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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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SPATIAL  S C A L E S O F SENSIBLE HEAT  REPRESENTATIVENESS  OF  FLUX  FLUX  VARIABILITY  MEASUREMENTS  SURFACE LAYER STRUCTURE OVER SUBURBAN  AND  TERRAIN  By H A N S PETER EMIL  Dipl.  Natw., The Swiss F e d e r a l  SCHMID  I n s t i t u t e o f Technology  Zurich, Switzerland,  '  A THESIS SUBMITTED  1984  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOHPY  in THE FACULTY OF GRADUATE STUDIES (Geography Department)  We accept  t h i s t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA June 1988 © Hans P e t e r Emil Schmid, 1988  (ETH).  In  presenting  this  degree at the  thesis in  partial  fulfilment  of  the  requirements  for  an advanced  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  department  or  publication  thesis for by  his  or  scholarly purposes may be granted by the head of her  It  is  understood  that  copying  or  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)  representatives.  my  ABSTRACT  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  s p e c i f i c surface sources or sinks upon an eddy-flux measurement i n the surface layer. These concepts are tested i n a suburban r e s i d e n t i a l area i n Vancouver, and  a  B.C.,  Canada.  digitized  data-bases  Remotely sensed  roughness  for  the  element  Fourier  surface  inventory  temperatures are  transforms  to  used  as  develop  representativeness c r i t e r i a f o r eddy-flux measurements. A set of sensible  heat  corresponding  flux source  recommendations  for  representativeness  of  measurements area the  at  six  c a l c u l a t i o n s are objective  sites used  evaluation  sensible heat  flux  of  to the  and  the  formulate spatial  measurements over  a  suburban area. The v a l i d i t y of the suggested evaluation methods i s confirmed by the observations. Internal boundary layer growth, estimated by the source  area  model, compares well with e x i s t i n g work. Some consequences of complex  surfaces  discussed.  on  the  surface  layer structure  are  briefly  - iv TABLE  OF CONTENTS  ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF TABLES  vii  LIST OF FIGURES  viii  LIST OF SYMBOLS AND ABBREVIATIONS  xiv  ACKNOWLEDGEMENTS PART I  1.  :  xx  INTRODUCTION  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  4.  SUMMARY  Spatial  31  Distribution  of  Surface  Temperature and  Roughness Elements 4. 1  33  SURFACE TEMPERATURE  4.1.1  Infrared Temperature  Remote  33 Sensing  of  Surface 36  - v 4.2  ROUGHNESS ELEMENTS  4.2.1  38  Data A q u i s i t i o n f o r the Roughness Inventory  39  5. S p a t i a l V a r i a b i l i t y : Homogeneity and Representativeness...46  6.  5.1  INTRODUCTION  46  5.2  HOMOGENEITY  48  5.3  REPRESENTATIVENESS  51  5.4  DISCUSSION  55  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  66  6.5  A RECIPROCAL PLUME MODEL TO ESTIMATE THE SOURCE  PATCHINESS  AREA 7.  73  The Source Area Model (SAM) 7.1  80  INTRODUCTION  7.2  AN  APPLIED  80 DISPERSION  MODEL  BASED  ON  METEOROLOGICAL SCALING PARAMETERS 7.2.1 The Gryning e t . a l . (1987) P.D.F. Model (GEA)  7.3  7.4 7.5  81 81  7.2.1.1  V e r t i c a l Dispersion  82  7.2.1.2  Lateral Dispersion  87  IMPLEMENTATION OF THE DISPERSION SUB-MODEL IN THE SOURCE AREA MODEL  90  THE FORTRAN-77 CODE OF THE SOURCE AREA MODEL  94  A STATISTICAL VERSION OF THE NUMERICAL MODEL (mini-SAM)  95  - vi P A R T III :  MEASUREMENTS A N D RESULTS  8. Evaluation of S p a t i a l V a r i a b i l i t y v i a Spectral Analysis..105  9.  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  Simultaneous Eddy C o r r e l a t i o n Measurements at Two 168  S i t e s : Five Configurationsons 9. 1  INTENTIONS  168  9.2  THE SITES  171  9.3  EQUIPMENT  9.4  INSTRUMENT COMPARISON  9.5  THE  178  INSTRUMENTATION AND DATA  SPATIAL  REPRESENTATIVENESS  185 OF  EDDY  CORRELATION MEASUREMENTS P A R T IV  10.  :  DISCUSSION A N D C O N C L U S I O N S  How Good are the SAM-Estimates ? - Comparison with E x i s t i n g Work  11.  199  The Structure of the Surface Layer over Complex Surfaces  12.  187  Summary of Conclusions  214 221 224  REFERENCES APPENDIX A :  Remotely Sensed Surface Temperatures  231  APPENDIX B :  V a l i d a t i o n 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  -  vii  -  LIST OF TABLES  2.1  Possible  framework  for  urban  climate  classification 7.1  Standardized  12 set  r e l a t i v e t o the 0 . 9  of  isopleth  dimensions  isopleth  96  7.2  Summary o f mini-SAM p o l y n o m i a l f i t  7.3  m i n i - S A M 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 d a t a f o r Cabauw SAM-runs  201  10.2  Computation  of  exponent  of  coefficients  internal  boundary  l a y e r growth  212  A. l  T e c h n i c a l d e t a i l s o f t h e remote s e n s i n g  B. 1  Statistics  for  'Prairie-Grass' B.2  B.3  99  Statistics  for  Hanford-30  data)  Statistics  for  Hanford-30  data)  CIC/Q  model  validation  flights....232 (with  data) CIC/Q  241 model  validation  (with 245  <r -model y  validation  (with 246  - viii -  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 3.1  Local  Nusselt  12  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 4.4  43  Same as Figure  4.3,  close-up  of Mainwaring  substation  44  5.1  Representativeness vs. sample size  6. 1  Internal  boundary  layer  interfaces  ...56 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  6.6  75  Integrated effect fraction : a) total integrated effect b) fraction 'P' of the total integrated effect.  77  - ix 7.1  The s c a l i n g regions of the atmospheric boundary layer  7.2  83  The source  area  characteristic  Is defined  dimensions  by the set of  of  i t s bounding  isopleth  92  7.3  Example plot of source area model r e s u l t s  93  7.4  mini-SAM v a l i d a t i o n s c a t t e r - p l o t : dimension a  100  7.5  Same as Figure 7.4 f o r dimension b  100  7.6  Same as Figure 7.4 f o r dimension c  101  7.7  Same as Figure 7.4 f o r dimension d  101  7.8  Same as Figure 7.4 f o r dimension x  102  m  7.9  Same as Figure 7.4 f o r area A  102  0.9  8.1  The F r i s b e e - f i I t e r  8.2  Daytime  107  temperature  sub-domains  selected f o r  Fourier-transforms 8.3  110  Nighttime temperature  sub-domain selected f o r  Fourier-transform 8.4  False  colour  Ill image  of  daytime  surface  temperature i n the study area 8.5  113  Dominant s p a t i a l scales i n suburban c i t y - b l o c k system  8.6  115  Same as Figure  8.4; s p e c i a l  colour coding to  show s t r e e t s and houses 8.7  Perspective  view  of  117 daytime  temperature  'topography' i n area (0) 8.8  Temperature (0)  variance  spectrum,  119 daytime,  area 120  - x 8.9  D i r e c t i o n a l d i s t r i b u t i o n of surface temperature variance : daytime, area (0)  8.10  121  Same as Figure 8.8 : daytime, area (1), 128x128 p i x e l 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) 8. 14  125  Normalized integrated r a d i a l variance  spectrum  of surface temperature :daytime, area (1) 8.15  Sectorial variance  break-up spectrum  of normalized of surface  125  integrated  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' 8.27  False  colour  142 image  of  nighttime  temperature i n the study area  surface 143  - xi8.28  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 f o r Fourier transforms  8.34  Perspective  view  of  roughness  150  element  d i s t r i b u t i o n ( 1 2 8 x 1 2 8 p i x e l 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 r a d i a l variance spectrum  of roughness elements : area SE 8.39  Sectorial  break-up  of normalized  153 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  - xii 8.49  Same as Figure 8.39 : area NE  162  9.1  Measurement s i t e s  172  9.2  Culloden s i t e (looking NE)  173  9.3  Argyle s i t e (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 s i t e (looking NE)  176  9.8  Memorial West s i t e (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 9.12  182  D a i l y v a r i a t i o n s of Q  at Sunset  (dashed) and  H  J  Culloden ( s o l i d ) 9.13  183  Scatterplot of Q  instrument comparison  186  H  9. 14  Scatterplot of Q  vs. Q  Hmob  9.15  Scatterplot of  9.16  Spread ^  of  186  Hsun  vs. 0.9-source area s i z e  Q„  H D I F F  (residuals of |Q„  vs. 0.9-source  H D I F F  area  189  size  - LOWESS-curve | )  189  9.17  Same as Figure 9.15, f o r c a l i b r a t i o n period  190  9.18  Same as Figure 9.16, f o r c a l i b r a t i o n period  190  9.19  Three examples  of source areas i n the study  area 10.1  u *22.S  195 vs. u  (at Cabauw)  201  »3.S  10.2  Cabauw SAM-results (JD 140,  3.5 m)  203  10.3  Cabauw SAM-results (JD 140, 22.5 m)  204  - xiii 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 s i 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 a i r c r a f t  232  A. 2  MSS scan-head  233  A.3  Plan of ground t r u t h s i t e s at Langara Community College  A.4  235  VZIP-image  of  temperatures  at  ground  truth  sites  236  A. 5  Daytime p i x e l value-temperature r e l a t i o n  238  B. l  Scatterplot  of  CIC/Q  model  validation  (with  ' P r a i r i e Grass' data) B.2  Effects Dyer  of and  Dyer  241  (1970) p r o f i l e s  Bradley  (1982)  compared  profiles  to on  p.d.f.-model r e s u l t s B.3  Effects  of k = 0.41  242 compared to k = 0.40  on  p.d.f-model r e s u l t s B.4  Scatterplot  of  242  CIC/Q  model  validation  (with  Hanford-30 data) B.5  Scatterplot  of  245 <r-model  validation  (with  y  Hanford-30 data)  246  - xiv -  LIST  OF SYMBOLS AND  Note :  ABBREVIATIONS  o Some symbols are only relevant f o r one  equation  and  are not l i s t e d here. o  The  Systeme  equivalents  Internationale  are  used  (SI)  throughout  units  this  and  work,  their unless  otherwise s p e c i f i e d . They are indicated i n brackets [ ] .  a  i s o p l e t h dimension  a, a i  2  A  (Figure 7.2)  constants i n non-dimensional  [m]  p r o f i l e s (Chapter 7)  - empirical constant (equation (3.5)) - f u n c t i o n of the shapefactor s (equation (7.6))  A  p  b b  source area contained i n P-isopleth [m ] i s o p l e t h dimension  f  (Figure 7.2)  [m]  slope of l i n e a r functional r e l a t i o n s h i p (Appendix E)  b^, b  slope of l i n e a r regression (Appendix E)  B  - empirical constant (equation (3.7)) - f u n c t i o n of shapefactor s (equation  c  - i s o p l e t h dimension  (Figure 7.2)  7.7))  [m]  - empirical constant (equation (7.8)) c  s p e c i f i c heat of a i r [J k g K ] _1  p  c  -1  a r b i t r a r y constants 1,2..  CIC  Cross-wind  Integrated Concentration [kg m  Cov(') d  covariance - i s o p l e t h dimension - zero displacement  (Figure 7.2) height  [m]  [m]  ]  - XV  -  - c o e f f i c i e n t of agreement (equation (E.5))  —2 —1 E E  water vapour f l u x density [kg m fc  s  ]  t o t a l e f f e c t (equation (6.15))  Exp.  peak exposure (Appendix B) [10  E(-)  expected value  f  - frequency  f  - constant i n LOWESS (Appendix E) f u n c t i o n of non-dimensional t r a v e l  y  kg s m  ]  time  (equation  (7.12)) F  - percentage of e f f e c t i v e f e t c h (equation (6.10)) - t o t a l irradiance (Appendix A) [W m  ]  g  a c c e l e r a t i o n of g r a v i t y [m s  ]  GHIS  Gryning et.al.  Gr  Grashof number (equation (3.6))  h  power-spectral density (equation (5.3))  H  integrated spectrum power (equation (5.10))  JD  J u l i a n Day  k  von Karman constant  K  eddy d i f f u s i v i t y  1  c h a r a c t e r i s t i c length scale  L  - Monin-Obukhov length [m]  (1987)  [m s ] 2  _1  - length of data^domain  [m]  (Chapter 5)  LAT  Local Apparent Time (solar time)  m  - empirical exponent  [m]  (equation (3.7))  - sample s i z e (Chapter 5)  - xvi - empirical exponent (equation (3.5))  n  - exponent of internal boundary layer growth (equation (6.4)) - number of data Nu  Nusselt number (equation (3.4))  0  observed v a r i a b l e (Appendix E)  P  empirical constant (equation (7.8))  p.d.f.  p r o b a b i l i t y density function  P  - predicted variable (Appendix E) -  fraction  of t o t a l  effect  contained  i n P-isopleth  (equation (6.16)) PBL  Planetary Boundary Layer  q  constant i n LOWESS (Appendix E)  Q  source strength [kg s ] 1  V ^ H D I F F  _2 sensible heat f l u x density [W m ] non-dimensional  inter-site  Q -difference  (equation  H  (9.1)) r  aerodynamic H  resistance to heat transfer [s m" ] 1  measure of representativeness (equation (5.7))  R Re Ri RMSE RMSD R2  Reynolds number Richardson number Root-Mean-Square-Error  (equation (E.2))  Root-Mean-Square-Difference  (equation (E.2))  r e s i d e n t i a l (2 levels) land-use c l a s s (Auer, 1978) c o e f f i c i e n t of determination (equation ( E . l ) )  R  2  shape f a c t o r (equation (7.9)) s surface types 1, 2 (Chapter 6) SI, S2  SAM t/T  Source Area Model y  non-dimensional t r a v e l time (equation (7.12))  T  absolute temperature [K]  u  wind-speed  u  -  XVI1  -  [m s" ] 1  roughness wind-speed  m  [m s ] - 1  U  uniform or bulk wind-speed  v  l a t e r a l wind-speed  VZIP  [m s" ] 1  [m s ] - 1  Vectrix-Zenith-Image-Processor  VAR( ')  var i ance  w  v e r t i c a l wind-speed  w  convection wind-speed  v  source weight per area (equation (7.18))  W  weight of a d i s c r e t e source (Chapter 6)  #  x^  ( i )  distance  from  [m s ]  leading  - 1  [m s ] - 1  edge  of  equilibrium  e f f e c t ) boundary layer (Chapter 6) X  L  (initial  [m]  along-wind width of evaporation s t r i p  (equation (6.8))  [m] X  F  x  F-percent e f f e c t i v e f e t c h (equation (6.10)) [m] upwind distance of maximum