an ele c t r i c air-compressor that can be run from a 12 V car battery. It has a maximum head-load capacity of 14 kg and is designed to withstand wind-speeds up to 12.2 m-s-1 without guys. The mast has an overall weight of 120 kg and is mounted on a road t r a i l e r equipped with extendable stabilyzer posts. In the retracted position the mast has a length of 4.56 m and can be pivoted into the horizontal for road transport. These specifications make this tower truly mobile. After some practice i t was possible to convert the tower from i t s transport position into an operational meteorological tower within 30 minutes, complete with an eddy correlation system and a net radiometer mounted on top. The salvage of the instruments from sudden strong winds or rain took about 3 minutes. A l l meteorological instruments used in this study have previously been described by Roth (1988) and Cleugh (1988) and the present section is heavily based on their work. - 179 -Sensible heat flux, or rather the kinematic heat flux (w9), was measured on both the mobile and the Sunset towers by two identical eddy-correlation systems consisting of a sonic anemometer and a fine-wire thermocouple probe (Campbell Sc i e n t i f i c CA-27T). The original version of this specific sonic anemometer was introduced by Campbell and Unsworth (1979). The thermocouple is a welded chromel/constantan junction of wires with a diameter of 12.7 fim. This type of thermocouple has a very large Seebeck effect of 58.7 fiV- °C _ 1 at 0 °C (Fritschen and Gay, 1979). The junction is mounted between the two sensor heads of the sonic anemometer, 20 to 30 mm from the sonic path (Roth, 1988). Both temperature and wind-speed are sampled at 10 Hz. The signals are amplified in a electronics-box after only about 1.5 m wire-distance from the sensor and then passed on to the datalogger. This electronics-box contains a reference temperature probe with a large enough thermal mass to make the temperature d r i f t component of the (w8)-covariance negligible. Two different data-loggers had to be used for the two eddy-correlation sets. At the Sunset site the voltages from the amplifier were sampled and accumulated over a period of 15 minutes, for which the covariance was computed and stored in a Campbell S c i e n t i f i c CR-21X data-logger. These eddy correlation Q -values were stored on cassette tape as 30 minute averages. H The data-logger at the mobile tower was a Campbell S c i e n t i f i c CR-5 logger equipped with a K18 eddy-correlation module. Here, the covariance was obtained for a period of 5 minutes and then averaged over 15 minutes, in which form the data were stored on - 180 -cassette tape and also printed on paper. The contribution to the covariance by eddies with a time-scale longer than 5 minutes is missed by this data-logger. The w'8'-cospectral analysis by Roth (1988) shows that a non-dimensional cut-off frequency of 0.022 (corresponding to a 5 minute averaging time computed with z = 20 m and u -= 3 m-s"1) could introduce a considerable error in the covariance. As he points out, however, the cospectral estimates in this frequency range are poorly defined and show large fluctuations. An instrument comparison of the two systems with 5- and 15 minutes correlation intervals showed no significant difference over a wide range of flux values (see next section). Figure 9.9 shows the sonic anemometer/thermocouple system mounted on the Sunset tower together with a Krypton hygrometer and two net-radiometers which were used for a different study (Cleugh, 1988). In Figure 9.10 the CA-27T is mounted on the mobile tower together with a net radiometer, which was used by Cleugh (1988). Note the electronics box on the t i p of the tower. The input data (u, 0, <p and <r^ ) for the SAM-runs were a l l obtained at the Sunset tower. Figure 9.11 (from Roth,1988) gives a schematic overview of the entire Sunset instrumentation as i t was operated in summer 1986 (see also Cleugh, 1988; Grimmond, 1988; Roth, 1988 and Steyn and McKendry, 1988). The instruments used in this study (other than the eddy-correlation system described above) included : a Met-1 cup-anemometer and wind-vane and a PT-100 resistance thermometer. The data for these were logged by a CR-21X data-logger and stored on cassette tape. F i g u r e 9 . 9 : Arrangement o f i n s t r u m e n t s on t h e S u n s e t tower . L e f t t o r i g h t : two net r a d i o m e t e r s , K r y p t o n h y g r o m e t e r , s o n i c a n e m o m e t e r / t h e r m o c o u p l e . Bi F i g u r e 9. 1 0 : Arrangement o f i n s t r u m e n t s on the m o b i l e tower . Net r a d i o m e t e r ( l e f t ) and s o n i c a n e m o m e t e r / t h e r m o c o u p l e ( r i g h t ) . - 182 -> NE SW •Q-V o 4 I s O — 22.0 19.0 18.4 13.9 12.1 8.9 .5 P — C (30.51 (27.5) (26.9) (22.4) (20.6) (17.4) Figure 9.11 : Schematic of the Sunset tower and instrumentation. Instruments used in this study include: sonic anemometer/thermometer (level 5, SW); Met-1 wind-vane and cup-anemometer (level 5, NE) (from Roth, 1988). Figure 9.12 : Daily variations of Culloden (solid) for the period from Q at Sunset (dashed) and H JD 212-214. - 184 -From the casette tapes the data were transferred to the. UBC-mainframe computer. The 15 minute and 30 minute values were averaged to give hourly means, determined every half hour. Data _2 for cases when Q at the Sunset site f e l l below 10 W-m were H excluded and not used for processing. The resulting time periods with valid and complete measurements, the data and some of the derived values (eg. L, u^, etc.) are summarized in Appendix D. This appendix also includes results from the corresponding SAM runs described below. Although the daily variations themselves are not the principal interest of this study, Figure 9.12 i l l u s t r a t e s the general characteristics using the data from the Culloden s i t e in comparison to the concurrently evaluated Q data from Sunset H (dashed lines). At night the fluxes are very small followed by a steep increase between sunrise and midday. Variations from hour to hour as well as between the two sensor locations are greatest around midday when convective a c t i v i t y is at i t s peak. Shortly after sunset, convection is once again damped so much that the sensible heat flux becomes very small. At f i r s t sight i t is questionable whether the differences between the Culloden-Q and the Sunset-Q in Figure 9.12 reflect H H real differences due to the different locations (i.e. different source areas) or whether the differences result mainly from instrument uncertainty. The instrument comparison for a wide range of flux conditions, described in the next section, addresses this question. - 185 -9.4 INSTRUMENT COMPARISON Between JD 255 (September 12) and JD 269 (October 6), 1986 the same two eddy correlation systems that were used in the f i e l d programme were mounted on the Sunset tower side-by-side, with a spacing of 15 cm between the two sonic paths, to avoid interference. The comparison of 184 hourly Q measurements with H the two systems showed remarkable agreement over a wide range of values, especially i f the much wider scatter encountered during the inter-site f i e l d programme is considered. Figure 9.13 shows the scatterplot of the instrument comparison during the calibration period alone and Figure 9.14 shows the calibration data (diamonds) together with the values from the inter-site f i e l d programme (stars). The RMSD between the two sensors from the inter-site measurements is more than four times as large as that during the calibration period. Further, the visual Impression gained by Figure 9. 14 indicates clearly that the diamonds and stars belong to two different data sets, where the amount of scatter and bias is governed by different phenomena. The principal axis analysis and the break-up of the RMSD into i t s systematic and unsystematic parts (see Appendix E) for the calibration period (Figure 9.13) confirms the unsystematic nature of the differences between the two sensors. Therefore, i t is suggested that the increase in scatter between the calibration period and the measurement period reflects a real effect of the differences among the individual sites. Furthermore i t is suggested that the strength of this site - 186 -350 12.7 (W m-2) 3.2 (W m-2) 12.3 (W m-2) 50 100 150 200 250 Q_H-MobMe (W m - 2) 300 350 Figure 9.13 Scatterplot of Q instrument comparison H 350 300 250 CM ' E 200 +j 150 OJ (/> c CO 100 I 50 -50 -50 * : intersite * : ca1i brat ion ' RMSDcaHb RMSDi ntersite R ^cal ib R intersi te 12.9 (W m-2) 45-9 (W m"2) 0.97 0.86 50 100 I S O 200 250 300 350 QH-Mobile (W m - 2) Figure 9. 14 : Scatterplot of Q vs. CJ & Hmob Hsun - 187 -effect may be measured by the absolute difference between the Sunset and the mobile Q values, normalized by the RMSD of the H calibration period (which appears to be f a i r l y constant for Q H values above 10 W-m , see Figure 9.13): This measure may be interpreted as a signal-to-noise ratio for the site-effect of the sensible heat flux measurements. It is this measure (or rather the range of v a r i a b i l i t y of i t ) which needs to be compared to the source area dimensions, in order to assess the representativeness of a sensible heat flux measurement. A l l the Q„ for the measurements of the f i e l d fiDIFF programme are included in the summary in Appendix D. An additional conclusion from the instrument comparison and the close agreement between the two sensor systems is that the error introduced by the 5 minute averaging cut-off for the covariance calculations of the CR-5 system i s insignificant compared to the random differences between the two sensors. 9.5 THE SPATIAL REPRESENTATIVENESS OF EDDY CORRELATION MEASUREMENTS As indicated at the beginning of this chapter, the specific source area level, for which representativeness estimates become relevant, is unknown. It has been shown in Chapter 7 (Table 7.1) that the dimensions of the various minor levels can be Q-' H D I F F (9.1) - 188 -approximated i f the dimensions of the 0.9 source area level are known. These 0.9 source area dimensions were computed for a l l complete sets of Q measurements using the necessary input data H gathered during the f i e l d programme and the mainframe version of SAM (see Chapter 7 and Appendix C). The results are included in the data and results summary in Appendix D. As mentioned, the v a r i a b i l i t y of the Sunset/mobile Q H differences should be studied in relation to the source area size. In order to do this, a method needs to be found to quantify this v a r i a b i l i t y : that which is reflected in the scatter of QIT from a large number of measurements. In other H D I F F words, i f a scatterplot of 0.9 source area sizes vs. Qjjjj I F F * s considered, i t is the changing spread in Qjjpjpj. along the source area axis which i s of interest. According to the hypothesis in Chapter 5, i t is anticipated that should be much more variable with small rather than large source areas. Chambers et.al. (1983) propose a method to examine the dependence of the spread of an ordinate variable on the abscissa value by smoothing absolute values of residuals. This was implemented for the present purposes as follows. F i r s t the ^ H D I F F V S ' S O U R C E a r e a - scatterplot is smoothed using a LOWESS routine with f = 2/3 and three robustness iterations. LOWESS is a non-parametric locally weighted regression scheme (described in detail in Appendix E). The result (Figure 9.15) shows the scatter plot and a sol i d line which is the varying position of the median of the QIT distribution within intervals of source H D I F F area size. In the next step the absolute residuals from this - 189 -Q 1000 1200 1400 0 . 9 - s o u r c e a r e a ( i n 1000 m 2) Figure 9.15 : Scatterplot of ^ H D I F F V S . 0.9-source area size. The LOWESS-curve is shown as a so l id S 2 l ine. The 6-10 ra area (dashed line) is equivalent to the 3.14-10 4 area of the 0.3-level . -o (0 QJ t_ CL (/> ZOO 400 000 000 1000 1200 0 . 9 - s o u r c e a r e a ( i n 1000 m2) 1 400 Figure 9.16 : Spread of H D I F F V S . 0.9-source area size (residuals of |Q„ - LOWESS-curve|) H D I F F - 190 -o zoo 400 aoo aoo 1000 1200 M O O 0 . 9 - s o u r c e a r e a ( i n 1000 m2-) Figure 9.17 : Same as Figure 9.15, for ca l ibrat ion period zoo 400 aoo aoo . 1000 1200 M O O 0 . 9 - s o u r c e a r e a ( i n 1 0 0 0 m 2) Figure 9.18 : Same, as Figure 9.16, for ca l ibrat ion period - 191 -smooth median curve are evaluated and plotted against the source area size. Once again a robust LOWESS curve (f = 2/3) is plotted through the residuals (Figure 9.16). This curve shows the position of the median of the residual distribution in relation to the source area size. Figure 9.16 demonstrates quite clearly that the spread of Q is wider for small source areas than HDIFF for large ones, as was anticipated by a visual examination of the scatterplot in Figure 9.15. An immediate reaction to this result might be that there are many more points at the small source area end (left) than on the right hand side of Figure 9.15 and that small source areas are related to large heat fluxes (see Chapter 7) so that more scatter in 0^^. in this region is due to the higher Q H-values and is therefore a t r i v i a l result. For this reason an identical analysis was performed for the Q measurements during the H calibration period, where the two sensors were placed very close together (see above). Figure 9.17 shows that the range of source area sizes during the calibration period was similar to that in the inter-site measurement period, but the Qj^ are generally much smaller : the LOWESS curve stays at a value of about unity over a wide range of source area sizes, as is to be expected from equation (9.1). The very clear decrease of the spread estimator curve towards larger source areas in Figure 9. 16 is almost entirely absent in Figure 9.18, for the calibration period. It is concluded that the effect of decreasing v a r i a b i l i t y of the measurements with increasing source area (as anticipated - 192 -from theoretical arguments in Chapter 5) is demonstrated in Figure 9.16. This result is also a confirmation of the suggestion that the spatial representativeness of sensible heat flux measurements is sensitive to the source area size, as computed by the source area model. It is the f i r s t proof that the source area model describes an effect which is real and may be observed by measurements. It is somewhat subjective to decide at what source area size in Figure 9.16 the spread of Q^^. becomes negligible. One might argue that the spread around the median curve in Figure 9. 15 is about equal to the random spread due to the instrument uncertainty (see Figure 9.17) when the LOWESS curve in Figure 9.16 reaches a value of unity (i.e. at a 0.9-level source area S 2 size of 6-10 m . This corresponds to a median of just over 2 in Figure 9.15. Accordingly, the Q variations due to the H s i t e differences (the non-representativeness component) start to be of the same order of magnitude as the random variations due to the instrument uncertainty. In other words, the site differences become undistinguishable from the instrument noise, 5 2 when the 0.9-level source area is larger than about 6-10 m . As noted already, this area is not equivalent to the representative sample size defined in Chapter 8, due to the continuous source weighting function which defines the source area. S 2 If the 0.9-level source area needs to be at least 6*10 m for a spatially representative heat flux measurement, how can this area size be related to the scale of representativeness of 200 m as determined from the spatial structure of the surface - 193 -characteristics found in Chapter 8 ? The 200 m result refers to the diameter of a discrete, 4 2 circular sample area, i.e. an area of 3.14*10 m . It was stated in Chapter 8 that the sample area (or source area in this case) needs to have a minimum axial dimension of 200 m in the direction across the house-rows. For a non-circular source area i t may be postulated that i t has a minimum size of 3.14-104 m2 and that i t s smallest axial dimension is 200 m. Because of the asymptotic character of the source weight function a source area that meets these size requirements can always be defined, i f a large enough effect level is chosen. The previous discussion, however, provides the additional constraint that the 0.9-level 5 2 source area needs to be 6-10 m . It was found in Chapter 7 that the size-relationship between the various source area levels is more or less constant. According to Table 7.1 the minimum area 4 2 size of 3.14-10 m is reached by the 0.3-level source area when the corresponding 0.9-level source area has a minimum size of 5 2 5 2 6-10 m .Therefore, a 6-10 m limit on the 0.9-level source area is entirely equivalent to a 3.14-104 m2 limit on the 0.3-level source area. The smallest axial dimension of the source area is most commonly the cross wind dimension, 2d (d is only the half axis, see Figure 7.2). In this way, the representativeness c r i t e r i a for a discrete sample resulting from the surface characteristics analysis are matched with the source area requirements reflected by the observation programme. This procedure could be seen as equivalent to a calibration of the source area model : i t shows both that the source area model is - 194 -in effect modelling an existing phenomenon and that the appropriate integrated effect level for the source area estimates is the 0.3 integrated effect fraction. As an i l l u s t r a t i o n to show how the source areas f i t into the study region, three cases have been picked from the scattered data in Figure 9.15 : The f i r s t point (1) is well on the representative side of Figure 9. 15, with a large source area and a small Q„ . The H D I F F outlines of the 0.3-, 0.5- and 0.9-level source areas, with respect to the Sunset site, for this data point are plotted onto a map of the study area (oriented in the appropriate wind direction) in Figure 9.19. The data for this case (1) were obtained on JD 222 at 8:00 (LAT) at Sunset, while the mobile tower was located at the Argyle s i t e (see Appendix D). It is interesting to compare case (1) with case (2) which has a source area of similar size, but is an 'odd point' in Figure 9.15, with a relatively large of over five times the value of case (1). The data for case (2) were recorded on JD 232 at 17:30 (LAT), with the mobile tower located at the Waverley site. The source area outlines are drawn only with respect to the Sunset site for c l a r i t y in Figure 9. 19. It is easy to imagine respective parallel source area outlines with respect to the mobile sites (although they would be a l i t t l e larger than the Sunset source areas because of the s l i g h t l y higher mobile tower). li.t:;-L > 11 . . . . —~ |li'"Y V"" " i a , i K n w p D O " ' / t " " " - . — -M 1 * S TV'1! V~; l / ' (D : Sunset Tower; © : Mounta in View C e m e t e r y ; (R) : Memor ia l P a r k ; (R) : K e n s i n g t o n Park © : Gordon Park ; © : ' H o t - C r o s s e d - B u n s ' ; © : L a n g a r a Community C o l l e g e Figure 9.19 : Source area outlines for three cases described in the text. Cross-hatched : 0.3-level; sol id line : 0.5-level; dashed l ine : 0.9-level . - 196 -The source area of the measurement in case (1) just barely misses the large intersection of 49th Ave. and Knight St. and reaches mainly over residential neighbourhoods. The 0.3-level area i s wide enough to include a minimal axial dimension of =<200 m as required by the representativeness c r i t e r i a . In contrast, a large portion of the 0.3-level source area of case (2) l i e s over the dry Mainwaring substation which contains the Sunset tower (Figure 9.19). Memorial Park is s u f f i c i e n t l y far away to play a minor role for the measurement at the the Sunset site. This is not the case for the mobile tower, which was located at Waverley si t e in case (2) (Figure 9.1). It was therefore closer and downwind of Memorial Park. A shift (which has to be imagined) of the case (2) source area from the Sunset s i t e to the Waverley si t e would make a considerable portion of the source area l i e over Memorial Park, whereas the substation would be entirely without influence. The different source area surface characteristics for the two sites in case (2) may explain the large Q^^pp excursion noted. A similar (imagined) shift of the source area in case (1) to the Argyle s i t e (see Figure 9.1) shows that the source area characteristics are much more similar between the Sunset and the mobile site here : hence CL^pp is expected to be small. A third example is given in case (3), where a small source area combines with the largest Qjjjjjpp recorded (Figure 9.15). The data in this case were recorded on JD 221 at 13:30 (LAT) with the mobile tower at the Waverley site (Appendix D). There is no large park, playfield or parking lot in the v i c i n i t y of - 197 -the source area of either the Sunset site or ( i f i t is shifted) the Waverley site (Figure 9.19), but the source area is much smaller than required by the representative c r i t e r i a . Therefore, the large QJJ^ may be expected, but is not necessary. In summary, i t is suggested that the scale of representativeness (as determined from the v a r i a b i l i t y of surface characteristics in Chapters 5 and 8) for sensible heat flux measurements can be related to the size and minimal axial dimension of the 0.3-level source area. Therefore the suggested c r i t e r i a for a s p a t i a l l y representative sensible heat flux measurement are : o the 0.3-level source area needs to be at least as large as a circular area with a diameter equal to the scale of representat iveness o the d-dimension of this 0.3-level source area needs to be at least half the scale of representativeness For the study area the scale of representativeness was identified as twice the mean spacing of house-rows or about 200 m (see Chapter 8). Obviously, these c r i t e r i a do not have the status of "laws". At best they should be seen as guidelines to assess the spatial representativeness of sensible heat flux measurements. Figure 9.15 shows quite clearly that a large number of measurement pairs of the Sunset and the mobile sites are within an acceptable range of each other even though the source area is much smaller than stated by the above c r i t e r i a . This sim i l a r i t y - 198 -is subject to considerable uncertainty however, as indicated by the wide scatter on the lef t hand side of Figure 9. 15. It is this uncertainty which the representativeness c r i t e r i a stated above help to evaluate. - 199 -PART IV : DISCUSSION AND CONCLUSIONS 10. How good are the SAM-Estimates ? - Comparison with Existing Work A l l numerical model results are questionable, i f they cannot be supported by observations. Unfortunately source area estimates are not simple entities that can be verified by direct measurements. However, in the present case, the SAM estimates can be validated indirectly by comparing the observational evidence of a changing source area with corresponding SAM results. The need to develop a model which estimates the source area of a turbulent flux measurement originally arose because of a concern about the spatial representativeness of such measurements. The suspicion that the spatial representativeness of turbulent flux measurements is related to the size of the source area was formalized in Chapter 9 into a hypothesis which can be tested. This hypothesis states that the v a r i a b i l i t y of the absolute differences between a large number of sensible heat flux measurements from two spatially separate sensors is inversely dependent on the size of the source area which may be estimated by SAM (see Chapter 9). The shape of the LOWESS curve in Figure 9.16 led to the acceptance of this hypothesis which is considered to be one of the fundamental conclusions of this - 200 -thesis. The fact that the results from the source area model were involved in this conclusion and that their interpretation is supported by observations is an indirect (but important) validation of SAM, at least in a qualitative sense. Comparison with Observations in Cabauw (the Netherlands) A more direct validation i s possible by applying SAM results to some observations of Beljaars et al. (1983). They discuss the results of some surface-layer turbulence observations in Cabauw, the Netherlands. These observations were conducted in poor fetch conditions, due to the presence of orchards and buildings upstream of the instrument tower. The surface in the immediate v i c i n i t y of the tower (within a radius of 300 - 500 m ) consisted of short grass with an estimated z « 0.02 m (see o below). Turbulent flux measurements at heights of 3.5 m and 22.5 m indicated a u # at 22.5 m which was consistently higher (by a ratio of 1:1.4) than that at 3.5 m. This discrepancy was not observed when the wind-direction was from an unobstructed sector (see Figure 10.1). Similar measurements of sensible heat flux and evaporation did not show this effect. The interpretation given by Beljaars et al. (1983) is as follows: o The 22.5 m level shows the integrated effect of a large upstream area. The shear stress measured at that level includes the form drag on obstacles far away and therefore i t is larger than the local surface drag. At the 3.5 m height, the shear stress tends towards the surface drag on the smoother near-tower terrain. - 201 -U K ( 3 . 3 ) Figure 10.1 : xi^ g vs. u > 3 (at Cabauw). Unperturbed measuring points are indicated by a c i r c l e ; the perturbed ones by a triangle (from Beljaars et.al. , 1983). Table 10.1 : Input data for Cabauw SAM-runs He i ght : 3-5 m / 22 .5 m 2 o = 0.02 m JD 1 kO 1 A1 160 161 17*1 1831 183II u.,(0 0. 30 0.21 0.53 0. 56 0.13 0.31 0.21 e ( 2 ) V 0.53 0.37 0.93 0.98 0.23 0.54 0.37 f 36 236 253 212 200 u(20m) 5.1 3-7 9.1 9-6 2 .3 5 .3 3-7 L -300 - 50 -800 -170 - 16' -135 -203 ( ' ' based on zo = 0 .02 m based on ( 3 V - u •;• 1 .75 - 202 -o Heat and vapour fluxes are not necessarily affected, since the ground temperatures and moisture values could be similar for both surface types. This problem is equivalent to the step change in roughness described in Section 6.2; and Beljaars et al. (1983) find their interpretations confirmed by the internal boundary layer growth estimates of Rao et al. (1974). The input data necessary for SAM-runs for seven unstable cases are given in Beljaars et al. (1983) and are presented in Table 10. 1. In order to examine the relative importance of the roughness elements, the SAM estimates were computed for the two measurement levels for each of the seven cases. It was assumed that the large obstacles have no effect at either measurement level, so that the model input value of z was that for the o grass surface (z = 0.02 m) in both cases . If i t can be shown o that a considerable portion of the source area (with the appropriate weighting) for the 22.5 m level includes the rougher orchards or buildings, but not for the 3.5 m level this hypothesis has to be rejected and the interpretations of Beljaars et al. 1983 are confirmed by the source area model. The results for three of these cases are presented in Figures 10.2 to 10.7. Note the different scaling of the alongwind- and cross wind-axes in the various plots : the shapes of the source area outlines are distorted. These plots indicate the strong sens i t i v i t y of the source area to the sensor height. The effects of s t a b i l i t y and u^ are secondary, but s t i l l strong, especially for the 22.5 m level (compare Figures 10.3 and 10.5). Variations SOURCE A R E A M O D E L - R E S U L T S SAM Calculations for Jul.Day. 140 at — L.A.T. at CABAUW Wind ol sensor level (sp./dir.) : — / 148.0 ; Zi : n/e ; SV : 0.630 Stability (Zs/L) :-0.012 ; U* : 0.300 ; Sensor height : 3.6 All data are in SI units F i g u r e 10 .2 : Cabauw S A M - r e s u l t s (JD 140, 3 . 5 m) SOURCE AREA MODEL - RESULTS SAM Calculations for Jul.Day. 140 at — L.A.T. at CABAUW Wind et sensor level (sp./dir.) : — / 148.0 ; Zi : n/e ; SV : 0 . 6 3 0 Stability (Zs/L) : - 0 . 0 7 6 ; U ' : 0 . 3 0 0 ; Sensor height : 22.6 Al l data are in SI units F i g u r e 10 .3 : Cabauw S A M - r e s u l t s (JD 140, 2 2 . 5 m) SOURCE AREA MODEL - RESULTS SAM Calculations for Jul.Day. 174 at — L A T . at CABAUW Wind at sensor level (sp./dir.) : — / 4 6 . 0 ; Zi : n/a ; S V : 0 . 2 3 0 Stability ( Z e / U : - 0 . 2 1 9 ; U # : 0 .130 ; S e n s o r height : 3.6 A l l data are in SI units F i g u r e 10 .4 : Cabauw S A M - r e s u l t s (JD 174, 3 . 5 m) SOURCE A R E A M O D E L - R E S U L T S SAM Calculations for Jul.Day. 174 at — L.A.T. at CABAUW Wind st sensor level (sp./dir.) : — / 46.0 ; Zi : n/a ; SV : 0.230 Stability (Zs/U : -1.406 ; U» : 0.130 ; Sensor height : 22.5 All data are in SI units F i g u r e 10.5 : Cabauw S A M - r e s u l t s (JD 174, 2 2 . 5 m) SOURCE AREA MODEL - RESULTS e SAM Calculations for Jul.Day. 161 at — L.A.T. at CABAUW - 1 Wind at sensor level (sp./dir.) : — / 263.0 ; Zi : n/a ; SV : 0.980 Stability (Zs/L) : -0.021 ; U» .: 0.660 ; Sensor height : 3.5 All data ere in SI units F i g u r e 10 .6 : Cabauw S A M - r e s u l t s (JD 161, 3 . 5 m) S O U R C E A R E A M O D E L - R E S U L T S SAM Calculations for Jul.Day. 161 at - L A T . at CABAUW Wind at sensor level (sp./dir.) : — / 253 .0 ; Zi : n/e ; S V : 0 . 8 8 0 Stability (Zs /L) : - 0 .132 ; U" : 0 . 6 8 0 ; S e n s o r height : 22.6 A l l data are in SI units F i g u r e 10 .7 : Cabauw S A M - r e s u l t s (JD 161, 2 2 . 5 m) - 209 -at the lower level are much smaller. The undistorted outlines of the 0.3-level and 0.5-level source areas of these three cases (JD 140, 161 and 1741) for both heights are plotted in the appropriate wind-direction on a map of the Cabauw tower and v i c i n i t y (Figure 10.8). On day 161 the wind direction was from an almost unobstructed sector. Judging from the source area outlines in Figure 10.8 for JD 161, i t is not surprising that Beljaars et al. (1983) did not find a stress discrepancy on this day : the source areas for both sensor levels contain a similar, f a i r l y uniform surface type, although their sizes d i f f e r by about an order of magnitude. The situation is different for JD 140 and 1741 : the a i r flows over the obstructed zone before i t reaches the sensors. The respective source area outlines in Figure 10.8 confirm the reason for the resulting stress discrepancy quite clearly. While the 3.5 m source areas are practically unaffected by the roughness elements (indicated by dotted and hached areas on the map), the corresponding 22.5 m source areas reach far into the obstructed zones (the 0.3-levels are plotted by dashed lines in Figure 10.8). The maximum source locations are indicated by small s o l i d c i r c l e s : for the 3.5 m measurement level these are very close to the tower (between 27 and 36 m) and are hardly distinguishable from the mast-marker. The maximum source locations for the 22.5 m level are much further away : for both JD 140 and 1741 they are right at the boundary between the obstructed and the unobstructed zones. - 210 -Figure 10.8 : Cabauw s i te and surroundings, with observed winddirections and estimated source area outlines for JD 140, 174 and 161. Day numbers are indicated at the end of the lines that mark the wind-direction during the different measuring runs. 0.3- and 0.5-level source area outlines are indicated by dashed and so l id lines respectively. The larger sets refer to the 22.5 m, the smaller sets to the 3.5 m sensor level . Maximum souce locations are indicated by large dots (o) (based on Beljaars e t . a J . , 1983). - 211 -A considerable difference in surface character between the source areas for the two measurement levels on JD 140 and 1741 is clearly indicated. The above hypothesis has to be rejected for JD 140 and 1741 and thus the v a l i d i t y of the SAM-estimates is supported by these observations from Cabauw. Estimation of the Growth of the Internal Boundary Layer The growth of the internal boundary layer (5 « x n) can be estimated from the variation of the maximum source location between the two levels, as follows : z = b-x n , (10.1) s m where b is a constant of proportionality (see also Chapter 6). After taking the log, the resulting sets of two linear equations for the seven cases in Table 10.1 are solved for the exponent n (Table 10.2).(This internal boundary layer growth evaluation is not dependent on the Cabauw data, they are used here simply for convenience). A plot of n versus the s t a b i l i t y parameter (-L) is given in Figure 10.9 and includes the corresponding estimates by Rao (1975). The general character of the s t a b i l i t y variation of n is similar for both Rao (1975) and the present results, although Rao's values are consistently s l i g h t l y lower. A hand-drawn curve through the present results indicates a neutral limit of n = 0.8 for the exponent in (10.1) which is in agreement with both theoretical results (Peterson, 1969) and f i e l d observations (e.g. Munro and Oke, 1974). - 212 -Table 10.2 : Computat ion o f exponent o f i n t e r n a l boundary l a y e r growth. JD z s x m n L 140 3.5 22.5 35-62 327-34 0.84 - 300 141 3.5 22.5 32.60 232.65 0.95 - 50 160 3.5 22.5 36.05 348.93 0.82 - 800 161 3.5 22.5 35.10 305.61 0.86 - 170 1741 3.5 22.5 27.27 147.52 1.10 - 16 1831 3.5 22.5 34.81 294.47 0.87 - 135 183II 3.5 22.5 35.29 313.25 0.85 - 203 1 5 " 1 4 " O - \ - SAM-values 1 3 " O Rao (1975) (R) neutral l im i t in Rao (1975) 1 2 ' \ n 1 1 1 + \ 1 0 - \ 0 9 -0 8 - — (R) 1 0° , 1 — —1 t — 101 102 103 1014 -L ( in m) F i g u r e 10.9 : Exponent o f i n t e r n a l boundary l a y e r growth vs . - L - 213 -With this result as another confirmation of the v a l i d i t y of the SAM-estimates i t may be concluded that these estimates show general qualitative and quantitative agreement with the few presently available observations. - 214 -11. The Structure of the Surface Layer over Complex Surfaces In Chapter 2 i t was noted that the surface layer above a rough surface (e.g. t a l l vegetation or buildings) involves a lower portion called the roughness sublayer, which is characterized by the influence of individual surface elements, and an upper porion c a l l the inert ial sublayer, where the flow is horizontally homogeneous and Monin-Obukhov scaling applies. Several authors have speculated on the height of the upper boundary of this roughness sublayer. An early suggestion by Tennekes (1973) states that turbulence sensors should be at a minimum height of 50 - 100 z . In a more o recent study of wind tunnel experiments over a regularly arranged rough surface, Raupach et al. (1980) estimate the height of the roughness sublayer in terms of the mean height of the roughness elements (h) and the mean inter-element spacing (D) as z = h + 1.5 D. This estimate applies to the momentum flux in neutral conditions. Garratt (1978 a,b) examines the height variations of the slope of the non-dimensional profiles for wind-speed and temperature over a very heterogeneous savannah surface in Australia. He interprets the height of the roughness sublayer (he c a l l s i t transition layer) as the level where such profiles converge to the familiar Dyer-Businger relations. His results indicate a z # which is higher for the momentum flux than for heat flux and show a variation in z # with s t a b i l i t y for both - 215 -fluxes : z0 is higher in unstable conditions than in near-neutral s t r a t i f i c a t i o n . The different zm for momentum and heat is interpreted to be the result of the different distributions of momentum sinks and heat sources especially for surface considered by Garratt (1978a,b). The connection of these z% estimates with the present work li e s in the interpretation of the nature of the roughness sublayer. As mentioned in Chapter 2, this interpretation i s summarized very well by Raupach and Thorn (1981). Flow in the roughness sublayer i s three-dimensional due to the individual wakes and "heat plumes" of the surface elements. A sensor close to these surface elements measures the effects of some surface elements preferentially compared to others. These inhomogeneities become more and more mixed and diffused with increasing height, until they disappear altogether. The height where horizontal differences become negligible is interpreted as the upper boundary of the roughness sublayer, above which Monin-Obukhov scaling can be applied. This interpretation of the nature of the roughness sublayer makes the height requirements for a sensor (i.e. z > z ) s * equivalent to the spatial representativeness requirements suggested in Chapter 9. Accordingly, i t is suggested that a sensor is located above zm i f i t s source area meets the representativeness c r i t e r i a stated in section 9.5. Such a suggestion would be in agreement with Garratt (1978 a,b) in having the roughness sublayer height change with s t a b i l i t y because the source area size for a given level varies with - 216 -s t a b i l i t y . The value of z0 w i l l also depend on the spacing and distribution of surface elements as suggested by Raupach et al. (1980). Judging from the variations of the scatter in Figures 9.15 and 9.16 and the finding that the minimum size of the 0.3-level 4 2 source area (for the Sunset area) should be 3.14'10 m S 2 (corresponding to 6.5* 10 m of the 0.9-level) to meet the representativeness c r i t e r i a , i t seems that the majority of the measurements taken for this study have been conducted within, rather than above, the roughness sublayer of the study area. This conclusion i s supported by the comparison of the sensible heat flux measurements at the Sunset s i t e and the various mobile sites (see Figure 9.14). On average, the Sunset si t e over-estimates in comparison with the sensible heat flux at the mobile sites by a considerable amount. The mean values for _2 the entire measurement period are 128.5 W-m (Sunset) and 99.6 _2 -2 W-m (mobile) resulting in an overestimation of 28.9 W-m or about 25% in the mean (calculated as [2-|Q - Q | ] / Hsun Hmob ' [Q + Q ]) . Since the mobile measurements involve several Hsun Hmob different sites, the mobile mean is more spatially averaged than the Sunset mean, and therefore is considered to be more spatially representative. A possible explanation for this overestimation is the preferential influence of the immediate surroundings of the Sunset tower, which consist of the non-vegetated and dry Mainwaring substation to the NW and a major road intersection and associated commercial buildings to the SE (see Chapter 9). - 217 -Such dry, non-vegetated surfaces may well shift the partitioning of the enthalpy flux to the sensible heat side compared with other site locations. This possible overestimation of sensible heat flux at the Sunset site and the conclusions from Figures 9.15 and 9.16 point to a roughness sublayer height above the suburban surface in the study area which is often higher than the sensor height on the Sunset tower so that the measured values are not entirely representative of the larger land-use area. In order to estimate the range of conditions when Sunset measurements are expected to be f u l l y representative, the source area calculations for the measurement period were examined in relation to s t a b i l i t y and lateral wind fluctuations (Figure 11.1). Fully representative cases (0.3-source area > 3.14*10 m) are plotted as squares and not f u l l y representative cases as dots. The hand-drawn curve separating the two groups can be interpreted as the condition in which z. = z at the Sunset tower. This does not imply that a l l * s the Q -data above this line are useless. It simply means that Hsun * J measurements obtained in 'not f u l l y representative conditions' are more l i k e l y to have been influenced by unusually hot, cold or dry surface elements and that the location of the source area should be monitored. Turbulence spectra obtained by Roth (1988), however, do not indicate important discrepancies with spectra obtained over smooth terrain. It is therefore suggested that the turbulence s t a t i s t i c s at that height may be independent of local geometry scale-lengths although they are not representative of the - 218 -0 -50 -100 -150 ? £ -200 -250 -300 -350 not f u l l y represen ta t i ve ( z A > z 5 ) ' . •* .