source [m]  max  X  a r b i t r a r y variable  z  height [m]  z  t  z  height of mixed layer, inversion height [m] sensor height [m]  s  Z Zz  T  Q  roughness length (temperature) [m] roughness length (wind) sublayer [m] depth of the roughness [m]  - xviii 1/T [ K ] -1  gamma function height of internal boundary layer [m] height of equilibrium boundary layer [m] emissivity a i r temperature, p o t e n t i a l temperature [K, or °C]  2 — 1 heat conductivity [m s ] wavelength [m]  2— 1  kinematic v i s c o s i t y [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 s t r e s s [N m ] - l a g (time or space) non-dimensional p r o f i l e (momentum, heat) normalized concentration [s m ] integrated non-dimensional p r o f i l e (momentum, heat) wind-direction  [rad, or degrees]  source weight source weight function (bar) mean of variable (circumflex) variable  denotes  (this  abbreviation,  the s p a t i a l  variability  i s meant  simply  as  independent  of  the  a  of a  shorthand statistical  - xix representation) (•)  ( t i l d e ) estimated value of v a r i a b l e  subscripts  :  caiib  calibration period  f  forced convection  film  film  1  laminar convection  H  heat  max  maximum  mob  mobile s i t e s  M  momentum  n  natural  R  radiative  sun  Sunset  sys  systematic  tot  total  unsys  unsystematic  temperature  convection temperature  site  -  XX  -  ACKNOWLEDGEMENTS  Without the encouragement, patience and shared wisdom of many i n d i v i d u a l s t h i s 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, i n 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 i n t e r e s t s . He provided guidance by putting my research, climatology and science into a greater context, rather than p o i n t i n g i n a p a r t i c u l a r d i r e c t i o n . His sense of humour - and the 'Great Laughter' - always was and always w i l l be appreciated. S i m i l a r l y , I am indebted to Dr. Douw G. Steyn f o r h i s open door and mind, both of which were often used to chase down problems concerning almost anything i n the universe. Informal discussions over a glass of beer or through ice-hockey face-masks have helped to shape many ideas f o r both t h i s thesis and future work. By convincing me to come to f i e l d s c h o o l i n Spring of 1985, he just might have changed my l i f e i n 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 e s p e c i a l l y f o r the f i n a l draft of t h i s thesis. Their help and encouragement Is greatly appreciated. Dr. K e i t h Knight from U.B.C.'s S t a t i s t i c s Consulting And Research Laboratory has helped to t r a n s l a t e some of my i l l - f o r m e d 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 t h i s work. While working on t h i s 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 h a l f years I was g r e a t l y influenced by many friends and f e l l o w Graduate Students. I p a r t i c u l a r l y want to thank Helen Cleugh and Sue Grimmond f o r t h e i r support during the 1985/86 fieldseasons and - together with Catherine Souch - f o r many hours of brainstorming over our morning coffee. Scott Robeson and Matthias Roth were and are invaluable as f r i e n d s and f e l l o w s c i e n t i s t s . I thank my parents and family f o r t h e i r continued support over the years and across so many miles and f o r sending a l l those 'hints' to remind me where 'Home' i s . Lastly, I want to thank Carolyn Porter f o r being so patient and f o r being my companion i n a l l walks of l i f e .  - 1 -  PART I : INTRODUCTION  1.  Academic and Spatial Context  RATIONALE  1.1 In  recent  boundary  years  layer  the  numerical  processes f o r  modelling  various  applications  c o n s i d e r a b l e a t t e n t i o n by r e s e a r c h e r s layer  meteorology.  spatial layer  With  resolution, processes  such and  progressively finer fundamental  similarity  not  consistent  areas  variable  the  of  is  thermal  not  provide  framework  heterogeneous Secondly, fluxes  over  incorporate  surface  variability  development  at  faces  for  two  mesoscale  hypothetical,  which This  areas  present  is  infinite,  surface  especially  for  for  o r c h a r d s and f o r e s t s ,  interaction,  regimes. a  layer  true  which  are  of s c a l e s i n respect to t h e i r  moisture  conceptual  This  crop-land,  and  does  improved  their  the  for  human-climate  meteorology  and  for  target  with  developed.  on a wide v a r i e t y  geometry,  boundary  fields  to  principal  agricultural main  of  in a l l  need  scales.  and smooth p l a n e s theory  received  :  of  urban a r e a s , the  spatial  most  are  homogeneous  models  atmospheric has  sophistication  account  difficulties  Firstly, models  increased  of  Present  consistent  transfer  i.e. highly  surface  surface  layer  theoretical  processes  over  or such  terrain. experimental very  data  heterogeneous  of  energy,  terrain  mass are  and  sparse,  momentum due  to  - 2 methodological problems i n the measurement process. Fluxes close to  a  complex  surface  features (roughness  are influenced  by  individual  surface  elements and sources or sinks). This r a i s e s  the d i f f i c u l t y of i n t e r p r e t i n g a f l u x measurement, as well as obtaining f l u x measurements representative of a large area. The assessment of the s p a t i a l v a r i a b i l i t y of surface f l u x e s i s thus g r e a t l y impaired by our i n a b i l i t y to i d e n t i f y a s p e c i f i c surface region i n 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 f l u x of  momentum, heat or mass. Interest i n the s p a t i a l these processes i s considered to be j u s t i f i e d However,  i t also a r i s e s  applications,  such  from  specific  v a r i a b i l i t y of  i n i t s own r i g h t .  problems  i n practical  as the estimation of regional evaporation,  obtaining an areal average of roughness windspeed or estimating the t o t a l pollutant emission of an area source. The  present  sensible heat  study flux.  number of reasons, in  nature.  sensible turbulent  It may  heat  flux  focusses  on the s p a t i a l  This s p e c i f i c  v a r i a b i l i t y of  f l u x has been chosen f o r a  some are fundamental and some are pragmatic be s a i d  that  the accurate  i s the "easiest"  fluxes : sonic  measurement of  of the various  anemometer/thermocouple  surface  systems are  very simple to use and the underlying technology i s f a r advanced compared to instruments f o r the measurement of other fluxes. In addition, two s i m i l a r eddy c o r r e l a t i o n systems f o r sensible heat f l u x were r e a d i l y a v a i l a b l e to perform simultaneous at two s p a t i a l l y separated s i t e s .  measurements  - 3 However, t h e s e n s i b l e heat f l u x is  significant  also  in  more  regime o f t h e atmosphere state :  i n stable  (and i t s s p a t i a l  fundamental  variability)  terms.  The  i s a p o w e r f u l c o n t r o l on i t s  thermal turbulent  c o n d i t i o n s , t u r b u l e n c e and m i x i n g a r e  damped  whereas t h e y a r e g r e a t l y enhanced i n u n s t a b 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 t o t h e t u r b u l e n t k i n e t i c e n e r g y . the  daytime  clouds,  in  sensible  atmosphere regime o f all  and  and the  the  heat  thus  absence flux  is  condensation  the  determines  boundary l a y e r .  of  to  main heat a great  processes  input  extent  As a consequence,  During or  into  the it  the  thermal  influences  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 t h u s f e e d s back upon  itself  in a  strongly non-linear  manner.  The s u r f a c e  sensible  heat f l u x i s a l s o one o f t h e p r i n c i p a l c o n t r o l s o f t h e h e i g h 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  estimates of t h i s  are therefore  an e s s e n t i a l  evaluates  volume o f a i r t h r o u g h w h i c h p o l l u t a n t s ,  the  flux  i n p u t t o any mixed l a y e r model w h i c h moisture  and heat a r e mixed and d i s p e r s e d . This  study  seeks  to  elucidate  the  spatial  scales  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 o v e r a complex s u r f a c e ,  of  viz :  a s u b u r b a n a r e a i n Vancouver, B . C . , Canada. I t a l s o s e t s out estimate  the  surface  c o r r e l a t i o n measurement  source  area  which  influences  an  eddy  i n unstable conditions. Further i t  tries  t o a s s e s s t h e consequences o f a complex s u r f a c e on t h e o f t h e l o w e r p o r t i o n o f the a t m o s p h e r i c s u r f a c e In  the  first  characteristic  instance, of  the  to  the  suburban  results  of  area  under  this  structure  layer. study  are  consideration.  only The  method by w h i c h t h e y were o b t a i n e d , however, may be a p p l i e d t o a  - 4 v a r i e t y of complex surfaces. Thus, i t s f i n d i n g s o f f e r p o t e n t i a l utility  to both the boundary-layer modelling  communities.  The  and remote  1.2  measurement  methods used are drawn from f i e l d s  micrometeorology, engineering,  and  agricultural  turbulent  meteorology,  d i f f u s i o n modelling,  heat  applied  such  as  transfer statistics  sensing.  STUDY OBJECTIVES The o b j e c t i v e s of t h i s work can be summarized as f o l l o w s :  1.  To develop a method by which the small s c a l e v a r i a b i l i t y of s e n s i b l e heat f l u x may be  2.  To  formalize  the concept  evaluated. of  "representativeness"  i n the  h o r i z o n t a l s p a t i a l sense and i n the context of s e n s i b l e 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 vertically. 4.  To develop a model which estimates the r e l a t i v e i n f l u e n c e of each surface element on the s e n s i b l e 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 w i t h i n the surface f a b r i c on the s t r u c t u r e of the unstable  surface  layer  of  the  atmosphere  over  complex  surfaces. 6.  To support the hypotheses and t h e o r e t i c a l f i n d i n g s of t h i s work by f i e l d  observations.  - 5 1.3  THE STUDY AREA The  observation programme f o r t h i s work was conducted  suburban area of Vancouver,  British  Columbia,  been the target area f o r a number of process climate  and micrometeorology  1979b; Kalanda, and  Cleugh,  with  other  1979; Steyn,  studies  suburban  boundary-layer  research  oriented urban  i n the past  work was completed  i n the same  comprehensive  It has  (e.g. Oke  1980; Oke and McCaughey, 1983; Oke  1987). The present studies  Canada.  in a  area,  which  surface-layer programme  together  and  (Cleugh,  in parallel form  a  coastal/urban  1988 ;  Grimmond,  1988 ; Roth, 1988 ; Steyn and McKendry, 1988). As a r e s u l t of the abundant "research heritage" i n t h i s area, most of the footwork i n respect to s i t e s e l e c t i o n and surface description (Kalanda, present  and  parameterization  has already  been  completed  1979 ; Steyn, 1980) and can r e a d i l y be adopted f o r the  study.  A brief  summary of the general  s e t t i n g of the  study area within the region and the s p e c i f i c c h a r a c t e r i s t i c s of the  main  suburban  site  (the Sunset  site)  are given  i n the  following. Vancouver i s located between the mouth of the Fraser River and Burrard Inlet on the S t r a i t of Georgia (see Figure 1.1). The lower Fraser V a l l e y forms an extensive lowland which i s bounded by the Cascade Range i n the south and by the Coast Mountains to the north, both of which reach elevations i n the order of 1500 m above sea l e v e l . Fraser  Valley  The climatology of Vancouver and the  i s discussed  by  Hay  and Oke  (1976).  lower  Weather  patterns i n winter are t y p i c a l l y characterized by the passage of  - 6 -  Figure 1.1  :  Map of the Greater Vancouver Metropolitan. Area  - 7 cyclones and t h e i r associated f r o n t a l disturbances, r e s u l t i n g i n high  precipitation  temperatures. their  cloud cover,  but  also r e l a t i v e l y  In contrast, the summers are  persistent  generally  and  small  anticyclonic pressure  commonly known f o r  high-pressure  gradients  and  mild  the  systems.  The  resulting  weak  synoptic flow are conducive to thermally induced d i u r n a l weather patterns.  The  sea-breeze, and  most common of these  meso-scale systems  which i s enforced by the e f f e c t s of  slope-wind  systems (Steyn and  Faulkner,  i s the  mountain/valley-  1986). Due  to  the  sea-breeze and the associated advection, the height of the mixed layer i s t y p i c a l l y confined to only about 500 m (Steyn and 1982; The  Oke,  Steyn and McKendry, 1988). main micrometeorological  s i t e of t h i s study,  tower, i s located i n south Vancouver (Figure 1.1) mainly r e s i d e n t i a l housing  (category R2,  the Sunset  i n an area of  a f t e r Auer, 1978)  with  occasional schools or commercial neighbourhood centres.The  mean  b u i l d i n g 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 a n a l y s i s by the same author r e s u l t e d in  a  mean roughness  length  of  0.5  Lettau,1969) and a zero-displacement The Power  m  (using  the  height of 3.5  method  of  m.  tower i s located i n the compound of the B.C.-Hydro  and  Authority's  The  Mainwaring  substation  (Figure  steel-frame structure has an o v e r a l l height of 27.5 base of the  m. Since the  of 5 m,  the  height  above ground needs to be corrected by t h i s amount  (see  Figure  9.11,  tower i s below an escarpment  1.2).  