•.»» '*• . •"• • Jj> ' * Q~0 _ • • * • ^ z * = z s 0 a •/-a • - • • • « • a • a f u l l y representa t ive (z... o • < --a a o • • : 0.3-source area <C 3- 1 it 10* m 2 a • a : 0.3-source area > 3- 14 10* m2 -o • -a i i i a 1 1 1 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Figure 11.1 : Representativeness conditions at Sunset tower - 219 -larger-scale surface conditions (Roth, 1988; personal communication). If this is the case, the interpretation of the top of the roughness sublayer given above needs to be re-examined for complex surfaces, where a wide range of spatial v a r i a b i l i t y scales co-exist. It seems appropriate to differentiate between two different mechanisms of spatial averaging and diffusion of horizontal inhomogeneities. Bluff-bodies create a turbulent wake, where mixing i s greatly enhanced by a process termed "wake diffusion" by Thorn et al. (1975) and Raupach (1979), so that horizontal inhomogeneities due to bluff-rough inhomogeneities are mixed very e f f i c i e n t l y . This i s not the case for other changes in surface character (such as changes in temperature or non-bluff roughness), where spatial averaging is therefore expected to be slower. Observations show that at the height of the sensor on the Sunset tower, flow distortions due to individual bluff-rough inhomogeneities do not affect the shape of the turbulence spectra (Roth, 1988). Since these spectra are normalized by the total flux, changes in heat flux on a larger scale than the individual element spacing are not reflected in them. The comparison of sensible heat flux at the different sites, however, does show these inhomogeneities. It follows that the lack of a continuum of spatial scales (as indicated by the variance spectra in Chapter 8, e.g. Figure 8.11) over very complex surfaces results in nested sublayers, each one referring to a different scale of variab i l i t y . Clearly, the term "roughness sublayer" refers to the sublayer due to the - 220 -bluff-rough v a r i a b i l i t y scale, which affects the transfer processes in the atmosphere directly. "Sublayers" due to other inhomogeneities on the other hand are only relevant when considering spatial representativeness. - 221 -12 Summary of Conclusions Following the format of the objectives outlined i n Chapter 1, the main conclusions of t h i s thesis are summarized as follows. o The small scale s p a t i a l v a r i a b i l i t y of sensible heat f l u x can be evaluated by an analysis of the structure of the d i s t r i b u t i o n s of the s p a t i a l surface temperature and of the dominant bluff-rough structures i n the area. It i s suggested that a measure f o r the s p a t i a l representativeness of a block sample i s given by the normalized integrated variance spectrum of the data i n the wavelength domain. For two dimensional data, the spectrum should be transformed into plane-polar coordinates and integrated with respect to angle, to y i e l d a representativeness curve r e f e r r i n g to c i r c u l a r samples with the diameter equal to the r a d i a l wavelength of the spectrum. o In the context of the sensible heat f l u x , the s p a t i a l representativeness of block-samples of both the surface temperature and the roughness element d i s t r i b u t i o n need to be examined. o The analysis of surface temperatures and roughness elements i n the study area indicates that a sensible heat f l u x measurement over a suburban r e s i d e n t i a l area, s i m i l a r to the one i n the study area, i s s p a t i a l l y representative, i f i t includes the averaged contributions of an area that extends across three - 222 -house-rows. Further work wi l l be needed to show whether this finding can be generalized, so that i t applies to the spatial structure of any complex surface. o A reverse-plume model provides an Eulerian approach to account for the surface area of influence of a turbulent flux measurement in the surface layer, based on the original suggestion by Pasquill (1972). A numerical source area model (SAM) to evaluate this area of influence was developed using the p.d.f.-plume model of Gryning et.al. (1987) as a basis. Observations which can be related to changes in the source area of a sensor or of a set of sensors, give indirect support to the va l i d i t y of the SAM-estimates both qualitatively and quantitatively. o It is concluded that the SAM-estimates are a useful tool to evaluate the spatial representativeness of a turbulent flux measurement. Observations indicate that the model may also be used for turbulent fluxes other than sensible heat. The model shows good agreement with theoretical and observational studies of internal boundary layer growth. The most original contribution of this work l i e s in the connection between the source area estimates for turbulent flux measurements and the s t a t i s t i c a l measure of the spatial representativeness of a given contributing area. This combination of turbulent diffusion modelling and two-dimensional spatial spectrum analysis is a new method for the evaluation of the spatial representativeness of turbulent surface layer flux measurements in an objective and quantitative fashion. In the - 223 -present case i t is applied over complex surfaces with small scale spatial v a r i a b i l i t y of temperature and roughness. Micro-meteorologists can use the source area estimates to refer with some precision to the surface to which their flux measurements relate, rather than pointing vaguely in the upwind direction with some coarse estimate of fetch derived from a measurement height. It i s f e l t that future work relating to some of the notions in this study promise interesting results. For example the application of the source area weighting function to model spatially weighted averages for turbulent fluxes (see e.g. Grimmond, 1988) or experiments (e.g. in a wind tunnel) to examine the relationship between spatial representativeness and the source area of turbulence measurements of different fluxes over complex surfaces. - 224 -REFERENCES Arnfield, A.J.: 1982, 'An Approach to the Estimation of the Surface Radiative Properties and Radiation Budgets of Cities', Phys. Geog. 3, 97-122. Auer, A.: 1978, 'Correlation of Land Use and Cover with Meteorological Anomalies', J. Appl. Meteorol. 17, 636-643. 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Environ. 12, 2125-2129. . Venkatram, A.: 1980, 'Dispersion from an Elevated Source in the Convective Boundary Layer', Atmos. Environ. 14, 1-10. Weil, J.C: 1985, 'Updating Applied Diffusion Models' , J. Clim. Appl. Meteorol. 24, 1111-1130. Welty, J.R. , Wicks, CE. and Wilson, R. E. : 1976, 'Fundamentals of Momentum, Heat and Mass Transfer', II ed. J. Wiley and Sons, New York, 789pp. Willis, G. E. and Deardorff, J.W. : 1976, 'A Laboratory Model of Diffusion into the Convective Planetary Boundary Layer', Quart. J. Roy. Meteorol. Soc. 102, 427-445. Willmott, C.J.: 1981, 'On the Validation of Models', Physical Geography 2, 184-194. Young, G.S. and Pielke, R.A. : 1983, 'Application of Terrain Height Variance Spectra to Mesoscale Modeling', J. Atmos. Sciences 40, 2555-2560. Zwick, H.H., McColl, W.D. and Edel, H.R.: 1980, 'The CCRS DS-1260 Airborne Multispectral Scanner (MSS)', Proc. 6th Can. Symp. Remote Sens., Halifax, Nova Scotia. - 231 -APPENDIX A : Remotely Sensed Surface Temperatures A summary of technical details of the f l i g h t s is given in Table A.l. The aircraft, a Falcon-20 (twin-jet) owned and operated by the Canada Centre for Remote Sensing (CCRS), is shown in a schematic in Figure A. 1. Both the multi-spectral scanner (MSS) and the push-broom imager (MEIS II) were operated during the day-flight, but only the infrared data from the MSS are used for this study. Figure A. 2 shows a cross-section through the scan-head of the Daedalus-1260 MSS. This MSS was origi n a l l y introduced by Zwick et.al. (1980). The rotating scanning mirror projects an instant f i e l d of view (IFOV) of 2.5 mrad over a set of folding mirrors to the Hg-Cd-Te detector (the beam is split-up by a dichoric and part of i t is passed to a spectrometer for the detection of near infrared and v i s i b l e light channels). This infrared detector has a sen s i t i v i t y window of 8-14 um. In the present application, the MSS scanned at 50 scans/second with a 73.7 degree swath width orthogonal to the flightpath. The 2.5 mrad IFOV is sampled and d i g i t a l l y recorded to 1 byte at 716 pixels per scan-line. At the beginning and end of each scan-line the IR-sensor is calibrated by two on-board black-body sources. A detailed description of the data format is given by Zwick et.al. (1980) and by Edel and McColl (1981). During both the day- and the night-flights, ground-truth measurements were performed on the grounds of Langara Community College, in the study area, using a PRT-10 hand-held infrared thermometer. Figure A.3 shows a plan of the area with the four - 232 -Table A . l : Technical detai ls of the remote sensing f l ights Dates : Aug. 25 (JD 237) ; Aug. 26 (JD 238), 1985 T imes : -14:15 (LAT) ; - 0 5 : 0 5 (LAT) A i rp1ane : Fa Icon-20 (see F igure A . l ) Height : ~1560 m above Ground Sensor : Daedalus-1260 MSS (see F igure A.2) Scan - ra te : Ground-speed : 50 per second 2.1m/scan 105 ms" 1 P r e - p r o c e s s i n g : o r t h o g o n a l i z a t ion Cutaway view of the Falcon-20 aircraft showing systems used in a typical visible/infrared remote sensing project. The pushbroom imager (MEIS) or the multispectral scanner (MSS) is the principal sensor on all projects flown with this aircraft. 1. Multispectral Scanner (MSS-Daednlus I 260) 2. Metric Camera (RC-101 • 3. Pushbroom Visible Imager (MEIS II) 4. Mission Manager's Console (camera controls and navigation data logger) 5. Inertial Navigation System 6. MEIS Console and Real-time Display for MSS and MEIS 7. MSS Console 0. High-Density Digital Tape Recorder 9. Navigation Data Logger Tape Recorder Figure A.1 : Schematic of Falcon-20 a ircraf t - 233 -SCAN HEAO CROSS SECTIONAL ILLUSTRATION Legend © INFRARED DETECTOR " (f) SPECTROMETER © GYRO © FOLDING MIRRORS (5) DICHROIC * H g - C d - T e s e n s o r (8-14 u) (§) SCANNING MIRROR © MOTOR (§) ENCOOER @ PARABOLIC MIRROR Figure A.2 : MSS scan-head - 234 -ground truth sites. These four sites included a typical range of surface types for a suburban area : irrigated grass, concrete, a gravel/tar roof and tarmac. For comparison, Figure A.4 is the VZIP image of the daytime radiation temperatures of the same area, with a specific colour coding to highlight the pixel values of the ground-truth sites. Ignoring emissivity errors for the moment, the PRT-10 measurements within each site showed an average standard deviation of only 0.8 °C (the PRT-10 was calibrated over a temperature controlled water surface). Due to the d i f f i c u l t y of identifying the exact ground truth site locations on the VZIP image, an uncertainty of ± 5 pixel values was estimated for the assignment of the temperatures to a pixel-value. The Stefan-Boltzmann law of thermal radiation for grey bodies can be written F = e-<r-T4 , (A.l) where F is s t r i c t l y an integrated radiance over a l l wavelengths, but for naturally occurring surface temperatures in this region F may be approximated by the radiance of the 8-14 um wavelength window. A survey of emissivities of building materials and natural surfaces occurring in the study area as given by Oke (1978) and Arnfield (1982) suggests a mean emissivity of e * 0.95 ± 0.02 , (A.2) mean i f the values are weighted according to the relative land-use fractions estimated by Oke et.ai. (1981). This result compares - 235 -Figure A.3 : Plan of ground truth s ites at Langara Community College - 2 3 6 -F i g u r e A . 4 : V Z I P - i m a g e o f t e m p e r a t u r e s at g r o u n d t r u t h s i t e s - 237 -well with the mean emissivity for R2-land-u.se zones in snow-free conditions of e = 0.944 reported by Arnfield (1982). The R2 emissivity error of ± 0.02 translates to a temperature error of ± 2 K, i f the true temperature i s 295 K. This is in agreement with the emissivity error estimates by Garratt (1978a), Carlson et.al. (1981) and Leckie (1980). Since only relative temperatures are of interest here, the PRT-10 temperatures were adopted without correction, with an assumed emissivity-related error range of ± 2 K. Following Leckie (1980) the error due to atmospheric attenuation (or radiation at night) i s assumed to be accounted for by the ground truth values. The mean values of these temperatures versus the corresponding pixel values of the four sampled surface types are plotted in Figure A. 5 for the day-flight. The error-bars indicate the uncertainties in the pixel value and the PRT-10 standard error (not the emissivity error). According to (A.l) the radiance is proportional to the fourth 4 power of the temperature. This T -relation is plotted between the two in-flight calibration points on Figure A.5. Compared to the assumed emissivity error of ± 2 K the difference from the linear T-relation is small for the limited temperature range of the present application. Therefore, the pixel value-temperature relation is evaluated directly from the ground truth data by structural analysis (see Appendix E) as : 60 50 (0 CL 2 lil < DC 30 20 10 rad.temp.[oc] = 15.9 + 0.17 pix 2 V R2 = 0.99 irrigated grass concrete gravel/tar roof tarmac in-flight calibration #1 in-flight calibration #2 20 (0 60 60 100 120 HO 160 180 200 220 2(0 PIXEL VALUE Figure A.5 Daytime pixel value-temperature re lat ion - 239 -rad.temp. [ C] = 15.9 + 0.17 (pixelvalue) . (A.3) day A s imilar analysis was performed for the nighttime data, with the result ing relationship : O rad.temp. [ C] = * n i g h t L J -1.565 + 0.13 (pixelvalue) (A.4) - 240 -APPENDIX B : Validation of the P.D.F.-Plume Model Two datasets were available to validate the plume model used for the SAM : "Project Prairie Grass" Experiment Data In this experiment SC"2 was released from a continuous point source at a height of 0.46 m and van Ulden (1978) reports CIC/Q values for distances of 50, 200 and 800 m. These values are estimated from samplers at 1.5 m height along arcs at intervals of 2° at 50 and 200 m and 1° at 800 m. The roughness length of the area is taken as 0.008 m. The data set is supplemented with L and u s values derived by Nieuwstadt (1978). CIC/Q was modelled for 34 runs in unstable conditions according to equation (7.3), using the Dyer and Bradley (1982) versions of the non-dimensional profiles <f> and <p (see Chapter M H 7). A scatterplot (in linear space) of the modelled versus the observed CIC/Q is given in Figure B. 1 and the summary s t a t i s t i c s of the model validation are shown in Table B. 1 (for an explanation of the s t a t i s t i c a l indices refer to Appendix E). The values in brackets result from a reduced data set, where the three triangles on the far right and the greatly overestimated star on the lef t In Figure B. 1 are rejected as outliers. These four data points account for more than half of the systematic error over a l l distances. The improvement of the validation s t a t i s t i c s in Table B. 1 by the omission of these points is - 241 -so MODELED CIC/Q vs. OBSERVED CIC/Q from PROJECT PRAIRIE-CRASS 70 s U) 60 O C 50 2 40 ? 30 at 50 m distance and 1.5 m height at 200 m distance and 1.5 m height at 600 m distance and 1.5 m height The CIC/Q was modeld with the parameters : p - 1.55 ; c - 0.4 ; k - 0.40 Flux profile relations by Dyer and Bradley (1982) »„ = (1-28 i/L)-'^ +„ = (1-14 z/i;-'/* 30 40 50 60 CIC/Q (in 10~3 s/m2) OBSERVED 70 80 Figure B.1 : Scatterplot of CIC/Q model val idat ion (with 'Pra ir i e Grass' data) Table B. 1 : Stat i s t ics for CIC/Q model val idation (with 'Prairie-Grass' data) S t a t i s t i c a l l d i s t . 5 0 m 2 0 0 m 8 0 0 n R 2 d 0 P R M S E t o t RMSE. RMSE. sys "unsys 8 0 ( 7 6 ) 0 . 9 2 ( 0 . 9 6 ) 0 . 9 7 ( 0 . 9 9 ) 2 0 . 4 ( 1 8 . 8 ) 1 9 - 7 ( 1 8 . 6 ) 5 . 8 ( 3 . 7 ) 3 . 4 ( ' 1 . 6 ) 4 . 7 ( 3 . 4 ) 2 9 ( 2 6 ) 0 . 6 5 ( 0 . 7 6 ) 0.84 ( 0 . 9 2 ) 43-9 (41.4) 40.4 ( 3 9 . 7 ) 8.9 ( 5 . 6 ) 6 . 9 ( 3-D 5 . 5 ( 4 . 6 ) 3 0 ( 2 9 ) 0 . 1 3 ( 0 . 2 5 ) 0 . 5 8 ( 0 . 7 0 ) 1 0 . 7 ( 1 1 . 0 ) 1 2 . 1 ( 1 1 . 7 ) 3 . 8 ( 2 . 8 ) 2 . 1 ( 1 . 8 ) 3 - 2 ( 2 . 1 ) 2 1 0 . 5 1 0 . 8 2 1 . 8 2 . 0 0 . 7 0 . 3 0 . 6 MODELED C I C / Q (Dyer k = .4 l ) vs. (Dyer ic B rad ley k=.40) at SO m distance and 1.5 m height at 200 m distance and 1.5 m height al 600 m distance and 1.5 m height The CIC/Q was modeld with the parameters p - 1.55 ; c - 0.4 10 20 30 40 50 60 CIC/Q (in 10-3 s/m2) [Dyer] eo Figure B.2 : Effects of Dyer (1970) profi les compared to Dyer and Bradley (1982) profiles on p.d.f.-model results MODELED C I C / Q (Dyer ic B r a d l e y v e r s i o n ) k = .41 vs. k = 0.40 0 10 20 30 40 SO 60 70 60 00 CIC/Q (in 10-3 s/m2) k = 0.41 Figure B.3 : Effects of k = 0.41 compared to k = 0.40 on p.d.f-model results - 243 -remarkable. This omission is jus t i f i ed because of the considerable uncertainty involved in both the concentration measurements and the derived u # and L values which are used to run the model. Figure B.2 shows that the use of the Dyer (1970) prof i les 2 —1/4 ( 0 = 0 = (1-16- z/L) ) as opposed to the Dyer and Bradley M H (1982) versions result in s l i g h t l y higher model-values and Figure B.3 i l lus trates that the effect of a von K£rm£n constant of k = 0.41 is insignificant compared to the results for k = 0.40. Hanford-30 Series Data The second data set used to validate the present implementation of the GEA model is from the "Hanford-30 Series", as published by Fuquay e t . a l . (1964). These data include the release times, the Richardson numbers (from data at 2.1 and 15.2 m), u at 2. 1 m, <r^ and cr^-u, values of peak "exposure" for arcs at distances of 200, 800, 1600 and 3200 m, the total source output of zinc sulfide in grams, and estimates of o- along the y sampling arcs for each run. Ten of these runs were made in unstable s t ra t i f i ca t ion . The conditions during the releases are summarized by Draxler (1984). The zinc sulf ide tracer was released at a height of 1.5 m in the gently r o l l i n g terra in with sagebrush vegetation of 1 to 2 m height near Hanford, Washington. The roughness length of the area is given as 0.03 m with a displacement height of 1.4 m (Doran and Horst, 1985). From this information the required quantities for the - 244 -model-validation could be obtained in the following manner. The Richardson number was taken equal to z / L in unstable conditions, following Golder (1972), where z refers to the geometric mean of the heights for which Ri was determined ( i . e . (7-50) 0 ' 5 = 18.7 ft or 5.7 m). The roughness wind-speed was determined according to u, = u(z) • k I ^ln((z-d)/z o) - «/»(Ri)j , ( B . D where the famil iar integration by Paulson (1970) was used for — 1 /4 — i/»(Ri) (with <j> (Ri) = (1-28RD ). <r -u was taken as equal to M <p cr^ , following Panofsky and Dutton (1984, p 159ff). The peak "exposure" is defined as the maximum time-integrated concentration along an arc, in [g*s*m ]. Therefore CIC/Q could be established from Exp. / . CIC/Q = • ar • / 2it , (B.2) Q-t where Exp. is the peak exposure and t the duration of the tracer release as given by Fuquay e t . a i . (1964). In Figure B. 4 the unstable runs from the Hanford-30 Series are compared with the model results for CIC/Q and a summary of the val idation s ta t i s t i c s is given in Table B.2. Two of the runs included in the data set overestimate the observed values consistently at a l l distances by an amount much larger than exhibited in the rest of the runs. It is probable that there is considerable error involved in the determination of the model inputs (L and u^ ) here and that the overestimation does not - 245 -MODELED C I C / Q vs. OBSERVED C I C / Q f r o m H A N F O R D - 3 0 SERIES 0 5 10 IS 20 25 30 35 40 45 CIC/Q (in IO'3 s/m2) OBSERVED Figure B.4 : Scatterplot of CIC/Q model val idat ion (with Hanford-30 data) Table B.2 : S tat i s t i cs for CIC/Q model val idation (with Hanford-30 data) S t a t i s t i c c 11 d i s t . 200 m 800 m 1600 m 3200 m n 35 ( 27) 9 ( 7 ) 9 ( 7 ) 8 ( 6 ) 9 ( 7 ) R2 0. 72 (0 84) 0.01 ( 0.2) 0 44 (0.49) 0 20 (0,35) 0 81 (0.60) d 0. 89 (0 96) 0.29 ( 0.6) 0 56 (0.72) 0 40 (0.73) 0 85 (0.84) 0 5- 9 (6 0 ) 17.8 (18.4) 3 3 (3-2 ) 1 3 (1.2 ) 0 59 (0.45) P 7- 8 (6 4 ) 23.2 (19-6) 4 5 (3-5 ) 2 0 (1.4 ) 0 72 (0.48) R M S E t o t 5. 9 (3 4 ) 11.3 ( 6.6) 2 2 (0.93) 1 3 (0.57) 0 34 (0.19) R M S E s y s 2. 2 (0 6 ) 7.4 ( 4.2) 1 4 (0.33) 0 7 (0.17) 0 23 (0.03) RMSE un s y s 5. 5 (3 3 ) 8.6 ( 5 .D 1 7 (0.87) 1 1 (0.55) 0 24 (0.19) - 246 -Table B.3 : S tat i s t i cs for <r -model val idation y (with Hanford-30 data) n R 2 d 0 P R M S E t o t R M S E s y s R M S E u n s y s 35 0.74 0.91 140.0 140.2 64.3 39-4 50.8 - 247 -necessarily portray a deficiency of the model i t s e l f . The s t a t i s t i c s for the reduced data set (without the outliers) are included in brackets in Table B.2. The values of these two runs are connected by dashed lines in Figure B.4. These outliers are responsible for a large part of the systematic error. Both Figures B.1 and B.4 as well as Tables B. 1 and B.2 show that the CIC/Q-model compares favourably with the observations at a l l distances. The concentration variation with distance is described quite well by the model, whereas the modeling of CIC/Q at a specific distance is somewhat less satisfactory. Note that the original version of the CIC/Q-model (van Ulden, 1978) was developed to f i t the Prairie Grass data, but is independent of the Hanford-30 Series. The c -observations from the Hanford-30 data were compared y with the model for lateral spread, using the linear relationship between cr and cr as described in Section 7.2.1.2 (<r = or -x/u). v y y v The results are plotted in Figure B. 5 and the validation s t a t i s t i c s are given in Table B.3. It is concluded that the agreement of the CIC/Q model with observations is good over distances up to a few km, whereas the equivalent for the lateral spread is at least acceptable for the present purposes. - 248 -APPENDIX C : SAM - Fortran - 7 7 Code 1 2 C 3 C ' T h i s i s the s h o r t e s t p o s s i b l e way to run SAM 4 C ' i n t e r a c t i v e l y . 5 C 6 CALL SAMSUB(AZS.AD.AZO.AUS,AQH,AZI.ASV.AUST,AL,0) 7 STOP 8 END 9 10 1 C ================================================================= 2 SUBROUTINE SAMSUBt AZS,AO,AZO,AUS,AQH,A2I,ASV,AUST,AL,IFLG) 3 C ================================================================ 4 C Source Area Model 5 C ================================================================= 6 C 7 C » > S A M « < 8 C 9 C * T h i s s u b r o u t i n e e v a l u a t e s the s o u r c e r e g i o n o f a t u r b u l e n t 10 C * f l u x s e n s o r at h e i g h t AZS. 11 C " The arguments are as f o l l o w s : 12 C * AZS : se n s o r h e i g h t AD : d i s p l a c e m e n t h e i g h t 13 C ' AZO : roughness l e n g t h AUS : windspeed at AZS 14 C * AQH : sens, h e a t f l u x AZI : m i x i n g h e i g h t 15 C ' ASV : l a t . wind s t d . dev. AUST: roughness windspeed 16 C * AL : Monin-Obukhov l e n g t h IFLG: f l a g 17 C ' 18 C * I f IFLG=1, the v a l u e s o f the arguments at e n t r y a r e used 19 C * as inp u t to the model. AZS, AD and AZO ar e n e c e s s a r y s i t e 20 C * i n f o r m a t i o n . M e t e o r o l o g i c a l input may be AUS and AQH or 21 C " AUST and AL, and ASV or AZI. F o r the l a t t e r c h o i c e s , the 22 C * model p r o v i d e s a p a r a m e t e r i z a t i o n o f AUS, AL and ASV. 23 C * 24 C * E l s e , the model goes i n t o i n t e r a c t i v e mode and the user i s 25 C * prompted f o r a l l i n p u t i n f o r m a t i o n . A p l o t t i n g o p t i o n , u s i n g 26 C * DISSPLA(10.0), i s a v a i l a b l e in t h i s mode. I f DISSPLA( 10.0) i s 27 C * not a v a i l a b l e , SUBROUTINE MONET needs to be r e p l a c e d f o r 28 C * c o m p i l a t i o n . 29 C * R e s u l t s a r e w r i t t e n to UNIT* 2, warnings, e r r o r messages and 30 C * i n t e r m e d i a t e r e s u l t s a r e w r i t t e n t o UNIT# 3. In i n t e r a c t i v e 31 C • mode UNIT# 6 = 'SINK* and UNIT# 5 = 'SOURCE*. 32 C 33 C * The 'source r e g i o n ' o r 'source a r e a ' i s d e f i n e d as th a t upwind 34 C ' r e g i o n on the ground which i s d i r e c t l y a f f e c t i n g t h o se a i r -35 C * p a r c e l s t h a t a r e e n c o u n t e r i n g the sensor by. t u r b u l e n t t r a n s p o r t . 36 C * The model makes use o f the analogy between the mechanisms of 37 C * d i f f u s i o n o f any s c a l a r and the plume b e h a v i o u r of a p a s s i v e 38 C * t r a c e r from a ground s o u r c e . ( R e c i p r o c a l plume model, see 39 C * C h a p t e r s 6 and 7 ) . 40 C * The b a s i s f o r t h i s model i s the p r o b a b i l i t y - d e n s i t y f u n c t i o n 41 C * plume-model by G r y n i n g e t . a l . (1987) ( h e n c e f o r t h : GHIS). 42 C * 43 C * 44 C 45 IMPLICIT REAL'8 (A-H.L.O-Z) 46 REAL * 8 XCUT(2),CFRAC(10,4),EF( 10,2),SN0(0:50,2),AUX( 11) 47 REAL*8 XLIMt2),PLETH(9,2,51),PDOTS(110),DUMP(2) 48 REAL'8 PROFX( 101 ,2),PROFY(101 ,2),PCHAR(10,7) - 249 -49 C 50 C " S p e c i f y input c o n s t a n t s 51 C 52 COMMON ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN.XBMIN 53 C 54 C=0.4 55 VK=0.4 56 C 57 C * I f IFL=1 : not i n t e r a c t i v e 58 C 59 IF( IFL .NE . 1) THEN 60 C 61 C " G i v e s i t e s p e c i f i c i n f o r m a t i o n : se n s o r h e i g h t ( Z S ) , ZO & D 62 C 63 WRITE(6,*)'GIVE SITE INFO. : SENSOR HEIGHT, ZO, D ; as i n : 64 WRITE(6,*)'ZSZSZS ZOZOZO DDDDDD' 65 REA0(5,*)ZS,ZO.D 66 C 67 c ZS=22.5 68 c Z0=.5 69 c D=3.5 70 c 71 WRITE(6,*)'DO YOU HAVE U* AND L ? (YES=1)' 72 READ(5,*)JJA 73 IF(JJA.EQ.1)THEN 74 WRITE(6,')'GIVE U* AND L : ' 75 READ(5.*)UST,L 76 END IF 77 c 78 c * Read J u l i a n Day, l o c a l apparent time, windspeed and d i r e c t i o n 79 c * ( a t se n s o r h e i g h t ) , s u r f a c e l a y e r temperature ( i n C e l s i u s ) , 80 c • s u r f a c e heat f l u x and boundary l a y e r h e i g h t . 81 c 82 WRITE(6,*)'GIVE J.DAY, L.A.T.. U, WD, Z I , SV, QH, TEMP. :' 83 WRITE!6,*)'DJD TH TM UUUUUU WDWDWD ZIZI SVSV QHQHQH TT' 84 READ(5,*)JD,JTH,JTM,US,WD,ZI,SV,QH,TT 85 c 86 IF(JJA.EQ.1)GOTO 5 87 c 88 c * C a l c u l a t e L and U* by a c a l l to MONOB 89 c 90 CALL MONOB(US,ZS,D,ZO,TT,QH,L,UST) 91 c 92 c * I f t h e r e o c c u r r e d an e r r o r i n MONOB, cut s h o r t . 93 c 94 IF(L.EQ.999.)GOTO 999 95 5 CONTINUE 96 ELSE 97 ZS=AZS 98 D=AD 99 ZO=AZO 100 US=AUS 101 QH=AQH 102 ZI=AZI 103 SV=ASV 104 UST=AUST 105 L=AL 106 IF(L.GE.0.)THEN - 250 -107 CALL MONOB (US , ZS , D , ZO, TT , QH , L , UST) 108 END IF 109 END IF 110 C 111 IF(SV.EQ.O.>THEN 112 SV=SIGV(L,ZI,UST) 113 END IF 114 C 115 CALL MINIM(ZBMIN,XBMIN) 116 WRITE(6,*)'MINIMUM UPWIND DISTANCE = ',XBMIN 117 c 118 c * Set up an array with a u x i l i a r y data that may be c a r r i e d over to 119 c * the plo t - s u b r o u t i n e MONET . Array AUXI10) contains the f o l l o w i n g 120 c * information : 121 c JD, L.A.T., Wind (sp./dir.),Zi.SV.ZS/L (stab.),U*,ZS & FLAG 122 c " i f FLAG=0, the run r e f e r s to the SUNSET s i t e , 123 c * i f FLAG=1, the run r e f e r s to a MOBILE s i t e . 124 c 125 AUX(1)=DFL0AT(JD) 126 AUX(2)=DFL0AT(JTH) 127 AUX(3)=DFL0AT(JTM) 128 AUX(4)=US 129 AUX(5)=WD 130 AUX(6)=ZI 131 AUX(7)=ZS/L 132 AUX(8)=UST 133 AUX(9)=ZS 134 IF(IFLG.NE.1)THEN 135 WRITE(6,*)'WHICH SITE ? (0 = SUNSET; 1 = MOBILE)' 136 READ(5,*)AUX(10) 137 ELSE 138 AUX(10)=0. 139 END IF 140 AUX(11)=SV 141 c 142 c * Write header for output d a t a - f i l e (#2) 143 c 144 WRITE( 2,*)'RESULTS FROM S A M - CALCULATIONS' 145 WRITE( 2 , ' ) ' =-==-==—=—==-==-===-===-=====— • 146 WRITE(2. 101)JD,JTH,JTM 147 WRITE12,122)QH,TT,SV 148 WRITE(2,121)US,WD,L,UST,ZI 149 101 FORMAT!//.'DATA FOR : JD*,I4.' ; at ' .12.' : ' . 12.' L.A.T.'./) 150 122 FORMAT('SURFACE HEAT FLUX : '.F6.2,' ; AIR TEMP. : '.F5.1. 151 #' SV ='.F5.2) 152 121 FORMAT('WINO (SPEED/DIR.) : '.F5.2,'/ '.F5.1,' L ='.F8.2. 153 #' U* = ',F5.3,' Zi ='.F5.0) 154 WRITE(2,128) 155 128 FORMAT(/,'(ALL DATA ARE IN SI-UNITS)'.///) 156 c 157 c 158 c 159 c * Find the point of maximum e f f e c t 160 c 161 CALL CMAXUMX ,CMX) 162 c 163 c * Specify the number of i n t e g r a t i o n steps f o r the e f f e c t - f i e l d 164 c * sec t i o n s ( 1/2 number in X and Y d i r e c t i o n ) . - 251 -165 C 166 C WRITE(6,")'GIVE HALF # OF INTEGRATION STEPS IN X DIRECTION :' 167 C READ(5 , * )N 168 C WRITE(6,*)'GIVE HALF # OF INTEGRATION STEPS IN Y DIRECTION :' 169 C READC5,* )M 170 C 171 N=10 172 M=25 173 C 174 C • S e l e c t the e f f e c t f i e l d l e v e l s and perform the section-wise 175 C * double i n t e g r a t i o n over the e f f e c t f i e l d l e v e l s . See ADDUP for 176 C • f u r t h e r d e t a i l s 177 C 178 CALL ADDUP(N,M,CMX,XMX,CTOT,CFRAC) 179 C 180 C • I f an e r r o r occurred in ADDUP, wr i t e to #3 and cut short 181 C 182 IF(CTOT.LE.O.)THEN 183 WRITE(3,*)'ADDUP WAS UNABLE TO INTEGRATE THE EFFECT FIELD' 184 WRITE(3,")'CTOT = ',CTOT 185 GOTO 999 186 END IF 187 C 188 C • Use a s p l i n e i n t e r p o l a t i o n to evaluate the is o p l e t h s that 189 C • contain decreasing f r a c t i o n s of the whole integrated e f f e c t 190 C • f i e l d (eg. 0.95, 0.90, 0.80, e t c . ) . NOTE : In CFRAC the 191 C • f r a c t i o n a l e f f e c t l e v e l s are given in % , the f r a c t i o n a l 192 C • i n t e g r a l s in decimals. 193 C 194 C * F i r s t set up the nodes for the s p l i n e , w r i t e them to comment f 195 C 196 WRITE(3,33) 197 33 FORMAT!//,'NODES FOR SPLINE : CUM. INTEGR. CONC. %',/) 198 J=0. 199 DO 503 1=1.9 200 IF(CFRAC(I,3).EQ.0.)GOTO 77 201 J=J+ 1 202 SN0(J,2)=CFRAC(I,1) 203 SN0(J,1)=CFRAC(I,4) 204 77 CONTINUE 205 503 CONTINUE 206 SNO(0,1)=0. 207 SN0(0.2)=100. 208 SN0(10,1)=1. 209 SN0(10,2)=0. 210 NM=J+1 211 DO 366 J=0,NM 212 WRITE(3,32)SN0(J,1),SN0(J,2) 213 32 FORMAT(19X.F12.5.F12.2) 214 366 CONTINUE 215 C 216 C * Set the desired i n t e g r a l f r a c t i o n s . EF (EF for e f f e c t ) . 217 c • Where E F ( i n t e g r a l f r a c t i o n , i s o p l e t h %) 218 c * Write subheader for comment l i s t i n g . 219 c 220 WRITE(3, 189) 221 189 FORMAT(/,'INTEGRAL FRACTION OF TOTAL EFFECT VS. ISOPLETH %') 222 WRITE(3 •)'-==-==-==-=====-===-========-===-==-====-==—==' - 252 -223 WRITE(3,')' ( i n t . f r a c . ) (cone. % of max.)' 224 WRITE(3,')' ' 225 C 226 X=0.95 227 J=0 228 NP=NM+1 229 DO 504 1=10,1.-1 230 IF(X.GT.SNO(NM,1))G0T0 88 231 j=j+1 232 C 233 CALL SPLYNE(SN0,NP,X,Y) 234 C 235 EF(I,1)=X 236 EF(I.2)=Y 237 C 238 C • W r i t e r e s u l t s i n t o c o m m e n t - f i l e 239 C 240 WRITE(3,1881X.Y 241 188 FORMAT(F20.5,F20.2) 242 88 CONTINUE 243 X=0.1'DFLOAT(I-1) 244 504 CONTINUE 245 NJ=J 246 C 247 C * Now the a r r a y EF c o n t a i n s round v a l u e s of the i n t e g r a l f r a c t i o n 248 C • v e r s u s the a s s o c i a t e d e f f e c t l e v e l i n % of the maximum. 249 C * The next s t e p i s to f i n d the l o c a t i o n o f the i s o p l e t h s f o r t h e s e 250 C * e f f e c t l e v e l s . T h i s i s done by i n v o k i n g s u b r o u t i n e CONC 251 C • w i t h the FLAG=2. So t h a t X and the e f f e c t l e v e l a r e the inp u t 252 C and Y(+) the o u t p u t . The maximum range o f X f o r each 253 C • e f f e c t l e v e l can be found w i t h CONC and FLAG=1. 254 c • The r e s u l t s w i l l be put i n t o a t h r e e d i m e n s i o n a l a r r a y : 255 c * 256 c • P L E T H ( [ n u m b e r ] , [ x / y ] . [ c a s e »]) ; di m e n s i o n (10,2,1000) 257 c • 258 c • The l i n e number ( i - v a l u e ) i n E F ( i , j ) c o r r e s p o n d s to the 259 c [number]-va1ue in PLETH. 260 c * In o r d e r to e n a b l e a g r a p h i c a l d i s p l a y o f the whole i s o p l e t h , 261 c * PLETH c o n t a i n s the x/y p a i r s f o r both pos. and neg. y ' s . 262 c * PDOTS c o n t a i n s o n l y h a l f the i s o p l e t h and i s used f o r the 263 c • c a l c u l a t i o n o f the ar e a bounded by each i s o p l e t h by c a l l i n g on 264 c • s u b r o u t i n e SIMPS. 265 c 266 WRITE(2,179) 267 179 FORMAT(//.' 268 # ' ) 269 WRITE(2.')' 270 WRITE(2,181) 271 181 FORMAT('S A M - RESULTS SUMMARY (ISOPLETH CHARACTERISTIC 272 #DIMENSIONS)') 273 WRITE(2.182) 274 275 182 #====--==—=-' ) 276 WRITE(2,*) ' i n t . e f f . a r e a a b c 277 # d xm' 278 WRITE(2,*)' 279 Y0=0. 280 NJM=NJ-1 -- 253 -281 XMIN=100. 282 XMAX=0. 283 DO 505 1=1,NJM 284 C 285 C • Chose the i s o p l e t h , w r i t e c h o i c e i n t o c o m m e n t s - f i l e 286 C 287 CP=EF(I,2)'CMX/100. 288 . WRITE(3,*)'ISOPL. #',I.' = ',EF(I,2) 289 C 290 c * Look f o r X-reach and w r i t e r e s u l t i n t o c o m m e n t s - f i l e 291 c 292 CALL C0NC(CMX,XMX,XC,XLIM,YO,CP,1) 293 c 294 XL=XLIM(1) 295 XH=XLIM(2) 296 IF(XL.LT.XMIN)XMIN=XL 297 IF(XH.GT.XMAX)XMAX=XH 298 WRITE(3,123)XL.XH 299 123 FORMAT('near X = ',F8.2,' f a r X = ',F9.2) 300 c 301 DX=(XH-XL)/100. 302 PLETH(I,1.1)=XL 303 PLETH(I,2,1)=0. 304 PLETH(1,1,51)=XL 305 PLETH(I,2,51)=0. 306 PD0TS(1)=0. 307 JJ=5 308 c 309 c ' C a l c u l a t e the l o c a t i o n o f the i s o p l e t h s 310 c 311 DO 506 J=2,101 312 XP=XL+DX * J 313 c 314 CALL CONC(CMX,XMX,XP,XCM,YP.CP,2) 315 c 316 IF(J.EQ.JJ)THEN 317 J J J = ( ( J J - 1 ) / 4 ) + 1 318 P L E T H d . 1,JJJ)=XP 319 PLETH(I,2,JJJ)=YP 320 c 321 K=52-JJJ 322 P L E T H d , 1,K)=XP 323 PLETHd,2,K)=YPM0.-1.) 324 JJ=JJ+4 325 END IF 326 IF(YP.GT.PDOTS(J-1))THEN 327 YYMX=YP 328 XYMX=XP 329 ENDIF 330 c 331 c * Put the +-ve YP's i n t o PDOT f o r use i n the a r e a c a l c u l a t i o n 332 c 333 PDOTSfJ)=YP 334 506 CONTINUE 335 WRITE!3,*)'YYMX = ',YYMX,' AT X = '.XYMX 336 c 337 c ' Compute the a r e a bounded by t h e s e i s o p l e t h s ; i . e . i n t e g r a t e 338 c • over the X-range from PDOTS( 1) to PD0TSM01) - 254 -339 C • using subroutine SIMPS, and m u l t i p l y by 2. 340 C • Array FF contains the i n t . f r a c . and associated area 341 C 342 CALL SIMPS(POOTS,XL.XH.101,FP) 343 344 C 345 C * Specify the c h a r a c t e r i s t i c dimensions of the i s o p l e t h s and put 346 C • them into array PCHAR(10,7) : (#;int.eff.,area.a,b,c,d,xm) 347 C • Write r e s u l t s into r e s u l t s - f i l e 348 C 349 PCHAR(I,1)=EF(1,1) 350 PCHAR(I,2)=FP'2. 351 PCHAR(I,3)=XL 352 PCHAR(I.4)=XYMX-XL 353 PCHAR(I,5)=XH-XYMX 354 PCHAR(I,6)=YYMX 355 PCHAR(I,7)=XMX 356 WRITE(2,199)(PCHAR(I,J),J=1,7) 357 199 FORMAT(F9.2,F12.1.4F10.3.F9.2) 358 505 i CONTINUE 359 C 360 C • C a l c u l a t e a normalized p r o f i l e in X - d i r e c t i o n and put the X.CONC 361 c ft values into array PROFX, to be used for p l o t t i n g purposes. 362 c 363 Y0=0. 364 PDX=DABS(XMAX-XMIN)/100. 365 XXP=XMIN 366 00 509 1=1,101 367 CALL CONC(CMX,XMX.XXP,DUMP.YO,CPX.0) 368 PR0FX(I.1)=XXP 369 PR0FX(I,2)=CPX/CMX 370 c 371 XXP=XXP+PDX 372 509 CONTINUE 373 c 374 c » C a l c u l a t e a normalized p r o f i l e in Y - d i r e c t i o n at the XMX, 375 c • put into array PROFY to be used for p l o t t i n g . 376 c 377 PDY=PCHAR(NJM,6)/50. 378 YYP=0. 379 DO 510 J=51.101 380 CALL CONC(CMX,XMX,XMX,DUMP.YYP,CYP,0) 381 PR0FY(J,1)=YYP 382 PROFY(J,2)=CYP/CMX 383 K=102-J 384 YYPN=0.-YYP 385 PROFY(K,1)=YYPN 386 PR0FY(K,2)=CYP/CMX 387 c 388 YYP=YYP+PDY 389 510 CONTINUE 390 c 391 IF(IFLG.EQ.1)GOTO 999 392 WRITE(6,*)'DO YOU WANT A PLOT ? (YES = 1 / NO = 2)' 393 READ(5,*)IESNO 394 IF(IESNO.NE.1)G0T0 999 395 c 396 c Display the r e s u l t s g r a p h i c a l l y by c a l l i n g on subroutine - 255 -397 C * MONET(PLETH,AUX,PROFX,PROFY) 398 C 399 CALL MONET(PLETH,AUX,PROFX,PROFY) 400 C 401 999 CONTINUE 402 C 403 C 404 RETURN 405 END 406 407 408 SUBROUTINE MONOB(UB,Z,D,ZO,TT,QH,L,USTR) C 409 c * This subroutine computes the Monin-Obukhov length (L) ft 410 c • iteratively via log-profile U* correction : 411 c • ft 412 c • U* = UB(z) k /[1n((z-d)/zo)-PSY((z-d)/L)+PSY(zo/D] (1) * 413 c • * 414 c • PSY = 1n[((1+x*2)/2)((1+x)/2)*2]-2arctan<x)+(PI/2) (2) * 415 c • a 416 c a x = (1-28(z-d)/L)A.25 = PHYA-1 (unstable) (3) * 417 c * 418 c * L = -RHO Cp TB U**3/(k g QH) (4) * 419 c ft • 420 c • In this application the constants take the following values • 421 c • related to the SUNSET site and for unstable conditions : a 422 c • • 423 c • k = 0.4 ft 424 c ft RHO = 1.204 « 425 c • Cp = 1010 • 426 c ft ft 427 c ft Equations (1) to (4) appear in the subroutine. • 428 429 c * • c 430 c 431 IMPLICIT REAL'S (A-H.L.O-Z) 432 REAL * 8 UB . WD. TT , QH, QHM. OL 1 . 0L2 . US. XZ . XO . PSYZ , P'SYO . D . OLD , QRD 433 c ft Specify the constants 434 RH0=1.204 435 CP=1010. 436 VK=0.4 437 G=9.81 438 c 439 c ft Ini t ia l value of U* is 0.1 ' 440 US=0. 1 441 c • Convert to absolute temperature * 442 IF(TT.LT.150.) THEN 443 TT=TT+273. 444 END IF 445 c • Exclude cases where QH <= 10. W/mi2 * 446 IF(QH.LE.10.)G0T0 305 447 c • Equation (4) ( » f irs t estimate) * 448 0L1 = ( - RHO*CP)*TT*US* * 3./(QH*VK*G) 449 N=0 450 c * Begin U" iteration ' 451 300 I CONTINUE 452 c • Equation (3) for (z-d) and zo * 453 XZ=( 1 -28. * (Z-D)/OLD * "0.25 454 X0=(1-28.'Z0/0L1)"'0.25 - 256 -455 C • Equation (2) for (z-d) and zo * 456 PSYZ=DL0G((( 1 + XZ"2. )/2. ) * (( 1 + XZ)/2. ) * *2. ) -2. *0ATAN(XZ)+1 457 PSY0=0L0G(((1 + X0* *2.)/2.)*((1 + X0)/2.)* *2.) -2. •DATAN(X0)+1 458 C * Equation (1) ( » U* correction) • 459 US=UB * VK/(DLOG({Z-D)/ZO)-PSYZ+PSYO) 460 C tl Equation (4) ( » second estimate) * 461 0L2=(-RH0*CP),TT*US**3./(QH*VK*G) 462 C * Evaluate improvement of estimate * 463 DD=0L1-0L2 464 0LD=0ABS(D0) 465 0L1=0L2 466 N=N+ 1 467 C • Maximum iteration steps set to 300 * 468 IF(OLD.GT.0.0001.AND.N.LT.300)GOTO 300 469 C 470 IF(N.EQ.300)THEN 471 WRITE(3,')'ITERATION FOR L UNSUCCESSFUL AFTER 300 STEPS' 472 L=999. 473 GOTO 306 474 END IF 475 C 476 L=0L2 477 USTR=US 478 C 479 GOTO 306 480 305 WRITE(3,205) 481 205 FORMAT (Tun is *** STABLE OR QH < 10 W/M2 •* • ' ) 482 306 CONTINUE . 483 c 484 RETURN 485 END 486 SUBROUTINE MINIM(ZBMN,XBMN) 487 c 488 c • This subroutine determines the minimum upwind distance, for 489 c * which an effect-level calculation is defined. 490 c 491 IMPLICIT REAL*8 (A-H.L.O-Z) 492 COMMON ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN,XBMIN 493 c 494 TOL=1.E-5 495 c 496 ZBA=D+1.E-8 497 PHYA=PHYMIN(ZBA,L) 498 PSYA=PAUL(PHYA) 499 QA=DL0G(C'(ZBA-D)/ZO)-PSYA 500 c 501 c • Find intervall where QB changes sign. 502 c 503 ZBB=ZS 504 IK=0 505 401 CONTINUE 506 PHYB=PHYMIN(ZBB,L) 507 PSYB=PAUL(PHYB) 508 QB=DL0G(C *(ZBB-0)/ZO)-PSYB 509 PROD=QA'QB 510 IF(PROD.GE.O.)THEN 511 ZBB=ZBB+1 . 512 IK=IK+1 - 257 -513 IF(IK.GT.500)GOTO 999 514 GOTO 401 515 ENDIF 516 C 517 C " Bisection begins 518 C 519 DO 500 1=1,100 520 ZBP=ZBA+(ZBB-ZBA)/2. 521 PHYP=PHYMIN(ZBP,U 522 PSYP=PAUL(PHYP) 523 QP=DL0G(C *(ZBP-D)/ZO)-PSYP 524 DIFF=DABS((ZBB-ZBA)/(2.*ZBB)) 525 IF(QP.EQ.O..OR.DIFF.LT.TOD THEN 526 ZBMN=ZBP 527 IF(ZBMN.L T.ZO)ZBMN=ZO 528 PSYQ=PAUL(PHYMIN(ZBMN,L)) 529 QQ=DLOG(C*(ZBMN-D)/ZO)-PSYQ 530 XBMN=XZB(ZBMN,PSYQ,L) 531 IF(XBMN.LT.ZO)XBMN=ZO 532 GOTO 402 533 ENDIF 534 C 535 PR=QA'QP 536 IF(PR.GT.O.)THEN 537 QA=QP 538 ZBA=ZBP 539 ENDIF 540 IF(PR.LT.O.)THEN 541 QB=QP 542 ZBB=ZBP 543 ENDIF 544 500 CONTINUE 545 WRITE (3,*)'MINIM : NOT FOUND AFTER 100 ITERATIONS' 546 WRITE (3.