i n Section 9.3,  on instrumentation), so that  the  maximum  F i g u r e 1.2  :  The Sunset tower, l o o k i n g e a s t  - 9 -  F i g u r e 1.3: View from l o o k i n g west  the top o f Sunset  tower,  F i g u r e 1.4: View from looking east  the  tower,  top o f Sunset  effective  height  (z-d) i s 19 m above the e f f e c t i v e  zero datum  (Steyn, 1980). Figure westerly  1.3 shows the view from the top of the tower i n the direction :  residential  it  indicates  housing outside  a  uniform  the substation  land-use  compound  of  i n this  d i r e c t i o n . About 100 m to the east of the tower there i s a large school b u i l d i n g (height about 20 m, see Figure 1.4). With strong flow  from  that  sector,  which  i s rare  i n summer  daytime, t h i s may r e s u l t i n some wake e f f e c t s .  and during  PART II : THEORY AND CONCEPTS  2.  Physical C o n t e x t  The  p r e s e n t  s t u d y  u n d e r s t a n d n ig of  the  is  d i r e c t e d  way  in  n ih o m o g e n e t e i s  i n f l u e n c e the  o lw e s t  of  l a y e r s  t e r r a i n .  It  l i m i t e d to  It  is  p h y s i c a l set  s c a l e s of  s y s t e m s and  and  n e c e s s a r y to  the  s t a g e for  the  the  o v e r  and  v e r t i c a l  p r o v d ie a t h i s  s u b u r b a n  s c a l e s .  s y n o p s s i of  l i m i t e d a s p e c t , in  d e t a i l e d a n a l y s i s of  s p a t i a l c o m p e l x t i y of  c o n s d ie r not s c a l e s ,  u p p e r s c a l e  and by  the  the s p a t i a l  p r o b e l m s  L e n s c h o w  (Ed.,  1 9 8 6 ) .  i d e a l i z e d  a r r a n g e m e n t  (Ed.,  F g iu r e of  the  of  is  the  and  the  c o n v e c t i v e l y d r v ie n m x i e d l a y e r ,  l a y e r  and  ( f r o m  b o u n d a r y l a y e r is  a t m o s p h e r e , the  p a ln e t a r y b o u n d a r y l a y e r  m o r e Oke,  l a y e r  c i t y . T h s i a r r a n g e m e n tr e f e r s p r m i a r y l i to w h e r e the  B e n ig c o n c e r n e d w t i h  b o u n d a r y  b o u n d a r y  is  a h e i r a r c h y of n e s t e d  1979) 21 .  u r b a n t e r r a i n it  ( 1 9 8 4 ) .  s u r f a c e  r e e l v a n c e  M c B e a n  the  s y s t e m , but  f o l o w i n g Oke  l i m i t of  r e v e i w e d  one  b e w te e n  C o n c e p t s  s t r u c t u r e ,  in  a v e r y c o m p e l x s y s t e m and  c o n c e p t u a l c o n t e x t of  i n t e r a c t i o n s  b e e n  a s p e c t of  b e t t e r  s u r f a c e  h e a t  p a r t i c u l a r l y  and  the  c o n v e n e i n t to  ( P B L ) .  f l u x of  a  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 .  B e c a u s e of  the  t u r b u e l n t  one  p r o v d in ig  s m a l s c a l e  r e l a t i v e l y s m a l h o r i z o n t a l  t h e r e f o r e u s e f u l  o r d e r to  w h c i h  a t m o s p h e r e ,  f o c u s s e s on  is  the  the  t o w a r d s  the  1984)  r e s e a r c h  h a v e  r e c e n t l y by s h o w s  s t r u c t u r e  an  o v e r  a  d a y m t i e  c o m m o n y l e q u a t e d w t i h the  w h o s e c a p p n ig i n v e r s i o n  o f r m s  - 12 -  (al "Plume' Mixed P8L  layer  UBL  /  , Surface  y  Ilil  L. Rural  layer  /  "  y-Wau.  Urban  Rural  UBL  Surface layer  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 f o r urban climate c l a s s i f i c a t i o n (from Oke, 1984). (a)  TUftAULCHT BOUNDARY LAYERS Mow  1,  Urban !•/•(  canopy (UCD  Turbulent  11.  Urban (UULt  (bl  Characteristic*  Hlohly turbulant, c o n t r o l l e d by r o u a h n a a a  5 H<i> t y p i c a l l y 10 2D-3Q^ t y p i c a l l y 20-40 *  IUMHCI  <*»Km  layer  H i g h l y t u r b u l e n t v e k e a and pluM«a , transition ton*  bounda r y  layer  Turbulant, Includes And n l m d l a y e r s  UKAAH  Urban  lbulldln9)  awrfact  D«p«nd * on a u r f a c e f l u * t * o f h e a t and montntun. . T y p i c a l l y - day 1 lut, n i g h t 0 . 2 >ua  I.  BuiIdinq  3.  Urban  ttatun•  Urban c l i m a t e phanocaane.  H  waka, p l u i M ahado*  Canyon  Slnole lAtlldlnq, t r c a or g a r d e n U r b a n ( t r e a t and bordering building* or treaa  ).  Block (n«IqfibOur-  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 , building cluster*, cuatulua, a t l n l - b r e e s e e  4.  Land-ua*  Residential, C(4MM(Cltk, 1 nilutir l a l ate.  L o c a l c l i e n t at I n c l . winds, cloud a o d l fIcation  •  and  MORPHOLOGY  unit*  City  Local  n<io  ten*  Urban  araa  O - hwtldtng  or 10at  »paclng  10*  3On  200*  O.Skn  Q.SkA  Skn  Skn  Jllu.  lit at l a l a n d , u r b a n 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  iUoiph»ft|  10»  Scale  Micro  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 Utate  D l M f i t l c n t of Lound*ry I m y i t > r i dapLh • of a f f a c t a d u n i t * i n tho* • of u i & u i i i r u c t u r * * or p l a n a r a a . building hatght,  01*enal oni*^ W L  Local  21k« ^ M*SO  d l M A t t o n a of  •orphologlcal  - 13 the l i m i t of the PBL modifies  the  (ideally)  (see e.g.  lowest  adjusted  layers  to  rural  impingement on the c i t y . the  depth  of  the  of  the  surface  urban surface  atmosphere, conditions  which  are  before  their  If the urban area i s extensive enough,  growing urban boundary  entire  Steyn, 1980). The  PBL.  layer w i l l Similarly,  eventually a  rural  include  boundary  the  layer  develops again at the downwind leading edge of the  urban/rural  transition  within  (see  Figure  2.1).  Spatial  variability  the  atmosphere at that scale has received considerable a t t e n t i o n i n the past by studies focussing on various processes  and  aspects  of climate and does not need further elaboration here (see Oke  e.g.  (1979a) f o r 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  P o l l u t i o n Study (RAPS) i n St.Louis (e.g. Ching et a i . , 1984). In the present study, s p a t i a l v a r i a b i l i t y within the land-use  3 scale i s of interest. With length scales up to the order of  10  m and time scales up to one hour, t h i s work looks at v a r i a b i l i t y in  the  micro-a and  micro-|3 scales, according  (1975) c l a s s i f i c a t i o n . The  to  the  lower portion of Figure 2.1  Orlansky i s 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  i n two  portion,  comprises the  termed the  range of v a l i d i t y and Thorn, 1981)  and  inertial  of  sublayer,  Monin-Obukhov s i m i l a r i t y  parts. The  theory  i s the r e s u l t of a double matching  upper  vertical (Raupach process  - 14 of upper-level and surface layer s c a l i n g laws (Tennek.es, 1981). In the i n e r t i a l sublayer, v e r t i c a l p r o f i l e s of wind, temperature and humidity do not depend on external scale lengths such as the characteristic  dimensions of the  surface  often r e f e r r e d to as the constant  flux  fluxes  both  vary  by  only  about  10  v e r t i c a l d i r e c t i o n s (Dyer and  %,  Hicks,  geometry.  layer,  It i s also  because v e r t i c a l  i n the  horizontal  1972). By contrast  and  i n the  lower portion of the surface layer, the p r o f i l e s are influenced by i n d i v i d u a l surface elements and the flow i s three (Raupach and this  layer,  roughness  Thorn,  1981). Several  including  sublayer.  turbulent The  spatial  averaging  spatial  atmospheric  term  process  dimensional  names have been coined  wake layer, transition  transition  inhomogeneities,  layer  or  to  the  layer : as  the  layer  refers  that applies i n t h i s induced  for  by  the  surface,  propagate upwards with eddy-motion, turbulent mixing r e s u l t s i n progressive spatial  horizontal  inhomogeneity  averaging is  in  with  transition  where i t i s greatest, and a height  increasing between  this  transition  zone  or  roughness  boundary of the i n e r t i a l sublayer (Figure 2.1) Table 2.1 nested The  the  The  surface,  i n the atmosphere where the  horizontal inhomogeneities disappear completely. of  height.  sublayer  The upper l i m i t i s the  lower  (Garratt, 1978a). This feature  i s examined i n the discussion of the present work. (from Oke,  scales and  rationale  1984)  summarizes the above framework of  the associated features of urban morphology.  behind  this  classification  of  the  vertical  structure and the relevant horizontal scales i s mainly based on phenomena of a i r f l o w and turbulent transport and  i s well s u i t e d  - 15 f o r the present study. The heat, which controls thermal influences on turbulence and diffusion surface. has  v i a buoyancy, The  energy  i s received  balance associated  been discussed extensively  by  Oke  and  transformed  with t h i s  at  the  partitioning  (1982),Cleugh  and  Oke  (1986) and Oke and Cleugh (1987) with s p e c i a l reference to the study s i t e . Rather than g i v i n g a review of current boundary  layer theory  and surface layer s c a l i n g the reader i s r e f e r r e d to such texts as Tennekes and Lumley (1972), Nieuwstadt  and Van Dop  (1981),  P a s q u i l l and Smith (1983), and Panofsky and Dutton (1984). These are considered to r e f l e c t  the  "state of the a r t " of  boundary  layer theory i n the recommendations by Weil (1985) and have been used as works of reference thoughout t h i s thesis.  3.  The  3.1.  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  INTRODUCTION  The  p r i n c i p a l e f f e c t of turbulence  i n the atmosphere i s the  mixing of atmospheric constituents and properties. V e r t i c a l  and  horizontal  are  d i s c o n t i n u i t i e s and  gradients  of  d r a s t i c a l l y reduced by t h i s mixing. Turbulent  any  kind  sensible heat f l u x  i s a process which i s l a r g e l y generated at the surface and transmitted  through  diffusion.  The  heat  mixing not  only  direction.  As  the  overlying  carrying eddies  i n the  a  atmosphere  vertical,  result,  are  but  surface  by  therefore  also  turbulent subject  i n the  turbulent  then  to  horizontal  fluxes  that  are  induced by a f i e l d of d i s c r e t e surface patches become s p a t i a l l y averaged  with  increasing  sensible  heat  flux  are  height.  thus  Atmospheric  measurements  inherently unable  to  detect  of the  d i s c r e t e nature of i t s s p a t i a l v a r i a b i l i t y at the surface. An obvious s o l u t i o n to t h i s problem i s to assess the variability  of  surface  sensible  surface  layer f l u x ) based  surface  f o r c i n g parameters f o r t h i s  relevant  surface  conditions  spatial  variability  spatial  differences  evaluated  on  heat  of are  i f an estimate  the  the  flux  spatial flux.  constitutes flux.  reduced  The  (or  The  height  of  variability  maximum  extent  f o r the s p a t i a l  turbulent  variability  the  with  any  spatial  to  averaging  of  possible  which may  the  these  then  be  process  by  turbulent mixing i s a v a i l a b l e . This s p a t i a l averaging process i s  - 17 discussed  i n Chapter  presented  i n Chapter  identification  of  5 6.  and  a  model  for  i t s calculation  is  In the following, the focus i s on  surface  parameters  that  may  be  the  used  to  evaluate the s p a t i a l scales of the v a r i a b i l i t y of the sensible heat f l u x f i e l d . The magnitude of the sensible heat f l u x 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 s o l a r energy  input into the Earth-Atmosphere system ( i . e . diurnal and  annual  cycles, f u r t h e r modulated by more i r r e g u l a r large-scale weather variability).  This  type  of  temporal  variability  is  spatially  coherent over scales at least an order of magnitude larger than the  longest  spatial  present study,  dimensions  except  (ca. 5  km)  i n conditions of p a r t i a l  i n areas where orography has a s i g n i f i c a n t cloud cover  (Hay,  considered  cloud cover  effect  but  s o l a r r a d i a t i o n due  the present  on the mean  variability  study area l i e s outside t h i s  into  the  system.  Some  surface  however, change temporally as a response solar  forcing  (e.g.  zone.  Therefore,  temperature  or  for  different  surface  soil  elements,  so  parameters  moisture).  relevant  distribution. surface  Obviously,  parameters  however,  governing  may,  Due  to  (e.g. thermal) can that  not  magnitudes of these surface parameters change, but spatial  spatial  to the change i n the  differences i n surface material the response vary  of  to o r o g r a p h i c a l l y induced cloud,  the s o l a r f o r c i n g i t s e l f does not introduce substantial variability  and  1984). The northern side of Burrard Inlet i n  Greater Vancouver i s subject to strong s p a t i a l incoming  i n the  the  only also  the their  distribution  sensible  heat  of  flux  variability  have  to  analysis based on  be  18  compiled  at  a  this distribution  specific  is strictly  time  and  the  speaking  only  v a l i d f o r the s i t u a t i o n at the time of the data acquisition or f o r times when the d i s t r i b u t i o n i s s i m i l a r to that s i t u a t i o n .  To  amend t h i s problem, i t i s suggested to compare two datasets that are expected to have d i f f e r e n t surface parameter d i s t r i b u t i o n s (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 f l u x i s conveniently given i n terms of a MoninObukhov s i m i l a r i t y  r e l a t i o n which,  i n i t s integrated version,  takes the form  Q  H  =  - ( 9 - 8 ) - k p c u , / (ln((z-d)/z )-i/» (z/L)) z  T  Q  (3.1)  .  H  This equation i s a semi-empirical r e l a t i o n and was developed f o r heat  transfer  over  ideal  surfaces  homogeneous planes which contain no  (i.e.  for  infinite,  large b l u f f bodies).  It i s  v a l i d only i n the i n e r t i a l sublayer portion of the surface layer i n which turbulent t r a n s f e r i s free of any  explicit  on  consequence  external  asymptotic  length double  s i m i l a r i t y and Specifically, Z/Z -» q  scales, limit  surface the  as  a  direct  matching  process  layer s i m i l a r i t y  lower l i m i t  of the  of  each  of  an  upper  layer  laws (Tennekes,  1981).  inertial  sublayer,  where  oo, has to be well above the roughness elements.  In contrast to ideal conditions, the present the  dependence  i n t e r a c t i o n of small-scale heat f l u x sources other  In  an  environment,  where  the  work considers or sinks  larger  with  roughness  elements (e.g. buildings) are l i k e l y to be of the same s p a t i a l  scales at which t h i s i n t e r a c t i o n occurs. Since the d i s t r i b u t i o n of  sources  and  sinks  is  three-dimensional  processes between the roughness elements, will  the  i n the canopy layer,  have to enter our considerations. Clearly,  under which the homogeneous r e l a t i o n  diffusion  (3.1) may  the conditions be used are not  met. Current  boundary  consistent  and  canopy layer conditions  layer  unequivocal  meteorology  not  provide  theory f o r transfer processes  (e.g. Baldocchi and  of  does  Hutchison,  horizontal heterogeneity.  a  in a  1987), or even i n  The  scenarios  where  a p p l i c a t i o n s of surface layer theory are of greatest p r a c t i c a l interest ( i . e . i n forests, a g r i c u l t u r a l crops, urban areas etc.) do not meet the ideal equation conditions i n general. Mainly  due  to the lack of anything better, the r e l a t i o n s h i p s developed f o r homogeneous modified  surfaces  with  especially  some  true  for  have  therefore  been  success,  in  such  modelling  purposes  Garratt (1978b), Shuttleworth and Wallace these  works heterogeneity  changes (Taylor, 1970)  i s limited  used,  and  situations. (e.g.  partly This  Taylor  (1970),  (1985) and others). In  to one  dimensional  step  or the transfer processes are looked at  as averages over larger areas (Garratt, 1978b; Shuttleworth Wallace,  is  and  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 i s of interest and concepts such as the f r i c t i o n v e l o c i t y and  especially  a  roughness  length  become  meaningless  applied to individual surface elements. Therefore not  be  used  to  discuss  this  semi-logarithmic  profiles  of  variability, wind and  since  temperature  (3.1)  when should  i t assumes above  each  - 20 surface  element  surfaces.  with  The  correlations  used  guidance  local  results in  biometeorologists most  no  of  the  advection  empirical  hydraulic  from or  engineering  1973)  (Monteith,  on  -  subject  neighboring  semi-empirical and  adopted  may  be  able to provide  of  small-scale  heat  by the flux  v a r i a b i l i t y and the relevant c o n t r o l l i n g v a r i a b l e s , since these c o r r e l a t i o n s r e f e r to the l o c a l heat f l u x from a s i n g l e element, rather than an average over a large area.  3.2  PRINCIPLES  The greatly  a n a l y s i s of convective heat simplified  by  using  t r a n s f e r problems has  non-dimensional  groups  of  q u a n t i t i e s involved. In t h i s way s i m i l a r t r a n s f e r s i t u a t i o n s  been the may  be compared : the c o r r e l a t i o n s of the relevant non-dimensional groups provide a q u a l i t a t i v e and quantitative expression f o r the heat  t r a n s f e r regime. For such an analysis the convective  flux  i s most conveniently expressed  i n terms of a  heat  temperature  difference between the surface and the f l u i d and an aerodynamic resistance r  to the t r a n s f e r : H  Q  H  =  p c  p  K  (9 -6 ) / r o z ' H  (3.2)  .  Note that, i n an ideal and homogeneous case, compatible  (3.1)  and (3.2)  through the appropriate expression f o r r  are  : H  r  =  ( l / ( k - u ))-(ln((z-d)/z  H  The fact that r  *  i n (3.2)  (z/D) T  (3.3)  H  i s a 'black box' and does not assume a  - 21 s p e c i f i c form of p r o f i l e , in  (3.3),  such as the semi-logarithmic p r o f i l e  i s the very reason  why  i t i s appropriate f o r the  present purposes. The most important non-dimensional r  group which contains t h i s  i s the Nusselt number : H  Nu  =  p c  l/(ic r )  P  The  Nusselt  conductive  number  may  thermal  be  interpreted  resistance  resistance of the f l u i d  .  (3.4)  H  to  as the r a t i o  the  of the  convective  thermal  (Welty et al., 1976) and Is therefore a  measure of the e f f i c i e n c y of the convective t r a n s f e r compared to the conductive transfer of heat i n the same medium. Just as the Reynolds viscous  number forces  i s a convenient associated  immersed i n a moving f l u i d , f o r comparing of  different  with  way  to compare  inertial  geometrically s i m i l a r  and  bodies  the Nusselt number provides a basis  rates of convective heat loss from s i m i l a r bodies scale,  exposed  to  different  wind-speeds  and  temperature differences (Monteith, 1973). There according  are two to  the  main  classes  forces  which  of convective drive  the  heat  flow.  transfer In  forced  convection the transfer i s driven by an e x t e r n a l l y imposed flow such as wind. Here, the turbulence regime which f a c i l i t a t e s the heat f l u x i n the boundary layer i s c o n t r o l l e d p r i m a r i l y by the geometry of the surface.  In t h i s case the dynamical  similarity  of d i f f e r e n t systems expressed by Re i s c l o s e l y r e l a t e d to the thermal regime expressed by Nu. In a i r , with a Prandtl number of about 0.71 (independent of temperature), the Nusselt number f o r  - 22 forced convection can be written, f o l l o w i n g Monteith (1973), Nu  =  A-Re  ,  n  as (3.5)  where values f o r A and n are tabulated f o r d i f f e r e n t  types of  geometry. Buoyancy induced arise  simply  (caused field,  by  flows,  because  of  density v a r i a t i o n s  differential  such  termed natural or free  heating  as g r a v i t a t i o n  within  processes)  (Jaluria,  1980).  convection,  in  the  a  fluid  body-force  In t h i s  case  the  Nusselt number i s a f u n c t i o n of the Grashof number (and s t r i c t l y also of the Prandtl number). Physically,  the Grashof  the r a t i o  inertial  square  of a buoyancy force times an  of a viscous force (Monteith,  1973).  number i s  force to  the  It i s c a l c u l a t e d  from  Gr  It  follows  that  =  in  a g l  3  (9 - 9 0 z  atmospheric  )/ v  .  2  '  free  (3.6)  convection  Nu  can  be  expressed as  Nu  =  B-Gr™  (3.7)  where B and m are tabulated f o r d i f f e r e n t  geometries.  Pure forced or pure free convection are c l e a r l y the cases.  In non-laboratory conditions both mechanisms w i l l  some importance and mixed  limiting  convection.  convection  are  the combined process The  criteria  excellently  for  described  i s commonly known as  forced, by  carry  free  Monteith  or  mixed  (1973).  As  - 23 indicated  above,  the  driving  force  buoyancy,  whereas forced convection  for  free  convection  i s generated  by  is  inertial  forces. The r a t i o of the strengths of these two l i m i t i n g regimes i s given by the r a t i o of t h e i r respective d r i v i n g forces which 2  can be written as Gr/Re . This r a t i o Richardson thumb  number  regarding  i n meteorology. the  prevalent  i s also known as the Monteith  transfer  reports  regime  Bulk  rules  derived  of  from  2  experimental  evidence.  When Gr/Re  > 16,  buoyancy  forces  are  much stronger than i n e r t i a l forces and the appropriate r e l a t i o n f o r free convection (3.7) should be used. Forced convection i s 2  dominant when Gr/Re  2  < 0.1.  For  intermediate values of Gr/Re ,  Nu should be c a l c u l a t e d both f o r forced and f o r free convection and the larger number should be used to estimate the heat f l u x . 3.3  ANALYSIS  The objective of t h i s a n a l y s i s i s to look at the c o n t r o l l i n g variables f o r the 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 and to  identify  distributions  the of  dominant these  surface  variables may  variables. then  be  The  used  spatial as  strong  indicators of the 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 . a start,  As  the l i m i t i n g cases of forced and free convection w i l l  be discussed. In the following, the s p a t i a l v a r i a b i l i t y w i l l denoted by a circumflex (-)  be  above the symbol f o r the variable.  It may be interpreted as the square root of the s p a t i a l variance or the s p a t i a l standard deviation of the variable. First,  l e t us consider the heat f l u x equation i t s e l f .  On the  - 24 right-hand-side of (3.2), p and c  may  p  be considered constant  in  space, they vary only s l i g h t l y with the bulk temperature i n the surface  layer. Therefore  p = 0,  c  » 0  p  and  they  introduce  no  v a r i a b i l i t y i n Q . On the other hand, the temperature d i f f e r e n c e H  between the surface and  some height  space. It i s even conceivable may  change  s i g n over  small  z i s strongly v a r i a b l e i n  that t h i s mean v e r t i c a l horizontal scales.  looked at as an analogy to Ohm's law, difference constant,  in  potential  i t controls  and,  the  (9 ~Q  flux  When (3.2)  everything  in a  is  ) corresponds to the  Q  if  gradient  linear  else  is  fashion.  held It  is  possible to s i m p l i f y t h i s v a r i a b l e f o r the present purposes, i f z  is  specified  vanish.  In t h i s  as  a  height  Instance  9  where  a l l horizontal  i s a constant  and  z that  the  spatial  variability  of  the  i d e n t i f i e d as a strong control of Q  gradients  ( 0 -G ) s e , so o z o  surface  temperature  is  independent of the t r a n s f e r H  regime. The assessment of r ^ i s more problematic. difference  signifies  the  a v a i l a b l e to the flux,  magnitude  energy.  of  the  fluid  the  r , or rather 1/r H  efficiency  of  If the temperature potential  i s a measure of  the  H  medium to t r a n s f e r the  Its function i n the  heat  context  sensible heat  of sensible heat  flux  is  very s i m i l a r to the function of a valve i n a pipe. Formally i n (3.2), f i s 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 i s a considerable degree of c o r r e l a t i o n 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 r Hf  .  A Re  K  With  the  p e l r  p  n  K  viscosity v,  the  (3.8)  p  1-u  A  heat  conductivity  K, p  and  c F  determined at bulk a i r temperature (3.8) may be simplified as  r  cJ  =  (3.9)  (A-l -u ) n_1  n  Hf  and when (3.9) is substituted into (3.2), the heat flux equation becomes  Q  = Hf  c -A (9 -8 2 0  ).l  B _ 1  .u  (3.10)  B  2  where p-c /c = c . Values for A and n for different geometries P 1 2 are given by Monteith (1973) : r  &  Re  n Flat plates (vert, or horiz.)  0.032  0.8  >  Cylinders  0.24  0.6  10 -5-10  2-10 3  4  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 t y p i c a l l y range of 10 m to 100 m. With the kinematic  i n the crude  v i s c o s i t y determined  at 25 °C, and allowing canopy layer wind-speeds between 1 and 10 m-s , the r e s u l t i n g Reynolds number range i s 1  6.5-10  5  <  Even though the buildings  Re  <  6.5-10  .  7  (3.11)  i n a suburban environment  resemble  rectangular boxes more than c y l i n d e r s , Figure 3.1 i s useful i l l u s t r a t e how the Nusselt  to  number can vary on a c y l i n d e r as a  function of angular distance from the stagnation point. It shows that  f o r high  maximum value. the  Re, the l o c a l  Nu may vary  up to 50% from the  This means that even with constant  v a r i a b i l i t y of the l o c a l  temperatures  heat f l u x extends to 50% o f  its  maximum and i s c o n t r o l l e d e n t i r e l y by the geometry of the heat source and i t s o r i e n t a t i o n r e l a t i v e to the flow. A large obstacle, such as a building, also a f f e c t s the f l u x 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 g r e a t l y reduced due to the s h e l t e r i n g e f f e c t of the b u i l d i n g and the trapping of a i r i n re-circulation cells.  This area i s known as the c a v i t y zone or  bubble (Oke, 1978). In strong flow conditions the c a v i t y length extends to about 1.5 times the height of cubic obstacles i n the down-wind d i r e c t i o n  and increases with  the r a t i o  of width to  height of the obstacle (Hosker, 1984). In weak wind conditions, however,  flow  separation  at  the  obstacle  edges  is  pronounced and the r e s u l t i n g sheltered c a v i t y zone i s l i k e l y  less to  0-Dci;rccl f'Oin  Figure 3. 1 : Local Nusselt numbers for crossflow about a c i r c u l a r cylinder : a) at low Re, b) at high Re (from Welty et al., 1976).  stagnation point  - 28 be  much  smaller,  or  even  negligible.  s t r a t i f i c a t i o n becomes more unstable,  In  addition,  as  the  a weakened s h e l t e r e f f e c t  i s to be expected due to enhanced v e r t i c a l mixing. After  this  direction,  sheltered  bubble  the disturbance  zone,  in  the  of the flow by the obstacle r e s u l t s  i n a turbulent wake zone which i s noticeable f o r a distance  downwind.  reduced,  the  turbulence  In  energy  this  wake,  removed from  l e v e l s are  along-wind  increased.  