*)'REL. DIFF IN ZB : ' .DIFF, ' QP = ',QP 547 402 CONTINUE 548 c 549 c ' Double check ZBMIN and XBMIN. 550 c 551 CALL CICZ(ZMIN.ZS.UB,S,XBMN.CIC,0) 552 ZBMN=ZMIN 553 PSYP=PAUL(PHYMIN(ZBMN,L)) 554 XBMN=XZB(ZBMN,PSYP.L) 555 999 CONTINUE 556 c 557 RETURN 558 END 559 c 560 SUBROUTINE CMAX(XMX.CMX) 561 c 562 c * This subroutine calculates the maximum CONC/Q at level ZS for 563 c ' a surface point-source, together with the downwind location 564 c * (X) of that point. 565 c * The sensor level ZS, and surface geometry parameters ZO and 566 c * D are treated as constants, residing in common memory space with 567 c • the main program (also wind at sensor level (US), U*. ZI & L) . 566 c 569 IMPLICIT REAL'8 (A-H.L.O-Z) 570 COMMON ZS.ZO,D,C,VK,US,UST.SV.L,ZBMIN,XBMIN - 258 -571 C 572 C C=0.4 573 C VK=0.4 574 C ZS=22.5 575 C Z0=.5 576 C 0=3.5 577 C 578 C * Write subheader into comment-file 579 C 580 CC WRITEO,')' 581 CC WRITE(3,')'THE MAXIMUM CONC(ZS) IS REACHED WITH THE DATA :' 582 CC WRITEO,') ' - - - ' 583 CC WRITEO, * ) ' ' 584 CC WRITEO, *)' X ZB ZS S CONC(ZS)' 585 CC WRITEO,')' ' 586 C 587 C 588 C * In i t ia l guess for CMX is at ZB = 0.6 ZS 589 c 590 ZB1=ZS'0.6 591 IF(ZB1.LT.ZBMIN)ZB1=ZBMIN 592 c 593 c • compute CIC(ZS) with ZB = ZB1 594 c • Adjust for lateral diffusion with LAT and Y=0 595 c 596 Y=0. 597 CALL CICZ(ZB1.ZS.UB1,S1,X1,C1,1) 598 C1=C1'LAT(Y,SV,.UB1 ,X1) 599 c 600 c * The search for a maximum in CMX begins. The method of "REGULA 601 c • FALSI" is used with a steplength of DAB in ZB. 602 c 603 DAB=ZB1 604 c 605 ZB2=ZB1+DAB 606 c 607 c * Compute CIC(ZS) with ZB = ZB2 608 c * Adjust for lateral diffusion with LAT and Y=0 609 c 610 CALL CICZ(ZB2,ZS,UB2,S2,X2,C2,1) 611 C2=C2*LAT(Y,SV,UB2,X2) 612 c 613 c * This is the beginning of the algorythm, an imitated "WHILE" 614 c * loop via IF-THEN & GOTO. 615 c Two values (ZB1.C1) and (ZB2.C2) are evaluated . Based on the 616 c * comparison of C1 and C2 it is decided in which direction the 617 c * search is to continue. Every time the search is changing 618 c direction, the steplength DAB is halved. 619 c LOOP 402 : Main bisection routine. 620 c 621 402 CONTINUE 622 DAB=DAB/2 623 C 624 C * The search is terminated when DAB < 1E-10 625 C 626 IF(DAB.GE.1E- 10)THEN 627 C 628 C • LOOP 400 : While C2 > C1 the search is continued to the right - 259 -629 C 630 400 CONTINUE 631 IF(C2.GT.C1)THEN 632 ZB1=ZB2 633 C1=C2 634 ZB2=ZB1+0AB 635 CALL CICZ(ZB2,ZS,UB2.S2.X2.C2,1) 636 C2=C2*LAT(Y,SV,UB2,X2) 637 C 638 C • In case of an "overshoot" skip out of LP-400, back to bisection 639 C 640 IFIC2.LE.C1)G0T0 402 641 GOTO 400 642 ENDIF 643 C 644 C * LOOP 401 : While C2 <= C1 the search is continued to the left . 645 C 646 401 CONTINUE 647 IF(C2.LE.C1)THEN 648 C 649 ZB1=ZB2-DAB 650 CALL CICZ(ZB1,ZS.UB1,S1,X1,C1, 1) 651 C1=C1'LAT(Y,SV,UB1,X1) 652 C 653 C • In case of an "overshoot" skip out of LP-401, back to bisection 654 C 655 IF(C2.GT.C1)G0T0 402 656 ZB2=ZB1 657 C2=C1 658 GOTO 401 659 ENDIF 660 ENDIF 661 C 662 C • End of "REGULA FALSI" algorythm. 663 C * Print last estimates with last steplengths as uncertainties. 664 C 665 EC=DABS(C2-C1) 666 EX=DABS(X2-X1) 667 XMX=X2 668 ZB=ZB2 669 S=S2 670 CMX=C2 671 C 672 CC WRITEO,')' ' 673 CC WRITEO,')'THE REMAINING UNCERTAINTIES ARE 674 CC WRITEO, *)'E-CIC : ' . E C , ' ; E-X : ',EX 675 CC WRITEO,')' ' 676 RETURN 677 END 678 C 679 c 680 SUBROUTINE CICZ(ZB.Z,UB,S,X,CIC,IFLAG) 681 c 682 c * Distance which plume has travelled to reach ZB is X , computed 683 c from GHIS (A1a). 684 c • If IFLAG=0. X is input and ZB is output, if IFLAG=1 , ZB is input 685 c • and X is output. 686 c • For that, compute PSY(C(ZB-D)/L) and give ZO, 0, K, P, C, using - 260 -687 C * 688 C * PSYM = PSYM(PHYA-1) from Paulson, 1970 689 C " PHY*-1 = XM = (1 - 28(ZB-D)A.25 ; K = 0.-4 (Dyer&Bradley. 1982) 690 C * (using the correlations for heat diffusion). 691 C * P = 1.55 ; C = 0.4 (GHIS.1987; 1983) 692 C * 693 IMPLICIT REAL*8 (A-H.L.O-Z) 694 COMMON ZS.ZO.D,C.VK,US,UST,SV,L,ZBMIN,XBMIN 695 IF(IFLAG.EQ.1) THEN 696 C 697 C * Want X for Z 698 C * compute PHYA-1 or XM 699 C 700 XM=PHYMIN(ZB.L) 701 C 702 C * compute PSYM using the Paulson(1970) formula 703 C 704 PSYM=PAUL(XM) 705 C 706 C ' compute X from GHIS (A1a) 707 C 708 X=XZB(ZB,PSYM.L) 709 IF(X.LT.XBMIN)THEN 710 X=XBMIN 711 ENDIF 712 ENDIF 713 IF(IFLAG.EQ.O) THEN 714 IF(X.LT.XBMIN)THEN 715 X=XBMIN 716 ENDIF 717 ZB=Z0FX(X,L) 718 ENDIF 719 C 720 C * compute S from GHIS (A2a) 721 C 722 S=SPAR(ZB.PSYM,L) 723 C 724 C * compute UB from GHIS (A3a) 725 C 726 UB=UBAR(UST.ZB.PSYM.L) 727 C 728 C * compute A from GHIS (A4) 729 C 730 A=AGAM(S) 731 C 732 C * compute B from GHIS (A5) 733 - C 734 B=BGAM(S) 735 C 736 C * compute CIC at Z when the mean plume height is ZB 737 C 738 CIC=CHYQ(A.B.ZB,UB,Z.S)'1.E5 739 C 740 IF(X.LE.O)THEN 741 WRITEO, *)'CICZ : X <= 0. IFLAG = ',IFLAG 742 ENDIF 743 C 744 RETURN - 261 -V 745 END 746 C 747 C 748 FUNCTION PAUL(XX) 749 C 750 C ' This function computes the PaulsonC 1970) PSY 751 C * for a given XX for unstable conditions. 752 C 753 REAL'8 XX,PAUL 754 C PAUL=0L0G( ( 1 + XX"2. ) * < 1+ XX) " 2/8) -2*DATAN( XX ) + 1 . 5708 755 PAUL=XX-1. 756 RETURN 757 END 758 C 759 C 760 FUNCTION XZB(ZB.PSYM.LD) 761 C 762 C * This function is equivalent to equation (A1a) in 763 C • GHIS, with P=1.55, C=0.4 (GHIS,1983). A1 = 14 764 C * XZB is the distance the plume has to travel to reach 765 C " height ZB. 766 C 767 IMPLICIT REAL'8 (A-H.L.O-Z) 768 COMMON ZS,ZO.D,C.VK,US,UST,SV,L,ZBMIN,XBMIN 769 ZZ=ZB-0 770 V=VK"2. 771 XZB=(ZZ/V)*(DL0G(C'ZZ/Z0)-PSYM)'( 1 -(5.425'ZZ/L)) " ( -.5) 772 RETURN 773 END 774 C 775 C 776 FUNCTION SPAR(ZB.PSYM.LO) 777 C 778 C • This function is equivalent to equation (A2a) in 779 C * GHIS, with C=0.4 (GHIS,1983), A1=14, A2=28 780 C * SPAR is the shape parameter for the exponent of the 781 C * vertical concentration prof i le . 782 c 783 IMPLICIT REAL * 8 (A-H.L.O-Z) 784 COMMON ZS,ZO,D,C,VK.US,UST,SV,L.ZBMIN,XBMIN 785 ZZ=ZB-D 786 787 PSYM=PAUL(PHYMIN(ZB,L)) 788 SP1=(1-7.'C*ZZ/L)/(1-14.*C'ZZ/L) 789 SP2=(1-28.'C'ZZ/D " (-.25)/(0LOG(C'ZZ/ZO)-PSYM) 790 SPAR=SP1+SP2 791 c 792 IF(SPAR.GT.1.9)THEN 793 SPAR=1.9+(1.-DEXP(-1.0*DABS(SPAR-1.9)))/10. 794 ENDIF 795 c 796 IFtSPAR.LT.O.)THEN 797 WRITE(3,*)'SPAR : '.SPAR.' SP1=',SP1,' SP2=',SP2 798 QQ=DL0G(C'ZZ/Z0)-PSYM 799 WRITE(3,*)'QQ=',QQ 800 ENDIF 801 RETURN 802 END - 262 -803 C , 804 C 805 FUNCTION UBAR(USTD.ZB.PSYM.LD) 806 C 807 C * This function is equivalent to equation (A3a) in GHIS, 808 C * with C=0.4 809 C * UBAR is the mean travel-speed of the plume 810 C 811 IMPLICIT REAL*8 (A-H.L.O-Z) 812 COMMON ZS,ZO.D.C.VK,US,UST,SV.L.ZBMIN,XBMIN 813 ZZ=ZB-0 814 UBAR=UST/VK*(DLOG(C'ZZ/ZO)-PSYM) 815 RETURN 816 END 817 C 818 C 819 FUNCTION AGAM(S) 820 C 821 C * This function is equivalent to equation (A4) in GHIS 822 C * AGAM is a function of the shapefactor S 823 C 824 IMPLICIT REAL*8 (A-H.L.O-Z) 825 S1=1/S 826 S2=2/S 827 AGAM=S'DGAMMA(S2)/(DGAMMA(S1))* *2 828 RETURN 829 END 830 C ; 831 C 832 FUNCTION BGAM(S) 833 C 834 C * This function is equivalent to equation (A5) in GHIS 835 C * BGAM is a function of the shapefactor S 836 C 837 IMPLICIT REAL*8 (A-H.L.O-Z) 838 S1=1/S 839 S2=2/S 840 BGAM=OGAMMA(S2)/DGAMMA(S1) 841 RETURN 842 END 843 C 844 C 845 FUNCTION DGAMMA(X) 846 C 847 C * This function approximates the Gamma-function according to 848 C * equation (A6) in GHIS. 849 C 850 IMPLICIT REAL * 8 (A-H.L.O-Z) 851 REAL*8 B(8) 852 DATA 8( 1 )/-.577191652/,B(2)/.988205891/,B(3)/-. 897056937/, 853 » B(4)/.918206857/,B(5)/-.756704078/.B(6)/.482199394/, 854 # B(7)/-.193527818/,B(8)/.035868343/ 855 C 856 DG=1. 857 DO 500 1=1.8 858 DG=DG+((X-1.)**I)'B(I) 859 500 CONTINUE 860 C - 263 -861 DGAMMA=DG 862 C 863 RETURN 864 END 865 C 866 C 867 FUNCTION CHYQ(A.B,ZB,UB,Z,S) 868 C 869 c * This function is equivalent to equation (3) in GHIS 870 c * CHYQ is the crosswind integrated concentration divided 871 c • by the source strength. 872 c 873 IMPLICIT REAL*8 (A-H,L,0-Z) 874 COMMON ZS.ZO.D.C,VK.US,UST,SV,L,ZBMIN,XBMIN 875 ZZ1=ZB-D 876 ZZ2=Z-D 877 CHYQ=A/(ZZ1*UB)*DEXP(-(B*ZZ2/ZZ1)* *S) 878 RETURN 879 END 880 c 881 c 882 FUNCTION SIGV(L.ZI.UST) 883 c 884 c • This function computes the V-standard deviation 885 c • SIGV (or sigma-V) according to eq.(18a) of GHIS. 886 c 887 IMPLICIT REAL*8 (A-H.L.O-Z) 888 SIGV=0.8*UST*((0.-ZI)/L)"*0.333 889 RETURN 890 END 891 c 892 c 893 FUNCTION LATfY.SV.UB.X) 894 c 895 c * This function computes the factor that the CIC wil l have 896 c * to be multiplied with to obtain a concentration value with 897 c • consideration of lateral spread and other than on the center-898 c • l ine. 899 c * The linear relationship between Sigma-V and Sigma-Y as 900 c * suggested by Pasquill & Smith (1983) for the diffusion very 901 c * close to the source is adopted here. 902 c * SV is the V-standard deviation e.g. from FUNCTION SIGV 903 c 904 IMPLICIT REAL*8 (A-H.L.O-Z) 905 c 906 c Compute the traveltime T = X/UB 907 c 908 T=X/UB 909 c 910 c * Compute the lateral diffusion parameter SIGY 911 c 912 SIGY=SV*T 913 c 914 c * Compute LAT 915 c 916 EX=(Y/SIGY)**2*(-0.5) 917 IF(EX.LT.- 180.)THEN 918 LAT=0. - 264 -919 GOTO 44 920 ENDIF 921 LAT=DEXP(EX)/(SIGY•2.5066) 922 44 CONTINUE 923 RETURN 924 END 925 C 926 C 927 FUNCTION ZOFX(X.LO) 928 C 929 c • This function iteratively determines the ZB value for 930 c • a given X in Eq. (A1a) of GHIS, using a secant root-finding 931 c • routine. 932 c • ZOFX is equal to ZB. 933 c • It cal ls on functions PAUL and XZB 934 c 935 IMPLICIT REAL'8 (A-H.L.O-Z) 936 COMMON ZS.ZO.D.C.VK.US,UST.SV.L.ZBMIN,XBMIN 937 JK=1 938 c 939 IF(X.LT.XBMIN)THEN 940 ccc WRITEO, *)'ZOFX : X < XBMIN' 941 ZOFX=ZBMIN 942 GOTO 402 943 ENDIF 944 c 945 T0L=1.E-5 946 ZB0=X/5. 947 IF(ZBO.LT.ZBMIN)ZB0=ZBMIN 948 ZB1=ZB0+1 949 XMO=PHYMIN(ZBO,L) 950 PSYMO=PAUL(XMO) 951 Q0=XZB(ZB0,PSYM0,L)-X 952 1=2 953 401 CONTINUE 954 IFM.LE.50) THEN 955 XM1=PHYMIN(ZB1,L) 956 PSYM1=PAUL(XM1) 957 Q1=XZB(ZB1.PSYM1,L) -X 958 IF(01.EQ.QO)THEN 959 WRITEO, •)'ZOFX : DIVIDE BY 0 ; X = ' ,X 960 WRITEO, *)'ZB1 : ',ZB1 961 Z0FX=ZB1 962 ENDIF 963 c 964 ZB=ZB 1 - Q1 * ( ZB 1 - ZBO) / (Q1 - 00) 965 IF(ZB.LT.ZBMIN)THEN 966 ZB=ZBMIN 967 IF(JK.GT.1)THEN 968 WRITEO, *)'ZOFX : TRIED TO SHOOT BELOW ZBMIN ' , JK, ' TIMES' 969 ENDIF 970 JK=JK+1 971 ENDIF 972 c 973 DIFF=DABS(ZB-ZB1) 974 IF(DIFF.LT.TOL) THEN 975 ZOFX=ZB 976 GOTO 402 - 265 -977 ENOIF 978 C 979 1=1+1 980 ZB0=ZB1 981 Q0=Q1 982 ZB1=ZB 983 GOTO 401 984 ENDIF 985 WRITE(3,*)'ZOFX : FAILED AFTER 50 ITERATIONS; OIFF =',DIFF 986 402 CONTINUE 987 RETURN 988 END 989 C 990 C 991 FUNCTION PHYMIN(Z,LD) 992 C 993 C * This function is equivalent to the reciprocal of a 994 C • empirical f lux-profile relation . 995 C • In this case for momentum flux as suggested by Dyer & 996 C * Bradley, 1982. 997 C 998 IMPLICIT REAL'S (A-H.L.O-Z) 999 COMMON ZS.ZO.D,C,VK,US.UST,SV,L,ZBMIN,XBMIN 1000 ZMIN=D+1.E-9 1001 IF(Z.LT.ZMIN)THEN 1002 WRITE(3,*) ' WARNING (Z-D)= M Z - D ) . ' ; IS SET TO +1.E-9 ' 1003 Z=ZMIN 1004 ENDIF 1005 PHYMIN=(1.-28.'C*(Z-DJ/L) " .25 1006 c 1007 RETURN 1008 END 1009 1010 SUBROUTINE CONC(CMX,XMX,XC.XCM,YC,CC,IFLAG) 1011 c 1012 c * This subroutine computes the concentration of a surface 1013 c * point source at coordinate location (XC.YC.ZS). or any 1014 c • third of the tr iplet (XC.YC.CC). when the other two are 1015 c * given and IFLAG set accordingly (see below). If the X -locat ion 1016 c • for a certain pair (CC.YC) is wanted, the result resides in the 1017 c * array XCM, with XCM(1) being closest and XCM(2) farthest from 1018 c * the source. 1019 c * The maximum concentration CMX at (XMX.O.ZS) is imported as a 1020 c * restraint for the secant search. 1021 G • * 1022 C • - • IFLAG=0 : XC.YC = Input; CC = Output 1023 c ft IFLAG=1 : CC.YC = Input; XCM(2)= Output 1024 c * IFLAG=2 : CC.XC = Input; YC(+) = Output 1025 c ft 1026 IMPLICIT REAL * 8 (A-H.L.O-Z) 1027 REAL'S XCM(2) 1028 COMMON ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN,XBMIN 1029 cc SV=SIGV(L,ZI,UST) 1030 c 1031 IF(IFLAG.EQ.0)THEN 1032 CALL CICZfZB.ZS.UB.S.XC.CIC.O) 1033 CC=CIC*LAT(YC,SV.UB,XC) 1034 c FF=DABS(XMX-XC) - 266 -1035 C 1036 CR=DABS(CC-CMX)/CMX 1037 IF(CC.GT.CMX.AND.CR.LE.0.01)CC=CMX 1038 ENDIF 1039 C 1040 IF(IFLAG.EQ.1.AND.CC.LT.CMX)THEN 1041 C 1042 T0LX=1.E-3 1043 DO 500 J=1,2 1044 C 1045 C • Search for XCM-points by going on either side of 1046 C 1047 N=2 1048 XP0=XMX 1049 CQ0=CMX-CC 1050 IF(J.EQ.I) XP1=XBMIN 1051 IF(J.EQ.2) XP1=XMX+20. 1052 C 1053 C * Bisection 1054 c 1055 CH=CMX-CC 1056 IF(CC.EQ.CMX)THEN 1057 XCM(J)=XMX 1058 GOTO 402 1059 ENDIF 1060 c 1061 c • Look for interval! where CQ changes sign 1062 c 1063 IF(J.EQ.2)THEN 1064 DXX=50. 1065 DO 551 1=1,300 1066 XP1=XP0+DXX 1067 CALL CICZ(ZB,ZS,UB.S,XP1.CQC.O) 1068 CQ1=CQC*LAT(YC.SV,UB,XP1)-CC 1069 c 1070 IF(CQ1 . EQ.OTHEN 1071 XCM(J)=XP1 1072 GOTO 402 1073 ENDIF 1074 c 1075 CQP=CQ0*CQ1 1076 IF(CQP.LT.O.)THEN 1077 c 1078 c • Found the interval 1. Start Bisection now 1079 c 1080 GOTO 451 1081 ENDIF 1082 c 1083 IF(I.EQ.300)THEN 1084 WRITEO,')'OUTER REACH IS > 15000 M' 1085 XCM(J)=15000. 1086 GOTO 999 1087 ENDIF 1088 c 1089 c * Update everything 1090 c 1091 CQ0=CQ1 1092 XP0=XP1 - 267 -1093 551 CONTINUE 1094 ENDIF 1095 451 CONTINUE 1096 C 1097 C * Bisection begins 1098 C 1099 NBB=1 1100 452 CONTINUE 1101 IF(NBB.LT.250)THEN 1102 PB=XP0+(XP1- XP01/2. 1103 RAT=DABS(XPO -XP1)/DABS(XPO) 1104 CALL CICZ(ZB.ZS.UB,S,PB,CPB,0) 1105 FPB=CPB'LAT(YC,SV,UB,PB)-CC 1106 IF(FPB:GT.CH)FPB=CH 1107 IFCFPB.EQ.O. .OR.RAT.LT.TOLX)THEN 1108 XCM(J)=PB 1109 C 1110 C * Found XCM(J) 1111 C 1112 GOTO 402 1113 ENDIF 1114 NBB=NBB+1 1115 PR0D=CQ0'FPB 1116 IF(PROD.GT.O .)THEN 1117 XPO=PB 11.18 CQO=FPB-1119 ENDIF 1120 IF(PROD.LT.O .)THEN 1121 XP1=PB 1122 CQ1=FPB 1123 ENDIF 1124 GOTO 452 1125 ENDIF 1126 C 1127 C * Iteration begins (secant method) 1128 C 1129 401 CONTINUE 1130 CALL CICZ(ZB,ZS,UB,S,XP1,CQC,0) 1131 C01=CQC*LAT(YC,SV,UB,XP1)-CC 1 132 IFIN.LE.50)THEN 1 133 CCC=CQ1-CQO 1134 IF(CCC.EQ.O. ) THEN 1135 WRITE(3. ') 'CONC(I) : DIVIDE BY 1136 WRITE(3.')'XP1 : '.XP1 1137 ENDIF 1 138 P=XP1 -CQ1*(XP1-XPO)/(CCC) 1139 DXP=DABS(P-XP1) 1 140 IF(DXP.LT.TOLX)THEN 1 141 XCM(J)=P 1142 GOTO 402 1143 ENDIF 1 144 C 1145 C * Update everything 1 146 C 1147 N=N+1 1148 XP0=XP1 1 149 CQ0=CQ1 1150 IF(P.LT.XBMIN)P=XBMIN - 268 -1151 C IF(P.LT.XMM)P=XMM 1152 XP1=P 1153 GOTO 401 1154 ENDIF 1155 WRITEO, *) ' SECANT SEARCH IN CONCU) FAILED AFTER 50 STEPS' 1156 WRITEO. ' ) 'DIFF =' .DXP 1157 402 CONTINUE 1158 500 CONTINUE 1159 ENDIF 1160 C 1161 T0LY=1.E-2 1162 IF(IFLAG.EQ.2.AND.CC.LE.CMX)THEN 1163 IF(CC.EQ.CMX)THEN 1164 YC=0. 1165 GOTO 412 1166 ENDIF 1167 C 1168 C * Search for YC 1169 C 1170 N=2 1171 YP0=0. 1172 C 1173 CALL CICZ(2B.ZS.UB.S,XC,CIC,0) 1174 CH=CMX-CC 1175 K=0 1176 C 1177 C * Search the interval where CQ changes sign, by stepping forward 1178 C 1179 DO 501 1=1,200 1180 CQO=CIC"LAT(YPO,SV,UB,XC)-CC 1181 IF(CQO.GT.CH)CQO=CH 1 182 IF(CQO.LE.0.)THEN 1183 YC=YPO 1184 C 1185 GOTO 412 1186 ENDIF 1 187 c 1188 YP1=YP0+50. 1189 CQ1=CIC'LAT(YP1,SV,UB.XC)-CC 1190 IF(CQ1.GT.CH)CQ1=CH 1 191 CQP=CQ0*CQ1 1192 IF(CQP.LT.O.)THEN 1193 c 1 194 c ' FOUND THE INTERVAL. STARTING BISECTION NOW. 1195 c 1196 GOTO 421 1197 ENDIF 1198 IF(I.EQ.200)THEN 1199 WRITEO,')'C0NC(2) STEPPED 200 TIMES, BUT NO BANANAS' 1200 YC=2.E+5 1201 GOTO 999 1202 ENDIF 1203 IF(CQ1.EQ.CQO) THEN 1204 K=K+1 1205 IF(K.GT.IO) THEN 1206 WRITEO,')'C0NC(2) FAILED TO FIND YC 1207 WRITEO, 100) YPO, CQO, YP1 , CQO 1208 100 FORMAT('YPO/CQO = '.2F10.5,' ; YP1/CQ1 = '.2F10.5) - 269 -1209 YC=2.E+5 1210 WRITEO,*)'C0NC(2): FORWARDSTEPPING UNSUCCESSFUL' 1211 GOTO 999 1212 ENDIF 1213 ENDIF 1214 C 1215 C ' Update everything 1216 C 1217 CQ0=CQ1 1218 YP0=YP1 1219 501 CONTINUE 1220 C 1221 C * Iteration begins (bisection) 1222 C 1223 421 CONTINUE 1224 NBB=1 1225 422 CONTINUE 1226 IF(NBB.LT.50)THEN 1227 YPP=YP0 1228 IF(YPO.EQ.O.)YPP=YP1 1229 PB=YP0+(YP1-YP0)/2. 1230 RAT=DABS(YP0-YP1)/DABS(YPP) 1231 FPB=CIC'LAT(PB,SV,UB,XC)-CC 1232 IF(FPB.GT.CH)FPB=CH 1233 IF(FPB.EQ.O..OR.RAT.LT.TOLY)THEN 1234 YC=PB 1235 C 1236 C * Found YC after bisection 1237 C 1238 GOTO 412 1239 ENDIF 1240 NBB=NBB+1 1241 PR0D=CQ0*FPB 1242 IF(PROD.GT.0.)THEN 1243 YPO=PB 1244 CQ0=FPB 1245 ENDIF 1246 IF(PROD.LT.O.)THEN 1247 YP1=PB 1248 CQ1=FPB 1249 ENDIF 1250 GOTO 422 1251 ENDIF 1252 C 1253 C • 20 Bisections did not do it . Try SECANT now. 1254 C 1255 411 CONTINUE 1256 C 1257 C * Iteration begins (secant) 1258 C 1259 IF(N.LE.50)THEN 1260 0C=CQ1-CQ0 1261 CPR=CQTCQO 1262 IF(DC.EQ.O..AND.CPR.GT.0.) THEN 1263 IF(CQ1.LT.O.)YP1=YP1 -100. 1264 IF(CQ1.GT.O.)YP1=YP1+100. 1265 N=N+1 1266 IF (N.GT.50)THEN - 270 -1267 WRITE(3.*)'C0NC(2) ERROR : THE CURVE IS FLAT HERE' 1268 YC=1.E+5 1269 GOTO 999 1270 ENDIF 1271 GOTO 411 1272 ENDIF 1273 P=YP1-CQ1*(YP1-YP0)/DC 1274 IF(P.LT.O.)P=YP1+10. 1275 DYP=DABS(P-YP1)/DABS(P) 1276 IF(DYP.LT.TOLY)THEN 1277 YC=P 1278 GOTO 412 1279 ENDIF 1280 C 1281 C * Update everything 1282 C 1283 N=N+1 1284 YP0=YP1 1285 CQ0=CQ1 1286 YP1=P 1287 CQ1=CIC*LAT(YP1,SV,UB,XC) 1288 IFCCQ1.GT.CH)CQ1=CH 1289 GOTO 411 1290 ENDIF 1291 WRITE(3,')'SECANT SEARCH IN CONC2) FAILED AFTER 100 STEPS' 1292 WRITE(3,*)'DIFF =',DYP 1293 412 CONTINUE 1294 ENDIF 1295 IF(CC.GT.CMX)THEN 1296 WRITE(3.')'THE SPECIFIED CONC. IS GREATER THAN CMAX' 1297 WRITE(3,*)'BY ',CR*100.,' %' 1298 ENDIF 1299 999 CONTINUE 1300 IFCCC.LT.O.)THEN 1301 WRITE(3,*)'CONC',IFLAG,' CC= ',CC 1302 ENDIF 1303 C 1304 RETURN 1305 END 1306 1307 SUBROUTINE ADDUP(N.M,CMX,XMX.CTOT,CFRAC) 1308 C 1309 C ft This subroutine defines a number of effect levels as %-fractions 1310 C • of the maximum. It then integrates the 'volume' under the 1311 C * effect level surface bounded by the previously specified iso-1312 C ft pleths. The integral over the whole surface (down to zero effect 1313 C ft level) is approximated by an extrapolation to zero by preserving 1314 C ft the curvature of the last three nodes. 1315 C ft The integrated bands are then translated into fractions of the 1316 C ft total . A cubic natural spline interpolation is used to map round 1317 C ft values integrated effect to a effect level. 1318 C • The results are carried over to the main routine in array CFRAC. 1319 C 1320 IMPLICIT REAL*8 (A-H.L.O-Z) 1321 REAL*8 CFRAC(10,4),XCM(2) 1322 C 1323 C ft Set strat i f icat ion limits in effect level dimension 1324 c ft CFRAC(i,1) contains fractions of CMX in 10% increments ( i .e . - 271 -1325 C * the lower limit of a band; e.g. 70 -> (70,80]) 1326 C * CFRAC(i,2) contains the associated effect level values 1327 C 1328 FRAC=10. 1329 DO 501 1=9,1,-1 1330 CFRAC(I.1)=FRAC 1331 CFRAC(I,2)=CMX * FRAC * 0.01 1332 C 1333 C * Update everything 1334 C 1335 FRAC=(11 -1)* 10 1336 501 CONTINUE 1337 C 1338 C=CMX 1339 D=CFRAC(1,2) 1340 YC=0. 1341 C 1342 CALL CONC(CMX,XMX,XC,XCM,YC,D,1) 1343 C 1344 AX=XCM(1) 1345 BX=XCM(2) 1346 CA=BX 1347 CB=AX 1348 C 1349 C " Start the loop to integrate the bands 1350 C 1351 DO 502 1=1,8 1352 IP=I+ 1 1353 C 1354 CALL DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT) 1355 C 1356 IF(XINT.LE.O.)THEN 1357 WRITE(3,*)'SOMETHING IN DU8SIM WENT VERY WRONG' 1358 WRITE(3,*)'XINT = '.XINT,' 1= ',1 1359 IFtXINT.EQ.O.)THEN 1360 XINT=1. 1361 GOTO 756 1362 ENDIF 1363 C GOTO 999 1364 756 CONTINUE 1365 ENDIF 1366 CFRAC(I,3)=XINT 1367 C 1368 C * Update everything 1369 C 1370 C=D 1371 0=CFRAC(IP,2) 1372 CALL CONC(CMX,XMX,XC,XCM.YC,D,1) 1373 CA=AX 1374 CB=BX 1375 AX=XCM(1) 1376 BX=XCM(2) 1377 502 CONTINUE 1378 C 1379 CALL DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT) 1380 C 1381 IF(XINT.LE.0)THEN 1382 WRITE(3,*)'ERROR IN THE FINAL ENTRY TO OUBSIM ' - 272 -1383 GOTO 999 1384 ENDIF 1385 CFRAC(9,3)=XINT 1386 C 1387 C * Evaluate CTOT by summation of the CFRAC(i,3)'s 1388 C 1389 CTOT=0. 1390 DO 503 1=1,9 1391 CT0T=CT0T+CFRAC(I,3) 1392 503 CONTINUE 1393 C 1394 c • Extrapolate CTOT to zero effect level, by conserving the 1395 c * curvature of CTOT-growth. 1396 c 1397 CTOT=CTOT+(CFRAC(9,3)** 2/CFRAC(8,3)) 1398 WRITEO, *)'TOTAL VOLUME approx. : '.CTOT 1399 c 1400 c * Evaluate the cumulative fraction of the total volume 1401 c * contributed by each concentration band, e.g. more than 60 1 1402 c * cone, contributes XX % of the total volume under the cone. 1403 c • surface. Put results into CFRACH.4) 1404 c 1405 V0L=0. 1406 DO 504 J=1,9 1407 V0L=V0L + CFRAC( J , 3) 1408 CFRAC(J,4)=V0L/CT0T 1409 504 CONTINUE 1410 c 1411 c • Write CFRAC to comments f i l e (#3) 1412 c 1413 CC WRITEO, 101) 1414 CC101 FORMAT(/,'PRINTOUT OF CFRAC',/) 1415 CC WRITE(3,*)'cone.% cone.val. int . frac . cum 1416 CC DO 505 1=1,9 1417 CC WRITEO, 100) (CFRAC (I, J) ,J=1 ,4) 1418 CC100 FORMAT(4F15.5) 1419 CC505 CONTINUE 1420 C 1421 C CTOT and CFRAC are output of subroutine 1422 c 1423 GOTO 998 1424 999 CONTINUE 1425 CT0T=-100. 1426 998 CONTINUE 1427 C 1428 RETURN 1429 END 1430 1431 SUBROUTINE SPLYNE(SNO,N,X,Y) 1432 C 1433 C • This subroutine uses a. natural cubic spline algorithm 1434 C • to compute the interpolated Y for any given X within 1435 c • the range defined by the nodes SNO(i.j). SPLYNE satisfies 1436 c • the boundary conditions Y''(SN0(1,0))= Y''(SN0(1,N-1)) = 0. 1437 c * N is the number of nodes. 1438 c • (Ref: Burden & Faires.1985; Algorithm 3.4) 1439 c 1440 IMPLICIT REAL'8 (A-H.L.O-Z) - 273 -1441 REAL * 8 SNO(0:50,2),H(0:50),A(0:50),L(0:50),Z(0:50) 1442 REAL'8 C(0:50),B(0:50),D(0:50),M(0:50) 1443 C 1444 NN=N-1 1445 N1=NN-1 1446 DO 501 1=0,N1 1447 IP=I+1 1448 H(I)=SN0(IP,1)-SN0(I,1) 1449 501 CONTINUE 1450 C 1451 DO 502 1=1.NI 1452 IP=I+1 1453 IM=I-1 1454 A(I)=(SNO(IP,2)*H(IM)-SN0(1,2)*(SNO(IP,1)-SNO(IM,1)) 1455 # +SN0(IM,2)*H(I))*3./(H(IM)*H(I)) 1456 502 CONTINUE 1457 C 1458 C * Solve tridiagonal linear system by Crout reduction method. 1459 C 1460 L(0)=1. 1461 M(0)=0. 1462 Z(0)=0. 1463 C 1464 DO 503 1=1,N1 1465 IP=I+1 1466 IM=I-1 1467 L(I)=2.*(SNOIIP,1)-SN0(IM,1))-H(IM)'M(IM) 1468 M(I)=H(I)/L(I) 1469 Z(I)=(A(I)-H(IM)*Z(IM))/L(I) 1470 503 CONTINUE 1471 C. 1472 L(NN)=1. 1473 Z(NN)=0. 1474 C(NN)=0. 1475 C 1476 00 504 J=N1 .0,-1 1477 JP=J+1 1478 C(J)=Z(J)-M<J)"C(JP) 1479 B(J)=(SNO(JP,2)-SN0(J.2))/H(J)-H(J)*(C(JP)+2*C(J))/3. 1480 D(J)=(C(JP)-C(J))/(3*H(J)> 1481 504 CONTINUE 1482 C 1483 C * Evaluate interpolated value Y for given X 1484 C 1485 DO 505 1=0,N1 1486 IP=I+1 1487 IF(X .GE.SNO( 1,1) . AND.X.LT.SNOdP, 1) ) THEN 1488 DX=X-SNO(I.1) 1489 Y=SN0(I,2)+8(I)*0X+C(I)*DX**2+D(I),DX**3 1490 GOTO 400 1491 ENDIF 1492 505 CONTINUE 1493 C 1494 C * If node-range not found, give error message 1495 C 1496 WRITEO,')'ERROR IN SPLYNE : GIVEN X LIES OUTSIDE NODES - RANGE' 1497 400 CONTINUE 1498 C - 274 -1499 RETURN 1500 END 1501 1502 SUBROUTINE SIMPS(DOTS,A.B.NPT.F) 1503 C 1504 C • This subroutine uses Simpson's Composite Algorithm to 1505 c • approximate the integral under a curve described by 1506 c • the nodes in DOTS. A and B are the lower and upper 1507 c • integration limits (values on ordinate, DOTS has only 1508 c • abscisse-values). NPT is the number of points (must be 1509 c • an odd number, to produce an even number of subintervals) . 1510 c » F is the approximated definite integral. 1511 c • (Ref : Burden & Faires, 1985; Algorithm 4.1) 1512 c 1513 IMPLICIT REAL'8 (A-H.L.O-Z) 1514 REAL'S D0TS(110) 1515 c 1516 M2=NPT-1 1517 M=M2/2 1518 H=(B-A)/DBLE(FL0AT(M2)) 1519 c 1520 XI0=00TS(1)+D0TS(NPT) 1521 XI1=0. 1522 XI2=0. 1523 c 1524 M21=M2-1 1525 DO 500 1=1,M21 1526 IP=I+1 1527 U=DABS(DBLE(FLOAT(NINT(DBLE(FLOAT(I))/2.)))-DBLE(FLOAT(I))/2. 1528 IF(U.EQ.O.)XI2=XI2+D0TS(IP) 1529 IF(U.NE.O.)XI1=XI1+D0TS(IP) 1530 500 CONTINUE 1531 c 1532 F=H*(XI0+2. 'XI2+4. ' X I D / 3 . 1533 c 1534 c 1535 RETURN 1536 END 1537 1538 SUBROUTINE DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT) 1539 c 1540 c • This subroutine uses a double Simpson's Composite Algorithm 1541 c to approximate the integral under a surface. 1542 c • For reference see Burden & Faires, 1985. 1543 c 1544 IMPLICIT REAL'8 (A-H.L.O-Z) 1545 REAL"8 XXM(2) 1546 c 1547 c • Step 1 : 1548 c 1549 H=(BX-AX)/(2'N) 1550 c 1551 c • Step 2 : 1552 c 1553 XJ1=0. 1554 XJ2=0. 1555 XJ3=0. 1556 c - 275 -1557 C • Step 3 : 1558 C 1559 00 501 I=0,2*N 1560 C 1561 C • (composite Simpson's method for fixed x) 1562 C 1563 X=AX+I*H 1564 CC=C 1565 C 1566 CALL C0NC(CMX,XMX,X,XXM,DX,D,2) 1567 IF(DX.EQ.2.E+5)G0T0 999 1568 IF(X.LT.CA.OR.X.GT.CB)THEN 1569 CX=0. 1570 C 1571 CALL CONC(CMX,XMX,X,XXM,CX.CC.O) 1572 C 1573 GOTO 402 1574 ENDIF 1575 C 1576 CALL C0NC(CMX,XMX,X,XXM,CX,CC,2) 1577 IF(CX.EQ.2.E+5)GOTO 999 1578 402 CONTINUE 1579 HX=(DX-CX)/(2*M) 1580 XK1=D+CC 1581 XK2=0. 1582 XK3=0. 1583 c 1584 DO 502 J=1,2*M-1 1585 Y=CX+J*HX 1586 c 1587 CALL CONCfCMX.XMX.X.XXM.Y.Z.O) 1588 UJ=DABS(DBLE(FLOAT(NINT(DBLE(FLOAT (J))/2.)))-DBLE(FLOAT(J))/2.) 1589 IF(UJ.EQ.O.)XK2=XK2+Z 1590 IF(UJ.NE.O.)XK3=XK3+Z 1591 XL=(XK1+2*XK2+4*XK3)*HX/3. 1592 502 CONTINUE 1593 c 1594 NN=N'2 1595 IF (I. EQ . 0 . OR . I . EQ . NN) THEN 1596 XJ1=XJ1+XL 1597 GOTO 401 1598 ENDIF 1599 UI=DABS(DBLE(FLOAT(NINT(DBLE(FLOATtI))/2.)))-DBLE(FLOAT(I))/2.) 1600 IF(UI.EQ.O.)XJ2=XJ2+XL 1601 IFtUI.NE.O.)XJ3=XJ3+XL 1602 401 CONTINUE 1603 501 CONTINUE 1604 c 1605 XINT=(XJ1+2*XJ2+4*XJ3)'H/3. 1606 GOTO 998 1607 c 1608 999 CONTINUE 1609 XINT=-100. 1610 998 CONTINUE 1611 c 1612 c 1613 RETURN 1614 END - 276 -1615 1616 SUBROUTINE MONET(PLETH,AUX,PROFX,PROFY) 1617 C 1618 C a This subroutine uses the OISSPLA language to present the 1619 C • results of the SANCTUM-mode1 graphically in a 3-D plot. 1620 C * Once the object f i l e is created, it has to be loaded and 1621 C • run together with the 'DISSPLA l ibrary. 1622 C 1623 C • Dimension data arrays (level 0) 1624 C 1625 REAL*8 PLETH(9,2.51),AUX(11),PR0FX(101.2) 1626 REAL'S PROFY(101,2) 1627 REAL*4 PRYX(101),PRYY(101),PRXX(101),PRXY(101) 1628 REAL*4 PLEX(201),PLEY(201),XSYM(5),YSYM(5) 1629 REAL*4 AX(2),AY(2),CX(2),DX(2),DY(2) 1630 COMMON /PAKRAY/ IPKRAYUOO) 1631 C 1632 c • In i t ia l i ze a device (rise up to level 1) 1633 c 1634 CALL DSPDEV(1 PLOT') 1635 c 1636 c * Set page size (XPAGE=11. , YPAGE=8.5) in inches (level 1) 1637 c 1638 XPAGE=11. 1639 YPAGE=8.5 1640 c 1641 CALL PAGE(XPAGE,YPAGE) 1642 CALL HWROT('AUTO') 1643 CALL HWSHD 1644 CALL SHDCHRf30.,1,0.01,1) 1645 c 1646 c • In i t ia l i ze coordinates for physical origin 1647 c 1648 XPHY = 0. 1649 YPHY = 0. 1650 c 1651 c • In i t ia l i ze dimensions for subplot area 1652 c 1653 XAREA =11. 1654 YAREA = 8. 1655 CALL PHYSOR(XPHY.YPHY) 1656 CALL AREA2D(XPAGE,YPAGE) 1657 c 1658 c * * ** Draw the caption (level 1) 1659 CALL HEIGHT!.15) 1660 CALL SWISSL 1661 JD=INT(AUX(1)) 1662 JH=INT(AUX(2)) 1663 JM=INT(AUX(3) ) 1664 US=AUX(4) 1665 UDIR=AUX(5) 1666 2I=AUX(6) 1667 XI=AUX(7) 1668 UST=AUX(8) 1669 ZS=AUX(9) 1670 FL=AUX(10) 1671 SV=AUX(11) 1672 XMES=4. - 277 -1673 YMES=1.1 1674 CALL MESSAG('SAM Calculations for Jul.Day. $',100, 1675 1 XMES.YMES) 1676 CALL INTNOCJD,'ABUT'.'ABUT') 1677 CALL MESSAG(' at $', 100,'ABUT','ABUT') 1678 CALL INTN0(JH,'ABUT','ABUT') 1679 CALL MESSAG( 1:$',100,'ABUT','ABUT') 1680 IF(JM.EQ.0)CALL MESSAG('00 L . A . T . at $',100,'ABUT', 'ABUT' ) 1681 IF(JM.NE.0)THEN 1682 CALL INTN0(JM,'ABUT','ABUT') 1683 CALL MESSAG(' L . A . T . at $'.100,'ABUT','ABUT') 1684 ENDIF 1685 IF(FL.EQ.O.) CALL MESSAG('SUNSETS',100,'ABUT','ABUT') 1686 IF(FL.EQ.1.) CALL MESSAG('MOBILES',100,'ABUT','ABUT') 1687 IF(FL.EQ.2.) CALL MESSAGCCABAUWS',100,'ABUT','ABUT') 1688 C 1689 CALL HEIGHT!. .12) 1690 YMES=YMES-.25 1691 CALL MESSAG('Wind at sensor level ( sp . /d ir . ) : $',100,XMES,YMES) 1692 CALL REALN0(US.1,'ABUT','ABUT') 1693 CALL MESSAGC / $',100,'ABUT','ABUT') 1694 CALL REALN0(UDIR,1,'ABUT','ABUT') 1695 IF(ZI.GT.O.)THEN 1696 CALL MESSAGC ; Zi : $',100,'ABUT','ABUT') 1697 CALL REALN0(ZI, 1 ,'ABUT' ,'ABUT') 1698 CALL MESSAG!' ; SV : $'.100,'ABUT','ABUT') 1699 CALL REALNO(SV,3,'ABUT'.'ABUT') 1700 ELSE 1701 CALL MESSAGC ; Zi : n/a ; SV : $',100,'ABUT','ABUT' ) 1702 CALL REALN0(SV,3,'ABUT','ABUT') 1703 ENDIF 1704 YMES=YMES- . 18 1705 CALL MESSAG('Stability (Zs/L) : $',100,XMES,YMES) 1706 CALL REALN0(XI.3, 'ABUT', 'ABUT' ) 1707 CALL MESSAGC ; U* : $',100,'ABUT','ABUT' ) 1708 CALL REALNOCUST,3.'ABUT','ABUT') 1709 CALL MESSAG(' ; Sensor height : $',100,'ABUT', 'ABUT') 1710 CALL REALNO(ZS,1,'ABUT','ABUT') 1711 YMES=YMES-.25 1712 CALL MESSAGCAll data are in SI un i ts$ ' , 100 , XMES . YMES) 1713 C 1714 CALL ENDGR(O) 1715 C 1716 C ' Increment physical origin and subplot area to f i t caption (1. 1) 1717 C 1718 YPHY = YPHY+1 .5 1719 YAREA = YAREA- 1.5 1720 CALL PHYSOR(XPHY,YPHY) 1721 C 1722 C " Define subplot area (rise up to level 2) 1723 C 1724 CALL AREA2D(XAREA,YAREA) 1725 C 1726 C 1727 C *'** Draw the t i t l e 1728 CALL SWISSB 1729 CALL ALNMES(.5,.5) 1730 CALL HEIGHT(.20) - 278 -1731 CALL MESSAG 1732 1 ('SOURCE AREA MODEL - RESULTSS',100,XAREA/2..YAREA+.2) 1733 CALL RESET('ALNMES') 1734 CALL RESET('HEIGHT') 1735 C 1736 C * Define the 3D plot volume 1737 C 1738 XV0L=12. 1739 YVOL=7.0 1740 ZV0L=7.0 1741 C 1742 CALL V0LM3D(XV0L,YV0L,ZV0L) 1743 C 1744 C * Define viewpoint of 3D box 1745 C 1746 CALL VUANGL(-60.,30.,30.) 1747 C 1748 c * Define user axis system (window) (rise to level 3) 1749 c 1750 CALL GRAF3D(0.,XVOL,XVOL,0.,YVOL,YVOL,0.,ZVOL,ZVOL) 1751 c 1752 c **** Define the Y-Z GRIFITI plane (jump back to level 1) 1753 c 1754 CALL GRFITKO. ,0. ,0. ,0. ,YV0L,0. ,0. .YVOL,ZVOL) 1755 c 1756 c * Set subplot dimensions (viewport) (level 1, rise to 2) 1757 c 1758 CALL AREA2D(YVOL,ZVOL) 1759 CALL GRAF(0.,YVOL.YVOL.0..ZVOL,ZVOL) 1760 CALL FRAME 1761 c 1762 c '*** Prepare to draw a message 1763 c 1764 CALL GRACE!10.) 1765 CALL HEIGHT(.22) 1766 CALL COMPLX 1767 CALL ALNSTY(.5.-5) 1768 c 1769 c • • * • Plot a message 1770 c 1771 MAXLIN = LINEST(IPKRAY.400,80) 1772 CALL LINES('Cross-Wind Profile of Effect Leve 1$',IPKRAY, 1) 1773 CALL LINES('at the max. eff. downwind distances',IPKRAY,2) 1774 NLINES = 2 1775 XMSG =3.25 1776 YMSG =6.25 1777 CALL STORY(IPKRAY,NLINES,XMSG,YMSG) 1778 CALL RESET('HEIGHT') 1779 CALL RESET('ALNSTY') 1780 c 1781 c • End the Y-Z plane (level 2, rise to 3) 1782 c ( i f you got this far, you might as well read to the end) 1783 c 1784 CALL END3GR(0.) 1785 c 1786 c * Prepare to draw the curve : set up the curve plane (back to 1787 c 1788 CALL GRFITI(0.,0. ,0. ,0. ,YVOL,0.,0. ,YVOL,ZVOL) - 279 -1789 XPHY2=XPHY+.25 1790 YPHY2=YPHY+.70 1791 CALL PHYS0R(XPHY2.YPHY2) 1792 XLEN=YV0L-2. 1793 YLEN=ZVOL-2.5 1794 CALL AREA2D(XLEN,YLEN) 1795 CALL CROSS 1796 C 1797 C * Set axis labels. 1798 C 1799 CALL DUPLX 1800 CALL YAXANGOO. ) 1801 CALL YREVTK 1802 CALL HEIGHTC.22) 1803 C 1804 C * Split up PR0FY(101,2) into two ID-arrays 1805 C 1806 DO 501 1=1,101 1807 PRYX(I)=PR0FY(I,1) 1808 PRYY(I)=PROFY(I,2) 1809 501 CONTINUE 1810 C 1811 C * Set the min. max and step for PRYX and PRYY 1812 C 1813 1814 CALL RNDLINfPRYX,101.XLEN.XMIN,XSTP,XMAX) 1815 CALL RNDLIN(PRYY,101,YLEN,YMIN,YSTP,YMAX) 1816 IF(XMAX.LE.200.)THEN 1817 XSTP=50. 1818 XMIN=-200. 1819 XMAX=200. 1820 ENDIF 1821 YPMIN=XMIN 1822 YPMAX=XMAX 1823 YPLEN=XLEN 1824 YSTP=.2 1825 CALL XINTAX 1826 C 1827 CALL XNAME('Cross-wind Distance (m)$',100) 1828 CALL YNAME(' $',100) 1829 C 1830 C * Define user axis system (window) (level 2, rise to 3 1831. C 1832 CALL GRAF(XMIN,'SCALE',XMAX,YMIN,YSTP,YMAX) 1833 CALL RASPLN(2.) 1834 CALL CURVE(PRYX,PRYY,101,0.) 1835 C 1836 C * End the current plane 1837 C 1838 CALL END3GR(0) 1839 C 1840 C ***** Begin X/Z-plane 1841 C 1842 CALL PHYSOR(XPHY.YPHY) 1843 CALL GRFITI(0..YVOL.O.,XVOL.YVOL,0.,XVOL,YVOL,ZVOL) 1844 CALL AREA2D(XVOL,ZVOL) 1845 CALL FRAME 1846 C - 280 -1847 C * Put in a message 1848 C 1849 CALL GRACE(10.) 1850 CALL HEIGHT!.22) 1851 CALL COMPLX 1852 CALL ALNSTY!.5,.5) 1853 C 1854 C * Plot a message 1855 C 1856 MAXLIN=LINEST(IPKRAY,400,80) 1857 CALL LINES!'Along-Wind Centerline Profile of Effect Levels', 1858 1 IPKRAY,1) 1859 NLINES=1 1860 XMSG=5.5 1861 YMSG=6.25 1862 CALL LSTORY!IPKRAY,NLINES,XMSG,YMSG) 1863 C 1864 C ' End this plane 1865 C 1866 CALL EN03GR(0) 1867 C 1868 C ' Prepare to draw the curve 1869 C 1870 CALL GRFITI(0.,YVOL,0.,XVOL,YVOL,0.,XVOL,YVOL,ZVOL) 1871 CALL OREL!.5,.25) 1872 XLEN=XV0L-1.5 1873 YLEN=ZV0L-2. 1874 CALL AREA2D(XLEN,YLEN) 1875 CALL CROSS 1876 CALL DUPLX 1877 CALL YAXANGOO. ) 1878 CALL HEIGHT!.22) 1879 C 1880 C * Split up PROFX!101,2) into two linear arrays 1881 C 1882 DO 502 1=1,101 1883 PRXX(I)=PR0FX(I,1) 1884 PRXY(I)=PR0FX(I,2) 1885 502 CONTINUE 1886 CALL RNDLIN!PRXX,101,XLEN,XMIN,XSTP,XMAX) 1887 CALL RNDLIN!PRXY,101,YLEN,YMIN,YSTP,YMAX) 1888 IF!XMAX.LE.2000.)THEN 1889 XSTP=250. 1890 XMIN=0. 1891 XMAX=2000. 1892 ENDIF 1893 XPMIN=XMIN 1894 XPMAX=XMAX 1895 XPLEN=XLEN 1896 YSTP=.2 1897 CALL XINTAX 1898 C 1899 C * Label the axes 1900 C 1901 CALL XNAME!'Upwind Distance (m)$',100) 1902 CALL YNAME('Re 1. Effect Level$',100) 1903 C 1904 C * Define user axis system (level 2, rise to 3) - 281 -1905 C 1906 CALL GRAF(XMIN,'SCALE',XMAX,YMIN,YSTP,YMAX) 1907 CALL RASPLN(2.) 1908 CALL CURVE(PRXX,PRXY,101,0.) 1909 C 1910 C * End this plane 1911 C 1912 CALL RESET('ALL') 1913 CALL END3GR(0) 1914 C 1915 C •*** Begin floor 1916 C 1917 CALL PHYSOR(XPHY.YPHY) 1918 CALL GRFITI(0.,0. ,0. ,XV0L,0.,0. ,XVOL,YVOL,0.) 1919 CALL AREA2D(XVOL.YVOL) 1920 CALL FRAME 1921 G 1922 C * Put in messages 1923 C 1924 CALL GRACE(10.) 1925 CALL HEIGHT!.25) 1926 CALL COMPLX 1927 CALL ALNSTY!.5..5) 1928 MAXLIN=LINEST(IPKRAY.400.80) 1929 CALL LINES! 1930 1 'Source Region Outlines with Various Effect- Integra 1s$', 1931 1 IPKRAY.1) 1932 NLINES=1 1933 XMSG=XV0L/2. 1934 YMSG=YVOL-.35 1935 CALL LSTORY(IPKRAY,NLINES.XMSG.YMSG) 1936 C 1937 C * Next Message 1938 C 1939 CALL HEIGHT I.22) 1940 CALL SIMPLX 1941 CALL ALNSTY!.5,.5) 1942 C 1943 MAXLIN=LINEST(IPKRAY,400.80) 1944 CALL LINES!'Isopleths are numbered 1 - 9 from centers' , 1945 1 IPKRAY. 1) 1946 CALL LINES! 1947 1 '(see inset and separate table for dimensions and area)$' 1948 1 IPKRAY,2) 1949 NLINES=2 1950 XMSG=XVOL/2. 1951 YMSG=.5 1952 CALL LSTORY(IPKRAY,NLINES,XMSG,YMSG) 1953 C 1954 C * End this plane 1955 c 1956 CALL END3GR(0) 1957 c 1958 c • Prepare to draw the curves 1959 c 1960 CALL GRFITI(0..0..0. ,XV0L,0.,0.,XVOL.YVOL,0.) 1961 CALL OREL(.70,.15) 1962 XLEN=XPLEN - 282 -1963 YLEN=YPLEN 1964 CALL AREA20(XLEN,YLEN) 1965 CALL CROSS 1966 CALL DUPLX 1967 CALL YAXANGOO. ) 1968 CALL XREVTK 1969 CALL HEIGHT(.22) 1970 CALL RASPLN(2.) 1971 C 1972 C ' Set up data arrays 1973 C 1974 DO 503 1=9,1,-1 1975 DO 504 J=1,51 1976 PLEX(J)=PLETH(I,1,J) 1977 PLEY(J)=PLETH(I,2,J) 1978 504 CONTINUE 1979 IF(I.EQ.9)THEN 1980 C 1981 C * Label the axes 1982 C 1983 CALL XINTAX 1984 CALL YINTAX 1985 CALL XNAME(' $',100) 1986 CALL YNAMEC $',100) 1987 CALL GRAFUPMIN, 'SCALE' , XPMAX , YPMIN , 'SCALE' .YPMAX) 1988 ENDIF 1989 CALL CURVE(PLEX,PLEY,201,0. ) 1990 503 CONTINUE 1991 C 1992 CALL END3GR(0) 1993 C 1994 CALL ENDGR(O) 1995 C 1996 C ***• Begin inset 1997 C 1998 CALL PHYSOR(XPHY,YPHY) 1999 CALL OREL(-.3,-1.5) 2000 C 2001 C * set inset dimensions and frame it 2002 C 2003 XINS=3. 2004 YINS=2. 2005 CALL AREA2D(XINS,YINS) 2006 CALL FRAME 2007 C 2008 C * Put in a t i t l e for the inset 2009 C 2010 CALL SWISSM 2011 CALL HEIGHT(.15) 2012 CALL ALNMESf.5,.5) 2013 CALL MESSAG 2014 1 ('Dimensions of IsoplethsS',100,XINS/2,YINS-.25) 2015 CALL ENDGR(0.) 2016 C 2017 C * Prepare to draw the symbol 2018 C 2019 CALL OREL(0.5,0.3) 2020 XLEN=XINS-.9 - 283 -2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 YLEN=YINS-1. CALL AREA2D(XLEN,YLEN) CALL CROSS Set up data-points DATA XSYM/125.,300..550.,300.,125./ DATA YSYM/O.,40. ,0 . , -40. ,0 . / DATA AX/O.,125./ DATA AY/0 . , 0 . / DATA CX/300.,550./ DATA DX/300.,300./ DATA DY/O.,40./ 5,XLEN,XMIN,XSTP,XMAX) 5,YLEN,YMIN,YSTP,YMAX) CALL RNDLIN(XSYM CALL RNDLIN(YSYM CALL XNONUM CALL YNONUM CALL XNAME(1 $',100) CALL YNAME(' $',100) CALL RASPLN( 1 .) CALL GRAF(0.,600.,600.,YMIN,YMAX,YMAX ) CALL CURVE(XSYM,YSYM,5,0.) Put in the dimension indicators etc. CALL THKCRV(0.03) CALL CURVE(AX,AY,2,0) CALL CURVE(CX,AY,2,0) CALL CURVE(DX.DY,2,0) CALL HEIGHT(0.1) CALL COMPLX CALL RLMESS('a$' , 100,75. CALL RLMESSf'b$',100,212. CALL RLMESSf'c$1 10. ) , 10. ) 100,425.,10.) CALL RLMESS('d$',100,320.,30.) CALL ENDGR(O) CALL ENDPL(O) END THE PLOT * Clean up the DISSPLA parameters and leave CALL RESET('ALL ' ) CALL DONEPL RETURN END - 284 -APPENDIX D : Data and Results Summary (1) : Culloden Site The tower was situated in front of the s ixth house south of 37 t h Avenue on the eastern side of Culloden Street (see Figures 9.1 and 9.2). This was the f i r s t s i te occupied. Operations started on July 31, 1986 (JD 212) at 5:35 LAT and ended on August 2 (JD 214) at 19:35 LAT. Due to the consistently very small absolute Q values at night i t was decided that the H measurements could be interrupted during the night at s i tes subsequently occupied. Weather conditions were generally sunny and warm, with light but sometimes gusty winds in late afternoon (see data summary in Appendix D). (2) : Argyle Site Figure 9.3 shows the mobile tower on the east side of Argyle rd Street, just south of 53 Avenue. This s i te was d irec t ly adjacent to the south-west corner of Gordon Park (see Figure 9.1). David Thompson Secondary School with a building height of about 20 m was approximately 50 m to the east of the tower. Data co l lec t ion started at 5:05 LAT on August 9 (JD 221) and ended at 19:50 LAT on August 11 (JD 223). At night, the measurements were interrupted as soon as the Q^-values became negative, commonly between 19:30 LAT and 20:00 LAT, and were restarted in the morning between 4:30 LAT and 5:30 LAT (see data summary in Appendix D). - 285 -Weather conditions were more variable than at the Culloden site with some wind from southerly directions and changing cloud cover. (3) : Waverley Site Measurements started at 12:10 LAT on August 18 (JD 230) and ended at 19:35 LAT on August 20 (JD 232). Figure 9.4 gives a visual impression of the site, which was located in the north-west corner of the U-shaped Waverley Street (see Figure 9.1). The f i r s t and the third day were very similar, very hot with clear skies or sparse high clouds and light winds. The second day (JD 231) was generally more windy (W to NW directions) with variable, fast moving medium-high clouds. In the afternoon of JD 231 (ca. 15:00 LAT) there were some instrument alignment problems and one element of the tower started to sag slightly. Shut-down during the nights were similar to the Argyle site (see data summary in Appendix D). (4) : Memorial East Site This s i t e was located just east of Memorial Park, on Ross rd Street just north of 43 Avenue (see Figures 9.1 and 9.5). To the west and north-west of that location the character of the park is f a i r l y open, with a baseball-diamond and other playgrounds. In the SW-direction, however, a group of high park-trees (up to about 40 m high) is located at a distance of 100-150 m from the mobile tower site (see Figure 9.6). Measurements started on August 24 (JD 236) at 10:05 LAT in light winds from southerly directions. The park was not irrigated on - 286 -this f i r s t day. The second day (JD 237) was very hot and sunny ,the park was irrigated and a slight sea breeze was setting in towards midday. Unfortunately there were data logging problems for the data at the main tower, so that source area calculations could not be performed for most of the day (see below). Similar problems occurred on JD 238, on which data collection was terminated at 13:45 LAT. (5) : Memorial West Site Figure 9.7 shows this s i t e looking north-east. On the western side of Prince Albert Street (see Figure 9.1) the terrain is characterized by high density residential housing as in Figure 9.8. Measurements started on August 31 (JD 243) at 6:00 LAT in cloudy and cool conditions. Even though the weather cleared in the afternoon, the following day (JD 244) was again characterized by a complete low stratus cloud cover. Due to the onset of light rain, the measurements were interrupted early in the morning. Data collection resumed on JD 246 (September 3) at 6:30 LAT in calm and generally sunny conditions, however, the data logging problems at the Sunset site continued until 16:30 LAT of that day, so that only the last few data-points could be used for SAM-runs (see data summary below). The measurement period ended on September 4 (JD 247) at 14:00 LAT again in and sunny conditions and only light winds. - 287 -Data and Results Summary The nomenclature of the various data columns i s as follows JDD : Julian day of 1986 HH: MM : Local apparent time (LAT) of the measurement. refers to hourly averages ending at this time. TT : Mean air-temperature in [K] UB : Mean wind-speed at the sensor level in [m/s] W-DIR : Mean wind-direction i n degrees (90 = E) SV : Equivalent to <r in [m/s] U* : Equivalent to u # in [m/s] L : Monin-Obukhov length in [m] QH-SUN : Sensible heat flux at the Sunset site in [W/m2] QH-MOB : Sensible heat flux at the mobile sites in [W/m2] (see Section 9.2) QH-DIFF : Equivalent to Qj^jpy. [nondimensional ] AR : Area of 0.9-isopleth in [1000 m2] A - XM : Dimensions a - xm of 0.9-isopleth in [m]. JOO HH: MM TT UB W-D1R SV U* 212 6: 30 288 .0 1 .46 122. 