t h i s wake region f a c i l i t a t e s  the  although  the The  wind-speeds  are  is dissipated  and  developed mixing  in  flow well  considerable  t r a n s f e r of heat and  reduces  the e f f e c t i v e aerodynamic resistance. In the  context  of s p a t i a l  variability  i t i s i n t e r e s t i n g to  note at t h i s point that because of the well developed mixing i n the  wake of  affected  by  a  large  the  flow  obstacle, around the  downwind, where a resistance  the  aerodynamic  obstacle  at  c a l c u l a t e d from  resistance  a great local  distance conditions  alone might be quite d i f f e r e n t from the e f f e c t i v e resistance to the wake. If the canopy layer contains i n more or  less regular spacing  area),  flow  the  and  mixing  and  The  are  i n a suburban  characterized  Wallace  i n t e r e s t i n g point  due  obstacles  generally good mixing,  Shuttleworth and  mean canopy airstream.  such large  buildings  conditions  cumulative wake i n t e r a c t i o n and what Thorn (1972),  (e.g.  is  causing  (1985) term  here  by  i s that  a in  such conditions the aerodynamic resistance seems to vary only at larger  spatial  variability,  due  scales to  the  compared  to  the  surface  'blurring' e f f e c t of  the  temperature mean canopy  flow. Claussen (1987) reports that t h i s b l u r r i n g i s also evident  - 29 on the upwind side of an obstacle momentum transport all  directions.  flow applies  or a roughness change due to  by pressure f l u c t u a t i o n s which propagate i n  This  a n t i c i p a t i o n of a surface  change by the  to momentum transfer, where pressure  fluctuations  have an e f f e c t , but only i n d i r e c t l y to s c a l a r fluxes. It seems that i n forced convection the aerodynamic r  i s c o n t r o l l e d by the dynamics  of the surface  resistance  canopy  layer  H  system. Parameters which introduce  spatial variability  the geometry of the i n d i v i d u a l surface spatial  dimensions)  regimes  (Burns,  and the l o c a l  1980).  Cumulative  include  elements (both shape and  wind-speed wake  and turbulence  interaction  indicates  that there i s no continuum of scales i n the s p a t i a l v a r i a b i l i t y of  the wind  field,  so that  the geometry  and the turbulence  regimes do not necessarily vary at the same s p a t i a l scales. The  effect  on  the heat  flux  Is that  both  9  and r Hf  0  (reflected  by the s p a t i a l d i s t r i b u t i o n of roughness  may assume r o l e s of equal importance on Q . While Q H  proportional  to 9 , i t i s inversely proportional q  elements)  i s directly  H  to r .  - 30 3.3.2  Free Convection  The  situation  convection.  i s quite  Here,  different  in  number  is a  the Nusselt  natural  or  function  free  of the  Grashof number (see equation (3.7)), so that we can write the corresponding resistance to natural convection :  p e l p  K  p  e p  l  ,  .  V  (3.12)  r K  B Gr  m  B  K  a g l  3  (9-9 ) u  z  The q u a n t i t i e s p, c , K, V, a and g may be considered p  and  will  be substituted  f o r by c  3  constants  f o r simplicity.  Equation  (3.12) becomes  r  =  c  Hn  3'  After s u b s t i t u t i o n into  (3.2),  the heat f l u x equation f o r the  case of natural convection follows as  Q  =  c  Hn  where p-cp/c (1973)  B - l - - ^ -9 ) 3  ±  1  4  0  (3.14)  m + 1  z  = c^. Values f o r B and f o r m are given by Monteith  f o r the Grashof  number  ranges  to be  suburban environment ( i . e . Gr i n the order of 1 0  B  expected 10  - 10 ) : 12  m  Horizontal f l a t surfaces  0.13  0.33  V e r t i c a l surfaces  0.11  0.33  Considering  in a  that B does not change very much, and that m stays  - 31 constant  for vertical  and  horizontal  surfaces,  Q  can  be  Hn  considered  proportional  to  1 33  (9 -9 ) 0 z  .  Thus,  in  natural  convection the heat f l u x i s almost completely c o n t r o l l e d by the surface temperature and i s v i r t u a l l y independent of geometry (B changes + 10%, see above). The conclusion i s that i n free convection 9  i s the dominant  o control over Q , with the geometry having a minor e f f e c t . H  3.4  SUMMARY  The main parameters introducing s p a t i a l v a r i a b i l i t y into the sensible  heat  convection  flux  and  f o r the  free  two  limiting  cases  convection are apparent  of  from  forced  equations  (3.10) and (3.14) respectively. In free convection the s i t u a t i o n i s quite clear, with temperature v a r i a b i l i t y being the dominant control, the  and geometry  choice of 9  q  having only a s l i g h t  to represent  e f f e c t . Therefore,  i s obvious i n natural or free  convection s i t u a t i o n s . The case becomes more d i f f i c u l t  i n forced  convection.  from  Equation  (3.10)  indicates  that  apart  the  temperature (which i s an important f a c t o r even here), the l o c a l wind  speed  and  both  the type  of  geometry  and  the  length  dimensions involved must be considered. The l o c a l wind speed i s of  course  itself  momentum-extracting  influenced roughness  by  the  elements,  geometry  which  of  the  illustrates  the  importance of the surface geometry even more. In in  the mixed convection s i t u a t i o n s which are to be expected  the  real  world,  i t would  therefore  seem  that  a  Q  - 32 representation by 0 ,  as  q  case,  loses  -  i s suggested  i t s justification  i n the free convection  when the  transport  process i s  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 s p a t i a l scales of sensible heat f l u x v a r i a b i l i t y may examining  be performed  the high r e s o l u t i o n s p a t i a l  temperatures elements.  and  the  Appropriate  distribution methods  to  adequately by  d i s t r i b u t i o n of of  dominant  establish  surface  roughness  the  necessary  datasets f o r the s p a t i a l d i s t r i b u t i o n of surface temperature and of roughness  elements are developed and discussed i n the next  chapter. Due to temporal contingency considerations described at the  beginning  of  this  chapter,  the  surface  temperature  d i s t r i b u t i o n should be evaluated at a time of highly developed convective  activity,  stable conditions, changing The  second  dataset acquired i n very  to enable an assessement  temperature  overall  with a  distribution  temporal  change  on of  of the temporally  the s p a t i a l the  d i s t r i b u t i o n i s assumed to be n e g l i g i b l e .  variability.  roughness  element  - 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 r e s o l u t i o n map  of  surface temperatures i s needed as a base f o r the evaluation of sensible heat f l u x s p a t i a l v a r i a b i l i t y and i n order to obtain an estimate of the dominant s p a t i a l scales of t h i s flux. Inevitably t h i s c a l l s f o r an appropriate d e f i n i t i o n of the surface and corresponding  surface  temperatures,  as  well  as  a  the  practical  method to measure these surface temperatures at a high  spatial  resolution. In  classical  surface  micrometeorology  i n respect  to turbulent  over  homogeneous  planes  sensible heat f l u x  the  i s usually  understood to be the v i r t u a l surface at the level (d + z ) above  T ground,  in  the  canopy  layer.  A  time-averaged  temperature  measurement at t h i s level i s also a s p a t i a l average of a kind, if  the  action  micro-advection  of  suitable  and  and  turbulence  and  the  resulting  across various surface elements i s considered  Taylor's hypothesis. surface  wind  the  f o r the  The  surface present  micrometeorological temperature problem,  definition  i s therefore  where a  in  of  the  clearly  not  spatially  resolved  temperature i s wanted. Consider  the turbulent sensible heat f l u x from an  individual  surface element. It i s generated at the top of the t h i n  laminar  - 34 boundary  layer,  through  which  the heat  i s transported by  molecular d i f f u s i o n . The temperature at the top of t h i s laminar boundary layer and  Wilson,  i s often c a l l e d f i l m 1976)  and  temperature  i s the appropriate  equation  (3.2),  if r H  is a  turbulent  heat  transfer.  With  resistance  (Welty, Wicks temperature f o r  involving  only the  the exception of very  surfaces (e.g. ice, glass, polished metal or s t i l l  smooth  water) t h i s  laminar layer i s extremely t h i n over natural surfaces and common b u i l d i n g materials. I f i t i s assumed that the laminar layer has the same thickness over a l l surface elements, equation (3.2) may be reformulated : Q H  =  p c (0 -0 ) / (r + r ) P R z ' H Hi'  (4. 1)  where 0 i s the r a d i a t i o n temperature of the surface and r i s R HI the  resistance  to molecular  laminar boundary layer. radiative than  temperatures  the  film  heat  transfer  across  the  thin  In daytime convective conditions these are expected to be considerably higher  temperatures  due  to  the  large  temperature  gradients across the laminar layer. Assuming that 0 « 0 for R film the l i m i t e d range of n a t u r a l l y occurring surface temperatures i t follows  that  also  0 « 0 . I n other words, R film  i f the surface  temperature  distribution  i s a strong indicator of the s p a t i a l  variability  of sensible  heat  radiative  surface  flux  temperature  (see Chapter  i s a suitable  3), then the  measure f o r t h i s  surface temperature. The going  same conclusion into  the argument  was  drawn by Garratt  explicitly.  (1978b) without  In h i s study of t r a n s f e r  - 35  -  c h a r a c t e r i s t i c s from a heterogeneous infrared  radiation  hand-held  calculated  The  consistency of  with these surface of  this  method.  In  used  (from both airplane-based and  radiometers) to obtain an o v e r a l l  temperature.  validity  temperatures  savannah surface he  heat  transfer  temperature their  effective  surface  coefficients,  values, support  work  on  surface  the  energy  c h a r a c t e r i s t i c s i n urban and r u r a l areas i n and around St. Louis, Dabberdt  and  Davis  (1978) argue  i n terms  of  a  climatonomic  approach to the surface energy balance that the temperature of the  material  f o r c i n g and  surface  and  the  reasonableness  temperatures  radiometers,  parameterization  schemes  sensible of  the  response  to  for  of  surface heat both  solar  response. (1981) use  airplane-  respectively)  atmospheric results  al.  (from  the  the  the secondary  (1978) and Carlson et  Davis  surface  satellite-based  including  primary  the surface energy fluxes  Both Dabberdt radiative  i s the  and  in  their  energy  flux. these  fluxes,  The  apparent  studies  lends  support to the use of the r a d i a t i v e surface temperature. In  these  examples  the  radiative  surface  temperature  was  obtained by remote sensing. Due to the f a c t that a remote sensor can only 'see' horizontal surfaces, one has to be c a r e f u l i n the use of such data f o r s p a t i a l averaging. The f a i l u r e to account f o r the contributions of v e r t i c a l surfaces to the sensible heat flux  could  resulting  be  average  during daytime, problem  important surface  especially temperature  i n built-up may  well  with the magnitude of the error  i s only of l i t t l e  areas : be  too  the high  unknown. This  concern f o r the present work, since  only the s p a t i a l scales of surface temperature  variability  and  - 36 not the absolute values are of interest. It  i s concluded  temperatures acceptable  provides method  simultaneous  4.1.1  that  infrared a  very  to obtain  surface  a  remote  sensing  convenient high  of surface  and  resolution  physically  inventory of  temperatures.  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 commissioned to the Canada Centre f o r Remote Sensing Ottawa,  Ont..  multispectral  The  instrument  line-scanner  used  (MSS)  was  sensing  a in  wave-band. Technical d e t a i l s of the f l i g h t s ,  were  (CCRS) i n  Daedalus-1260 the  8-14  the data  um  format,  ground-truth measurements and data c a l i b r a t i o n are described i n Appendix A. The following provides only a b r i e f o u t l i n e of the f i e l d campaign f o r contextual purposes and concentrates on the processing of the data. The  flights  aircraft average  were  travelling height  scans/second  performed  using  at a ground  speed  of 1560 m above ground.  a  Falcon-20  twin-jet  of 105 m-s"  1  and an  The MSS recorded  50  with 716 pixels/scan. The data were d e l i v e r e d i n  binary form on computer compatible  tape. They were preprocessed  by CCRS f o r instrument c a l i b r a t i o n , yaw- and r o l l - c o r r e c t i o n and orthogonalization (see Appendix A). The  times of the f l 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.),  minimum convective a c t i v i t y ,  August just  25  and  at  the  before sunrise,  P.D.T. (5:05 L.A.T.) on August 26,  1985.  expected  about  6:20  The weather conditions  were ideal f o r both f l i g h t s , with exceptionally c l e a r skies and calm winds. Ground truth measurements with hand-held radiometers were  obtained  f o r various surface types  on  the  grounds  of  Langara Community College, i n 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 of  the  study  area  distance measurements on a 1:25000  provided  the  average  pixel  dimensions  3.26 m (3.28 m) i n the N/S d i r e c t i o n and 3.21 m (3.23 m) E/W  direction  dimensions different  f o r the day  (night) f l i g h t .  The  different  f o r the day and night f l i g h t s r e s u l t f l y i n g heights. (Note : there i s a  map as  i n the pixel  from s l i g h t l y  16%  overscan  on  average, see Appendix A). Using the c a l i b r a t i o n integer p i x e l  curves from Appendix A,  the  one-byte  values were converted to Celsius temperatures  as  four-byte r e a l values. In t h i s form they were stored on tape, ready  f o r further  analysis  of  the  spatial  temperature on the mainframe computer.  