0 .63 0 23 2)2 7: 0 288 .2 1. .49 141. 0 .42 0. 24 212 7: 30 288 .6 1 .67 151. 0 58 0. .27 2(2 B : 0 289. . 1 1 .85 161. 0 .77 0 .30 212 8: 30 289 .6 1 .74 171. 0 .76 0 28 212 9: 0 290. . 1 1 .62 182. 0 .75 0. .26 212 9: 30 290 6 1 .72 192. 0 78 0. .28 212 10: 0 291 .2 1 .82 202. 0 .81 0 .31 212 10: 30 291 .7 2 oa 212. 1. .00 0. 34 212 11: 0 292. . 2 2 .35 221 . 1. .20 0. 36 2 12 11: 30 292. 6 2 .28 238. 1 .25 0. 36 212 12: 0 293. . 1 2 .21 256. 1. .30 0. .37 212 12: 30 294. . 1 2 . 16 246. 1 .40 0 36 212 13: 0 295 0 2 . 1 1 236. 1 .51 0, .36 212 13: 30 295. 0 2. 67 247. 1. .49 0. 43 212 14 : 0 295. 0 3. .22 259. 1 .31 0. .46 212 14 : 30 295. .3 3 .36 255. 1. .20 0. .48 212 15: 0 295. 6 3 .49 252. 1. 08 0. 50 212 15:30 295. 6 3. .47 252. 1. .03 0. 49 212 IG: 0 295. 6 3 .46 252. 0. 97 0. 48 212 16: 30 295. 3 3 .41 253. 1 .03 0. 48 212 17: 0 294. 9 3 .37 254. 1. . 10 0. 47 212 17: 30 295. 0 3. .20 264. 1. 00 0. 45 212 18: 0 295. . 1 3 .04 275. 0. 91 0. 42 212 18: 30 294. .8 2 .76 277. o, .90 0. .38 212 19: 0 294. .4 2 .48 279. 0. 87 0. 34 212 19: 30 293. .8 2. . 14 283. 1. .04 0. 30 212 20: 0 293. 1 1 81 287. 1. .11 0. 25 213 7: 0 290. 5 1. .22 206. 0. 35 0. 18 213 7: 30 291, . 1 1. .50 208. 0, .50 0. .24 213 8: 0 291. 6 1 .78 211. 0 67 0. .29 213 8: 30 292. 2 2 .20 229. 0. 83 0. 33 213 9: 0 292. 8 2. 61 246. 1. .00 0. 38 213 9: 30 293. 4 2 .54 244. 1 06 0. 38 213 10: 0 293. .9 2. .46 241 . 1. . 11 0. 39 213 10: 30 294. 4 2 85 260. 1. 53 0. 43 213 11: 0 295. 0 3. 24 279. 2. .02 0. 47 213 11: 30 295. 3 3 .37 263. 1. .57 0 49 213 12: 0 295. 6 3 .50 247. 1 09 0. SO 213 12: 30 296. 3 3. 31 245. 1. 10 0. 48 213 13: 0 296. 9 3. 12 244 . 1. . 10 0. 46 213 13: 30 297. 1 3. .37 246. 1. . 12 0. 50 213 14 : 0 297 . .3 3 .63 249. 1. . 12 0. .52 213 14: 30 297. 6 3. .51 244. 1. 08 0. 51 213 15: 0 298. 0 3. .39 239. 1. ,05 0. SO 213 15: 30 298. 3 3. 22 245. 1. 01 0. 47 213 16: 0 298. .5 3. 05 2S2. 0. 97 0. .44 213 16: 30 298 . 9 3. .09 264. 1 .00 0. 44 213 17: 0 299. 2 3 . 14 276. 1. .02 0. 42 213 17: 30 299. 4 3. .02 278. 1. . 19 0. 41 213 18: 0 299. 5 2. 90 279. 1, .35 0. 39 213 18: 30 298. 9 2. 47 270. 1, 10 0. 31 213 21: 30 293. 3 2. 21 142. 0. 59 0. 28 214 6: 30 291. 8 1. 33 321 . 0. 69 0. 19 214 7: 0 292. 3 1. 01 288. 0. 84 0. 16 214 7: 30 292. 4 1 . 13 293. 0. 78 0 18 214 8: 0 292 . 6 1. 25 297 . 0. 68 0. .22 214 8: 30 293. 5 1. 25 310. 0. 87 0. 22 L OH-SUN QH-MOB OH-DIFF AR -25, .73 43. .95 13. .75 2. 38 225 49 -24. .54 50. 25 31. . 13 1. 51 136. 67 -23, .16 77 . .40 55. .00 t. 76 158. .43 -24, .92 94. .20 80 .25 1. 10 203.81 -23, .83 84. . 10 73 .63 0. 82 205. 61 -25. .26 62. .20 74. SO 0. 97 233. .05 -23, 86 81. 35 84. 63 0. 26 211. ,61 -17, . 19 160. .45 122. .75 2. 97 137. .09 -22, .90 153, 75 IBS .25 2. 72 213. .76 -34. . 19 121 . 35 182. SO 4. 82 369. .45 -25 43 173 to 153. .25 1. 56 281. .59 -19, .93 229 .55 210 .75 1. 48 218. .24 -18 .89 233. .80 2SB. .00 1. 91 226. 93 -18 70 222 00 256 .88 2. 75 242. .42 -25. ,70 275. 85 248. .50 2. 15 283. .22 -36, .74 283 40 228. . 13 4. 35 325. .88 -49, .67 208. . 10 225 .25 1. 35 399. . 14 -63. .76 208. .50 204. .38 0. 32 369. OS -54. ,91 198. 90 174. .50 1. ,92 366. .09 -65. .49 153. ,9S 147 .00 0. SS 407. ,07 -61, ,00 162. .55 156. ,00 0. 52 408 .07 -62. . 15 152. .65 144. . 13 0. 67 452. 90 -62. .22 130. .55 104. .63 2. 04 430. .00 -62. 91 110. 25 92 .50 1. 40 421. SS -73. .84 66. ,00 61. OO 0. 39 522 .68 -73, 64 48 ,00 27, . 13 1. 64 567. 36 -56. ,48 44. .SS 10 .00 2. 72 618. .69 -74. 31 18. .35 -2 .50 1 . 64 984. 88 -35. 68 IS 85 22, .25 0. SO 225. .23 -24, .45 52 .05 46 63 0. 43 161. .37 -24. .24 88 .30 70 .88 1 37 178 .09 -39, ,42 80 .75 95, .25 1. , 14 326 .33 -48. .93 98. 80 117. ,25 1. 45 416. 16 -35 .39 146 .20 139 . 13 0 56 319. .88 -28, ,89 180. .05 148 .25 2. .50 263. .53 -37. .69 188. .95 144. .63 3. 49 437 .73 -44. 99 214 85 178 .38 2. ,87 627 .89 -46. .76 228, .95 196 .88 2. .53 485, .97 -48. .73 241 .95 223 .75 1. .43 343. .26 -46, 31 220. .65 223. .88 0. 25 343. .79 -40, .35 226. . IS 194. .63 2. .48 317. .42 -42. . 11 268 .00 207 .25 4. 78 309. .53 -51, . 19 253 .05 205 .50 3. .74 354 03 -46, . 16 265, .65 169. .SO 7, 57 317, . 17 -42, .71 268. .05 167, .75 7. 90 293 .71 -46. 28 204. .70 188 .75 1. 26 322 .76 -46, .44 173 .25 146 . 13 2. . 13 331 .34 -53. . 13 148 .80 102 .25 3. ,67 384 .65 -82 .98 84. . to 109 .75 2. .02 585 .68 -82, .95 74, .90 91 . 13 1. ,28 701 .44 - B l .65 67 . .80 43 .25 1. .93 826 .75 -160. .82 17 . 15 26 .25 0. .72 1314. .59 -168, .66 11. .35 -4 .25 1. 23 798 .05 -45, .20 14, .60 6 .00 0 68 531 .87 -31 , .67 10 .80 6 .88 0. .31 537. .09 -30 .64 IS 90 12 .25 0 29 425 .58 -16 .67 54 ,80 26 .50 2, .23 156 .21 -16 . 11 SB .00 40 .75 1. .36 191 .34 A B C 0 XM 13. .72 348 .42 296. 80 224 .22 47 .02 13. .42 337 . IB 287 .23 140. 23 45. .84 13 .04 307 . 16 295, . 12 167. .97 44 .41 13. .52 334 .40 296 .55 207 .03 46. .22 13. .23 324 .65 287. .90 214. 84 45. . 12 13. 60 325 . 10 312. .35 233. .59 46. .56 13. ,23 331 .43 282 . , 33 221 . 09 4S. 15 11. . 11 257 .42 228 .28 180. .47 37, .24 12. ,96 310 .44 286 56 228 91 44. . 14 IS ,51 402. .57 371 .61 305. .47 54. , 14 13. 65 338 .95 300 .57 282 .03 46. .73 12 .06 297 .55 243 .45 25B. 59 40. .76 II 73 275 .67 244 . .46 278. 91 39. .47 11 . 65 268 .75 248 .08 299. 22 39. .23 13. 72 334 .28 308. .56 282 . .03 46. .99 15, .94 405 .35 405 .35 257 .03 55 .94 17. .68 492 .88 473 .55 264 84 63 .40 18. , 12 513 . 15 493 03 235 . 16 65 .31 18. ,23 509 . 14 509. . 14 230. .47 65. .82 19. , 14 5S6. .06 556. .06 23S. 16 69 .94 18. 79 568 .87 504 .47 244. .53 68 .30 IB. 88 553 . 16 531. .46 267. ,97 68 .73 18 .88 531 .86 553. 56 253. 91 68 .76 18. 94 577 .03 511. .71 249. .22 69. .01 19 .74 622 .42 551 .96 286. .72 72 .64 19. .70 612 .29 565 .20 310. . 16 72 .58 18 38 506 .11 526. .77 382. 81 66. .49 19. .76 577 .42 600. .99 S3S. 94 72. .78 IS. .77 429 .46 365 .84 182 03 55. .21 13 40 329 .45 292 . 16 166. .41 45. .75 13 .34 315 .95 303. .56 183. ,59 45. .54 16 .33 451 .52 400. .40 246. 09 57 .70 17. ,59 479 .95 479. .95 277. .34 63 ,04 15. .72 403. . 13 387. .32 258 .59 55 .00 14. .47 360 .80 333 .04 242. .97 49 .91 16. 08 404 .38 420. .89 339. .06 56. .58 17 . 12 458 .06 458 .06 439. 06 60. .98 17 34 468 . 19 468. . 19 332. .81 61 .93 17, .57 507 .88 450 .38 230. .47 62 .94 17 .28 465 .37 465. .37 236. 72 61 .69 16. .47 431. .65 431 .65 235. 16 58. .28 16. 73 451 .05 433 .36 224. .22 59. .34 17. 84 491 .70 491 .70 230. .47 64 . 13 17 .27 501 .72 427 .39 219. .53 61. .61 16 .81 472 . 19 418 .74 211. .72 59, .70 17 .28 483 .98 446 .75 222. 66 61 . .67 17, .31 456, .47 475. . 10 227 .34 61 .76 IB OS 500 .25 500. .25 246 .09 65 03 20 .26 619 .45 619 45 303. .91 75 . 17 20 25 619 .83 619 .83 364 . 06 75 . 16 20 . 18 640 .02 590. .79 432 81 74 .82 22 .65 799 .94 799 .94 529. .69 87 .38 22. .81 794 . 18 826. .59 317. . 19 88 . 11 17 . IS 495. 60 422. . 18 373. .44 61 .09 15. 04 398 . 18 339. . 19 467 . 19 52 .22 14 .85 388 .77 331 . . 17 379. .69 51 .39 10. .92 246 .81 227 . .82 210 . 16 36 .53 to .70 249 .24 212, .32 264. .84 35 .74 JOO KH; : MM IT UB W-DIR s v U* 214 9: : 0 294, .4 1, .24 324. i . OS 0. 21 214 9: :30 295, .3 1. 67 262. i . . 11 0. 28 214 10: : 0 296 .2 2. 10 201. i . .03 0. 34 214 10: :30 295. .9 2. .71 215. i . . 17 0. 42 214 11 : 0 295. .6 3, ,32 230. i . .24 0. 47 214 11 : 30 296 .0 3 .00 235. i , . 10 0. 43 214 12: : 0 296 .4 2. .69 240. 0, .97 0. 42 214 12: :30 297. . 1 2. .94 230. 1 00 0. 45 2 14 13: : 0 297 .8 3. . 19 220. 1. 02 0. 48 2 14 13: :30 298 .2 3. 12 225. 0 99 0. 47 2 14 14 : 0 298 .6 3. 05 230. 0. .96 0. 46 214 14 : 30 298 .7 3. 10 234. 0. 94 0. 46 214 15: : 0 298 a 3. . 14 237. 0. 93 0. 46 214 15: 30 298 .4 3. 26 235. 0. .98 0. 46 214 16: : 0 298 . 1 3. 38 233. I. .03 0. 49 214 16: 30 298 .4 3. .34 255. 1. . 17 0. 48 214 17; : 0 298. 6 3. ,29 277. 1. ,30 0. 46 214 17 : 30 298. .8 3, .09 266. 1. ,03 0. 42 214 18: : 0 299. .0 2. .88 254. 0, .79 0. 39 214 18: :30 298 .6 2. 78 260. 0. 90 0. 37 214 19: : 0 298. .2 2. 67 265. 0 .99 0. 33 221 10: :30 296. .9 2. .54 208. 0 77 0. .41 221 11: : 0 297, .4 2. 66 212. 0 .79 0. 42 221 I I : ;30 297, 8 2 67 219. 0. 86 0. 42 221 12: : 0 298, .2 2. 67 226. 0. .92 0. 42 221 12: :30 298. 8 2. 77 226. 1. .03 0. 43 221 13: : 0 299. .3 2. 67 226. 1, . 15 0. 45 221 13: :30 299. .6 3. 25 219. 1, . 19 0. 49 221 14: : 0 299. .8 3 .64 211. 1. .20 0. 53 221 14 : 30 300. . 1 3. 31 214. 1. . 12 0. 48 221 IS: : 0 300. ,4 2. .97 218. 1. .04 0. 44 221 IS: :30 300. 7 2. 98 220. 0 .97 0. 43 222 6: :30 287 .7 4. 54 125. 1. . 1 1 0. 55 222 7 : 0 287 , 4 4. 45 129. 1. . 13 0. 54 222 7 ; 30 287. . 4 4 . 24 140. 1. . 16 0. S3 222 8: : 0 287 , .4 4, 02 152. 1. . 17 0. .50 222 8: ;30 287. .6 4. 03 156. 1. . 13 0. 52 222 9: : 0 287. 9 4. 03 160. 1. .09 0. 53 222 9: 30 288. 0 3. 72 156. 1, . 14 0. 49 222 10: : 0 288. . 1 3. 42 152. 1. . 17 0. 47 222 10: 30 288. 4 3. 36 145. 1. . 16 0. 47 222 11 : 0 288. .6 3. 30 139. 1. . 15 0. .47 222 1 I : :30 288, .8 3. ,50 147. 1. .25 0. 50 222 12: : 0 288. 9 3, 70 156. 1. 36 0. 52 222 12: 30 288. .9 3. 61 157. I. .24 0. 50 222 13: : 0 289. 0 3. 52 158. 1 . 13 0. .48 222 13: :30 289. ,2 3. s o 150. 1 .06 0. .47 222 14: : 0 289, 4 3. 48 143. 1 .00 0. 49 222 14: ;30 289. .6 3. .36 143. 1. .02 0. 48 222 IS: : 0 289. .7 3. 25 144. 1. .03 0. 47 222 IS: 30 289. 8 2. .79 147 . 0 .87 0. .42 222 16: : 0 289. .9 2. .34 ISO. 0 .71 0. 36 222 16: :30 289. .9 2. 64 148. 0 80 0. 39 222 17: : 0 289. .9 2. 94 146. 0 .89 0. .44 222 17 : 30 289 .9 3. 00 144 . 0. .78 0. 43 222 18: : 0 289. 9 3. 06 142. 0 .67 0 42 222 IB: :30 290, 0 2. 94 135. 0 .71 0. 4 1 222 19: : 0 290. . 1 2. .82 127 . 0 .74 0. 39 L OH-SUN OH-MOB 0H-D1FF AR -19. . 16 43. 30 72. 68 2. 33 297. 36 -20, 85 93. 10 114. 13 1. 66 259. 40 -23. 06 159. 00 125. 38 2. 65 221 . 72 -32. 40 204. 10 124. 25 6. 29 291 . 84 -55, 55 171. 35 135. 25 2. 84 464. 66 -47, .91 156. 35 156. 25 0. 01 396. 59 -31. .22 211. 40 169. 13 3. 33 233. 26 -34, 19 241. 60 233. 00 0. 68 246. 38 -38 .56 259, 10 264. 13 0. 40 267. 72 -39, . 13 237. 60 231. 00 0. 52 270. 26 -37, .72 234. .55 221. 00 1. 07 257. 58 -37, .95 244 . . 15 166. .38 6. 12 252 . 27 -46, .62 188. 20 160. .88 2. . 15 305. .49 -52. .79 176. , 10 168. 00 • 0. .64 359. 57 -50. .26 210. .25 156. .88 4 . 20 339. 85 -48, .27 215 15 142 00 5. .76 379. 10 -56. 58 164. . 15 107. 25 4. 48 504. .34 -71. 58 98. ,00 95. 50 0. ,20 529. 62 -73, .96 75. 90 67. 00 0. .70 448. 05 -97 .38 46, .95 40. .00 0. 55 657, 99 -173. .43 19. .65 19. ,38 0. 02 1 160. ,79 -24, .89 250. .95 167, .50 6. .57 149. , 12 -26, .91 256. .40 175, .00 6. 41 161. 41 -28 .70 235, .55 164. .75 5 58 188. . 18 -27 ,04 258, . 15 154! .63 8. . IS 189. ,03 -28. .48 267 00 183. .88 6. .55 218, 43 -28, .07 304 OS 188. .50 9, . 10 229, .91 -34 .93 318, .75 165 .75 12 .05 275 .24 -48, .36 279 . 10 158. .75 9, .48 353. .45 -44, .73 235 .05 135 .63 7 .83 340 .34 -43, .02 179. .80 135 .25 3. .51 333. .04 -48. .42 153 30 154 .00 0 06 352 82 -281, 31 51 .45 27. .38 1 .90 979 .97 -240. . 14 58 .60 29. .75 2 .27 947 .66 -184 .87 70 .00 37 .25 2 .58 874. .02 -174 .80 64 .00 53. .50 0 .83 905. .90 -117. .76 107 . 15 70 .38 2 .90 667 .89 -108 .05 120 . 10 86 .25 2 .67 596 .69 -97. .01 109. .05 77 .00 2 .52 628 .47 -67 .71 138 .35 107 .38 2 .44 513 .62 -60 .20 154 .70 137 .00 1 .39 464 .54 -50 .97 185 .65 142 .38 3 .41 401 .32 -49 .09 233 .85 150 .88 6 .53 394 .01 -56 .64 225 .55 126 .25 7 .82 466 .99 -68 .74 159 .80 101 .25 4 .61 516 .57 -77 . 13 126 .45 88 .75 2 .97 538 .43 -77 .70 123 . 15 78 . 13 3 .55 518 . 14 -61 .61 167 .00 1 13 .38 4 .22 390 .77 -53 .38 184 . 10 127 .88 4 .43 360 .64 -49 .28 186 .80 109 .63 6 .08 347 .48 -36 .94 179 .70 112 .25 5 .31 249 .76 -32 . 17 130 . 10 99 .88 2 .38 205 .42 -39 .77 136 .65 1.14 .00 1 .78 268 .70 -39 .30 192 .05 113 . 13 6 .21 260 .01 -49 .60 145 .70 79 .88 5 . 16 290 .46 -68 .79 97 .55 68 .25 2 .31 334 .20 -57 .61 110 .90 60 . 13 4 .00 314 .88 -69 .05 76 .00 32 .50 3 .43 400 . 18 A B C 0 XM 11. 81 273. .44 252 .41 360. .94 39 .81 12. 36 290. 67 268 31 296. .09 41 .85 13. 01 317, .71 281 .75 236 .72 44 .31 15. 18 381. .86 366. .89 249 .22 52 .79 18. 29 543. 12 481, .63 291 .41 66 .09 17. 47 484 . . 17 465. . 18 267 .97 62 .52 14. 95 373. . 13 358 .50 203. .91 51. .86 15. 51 402 .57 371 .61 203 .91 54 . 14 16. 22 443. .83 393 .59 205. .47 57 . . IS 16. .30 457 .00 389 .30 205. .47 57 .52 16. 08 413. .03 4 13 .03 199. .22 56. .60 16. . 13 446. .55 380. .40 196. .09 56. .75 17. 32 486. 08 448 .69 210. . 16 61 . .85 18. 01 499. .80 499. .80 230. .47 64 . .87 17. ,74 496. .58 477, . 10 224. .22 63 .69 17 .52 485. .03 466 .01 255 .47 62. .70 18. 38 527, . 18 506. .51 314. .06 66 .53 19. .57 579. .90 579 .90 292, .97 71 .95 19 .74 587. .99 587 .99 244. .53 72 .67 20 .93 689 .87 636 .80 320 .31 78. .47 22 .87 869. . 13 770 .74 457, .81 88 .52 13 .51 328 08 302. 84 151. . 17 46. . 19 14. .02 344, .26 317 .78 155 .86 48. . 14 14 .42 352, .40 338. .58 174 22 49. .75 14 OS 338. .89 325. .59 182. .03 48. .26 14. .36 371 .37 316. .35 203. 91 49. .55 14. .28 368. .27 313 .71 216. .41 49, . 19 15 .64 400 .62 384. .91 224, .22 54 .68 17 .52 495. .78 457 .65 238, 28 62. .75 17 .08 457. .23 457. .23 238. .28 60 .84 16 .85 456. .44 438. .54 238. .28 59. .87 17 .53 477. . 12 477 . 12 236. .72 62. .78 23 .95 882 .54 995 .21 335. .94 94 . .67 23. .65 865. .35 937 .47 339. 06 92. .88 23 .04 786. .52 886. .93 335. .94 89. .45 22 .89 788 .68 654 .41 354 . .69 88 .64 21 .66 730 .28 701 .64 300, .78 82. .11 21 .35 691. .49 691, .49 277 .34 80 .50 20 .92 676 . 17 649. .66 305, .47 78 .39 19 .32 586. .73 541 .60 292. .97 70 .70 IB .71 533 .82 533. .82 278 .91 67. .99 17 .81 490 .88 490 88 261. .72 64 .03 17 .62 479 .58 479 .58 263. .28 63. . 12 18 .40 527 .58 506. .90 289. .84 66 .55 19 .39 590. . 12 544 . .73 292. .97 71 . 04 19 .93 600 .07 600. .07 288 .28 73. .59 19 .97 625 .76 577 .63 277. .34 73. .75 18 .83 528 .71 550. .29 232 03 68, .53 18 .07 501 .46 501 .46 230 .47 65 . 14 17 .64 500 .02 461. .56 232 . 03 63 .21 15 .97 407. .36 407. .36 196. .09 56. .08 15 . 13 365. .48 380. .40 175 .78 52 .61 16 .38 428 .00 428 .00 200 .78 57 .92 16 .33 415 . 11 432. .05 196. .09 57 . .63 17 .66 503 .35 464 .63 192. 97 63. .37 19 .38 557 .62 580 . 38 IBB. .28 71 .05 18 .47 512 .03 532. .93 192. .97 66 .95 19 .39 616 .67 525 .31 225 .78 71 . 14 JDD HH: MM TT UB W-OIR SV U» L OH-SUN OH-MOB OH-DIFF AR A B C D XM 222 19: 30 289 .9 3 .03 119. 0 79 0 .39 -136 .74 37 .80 9 .63 2. .22 6B3 97 22. 16 772 .01 74 1 .74 291. ,41 84. .74 223 8: 30 287 .5 1 .86 148. 0 .66 0 .28 -37 . 14 52 .40 40 .25 0. .96 284. 68 16. 00 449, .01 367 .37 224. ,22 56. 21 223 9: ; 0 287 .8 2 .01 149. 0. 62 0 .30 -41, .23 56 .80 54 .25 0. .20 279. 19 16. 61 4S4. .00 419 .07 205. 47 sa . 82 223 9: 30 288 . t 2 . 15 148. 0. .69 0 .33 -30 .74 107 .35 63 .88 3. .42 207. 21 14 . 85 376 .23 347 .29 183, .59 51 .47 223 10: : 0 288 .3 2 .30 148. 0. .75 0 .36 -30 .92 130 .35 71 .50 4 63 207. ,41 14. .89 391. .82 333 .77 183. 59 51. 61 223 10:30 288 .4 2 .23 14S. 0 80 0 .34 -32 .26 t i t .60 68 .50 3. 39 245. 26 15. 15 373 .35 373 .35 210. . 16 32. 68 223 t t : 0 288 .6 2 . 17 143. 0 84 0 34 -30 .98 109 .30 58 .38 4. 01 246. 68 14 . .90 392 .68 334 .51 217, 97 51 .66 223 11: 30 288 .9 2 . 19 141. 0 80 0 .34 -32. .23 106 .05 47 .63 4 . 60 244. 78 IS. 15 380 .21 365 .30 210. . 16 52. 66 223 12: 0 289. .3 2 .22 139. 0. 76 0 .34 -31 , .87 112 .43 78 SO 2. 67 229. .69 15. 08 392. .11 347 .72 199. .22 52. 37 223 12: 30 289 .9 2 .39 147. 0 .87 0 .36 -34 .82 123 .25 too 38 1. .80 273. ,84 15. 61 392 .34 392 .34 222, .66 54. 60 223 13: 0 290. 4 2 .56 155. I. .00 0 38 -37 .96 133 .65 117 .75 1. .25 325. 65 16. . 13 430 .83 397 .69 252. .34 56. 76 223 13: 30 290. .7 2 .65 156. 1. 05 0 .40 -34. .01 174 .43 124. .75 3. 91 289. ,38 15. ,48 393. .58 376. . 15 239. .84 54. 01 223 14: 0 291 . 0 2 .74 158. 1. . 11 0 41 -37 .52 167 .00 116 .00 4. 02 331 ,89 16. 06 42S .28 395 .33 236 .59 56. .47 223 14: 30 291. .4 2 .58 152. 1. . 12 0 39 -34, .36 158 .95 135 . 13 1 88 319 31 15. 54 396 .07 380 54 263. 28 54 . .26 223 IS: 0 291 . 7 2 .42 147. 1. 12 0 .39 -24. .49 218 .50 168. .25 3. .96 224. 22 13. ,40 343 .4 1 280 .97 230. .47 45. ,79 223 IS: 30 291 . a 2 .36 154 . 1. .48 0 .39 -22 .68 228 . IS 137 . 13 7. , 17 272. .92 12. 80 314. .25 276. .67 294. 53 43 .90 223 16: 0 292. . i 2 .30 162. 1. .83 0. .37 -26. .37 167 .90 101 .75 5. .21 416 08 13. 88 346. .76 307 SO 407, .81 47 .63 223 16: 30 292. 0 2 .52 tSB. 1. .40 0 .37 -41 .45 112 .70 100. SO 0. .96 SIS. 27 16. 63 447, .31 429 .77 376. ,56 58. 95 223 17: 0 291. 9 2 .75 154. 0. .86 0 39 -52. .25 104 .90 64 . .38 3. .19 369. IB 17. 96 306 .91 467 .03 236. .28 64. ,63 223 t7 : 30 291. 8 2 57 152. 0. 79 0 .37 -52 .83 84 .23 41 .38 3. 38 359. .76 18. 02 53B. .95 459 . to 232, 03 64 .89 223 18: .0 291 . .6 2 .39 151. 0. .73 0 .34 -52 .49 68 .33 35. .75 2 .57 360. 08 17. 98 S07, .75 487. .84 232. 03 64. 74 223 18: 30 291. .4 2 .27 153. 0. 60 0 32 -S4. .07 56 . 10 23. .75 2. 55 323. 84 IB. . 13 526. .53 486. .02 203. ,47 65. 45 223 19: 0 291. 2 2 . 14 153. 0. .49 0. 30 -60, .27 40 .30 8. .38 2. .51 307. .70 18. .71 544, .51 523 . 16 185. . 16 68. .01 223 19: 30 290. .9 1 .99 156. 0. .50 0. 27 -69 02 26 .80 6. .75 1. .58 388. SO 19. 40 592, .60 547 .01 219. 53 71. . 13 230 13: 30 296. a 2 .00 190. 1. 06 0 36 -14, .44 285 .63 179 .38 8. .37 124. .96 to. 01 228 .87 194 .96 188. 28 33. .25 230 14: 0 297. 0 2 .26 199. t. 20 0. ,37 -21 . .14 227 . 10 tst .50 3, .93 215 61 12 .45 299 .29 265 .41 244. S3 42. . 18 230 14: 30 297. . 1 2 .61 207. 1. , 14 0. .40 -3 t , .00 195 .65 139 .25 4 .44 284. 80 14. 90 377 .94 348 .87 250. ,78 51. .68 230 IS: 0 297. 3 2 .96 215. 1. .02 0 .45 -33. 48 254 .50 153 .68 7. 92 245. .49 ts . 38 381. .91 381 . .91 203. ,47 S3 ,61 230 15: 30 295. 9 3 .53 219. 1. 05 0 51 -50, .54 235 .85 173 . 13 4. .94 334. 01 17. ,77 487 .67 487 .67 219. ,53 63. 82 230 16: 0 294. 5 4 .11 223. 1. 04 0. .56 -79 . 10 198 . IS 190 .63 0 .59 432. 74 20. .05 594 .41 618 .67 228. 91 74 . 14 230 16: 30 294 . 8 3 .57 232. t. 09 0, 49 -67. 40 162 .00 172, 50 0 83 458. 60 19. 28 586. .29 541 . 19 261. ,72 70, 59 230 17: 0 295. 0 3 .04 240. 1. 08 0 .43 -60. 38 1 16 .75 115 .75 0. 08 473. 54 IB. 73 566. .29 502 . 18 28S. , 16 68. 06 230 t7: 30 295. . 1 2 .54 242. 0. 90 0. .37 -44, .57 104 .93 73 . 13 2 .31 352 68 17 06 456 .02 436 .02 247. .66 60. .75 230 18: 0 295. . 1 2 .04 245. 0. 72 0. 30 -39, . 10 65 .80 50 .88 t. . 18 309 .29 16. 28 432 .41 415 .43 233. 59 57, SO 230 18: 30 294. 9 1 .59 241. 0. 88 0. .23 -44 . 07 26 . IS IB .88 0 57 549. 21 17. 00 462. .24 444 . . 12 389. ,06 60. ,47 231 6: 30 286. t 3 .31 105. 0. 87 0, .39 -364, .20 14 .60 10 .75 0 .30 1188. .03 24. .38 1051. .80 832 .73 389. 06 97 . 18 231 7: 0 286. 4 3 .43 114. 0. 98 0. 44 -121. 05 63 .45 38 .50 t. .97 696. .95 21. .76 723 .59 723 .59 310. . 16 82 .61 231 7: 30 286. 2 3 .35 125. 0. 95 0. 45 -78. .90 104 .65 52 .63 4. .10 491 .74 20. 03 606 . 13 606. . 13 260. . 16 74, .09 231 8: 0 286. 0 3. 27 137. 0. 92 0. 46 -61 .41 137 .55 63 .00 5. .71 383. 45 IB. .62 571. .83 507 . to 228, .91 68 .45 231 8: 30 286. 4 3. .04 147 . 0. 85 0. 44 -44, .33 175 .90 97 .38 6. . 18 278. .99 17, 03 445. .71 463 .90 196. .09 60 .63 231 9: 0 286. 7 2 .81 158. 0. 77 0. 41 -43, 03 145 .30 125 .25 t. .58 264 .52 16. 85 465 .39 429 .39 189. .84 59 .88 231 9: 30 287. 8 2. 42 167. 0. 81 0. 36 -36. 68 1 17 .65 128. 63 0. 87 268. .54 IS. 93 404 , .95 404 .95 211. .72 55 .90 231 10: 0 288. 8 2 .04 176. 0. 80 0. 34 -20, .37 172 .00 168 .63 0 .27 150. .26 12. 20 280. .49 269. .49 174. .22 41 .28 231 10:30 289. 6 2 .08 202. 1. 08 0. 36 -16 .62 251 . 10 203 .50 3. .75 150, .59 to. .90 255 .22 217 ,4 1 203. 91 36 .46 231 11: 0 290. 5 2. . 12 228. 1. 38 0 37 -16, 04 282 .20 213 . 13 5. .44 179 .74 10 68 243. .97 216 .35 249. 22 35. .64 231 11: 30 291. 1 2 73 243. 1. 41 0. 43 -26 .89 271 .55 292. .25 1. .63 281. .46 14. 01 337 .84 324 .59 271. 09 48 . 12 231 12: 0 291 . 6 3. 33 259. 1. 29 0. 50 -36 .96 307 . IS 304 .75 0 . 19 310. .45 15. 98 423 .25 390 .69 244 . ,53 56. .09 231 12: 30 291. 9 3. .66 253. 1. 23 0. S3 -47, .50 283 .50 259 .75 1. .87 357, .67 17 , ,42 491 .55 453 .74 242. 97 62 .31 231 13: 0 292. 3 3. .99 248. 1. . t3 0. 56 -62. .34 250 . to 312 .00 4 .87 390, . IS 18. .89 531 .49 553. . 19 230. .47 68 .80 231 13: 30 293. 2 3 99 260. 1. 23 0. .56 -58, .25 275 .90 297 .25 1, .68 401 .22 18 54 545 . 10 503 . 17 246 09 67 .21 231 14: 0 294. 1 3. 99 273. 1. 34 0. .56 -57, SO 281 .85 224. SO 4, .52 432. .50 18. ,48 531 .31 510. .48 266. .4 1 66. .61 231 14: 30 294 . 6 4. .20 286. t. 42 0. 58 -63 .38 2B7 .30 220 63 5 .25 479. .39 18 98 568 .65 524 .91 282. .03 69 . 18 231 15: 0 295. 2 4 . .41 299. 1. 50 0. 60 -76 .59 256 .45 199. .00 4 .52 568 .60 19. .90 598 .06 598 .06 305. .47 73 .44 231 IS: 30 295. 7 4 . 87 299. 1. 38 0. 64 -96. 63 252 .35 163 .00 7 .05 581. .77 20. 91 648 .86 675 .34 282. 03 78 .31 231 16: 0 296. 2 5. .33 300. 1. 20 0. .68 -126. .33 232 .90 129 .88 a .11 564 .98 2t .92 718 .60 747 .93 247 . .66 63 .37 231 16: 30 296. 4 5. .35 294 . 1. 19 0. 66 -189 .4 1 140 .70 101 .38 3 . 10 729 .69 23 .09 626 .25 859 .98 278. .91 89 .79 231 17: 0 296. 6 5. 37 289. 1. 19 0 65 -254. 68 98 95 86 . 13 1 .01 850 .52 23 76 933 .28 896 .68 300. 78 93 .57 231 17: 30 296. 4 5. 01 293. 1. 15 0. 60 -264 . 10 76 85 57 .50 1 .52 90S. . tt 23 84 922 .96 922 .96 317. . 19 93 .98 J D D HH: MM IT UB W-DIR SV U» L OH-SUN OH-MOB OH-DIFF AR A B C 0 XM 231 18: 0 296 .2 4 .64 298. 1. . to 0 .56 -287 .03 55 .20 32. . 13 1 . .82 957. . IB 24 . 00 960. . 13 922 .47 329. .69 94. .88 231 18: 30 295 .6 4 .29 303. 1. .38 0 .51 -348. .40 34 .60 12. 63 1. ,73 1423. . 11 24. 32 965. 36 1004 .76 467. . 19 96. ,78 232 7: 30 290 .6 1 . 15 316. 0 .87 0 .20 -17 .26 40 . IS 21 .63 1. 46 229. 05 I I . . 14 262 .95 223 .99 300 .78 37, .34 232 8: 0 290 .9 1 .56 319. 0 .80 0 .26 -19 .23 84 .75 29 50 4. 35 183 63 11. 84 279 .56 247 .91 222 .66 39. .90 232 8: 30 292. .2 1 .45 275. 1. . 11 0 .25 -16 .81 84 .35 40. .88 3. .42 225. 92 10. 98 252. .46 223 .88 302. .34 36 .72 232 9: 0 293 .5 1 .34 232. 1 .36 0 .25 -12 .31 109 .70 SO 63 4 65 187. .95 9. 02 200 35 170 .66 323 .44 29 .74 232 9: 30 293 .9 1 .82 223. 1 .29 0 .31 -19 . 14 136 .95 97. .25 3. . 13 246. .92 II .81 288 .99 236 .45 300. .78 39 .79 232 10: 0 294 .2 2 . 29 214. 0 .92 0 .36 -28 86 145 .60 148 38 0. .22 236 93 14 46 354 .25 340 .36 217 .97 49 .88 232 10: 30 294 6 2 .50 224 . 0 .93 0 .39 -30 .05 178 .55 164 . . 13 1. . 14 230. .55 14 , 71 363. .40 349 . 15 207 03 SO .90 232 11: 0 294 .9 2 .70 234. 0. .92 0 .41 -32 .76 197 .90 191 .38 0 51 237 .37 IS. ,25 399. .01 353 .84 202 .34 53 .07 232 11: 30 295 .5 2 .86 257 . 1 . 11 0 .44 -30 .52 262 .00 237. .75 1 .91 247 .31 14. .82 359 .54 359 .54 219. .53 51 .29 232 12: 0 296 . 1 3 .01 279. 1 31 0 .46 -34 .61 253 .80 219 .25 2 .72 320 .07 IS. .58 390. .33 390 .33 261. .72 54 .44 232 12: 30 296 .6 3 .33 285. 1. .39 0 .49 -41. .62 262 .60 208. .25 4 . 28 385. 38 16. ,67 4S5. .71 420 .66 282. 03 59 .05 232 13: 0 297. . 1 3 .66 291 . 1 .47 0 53 -49. . 12 274 .95 248. 88 2 .05 438. 38 17 61 499. 59 461 . 16 292. 97 63 . 13 232 13: 30 297 6 3 .84 296. 1 .36 0 .53 -63 . 13 223 .05 241 .75 1. .47 SOI .97 IB. .95 545. .97 545 .97 294, .53 69 .09 232 14: 0 298. .2 4 .01 301. 1 .23 0 .•55 -68 . 11 228 .95 225. . 13 0 .30 464. .88 19. .34 566. .57 566 .57 263 .28 70 .83 232 14: 30 298. .7 4 . 18 294. 1. .38 0 56 -85 .35 190 .65 212. .50. 1, .72 606. .72 20. 38 652 . 19 602 .02 311, .72 75 .76 232 IS: 0 299. .2 4 .35 287. 1. .54 0 .57 -105, .34 162 .35 175 63 1. .05 772 .27 21. .25 726 . 15 643: .94 364 . .06 60 .01 232 15:30 299. S 4 .25 294. 1. .35 0 .56 -96 .91 169 .35 133. . 13 2. .85 652. .50 20. .92 676 .56 650 .03 317, . 19 78 .37 232 16: 0 299. .8 4 . 14 301. 1. . IB 0 SS -86 .21 183 .40 102. .00 6, .41 532. .76 20. .42 617. .33 642 .52 271 .09 75 .97 232 16: 30 299. .8 4 .37 300. 1, . 17 0 .58 -91 .55 198 .90 92 . 13 8. .41 524. .38 20 68 647. .61 647 .61 260. . 16 77 .22 232 17: 0 299. .8 4 .60 299. 1 . 14 0 .59 -119 ,65 162 .65 68 25 7. .43 600. .71 21. ,71 706. 36 735 . 19 267. .97 82 .40 232 17: 30 299. .6 5 .05 304. t, . IS 0. .61 -237 .83 90 .35 36 .50 4. .24 8S2. .56 23 .62 899, .81 899 .81 305. .47 92 .76 232 18: 0 299. .5 5 .50 309. 1 . 13 0 63 -642 .79 36 .65 9 .00 2. . 18 1122 .85 25 .02 1069 .00 1112 .63 332 81 101 . 18 236 11: 0 292. 6 3 .50 164. t. . 12 0 SO -50, .72 226 . 15 216. . 13 0 .79 363. .91 17. .80 478. .33 497 .85 238. .28 63. .91 236 11: 30 292. 7 3 .45 161. 1. .11 0. SO -47 .SS 237 .70 233 SO 0 .33 340 .80 17. .43 490. .74 452 .99 232. .03 62 .34 236 12: 0 292. .8 3 .40 158. 1, . It 0. .49 -48 .33 222 .30 237 .00 1. . 16 353 .93 17, .52 486. .24 467. . 18 238. .28 62 .74 236 12: 30 293. .4 3 .24 157. 1. .02 0 48 -42 .57 231 .50 275 .00 3. .43 29S. .34 16 .80 453. . 15 435 .38 213. .28 59. .61 236 13: 0 293. .9 3 .07 156. 0 93 0 47 -33 38 282 .00 297. . 13 1 . 19 214 .09 15 36 374 .05 389. .32 178. .91 S3 .54 236 13: 30 294. 3 2 .80 157. 1 .25 0. .44 -28, .75 267 .80 242. .50 1. .99 262. .40 14. .43 374. .00 318. .59 242. ,97 49. .79 236 14: 0 294. .7 2 S3 158. 1 .49 0 .41 -22 .67 283 .85 219 . 13 S. . 10 261 .07 12. .90 314 .03 278 .48 282. .03 43 .89 236 14: 30 295. .0 2 .47 158. 1. .26 0. .41 -20 .08 318 .85 212 ,88 8 .34 192 .30 12. . 12 282. .60 260 .86 225 .78 40 .94 236 IS : 0 295. .3 2 .42 158. 1. 04 0. 40 -21 .54 269 .35 169 .88 7 .83 176 .31 12. .57 308. .49 262 .79 . 197. .66 42 .64 236 IS: 30 295. 6 2 .28 183. 1. . 14 0. 37 -22 .33 213 .35 129 63 6. .59 217 .56 12. 81 298. 85 287. . 13 236. 72 43 S2 236 16: 0 296. 0 2. 13 209. 1. 21 0 .35 -20 .68 196 .00 123 63 5 .70 223 68 12 31 283. .01 271 .91 257. 03 41 .65 236 16: 30 295. 3 2 .89 233. 1 .21 0, .43 -38 .92 188 .40 113 .00 5, .94 360 .48 16. 26 422. .71 422 .71 272. .66 57, .38 236 17: 0 294. 5 3 .65 257. 0. .99 0. 50 -77. . 19 143 .50 87 .38 4 , .42 455. .25 19. .92 601. .26 601 .26 242. 97 73 .61 236 17: 30 294. 0 3 .28 260. 0 .84 0. .44 -89 .21 85 .40 50 .38 2. .76 484 . .70 20. .59 ' 664. .34 613. .24 244. 53 76. .68 236 18: 0 293. 5 2 .91 263. 0. .70 0. 39 -86 .89 61 .70 19. .63 3, 31 448 . 14 20. .46 619 .30 644 . .58 227 . .34 76 . 14 236 18: 30 293. 4 2. .48 281. 0 .70 0. .34 -69, .22 52 . 10 2, . 13 3. .94 433. . 14 19. .42 604. .44 536. .01 244. .53 71 . 19 236 19: 0 293. 3 2 .05 300. 0. 66 0 28 -65, 01 32 . 10 -0 , . 13 2, .54 472. . 16 19 . 10 576. .21 531 .88 274. 22 69, .77 237 7: 0 290. 2 0 .92 316. 0. 71 0 16 -16, 78 21 .45 1 . 13 1 .60 225. .71 10 .96 252. .44 223 .86 302. .34 36. .68 237 7 : 30 290. 7 1. . 18 252. 0 68 0. 20 -17 .77 41 .45 13. .38 2 .21 184 .77 11. .33 278. .43 218 .77 23». 28 38. .02 237 8: 0 291. 2 1. 45 188. 0. 55 0 23 -26 .04 42 .75 28 .75 1 . . 10 198 .03 13. .81 349. .79 297, .97 196. 09 47, .32 237 8: 30 291 . 9 1. .48 173. 0 76 0. .24 -21, .03 63 .20 55 .63 0 .60 209. .94 12 .41 298. .42 264 , .63 238. 28 42. .06 237 9: 0 292. 5 1. .51 158. 0. 97 0. 25 - IB, .91 79 .30 75 00 0 . 34 227, .72 t l . 72 276. 51 245. ,21 278. 91 39 .50 237 9: 30 293. 3 1 . .42 196. 0. 90 0. 26 -11. ,74 140 .75 61 . 88 6. .21 112 . 10 8. .74 192. . 15 163. .68 200. 78 28. .73 237 10: 0 294. 1 1. 34 234. 0. 83 0. .26 -8 .80 189 .35 81 . 13 8 .52 69 . 17 7. .07 149. .82 122 .58 161 . 72 22 .94 237 10: 30 294 . 4 1 .91 225. 1. 06 0. 32 -18, 34 169 .35 131 .38 2 .99 186, .90 11 53 279. 99 229. 08 235. 16 38 .77 237 11: 0 294. 6 2 . 49 215. 1. 21 0. 40 -26. .48 213 .50 153. .75 4 .71 256, .03 13. .91 334. ,71 321. .58 249. 22 47. ,73 237 1 1: 30 294. 9 2. .94 232. 1. 40 0. 45 -31 , .33 273 . 10 166. SO 8. .39 314. .42 14. .98 373. .36 358 . ,72 274. 22 51. .94 237 12 : 0 295. 2 3. .39 248. 1. 58 0, 50 -41 .78 274 .20 174 .00 7 89 431 .93 16. .69 430. .61 448 . . IB 314. 06 59. 15 237 12: 30 296. 6 2. 90 284 . 0. 68 0. 46 -27 .90 313 .70 213 . 13 7 92 131 .93 14 .25 359. .52 318. 82 124 . 61 49. .04 246 17: 0 296. 7 1. 95 272. 1. 47 0. 32 -22, . 13 135 .80 94. .00 3 .29 321 . .04 12. 75 296. 98 285. 33 351. 56 43. 30 246 17: 30 296. 1 2. 35 279. 1. 42 0. 34 -44. 50 83 .60 61 , 38 1. .75 606. .30 17. 04 464. 73 446. .51 426. 56 60. 71 246 18: 0 295. 4 2. 75 2B5. 1. 26 0. 37 -87 .20 52 . 15 38 .50 1. 08 850 .96 20 48 632 75 632 .75 432. 61 76 .21 246 18: 30 294 . 6 2. 49 290. 0. 97 0. 33 -96 .80 33 .55 21 .50 0. 95 792 58 20. 91 675. 36 648. 87 385. 94 78. 35 246 19: 0 293. 8 2. 22 29S. 0. 70 0. 30 -88. 89 26 .55 0 .88 2 .02 592. .07 20. .56 638. .39 638 . 39 297. 66 76 63 ODD HH; MM TT ua W-01R sv U« L 0H-! SUN OH-MOB QH-DIFF AR A B C 0 XM 247 7 : 0 289. 6 1. 31 148. 0. 28 0. 20 -33 .27 21 .70 - 1 , ,63 1. .84 150. .90 15 .34 395. 90 365 .45 126. .95 53. 45 247 7; :30 289. 9 1. 82 170. 0 . 50 0. 27 -39 .48 45 .25 13. .50 2. .50 240 .03 16 .35 425. 58 425 .58 180. 47 57. 74 247 a: : 0 290. 1 2. 33 193. 0 . 78 0. 33 -60 .62 51 .40 37, .75 1. .08 446. .20 18. ,75 545. ,35 523 .97 267. 97 68. 15 247 8: 30 290. 8 2. 13 201. 0 . 90 0. 32 -38, 64 75 . 10 57 .63 1. .38 355 55 16 .23 4 19. 51 419 .51 271. .09 57. 20 247 9: : 0 291 . 5 1. 93 210. 0. 98 0. 31 -24 31 112 .00 72 .75 3 .09 244 81 13. .36 329. , 19 291 .93 252. 34 45. 61 247 9: 30 291. 7 2. 11 214. 1. 14 0. 33 -28, .26 116 .60 69 .88 3. .68 313. .95 14. .32 356. 52 329. . to 292. ,97 49. 36 247 10: : 0 291. 9 2. 29 217. 1. 31 0. 36 -26 .79 161 .65 85 .63 5. .99 312 86 13 .98 351. .26 311 .50 302. .34 48. 02 247 10: :30 292. 5 2. 19 193. 1. 21 0. 36 -21 .72 195 .20 121 . 13 5. .83 230. .03 12 .63 298. .97 275 .97 255. ,47 42. 84 247 1 1: : 0 293. 1 2. 09 169. 1. 13 0. 34 -23. . 14 154 .30 147 .38 0 .55 245 . 19 13 03 312. 97 288 .90 260. . 16 44. 39 247 11: 30 293. 7 1 . 93 178. 1. 29 0. 33 -18 83 167 .40 175 .25 0 .62 227. 63 11. .70 280. 21 238 .70 280. 47 39. 40 247 12: : 0 294 . 3 1 . 77 1B7. 1. 42 0. 32 -13 .35 222 .55 203 .00 1. .54 170 .02 9. .52 210. 50 186 .67 272. .66 31 . 50 247 12: 30 294 . 6 1. 98 225. 1. 25 0. 34 - IB 01 194 .30 191 . 13 0 .25 202. .90 11. .42 270. .93 230 .79 258. .59 38. 34 247 13: 0 295. 0 2. 18 264. 1. 01 0. 36 -19 ,89 222 .45 148. . 13 s. .85 174 .29 12 06 275. 89 265 .07 205. 47 40. 71 247 13: 30 295. 3 2. 50 248. 1. 07 0. 41 -23 53 259 .30 204 .25 4 .34 196 .03 13 . 14 310. .70 298 .51 205. .47 44 . 81 247 14: : 0 295. 5 2. 83 232. 1. 11 0. 43 -33 .50 220 .95 250 .25 2 .31 279. .64 15 .39 381, .91 381 .91 233. 59 53. 63 - 293 -APPENDIX E : Statistical Indices and Methods E.1 MODEL VALIDATION STATISTICS For a detailed discussion of the s t a t i s t i c a l indices used to validate models, refer to Willmott (1981). In the following, observed values are denoted as (0 , 0 0 ) and model 1 2 n predictions as (P^ P ). The observed and predicted means are 0 and P respectively. Coefficient of Determination : R •2 . R is the square of the Pearson's Product-Moment Correlation Coefficient : R 2 = it n* ( I ° i * P i ] ~ n 2 * ° * P 1=1 n- £ o 2 - (n-0) 5 1=1 n- [ P 2 - (n-P)2 1=1 (E. 1) R describes the proportion of the total variance explained by the model. Standard Error (Standard Difference) : RMSE, (RMSD) The root-mean-square error is defined as -•0.5 RMSE = 1 — y (p - o f n L i l 1=1 (E.2) - 294 -If P and 0 are variables with comparable uncertainties, this index is referred to as the root-mean-square difference (RMSD). Systematic and Unsystematic Error : RMSE , RMSE sys unsys RMSE sys — V (p - o y n L i r i = i o.s (E.3) RMSE unsys - y (p - p y n L l i i=i o.s (E.4) where P is the ordinary least-squares estimate of P . Note that RMSE2 = RMSE2 + RMSE2 sys unsys Index of Agreement : d This descriptive s t a t i s t i c ref lects the degree to which the variate is accurately estimated by the simulated variate (Willmott, 1981) : d = 1 - i = i (E.5) [ c i P i - oi + ioi - oir 1 =1 Thus, d specifies the degree to which the observed deviations about 0 correspond, both in sign and magnitude, to the predicted deviations about 0, where 0 is considered to be error-free. - 295 -E.2 STRUCTURAL ANALYSIS / PRINCIPAL AXIS Ordinary least-squares regression analysis should be restricted to predictive situations. For comparisons with theory or among f i t t e d lines, the related technique termed functional analysis or structural analysis should be used (Mark and Church, 1977). If ordinary regression analysis has been performed, the slope of the linear functional relation may be obtained from b f = 2b r [b 2 / R2- X] + / [ b 2 / R2- X]2+ 4X-b: r V r i (E.6) where b is the slope of the regression line and X is the ratio r 2 2 of the error-variances X = E /E . The assumption that y x X = VARCJO/VARCX) yields the reduced major axis and i f X = 1, the equation for b f becomes the principal axis solution (Mark and Church, 1977). E.3 LOWESS LOWESS stands for local l y ueighted scatter-plot smoothing (Cleveland, 1979). This method is summarized by Chambers et.al. (1983). Here, the steps followed to obtain a LOWESS curve through a scatter-plot of n (x/y)-pairs are : 1. Sort the data to increasing x (x^ i = l,n). 2. Select " f " as the fraction of points used in a weighting "neighbourhood", so that "q" i s that number of points : - 2 9 6 -q = nearest integer of (f-n). 3 . Determine the q nearest neighbours to (including x^, 4 . d = |x - x | i 1 q-est neighbour 5 . Assign a weight t ^ j to each neighbour (k = l,q) according to ' i ( k ) 3 -x -x 1 i(k) d l W J 1 -6. Perform a weighted linear regression on the q neighbours of x . i 7 . LOWESS estimator of y : y = a + b -x . M Ji l l i 8 . Perform steps 3 - 7 for each x^ 9 . Decide on the number of "robust iterations" for outlier rejection (e.g. 3 ) . 1 0 . Compute the absolute residuals : r ^ |y - I , (i = l,n) 1 1 . Determine the median r (by sorting r^ : median (r^ = m. 1 2 . Compute a robustness weight for each point : , (i = l,n) if r £ 6-m i if r < 6-m i w = 0 i w = i 2 1 r i 1 -6-m 1 3 . Go to step 3 and multiply the t 's with the appropriate w^s. Thus steps 3 - 7 update the y^s. 1 4 . Perform steps 9 - 1 3 for the number of robustness iterations specified. 15.Plot a smooth line through the ( X J . Y J ) using a cubic spline - 297 -The l i s t i n g o f the FORTRAN-77 c o d i n g o f the LOWESS-version used i n the present work i s i n c l u d e d i n the f o l l o w i n g : i n t e r p o l a t i o n . 1 2 SUBROUTINE L O W E S S ( X Y Y . N . F , I T E R . X Y T . W ) 3 C 4 C * T h i s s u b r o u t i n e p e r f o r m s a LOWESS s m o o t h i n g w i t h o p t i o n a l 5 C * o u t l l e r r o b u s t n e s s on a t w o - d i m e n s i o n a l d a t a s e t . LOWESS 6 C • ( L O c a l l y w e i g h t e d S c a t t e r p l o t Smoother ) Is a n o n - p a r a m e t r i c 7 C * r e g r e s s i o n method d e v e l o p e d by C l e v e l a n d ( 1 9 7 9 ) . 8 C • 9 C * a r g 1: XYY i s the (3 .N ) a r r a y w i t h the d a t a - x and - y in 10 C * co lumns 1 & 2 (and the smoothed y ' s i n co lumn 3 on e x i t ) . 11 C • a r g 2: N i s the number o f d a t a p o l n t s . 12 C * a r g 3: F i s the d e g r e e o f l o c a l i t y f o r the s m o o t h i n g . I .e . 13 C * the f r a c t i o n o f N i n e a c h r e g r e s s i o n s t r i p . 14 C * a r g 4: ITER i s the number o f o u t l i e r r o b u s t n e s s i t e r a t i o n s 15 C * d e s i r e d ( ITER=0 w i l l i g n o r e the r o b u s t o p t i o n ) . 16 C * a r g 5: XYT i s a (3 .N ) w o r k a r r a y ( c o n t a i n s the X .Y & w e i g h t s 17 C * o f the l a s t s t r i p ) . 18 C * a r g 6: W i s a (N) w o r k a r r a y ( c o n t a i n s the r o b u s t w e i g h t s ) . 19 C 20 C » I f F i s t o o s m a l l . LOWESS becomes u n s t a b l e . T h i s w i l l 21 C * a u t o m a t i c a l l y be c o r r e c t e d by i n c r e a s i n g the F . A no te o f 22 C * t h i s , i n c l u d i n g the r o b u s t i t e r a t i o n s t a t u s i s w r i t t e n on 23 C • u n i t 6 . 24 C * 25 C * T h i s s u b r o u t i n e c a l l s on a s o r t i n g r o u t i n e ISORT, wh ich i s 26 C * a F o r t r a n c a l l a b l e r o u t i n e on the UBC-MTS-G ' L I B R A R Y . 27 C 28 C 29 IMPLICIT REAL*4 ( A - H . L . P - Z ) 30 R E A L M X Y Y ( 3 . N ) , X Y T ( 3 . N ) , W ( N ) 31 WRITE(6,* ) '>>>> LOWESS' 32 C 33 C * F = FRACTION O F POINTS TO BE INCLUDED IN LOWESS STEPS 34 C « 10= N * F = NUMBER OF POINTS CORR. TO F 35 C 36 IQ=NINT(N*F) 37 I F ( 1 0 . L T . 3 ) 1 0 = 3 38 C « SORT XYY T O INCREASING X (COLUMN 1) 39 CALL I S O R T ( X Y Y . 3 . N . 1 , N . 1 . 3 . 1 ) 40 C 4 1 6 10 CONTINUE 42 00 503 I=1.N 43 X Y Y ( 3 . I ) = 0 . 44 X Y T ( 1 , I ) " O . 45 X Y T ( 2 . I ) = 0 . 46 X Y T ( 3 . I ) = 0 . 47 503 W(I)=1. 48 C 49 C • START ITERATIVE ROBUSTNESS LOOP 50 C 51 IITR=0 52 F F = ( I O / F L O A T ( N ) ) 53 W R I T E ( 6 , • ) ' N * F = ' . 1 0 . ' F = ' . F F 54 600 CONTINUE 55 W R I T E ( 6 . * ) 'NUMBER OF ITERATION = ' . I I T R 56 C 57 C * START LOOP FOR EACH POINT 58 C - 298 -59 00 500 1=1.N 60 C 61 C * DETERMINE 10 NEAREST NEIGHBOURS TO X Y Y O . I ) 62 C 63 X Y T ( 1 , 1 )=XYY( 1 , 1 ) 64 X Y T ( 2 , 1 ) = XYY ( 2 , 1 ) 65 XYT(3 .1 )=W( I ) 66 C 67 K1=I+1 68 K 2 - I - 1 69 DO 501 K=2, I0 70 I F ( K 1 . L E . N ) T H E N 71 D 1 = A B S ( X Y Y ( 1 , 1 ) - X Y Y ( 1 . K l ) ) 72 ELSE 73 D 1 » A B S ( X Y Y ( 1 . 1 ) - X Y Y ( 1 . N ) ) 74 ENDIF 75 I F ( K 2 . G E . 1 ) T H E N 76 D 2 = A B S ( X Y Y ( 1 , I ) - X Y Y ( 1 . K 2 ) ) 77 I F ( D 2 . L E . D 1 ) T H E N 78 X Y T ( 1 , K ) » X Y Y ( 1 , K 2 ) 79 X Y T ( 2 . K ) " X Y Y ( 2 . K 2 ) 80 XYT(3 .K)=W(K2) 81 K2=K2-1 82 ELSE 83 X Y T ( 1 , K ) » X Y Y ( 1 , K 1 ) 84 X Y T ( 2 . K ) = X Y Y ( 2 , K 1 ) 85 XYT(3,K)=W(K1) 86 K1=K1+1 87 ENDIF 88 ELSE 89 X Y T ( 1 , K ) = X Y Y ( 1 , K 1 ) 90 X Y T ( 2 . K ) = X Y Y ( 2 , K 1 ) 9 1 v XYT(3,K)=W(K1) 92 K1=K1+1 93 ENDIF 94 501 CONTINUE 95 D = ABS(XYY( 1 , I ) - X Y T ( 1 , 1 0 ) ) 96 I F ( D . E O . 0 ) T H E N 97 10=10+1 98 GOTO 610 99 END IF 100 C 101 C * CALCULATE THE WEIGHT FOR EACH XYT(K) AND SUMMARY STATISTICS 102 C * FOR THE WEIGHTED REGRESSION . 103 C 104 SXY=0. 105 SX=0. 106 SY=0 107 SX2=0. 108 ST=0. 109 DO 502 K = 1 . 1 0 110 U = ( A B S ( ( X Y Y ( 1 , I ) - X Y T ( 1 , K ) ) / D ) ) * * 3 111 IF (U . L T . 1 . )THEN 112 T = ( 1 . - U ) * » 3 113 ELSE 114 T=0. 115 ENDIF 116 T K = T * X Y T ( 3 . K ) - 299 -1 17 XYT(1,K)=XYT(1.K) 1 18 XYT(2,K)=XYT(2,K) 1 19 C 120 SXY=SXY+(XYT(1,K)*XYT(2,K)*TK) 121 SX=SX+(XYT(1,K)*TK) 122 SY=SY+(XYT(2,K)*TK) 123 SX2=SX2+((XYT(1,K)**2)*TK) 124 ST=ST+TK 125 502 CONTINUE 126 C 127 c * CALCULATE THE REGRESSION COEFFICIENTS AND THE LOWESS 128 c 129 SDIV=(SX2/SY)-SX**2/(ST*SY) 130 IF(SDIV.E0.0..OR.ST.EO.O.)THEN 131 10=10+1 132 GOTO 610 133 ENDIF 134 BW=((SXY/SY)-(SX/ST))/((SX2/SY)-SX**2/(ST»SY)) 135 AW=(SY/ST)-(BW*SX/ST) 136 XYY(3,I)=AW+(BW*XYY(1,I)) 137 500 CONTINUE 138 c 139 IF(IITR.LT.ITER)THEN 140 c 141 c * COMPUTE MEDIAN OF ABSOLUTE RESIDUALS :RM 142 c 143 DO 510 1=1.N 144 510 W(I)=ABS(XYY(2.I)-XYY(3.I)) 145 CALL ISORT(W,1,N,1,N.1.3,1) 146 M=NINT(I/2.) 147 RM=W(M) 148 c 149 c * COMPUTE THE ROBUST WEIGHTS 150 DO 511 1 = 1 ,N 151 U=ABS((XYY(2.I)-XYY(3,I))/(6.*RM)) 152 IF(U.GE.1.)W(I)=0. 153 IF(U.LT. 1 . )W(I) = (1.-U**2)**2 154 511 CONTINUE 155 IITR=IITR+1 156 GOTO 600 157 ENDIF 158 c 159 RETURN 160 END 

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