scales  of surface  - 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 d i s t r i b u t i o n of roughness elements  therefore  necessarily  involves  an inventory  of the  dominant bluff-bodies i n the area. To  s i m p l i f y the data structure f o r such an inventory, the  surface  elements  i n the study area  were c l a s s i f i e d  Into  five  categories of the most common surface cover types i n the mostly residential  neighbourhoods  of the study  area.  These  a  buildings,  classes  are : 1. Houses  : including churches,  few  larger  schools,  apartment  and  such  as  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 fields  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 contains  (see below). Since  the r e s u l t i n g map  already  the locations and approximate horizontal dimensions of  - 39 these roughness elements, make  the  mapped  each category was assigned a height to  information  three-dimensional.  Residential  houses are l i m i t e d by municipal regulation to heights less than 35 f t (or 10.7 m). height average  of 8.5 m of  10 m  Considering the estimate f o r mean b u i l d i n g  i n the same area by was  chosen  as  the  Steyn  (1980),  assigned  a  height  rounded 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 v a r i a t i o n i n height. A small number of park trees reach a height of 30-40 m, much higher  than  the  most boulevard- and houses,  garden-trees  whereas shrubs  are  not  are considerably  lower. After examination by eye on s i t e and using photographs,  a  height of 15 m seemed reasonable as the assigned height f o r the 'Trees' category. Streets and other open spaces are important  on  the map  To  because they are areas without  distinguish  between  the  categories  large bluff-bodies. 'Streets'  and  'Grass',  'Grass' was assigned a height of 0.2 m and 'Streets' of 0.1  4.2.1  m.  Data A q u i s i t i o n f o r the Roughness Inventory  The area covered by the roughness element inventory i s shown i n Figure 4.1.  The data were compiled i n a stepwise process.  airphoto mosaic of the area (scale : 1:2500), dated A p r i l , and a v a i l a b l e from Vancouver C i t y H a l l , was g r i d c e l l s of 15 by 15 cm the  inventory area  divided into  (equivalent to 375 by 375 m),  i s made up  1987  square so that  of a block of 5 "x 4 = 20  c e l l s . For each g r i d c e l l and each of the f i r s t four  An  grid  categories  j r r j ^ £ v f r.v:j-;| |i~.  ~  :  • — ) N.I!;  TR7rr;-k;K(:  // .'77—" - V " K E  vi—""•","'r T i : A  Tl—i—-«»r»."!«i 1  ll--"|<^[:;|-:lij(''/!l2..~  .  .j  < j I i i i j ! : ~"  __r^ui>i*L**i_L_:_L.i>Jij^„'  rt 11 J. ^ (S)  : Sunset  ©  : Gordon Park  Figure 4.1  :  Tower; ;  ©  : Mountain  View C e m e t e r y ;  ®  : 'Hot-Crossed-Buns'  Area of roughness element inventory  ;  ©  : Memorial  (J)  : Langara  Park;  (R) : K e n s i n g t o n  Community  College  Park  1M  -  Figure  4.2  :  The  41  -  video-framestore-VZIP  configuration. Graphic information is digitized by the video camera (left) and s t o r e d as a r a s t e r - i m a g e (framestore u n i t and monitor i n the c e n t r e of the p i c t u r e ) . It can then be sent to the PC-based VZIP system (right).  - 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.  on-screen editing facility of VZIP was used to images of noise, 4.2).  'clean'  The the  introduced by the video camera (see Figure  After the 20 frames for the first four categories were  combined to image-arrays of 818 x 836 elements, the greyshades from  the  framestore  transformed  scan  into one-byte  (ranging integer  from  values  0 of  to  255)  ten times  were the  appropriate height. The four classes were then added together ('sandwiched') value  of  and all remaining zeros were assigned to the  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, categories  the  alignment  and the  matching of  the  different  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  -  t(l  43  -  500m  F i g u r e 4.3 : VZIP-image o f roughness element inventory, total 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 garages as b l u e .  -  44  -  - 45 a 1:25000 map E/W  as 1.77  m i n the N/S  d i r e c t i o n and 2.15  m  in  the  direction. The  l a s t step of t h i s data a q u i s i t i o n and preparation process  involved  the  computer and  transfer of the the  final  t r a n s l a t i o n of the  image f i l e  to the  one-byte integer  four-byte r e a l values, to prepare them f o r further  mainframe data  analysis  the s p a t i a l structure of the roughness element d i s t r i b u t i o n .  into of  - 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 i n which the  atmospheric  properties  geometric,  thermal,  surface  and fluxes are c l o s e l y r e l a t e d to the  moisture  (etc.)  c h a r a c t e r i s t i c s of the  (Oke, 1978) i s known as the atmospheric surface  layer.  Most surface layer theory therefore r e f e r s to some d e s c r i p t o r of these  surface  effects  c h a r a c t e r i s t i c s i n order  into  the  parameterizations. generally  appropriate Atmospheric  concentrated  atmosphere,  assuming  on  to incorporate  scaling surface  the  laws layer  vertical  h o r i z o n t a l l y uniform  and  flux  theory  has  structure surface  their  of the  conditions.  S t r i c t l y speaking, t h i s assumption of homogeneity i s never met i n any f i e l d  s i t u a t i o n , since even the most extensive  grass f i e l d of  bare  is still soil.  variabilities turbulent  One  variabilities turbulent  might  conditions  when  the  (Monin and Yaglom,  roughness that  Garratt, et  that  become  linear  elements  1971).  a very  al.,  spatial  irrelevant f o r  scales  of  eddies  these i n the  In addition,  the common  sensor should be placed f a r above  (e.g. z  = 100-z ;  min  ensures  however,  i.e. they are r e l a t e d to the Kolmogorov  postulate that any turbulence the  argue,  approach the s i z e of the smallest  atmosphere,  microscale  made up of i n d i v i d u a l grasses on a matrix  i n surface  processes  and f l a t  vague d e f i n i t i o n  Tennekes,  1973)  0  of ' i d e a l '  s i t e s (e.g.  1979) i s usually s u f f i c i e n t f o r the homogeneity  - 47  -  requirements. Over non-ideal s i t e s the surface v a r i a b i l i t y i s r e f l e c t e d i n horizontal important  inhomogeneities at  low  levels  in and  the  atmosphere,  become  which  increasingly  greater heights (see also Chapter 6). Raupach et Raupach  and  Shaw  (1982)  t e r r a i n wind p r o f i l e s  show  converge  that  even  al.  over  may  be  blurred  at  (1980) and  inhomogeneous  upon the one-dimensional  form,  a f t e r s p a t i a l averaging, so that some homogeneity condition i s again v a l i d . Obviously, t h i s s p a t i a l averaging process needs to incorporate a large enough domain, to be representative of the surface  character. In the case  inhomogeneities, Raupach et  al.  such  as  (1980),  the  the  of a very regular pattern of experimental  in  determination of a representative  averaging domain seems f a i r l y t r i v i a l . such as  configuration  In many f i e l d s i t u a t i o n s ,  the suburban area i n the present study,  however,  the  surface inhomogeneities occur across a v a r i e t y of scales and the assessement  of  representativeness  and  homogeneity  becomes  d i f f i c u l t and subjective. In  the  following  'homogeneity* addition,  a  relationship  and  sections  concept  'representativeness'  mathematical between  a  tool  spatial  is  is  designed  scale  and  q u a n t i t a t i v e l y . Even though two-dimensional  of  the  terms  developed. to  examine  In the  representativeness inhomogeneity  i s of  concern here, i t i s convenient to discuss some p r i n c i p l e s i n one dimension  by  using  the  methods  of  geophysical  time  series  analysis. The r e s u l t s are e a s i l y expanded to two dimensions, i f these dimensions are l i n e a r and orthogonal.  - 48 -  5.2  HOMOGENEITY Curiously, the term 'homogeneity' i s not commonly defined i n  standard s t a t i s t i c s texts (e.g. Larsen and Marx, 1981), homogeneity within a dataset many s t a t i s t i c a l analysis  the term  interpreted  statistical  i s a fundamental p r e r e q u i s i t e f o r theorems and laws.  'stationarity'  as homogeneity  stationarity  time  analyses,  i s well  i n time.  In time s e r i e s  defined  Priestley  i n terms of a random process properties of a random process  ( i . e . they  although  and may be  (1981)  defines  i n time : i f the do not change over  are the same at a l l times),  the process i s  c a l l e d stationary. To express t h i s i n a more formal way, suppose some process {XJ  i s measured at equally spaced points ( i n time  or along a l i n e transect i n space). The r e s u l t i s a data s e r i e s X^  X  . . . , X . The process  2 >  i s stationary, i f the mean of { X } F C  r  i s independent of the index t and the covariance between X and FC  X  F  C  +  T  depends only on the distance between the points  | T | . That  is,  E(X )  =  T  C0V(X ) T  where  g  i s some  f o r a l l t;  X  =  function  g(|T|)  of  |x|  (5. 1)  ,  (Knight,  (5.2)  1987,  personal  communication). I f (5.1) i s seen as the expected mean of a block sample of { X } , T  centered  at t, then  (5.1) can obviously  only  hold f o r blocks above a c e r t a i n size. The f i r s t p r e r e q u i s i t e f o r stationarity to  i n time or homogeneity i n space therefore r e l a t e s  the notion  of representativeness,  which w i l l  be discussed  - 49 extensively i n the next section. Equations (5.1) and (5.2) describe {X } i n the time domain or fc  i n the space domain. Alternately, {X } may be characterized by fc  its  spectral  approach  properties.  decomposes  This  the  frequency  variability  or wavenumber  of  {X }  into  fc  domain  frequency  components by means of a s p e c t r a l measure (or a s p e c t r a l density function,  i f no exact  periodic components e x i s t , and  ignoring  the d i s c r e t e nature of the s e r i e s f o r the moment). This s p e c t r a l density  function  covariance  results  from  the Fourier  transform  of the  function i n (5.2), which may be written as  (5.3)  h(f)  following P r i e s t l e y (1981). For r e a l data the s p e c t r a l  density  function  defined  h(f) i s a  non-negative,  symmetric  function  between -0.5 £ f s+0.5, where f i s i n cycles per space between datapoints,. At. The l i m i t i n g frequency f = 10.51 corresponds to the  Nyquist  frequency,  unambiguously detected  the  highest  frequency  which  so  that  transform,  X = 0)  also  be  (Kanasewich, 1981). The function h(f) may  be determined d i r e c t l y from the data s e r i e s ( a f t e r it  can  as  known as  the  square  the power  modulus  of  spectrum.  transforming i t s Fourier  From  (5.3) i t  follows that h ( f ) - d f i s the average (over a l l r e a l i z a t i o n s ) of the contributions to the t o t a l variance  of {X } from components  between f and f+df, so that, f o r r e a l valued  fc  {X >, t  - 50 0.5  VAR(X )  •  fc  <r  =  2  2- ^ h ( f ) Af  (5.4)  o (e.g. P r i e s t l e y ,  1981). Therefore  may  as  be  viewed  frequency  the  components  spectrum  (e.g.  Young  the s p e c t r a l density f u n c t i o n  decomposition  and  is  and  Pielke,  of  sometimes  the  variance  called  1983).  In  the  the  into  variance  case  of  a  d i s c r e t e data s e r i e s (and thus a d i s c r e t e Fourier transform) a l l the integrations above need to be replaced by summations and  the  lower i n t e g r a t i o n l i m i t  the  Nyquist  needs to be  replaced by  frequency.  Given the preceding, the  i n (5.4)  we may  spectral distribution  series  may  be  variation  defined  occurs  frequencies  at  of  to  consider homogeneity i n terms of variance.  be  homogeneous,  relatively  (the lowest  Qualitatively,  frequency  high  i f most  (spatial  or  a  data  of  its  temporal)  being the s i z e of the e n t i r e  data domain). Thus the homogeneity of a d i s c r e t e data s e r i e s i n space or  i n time  d i s c r e t e variance variance the  at  low  is directly spectrum.  frequencies  contributions  at  reflected  i n the  shape  of i t s  I f the contributions to the are r e l a t i v e l y  higher  frequencies,  small, the  total  compared data  to  series  q u a l i f i e s as homogeneous. Quantitative measures of homogeneity may  be  designed  based on  communication).  The  which  contributions  variance  scale  this  notion  (Knight,  corresponding start  to  r e l a t e d to the term 'representativeness' i n the next section.  to  1987,  personal  the  frequency  at  become  important  is  and  will  be  discussed  - 51 In summary i t i s concluded  that  the degree of homogeneity  within a data s e r i e s i s a property of the e n t i r e s e r i e s and i s dependent  on  the  relative  contributions  of  the  (spatial  or  temporal) frequency components to the t o t a l variance.  5.3  REPRESENTATIVENESS  The  concepts  closely  interlinked  distribution found  of  to  representativeness and  both  of v a r i a b i l i t y . be  a  property  are  related  However, of  and  homogeneity to  the  whereas  the  are  frequency  homogeneity i s  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 c e r t a i n  scale  portion  or,  i n other  words,  over  a  of the data.  r e l a t i o n s h i p between the two terms can be expressed to the d e f i n i t i o n  of homogeneity given  dataset  completely  is  unambiguously alternately,  not  subset  the representativeness of the subset but  not  on  homogeneous  i t s location, at  representativeness.  least It  to  the  should  completely  not  exist,  and (ii)  e x i s t s i n a dataset and  depends only on the s i z e ,  dataset  the  a  does  i f a representative subset  by reference  previously : ( i ) i f a  homogeneous,  representative  The  must  necessarily  degree  of  be  noted  the  be  subset's  that  the  representativeness of a block sample r e f e r s to the s i z e of the sample and not to a s p e c i f i c r e a l i z a t i o n of t h i s sample. It i s of course  possible that the s t a t i s t i c a l  properties of a small  sample are c o i n c i d e n t a l l y s i m i l a r to the e n t i r e data domain, but  - 52 i n t h i s case the s i m i l a r i t y  i s conditional on the p o s i t i o n of  the sample. A subset can only be considered t r u l y representative of  a dataset,  i f there  i s an i n t r i n s i c  similarity  and only  n e g l i g i b l e dependence on the s p e c i f i c configuration. Consider a block sample of m consecutive observations X , X , ...,  X . The mean of t h i s s p e c i f i c sample i s defined as 1  X  ra  m  -  =  V Xt  (5.5)  L,  m  t=l Over a l l r e a l i z a t i o n s expected X  of such  a block  of m-observations  the  i s equal to X f o r a l l observations,  ID'  E ( X )  =  III  One  measure  observations  of  E(x ) = f  the  (5.6)  X  t  t  representativeness  i s the r e l a t i v e  decrease  of  a  block  i n the v a r i a t i o n  of  m  of X m  from  the variance  of X . FC  Thus  i t i s 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 )  i s the v a r i a b i l i t y  among  a l l the  different  on  r e a l i z a t i o n s of the block sample, the quotient on the r i g h t of (5.7) must n e c e s s a r i l y be smaller than unity and R(m) i s seen to vary between zero and unity. It  i s again  convenient  to examine  this  problem  i n the  - 53 frequency total  or wavenumber domain. The frequency components of the  variance  contributions shorter  (i.e.  of  than  or  of  VAR(X ))  can  that  correspond  t  frequencies equal  to  the  be  blocklength  of  divided  into  to  wavelengths  the  sample  and  components with longer wavelengths. The expected variance within any one be  r e a l i z a t i o n of the m consecutive observations may  interpreted as  spectrum  and  the  the  high-frequency  variance  among  portion of  block  the  then  variance  sample-means  as  the  low-frequency part, following Young and Pielke (1983), so that  E(VAR(X) ) m  +  VAR(X) m  =  VAR(X ) t  (5.8)  f  If n i s the t o t a l number of observations and  At  the  interval  that  each  i n the data domain  observation  occupies,  then  L = n-At  i s the length of the domain. Since f i s defined i n the  previous  s e c t i o n as  observations Therefore  frequency  (At), the  wavelength  the frequency  observations,  X  i n cycles per  corresponding  = m-At  is  f  is  m  1/n.  s u b d i v i s i o n of the  Frequency  f  = 1/m  f  as  X = At/f.  blocklength of and  the  the  delimiter  variance density spectrum and,  equation (5.4), (5,8) may  between  m  lowest  m  is  m  to  to the  = At/A  m  frequency  relates  interval  for  the  considering  be translated to the frequency domain  as  0.5 2  • ^h(f) f  where  the  -df)  (f  Af  +  2  m • ^h(f)  =  VAR(X )  (5.9)  t  1/n  m  d i s c r e t e nature  Af  of  the  transforms  is  ignored  for  - 54 s i m p l i c i t y and each term corresponds to the respective term i n (5.8). (Note that h(0) = 0 since X  fc  of  the  data  spectral  as  described  distribution  = 0 a f t e r the  above).  function,  The  transformation  integrated  H(f),  is  variance  introduced  for  convenience, following P r i e s t l e y (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. I f h(f) = h(At/A) = h (A), (5.9) becomes A  2 • ^\(A)  AA  +  2 • ^\(A) A  2At  AA  =  VAR(X )  (5.11)  t  +AA m  Note that transition  the  integration limits  from  frequency  to  have been reversed  wavelength  i n this  presentation.  The  equivalent to the integrated spectrum then follows as  H^U)  =  2 •  A  ^  '  (5.12)  2At  so that  ^(L)  and  =  H(0.5)  =  VAR(X ) fc  (5.13)  - 55 A m  2 • ^J^U)  AA  =  •  (5.14)  2At  After s u b s t i t u t i o n into (5.8) and rearranging, i t follows that  H.(A  V A R ( X  )  A m  =  V A R ( X  T  )  m  1  )  V A R ( X  Thus the normalized  T  3  R(m)  .  (5.15)  )  wavelength-integrated  variance  distribution  function i s i d e n t i c a l to the measure of representativeness f o r a block sample of s i z e m, as introduced i n (5.7). In  practice  values  are  the  obtained  R(m) from  Fourier transform. R(m) wavelengths  smaller  for a the  discrete square  i s twice the sum  or  equal  to  A ,  data  moduli  series of  of  the  real  series'  of the components with divided  by  the  total  m  variance.  5.4  DISCUSSION  It  has been shown that the homogeneity of a dataset and  existence of a representative block sample subset related  and  dependent on  are  the  closely  the d i s t r i b u t i o n of variance  i n 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  i d e n t i f i e d as the normalized  spectrum  i n the  wavelength domain.  versus A  i s given i n Figure 5.1.  A  integrated variance  schematic  plot  Apart from showing  of  the  R(m) shape  - 56 -  J  1  |  |  |  I  I  I  L  sample-length 3  m  Figure 5.1 : Representativeness vs. sample s i z e (schematic). The shape of the normalized integrated variance spectrum curve determines the homogeneity of the ( i n t h i s case hypothetical) data.  of the representativeness-sample  size relationship,  Figure 5. 1  also i l l u s t r a t e s the homogeneity of the (hypothetical) data. In s e c t i o n 5.3 i t was indicated that a dataset an  intrinsically  dataset  will  representative subset  always  have  a  consequence i s that  the dataset  subset  Thus  of  itself.  R  i s homogeneous, i f  exists.  finite  variance,  i s always  becomes  Since  unity  a  a finite  a  trivial  representative  for A  = L.  This  m  c r i t e r i o n alone Section  5.2,  would q u a l i f y a l l datasets  however,  i t was  stated  as homogeneous. In  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  i s therefore  indicated  by the  curvature of the R-A r e l a t i o n i n Figure 5. 1. I f the curvature i s convex  (negative),  the  data  are  homogeneous  because  the  integrated variance increases quickly at small scales and slower at large scales. I f there i s mostly  large scale v a r i a b i l i t y the  curve w i l l be concave ( p o s i t i v e ) . A s t r a i g h t corresponds  l i n e i n Figure 5.1  to a random data structure, as i n 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 i n t e g r a t i o n of the variance spectrum i n the wavelength domain, i t follows from the  above  proportional (i.e.  discussion  that  homogeneity  may  be  considered  to the mean d e r i v a t i v e of the variance  the average  slope  of the variance  spectrum  spectrum  across a l l  wavelengths). For s p a t i a l v a r i a b i l i t y i n 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.  orthogonal,  this  expansion  is  Since the two  straightforward.  indices are The  spectrum i n the wavelength domain can be transformed  variance into polar  coordinates and, a f t e r i n t e g r a t i o n with respect to angle, can be plotted  against  radial  dataset  i s lost  i n t h i s way).  estimates  then  wavelength  refer  to  (any  The  directionality  resulting  circular  in  the  representativeness  sample  blocks  of  varying  radius. In the  context  homogeneity are  of  of  the  interest  surface  characteristics  process  of  turbulence.  present  surface  i n respect  and  the  turbulent  Clearly,  work representativeness to  the  subsequent  fluxes by  the  i t i s highly u n l i k e l y  'sampling'  and of  spatial  averaging  mixing  a c t i o n of  that  these  surface  c h a r a c t e r i s t i c s are purely i s o t r o p i c - e s p e c i a l l y i n a suburban area with preferred street 'block  sample'  which needs  directions. to  be  Also, the shape of  considered  i s probably  the not  c i r c u l a r . These points are further discussed i n Chapter 9. A  suitable  definition  d i f f u s i o n and approximations are presented  for  a  sample  involving  turbulent  f o r i t s dimensions and o r i e n t a t i o n  i n Chapters 6 and  7.  Together with t h i s  present  chapter, they provide the basis f o r an objective assessement of the representativeness of a turbulent f l u x measurement and f o r an  evaluation of  the  homogeneity  measurement i s conducted.  of  the  area  i n which  the  - 59 6.  -  Estimating the Source Area of a Turbulent F l u x  Measurement  over a Patchy Surface  6.1  INTRODUCTION  In any  measurement procedure  i t i s of  fundamental  that the quantity measured by the sensor i s t r u l y the of  interest.  In  the  context  of  the  measurement  turbulent fluxes, t h i s i s not a t r i v i a l originates  at  the  c h a r a c t e r i s t i c s ) but above,  i n the  quantity  surface  (and  quantity  of  surface  concern, since the f l u x is  influenced  by  its  the sensor i s commonly s i t u a t e d somewhere  atmosphere. Over an  sensed  concern  refers  only  to  inhomogeneous surface,  those  surface  the  elements  that  influence the measurement. The region of these surface elements may  be termed the Source Area. In the following, t h i s notion i s  used to develop a tool to estimate  the dimensions of the source  area to which a s p e c i f i c turbulent f l u x measurement r e f e r s . This i d e n t i f i c a t i o n of the contributing surface elements i s a  first  step  large  to  obtain  truly  representative  flux  values  for a  area. The  source  area  of  an  eddy  c o r r e l a t i o n measurement  of  sensible heat f l u x i s defined as the surface area containing the heat  sources  which  can  conditions. which  the  be  and/or  sinks  which  c a r r i e d past  the  influence sensor  those  under  air  given  In other words, i t i s that portion of the instrument  'sees'  in  an  aerodynamical  parcels external surface  sense.  The  distances of the upwind, downwind and l a t e r a l boundaries of t h i s source  area  from  the  sensor  l o c a t i o n are  dependent  on  the  - 60 characteristics development  of  the  flow  and  on  the  boundary  layer  i n the atmospheric layer between the surface and the  sensor l e v e l . The adjustment of the flow to a new set of surface conditions and  the development  of the boundary  layer  leading edge has received considerable a t t e n t i o n primarily  i n the case of a  subsequent  modification  wind-speed  of  discrete the  roughness  Reynolds  after  a  i n the past, change  stress  and  T(Z)  a  and  (u(z) and w(z)) throughout the surface layer. Two of  the three equations required f o r a mathematical s o l u t i o n of t h i s problem are provided by the equations of conservation of mass and  horizontal  momentum.  Various  assumptions,  r e s t r i c t i o n to neutral s t r a t i f i c a t i o n , the set of equations ( E l l i o t , Peterson,  1969;  Taylor,  such  as  a  have been used to close  1958; Panofsky and Townsend, 1964;  1970).  Pasquill  (1972)  reviews  and  discusses some of these studies. Here, only the general concepts r e l a t i n g to conditions i n the v i c i n i t y of an i d e a l i z e d single cross-wind d i s c o n t i n u i t y w i l l be reviewed. Subsequently these concepts are adapted to describe the  more  complex  problem  of  a  two-dimensional inhomogeneity. F i n a l l y ,  patchy  surface,  an a l t e r n a t i v e  i.e.  solution  i s developed i n an attempt to solve t h i s problem, following the original  proposition  patchy roughness.  by  Pasquill  (1972)  for a  surface  with  - 61 6.2  ONE-DIMENSIONAL CROSS-WIND DISCONTINUITY  In the the  case  degree  described  of a single,  of  readjustment  in a  interfaces  discrete  statistical  to  cross-wind discontinuity,  the  manner  new  and  conditions  various  boundary  interfaces,  6.1).  G i v e n t h e 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  following  Munro  d i s c o n t i n u i t y to surface  marks  the  interface  be  layer  may be d e f i n e d a c c o r d i n g l y . H e r e , we d i s t i n g u i s h two  such  the  can  beginning the  flow  of  the  i s not  e q u i l i b r i u m may r e f e r  and  S2, the  Oke  first  (1975)  (see SI  of these  modifications  by  Figure across  interfaces  S2.  Below  this  i n e q u i l i b r i u m w i t h S I anymore.  This  to the mechanical t u r b u l e n c e ,  t h e r m a l s t a t e o f t h e atmosphere,  humidity or  as the c a s e may be.  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 t h e h e i g h t ,  The h e i g h t  up t o w h i c h t h e  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 t h a n an arbitrary  factor  Reynolds s t r e s s the  stress  (e.g.,  i n Peterson  boundary  differs  from  Experimental studies  layers the  the  interface  i s d e f i n e d by p o i n t s  upstream  value  by  i n both laboratory conditions  1968) and a g e o p h y s i c a l s i t u a t i o n growth o f 5(x)  (1969)  p r o p o r t i o n a l to  only  which  0.1  %).  (Schlichting,  (Munro and Oke,  the  at  for  1975) show a  / 5 power o f x ,  where x  is  t h e 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 profile Peterson  second is  interface  completely  (1969)  g e n e r a l l y about  reports ten.  with t h i s estimate  5'(x), adjusted evidence  is and  the in  that  one  below  measurements  the  with  S2.  equilibrium the  ratio  of  Munro and Oke (1975) f i n d a f a i r  in their  which  over  a  5/5*  is  agreement  change  from  - 62 -  .-o(x)  z  S 1 - e q u i 1 i b r i um 1 a y e r y '  /' / '  t r a n s i t i o n zone  / '  Wind  EJF  /  / SI  ^  &'(x)  —  S 2 - e q u i 1 i b r i um 1 a y e r  ( D  -  S 2  x  Figure 6.1 : Internal boundary layer interfaces with one-dimensional d i s c o n t i n u i t y . The degree of adjustment of the a i r to the downwind surface-type i s indicated by a i n i t i a l modification interface (IMIF) and an equilibrium interface (EIF).  Figure 6.2 : The source area i n 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  i n t r i g u i n g l y simple  suggested  by  Miyake,  a l t e r n a t i v e approach was  as  quoted  i n Panofsky  originally  (1973).  In  this  model, the speed, at which information about changes i n surface conditions standard  i s propagated upward i s assumed proportional  deviation  of  the  vertical  «  <r/u  the  velocity fluctuations,  that the interface 5(x) i s described  do/dx  to  so  by  .  (6.1)  w  (1981) assumes that <r  Hajstrup  can be adequately described  by  w  the  r e l a t i o n obtained  over homogeneous t e r r a i n ,  so  that  (6.1)  can be written dS/dx  «  k <f> ( z / L ) / ( l n ( z / z ) - tf> (z/L) 0  w  where  or /u w  =  *  d> (z/L)  and  w  r e l a t i o n s f o r <f> (z/L) can the Panofsky e t . a i . 0  =  one be  ,  (6.2)  M  of  used.  the  common  In BJajstrup's  empirical case he  used  (1977) r e l a t i o n :  (1.5 + 2.9  (z/L) ' ) ' 2  3  1  2  .  (6.3)  w  This use instant who  of the homogeneous r e l a t i o n  equilibrium with the  used an  recognizes  identical  i n e f f e c t assumes  downwind surface.  assumption  i n h i s mixing  Taylor length  (1970), model,  i t as a major weakness and a source of error of h i s  model, but concludes that i n the absence of anything homogeneous r e l a t i o n s may solution.  an  Rao  (1975)  on  better  the  serve as a f i r s t approximation to the the  other  hand  employs  w-model with also rather crude assumptions f o r w.  Panofsky's  It seems that  some sort of assumption f o r the d i f f u s i o n c h a r a c t e r i s t i c s i n the  - 64 t r a n s i t i o n zone are i n e v i t a b l e as a working basis. The canopy layer d i f f u s i o n discussion between Raupach (1979), Thorn  et.al.  (1975) and Garratt  here.  While  a  (1978a, b)  diffusion profile  support within  the case  the canopy  accurately parameterized i n c e r t a i n cases, highly  dependent  on the s p e c i f i c  i n point layer  may  be  i t s character remains  s i t u a t i o n , with  only  little  value f o r general a p p l i c a t i o n . Rao's (1975) r e s u l t s are i n t e r e s t i n g , at least q u a l i t a t i v e l y . He describes the growth of the i n t e r n a l boundary layer as 5  ec x  n  ,  (6.4)  where n depends only on the Monin-Obukhov length L f o r a given set of, i n h i s case, upstream conditions. ranges  from  0.8  (1958),Peterson  f o r neutral (1969)  and  In h i s c a l c u l a t i o n s n  conditions  (confirming  the r e s u l t s  from  (1970)), up to n = 1.5 f o r a f r e e convection  Munro  Elliott and Oke  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  the source area f o r a point terms  of  equilibrium  an  initial boundary  growing  internal  at height  modification layer  z  g  one-dimensional case the source area  layer,  can be described i n  interface  interface  boundary  (IMIF)  (EIF).  and an In  the  i s l i m i t e d upstream by a  region which i s too f a r away to have a considerable the flow at the reference point (see Figure 6.2)  e f f e c t on  - 65 The  upwind  leading  edge of t h i s  i s marked as -x ,  region  e  denoting  a distance from the point of reference of |x | i n the e  upwind d i r e c t i o n . This l o c a t i o n i s defined by 5*(|x  |)  =  z  e  (6.5) s  which states that the equilibrium boundary layer i n t e r f a c e with leading edge at x = -x , reaches a height  its (i.e. the  a f t e r a distance surface  Ix I).  at x = 0  of z  e  s  Thus, being  i n equilibrium  downstream from -x , the p r o f i l e  below 5' i s not  a f f e c t e d by the surface e f f e c t s of the upstream area. reaches a height z  with  Since 5'  at x = 0, the sensor i s not a f f e c t e d by the  s  surface upstream of -x . e  The  downstream boundary of the source area  s i m i l a r fashion.  It i s postulated  that  downstream  r e l a t i o n i s therefore  g  i  the surface  in a  on the conditions at (x = 0,  from that boundary has no e f f e c t z = z ) . The corresponding 6(1x1) = z  i s defined  (6.6)  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  x = -x , reaches the height  at  i  IxJ.  z  s  i t s leading edge  at x = 0, a f t e r a distance  Thus, the surface conditions downstream from x = -x  not f e l t  up to a height z  are  at x = 0.  s  In summary, the source region of a reference point at height z  s  i n one-dimensional  patchiness  i s defined  by  an  upstream  equilibrium boundary layer interface, i n t e r s e c t i n g at the point of  reference  with  an  initial  interface (see Figure 6.2).  modification  boundary  layer  - 66 -  6.4  THE SOURCE AREA IN TWO-DIMENSIONAL PATCHINESS  Taking arises  the preceding example  where a s e r i e s  encountered rapidly  a  further,  a  situation  of one-dimensional d i s c o n t i n u i t i e s  i n the source area.  very complex,  step  seeing  In t h i s  that  case,  several  things  different  are  become internal  boundary layers are superimposed upon each other. Each interface w i l l be a f f e c t e d by a s l i g h t l y d i f f e r e n t and therefore w i l l (6.4)  and  Rao,  have a d i f f e r e n t  1975),  making  l o c a l s t a b i l i t y regime  growth rate  i t necessary  (see equation  to  use  a  bulk  even  more  s t a b i l i t y parameter as an average. The  internal  boundary the  layer  unsuitable  if  problem  is  patchiness,  as i n a more t y p i c a l  the concept of internal boundary  approach  becomes  extended  to  real-world  two-dimensional  surface.  Clearly,  layers cannot be continued to  two-dimensional d i s c o n t i n u i t i e s because the extent of the source area i s not only c o n t r o l l e d by v e r t i c a l d i f f u s i o n and the mean wind, but also by l a t e r a l d i f f u s i o n ( i . e . i n the y - d i r e c t i o n ) . A more s u i t a b l e procedure f o r t h i s case was f i r s t suggested by  Pasquill  (1972).  In h i s approach  the  developing zone  of  influence downwind from a surface element i s treated as a plume. Thermal plumes,  c o n s i s t i n g of temperature anomaly zones i n the  boundary  have been  layer,  identified  by Holmes  (1970) from  a  s e r i e s of airborne temperature transects over a patchy area i n southern Alberta, Canada (see Figure 6.3). Both 'hot plumes' and 'cold plumes' have been observed to s t a r t at the leading edge of  - 67 -  b  a ->  <  >  •  *  WIND ' Dimensions in k m  Height  50 m  a  b  C  a  !>  C  unstable  0-2  0-8  0-1  Q-l  0-4  0-1  neutral  0-6  2-8  0-4  0-3  l'S  5  40  6  2-9  28  0-2'  0-3  0-9  01  0-6  stable unstable 100 m  200 m  400 m  G cnerally rough Zo - 1 m  G cnerally smooth Zo - 3 c m  Stability  0-3  1-5  :  '  0-2 3-4  neutral  1-4  7  1  0-7  4-4  stable  13  (270)  21  10  (130)  14  unstable  OS'  3-2  0-J  05  2-1  0-3  3  25  3  1-6  17  2  6  0-7  90  8  neutral unstable  1-S  9  1-2  11  neutral  10  130  12  6  Figure 6.4 : Estimates of source area dimensions ( P a s q u i l 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 f o r a considerable distance downwind (the height exaggeration  in  Figure  deceiving :  to  D  the  distance from  A  6.3  i s about  is  somewhat  40  km).  In  P a s q u i l l ' s approach  the patchy surface i s thus regarded as  am  array of elementary  'sources' from which a property i s emitted.  P a s q u i l l termed t h i s source area the ' e f f e c t i v e fetch'. This use of the term 'fetch' i s not to be confused with the t r a d i t i o n a l meaning of the word i n meteorology,  where i t denotes a distance  rather than an area. In the following,  both  'source area'  and  ' e f f e c t i v e fetch' w i l l be used as.synonyms. To  become  familiar  with  this  alternative  concept  of  elementary sources and t h e i r plumes, rather than boundary layer interfaces, i t i s convenient to take a step back and again look at  the  one-dimensional  two-dimensional  problem,  before  proceeding  to  patchiness. Gash (1986) estimates the e f f e c t of  a l i m i t e d f e t c h f o r an evaporation measurement by considering an upwind continuum of elementary infinitesimal height  z  strip  above  of  the  width  l i n e sources, each occupying Sx.  For  zero-plane  a  sensor  displacement,  mounted water  an  at  a  vapour  s  diffusing  from  concentration  a  distance x  p (x.z ).  on  average  Applying  a  diffusion  equation  be  solution  sensed by  with  Calder  s  V  (Gash,1986) to  will  the  basic  f o r an  infinite  cross-wind l i n e source (Q) of passive p a r t i c l e s i n conditions of neutral s t r a t i f i c a t i o n , and assuming a uniform windfield, gives Q p (x.z ) V  s  =  .  -Uz e  3  /ku^x (6.7)  k u^x where U  i s the uniform wind-speed, assumed to apply over  the  - 69 whole n e u t r a l l y buoyant boundary layer. I f E = Q/Sx and l e t t i n g 5x  approach  zero,  (6.7)  may  be  integrated  concentration caused by a f i n i t e s t r i p of width X  to  give  the  Gash (1986)  l >  then considers the v e r t i c a l concentration gradient and obtains, after  d i f f e r e n t i a t i o n with  respect  to z  and i n t e g r a t i o n with  s  respect  to  x,  gradient at z  an  equation  f o r the  vertical  concentration  as caused by the d i f f u s i o n from a s t r i p of depth  s  x : L ^v^ s^  ^  Z  -Uz /ku x  s  3z As  X  l  = s  k u z  (6.8)  e  * s  approaches  gradient  -  „,  .  * L  infinity,  equation  (6.8) takes  f o r evaporation  the form of the f l u x  i n homogeneous  and neutral  conditions and unlimited fetch, (z )  dp V  E (6.9)  3 =  dz  He  then  considers  ku z  s  * s  humidity  measured  at z  with  an  infinite  s  upwind f e t c h and a gradient given by (6.9). A percentage of that gradient  (and therefore  diffusion  a  distance  measurement. He terms that  distance  fetch*.  from  of the f l u x ) w i l l  within  be the r e s u l t  x^, from  the  point  the 'F percent  of of  effective  It i s given by the r a t i o 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  s u i t a b l e choice  of U/u , #  an  estimate  of the  - 70 effective  f e t c h f o r a measurement at height z  under  neutral  s  conditions i s given. Gash continues the argument to estimate the fractional  error  step-change  i n evaporation rate at  For  the  of  present  an  evaporation  study  two  measurement  due  to  a  x. p  points are  of  importance.  Gash  (1986) presents the e f f e c t i v e f e t c h i n terms of a percentage, r e a l i z i n g that the t o t a l e f f e c t sensed by the instrument i s due to a l l elementary sources upwind, but that the elementary source weight  decreases  with  distance,  as  represented  concentration i n (6.7). The second point Gash  (1986) :  inhomogeneity  the  diffusion  i s one  calculations  the  noted also  over  (and therefore also an energy input  are not f u l l y  by  a  by  surface  inhomogeneity)  i n t e r n a l l y consistent. Although d i f f e r e n t  fluxes  from the two surface types are considered f o r the evaluation of the  concentration  conditions effect, mentioned (1975),  are  distribution,  assumed  the energy  to  be  the  stability  horizontally  balance constraint  and  uniform.  i s violated.  flow  Thus,  in  Again,  as  i n respect of the assumptions of Taylor (1970) and Rao homogeneous  assumed, even though  horizontal  flow  and  stratification  the energy  input into  the system  are  at the  surface i s not. Atmospheric s c i e n t i s t s are f a m i l i a r with assumptions  of t h i s  kind. A famous example i s the set of Boussinesq approximations f o r the Navier-Stokes equations, where the e f f e c t s of buoyancy and  compressibility  compressibility (e.g.  Nieuwstadt  are  everywhere and  accounted except  van Dop,  for  by  neglecting  i n connection with  1981).  As  gravity  i n the present case,  only the most d i r e c t e f f e c t s of the phenomenon i n question are  - 71 considered, and a f i r s t order approximation i s obtained. Keeping these l i m i t a t i o n s f o r the simple one-dimensional case i n mind, we proceed to two-dimensional by P a s q u i l l  patchiness and the o r i g i n a l  work  (1972).  P a s q u i l l ' s analysis f o r the e f f e c t i v e f e t c h or source area i s developed appear  f o r a 'momentum plume',  as individual  particular  sinks  where the surface  rather  than  elementary area i n i s o l a t i o n ,  sources.  elements  Regarding  the momentum  a  deficit  which t h i s 'sink' w i l l produce at a given height w i l l r i s e to a maximum  at a  certain  distance  continuously as distance  downwind  i s further  and then  increased.  fall off  The f u n c t i o n a l  form i s p r e c i s e l y that contained i n the t h e o r e t i c a l ground-level concentration familiar and  distribution  reciprocal  elevated  relation  sources  from  an elevated  source,  f o r the d i s t r i b u t i o n  i s adopted  (Smith,1957)  from and,  convenience the Gaussian form of v e r t i c a l d i s t r i b u t i o n as a working approximation (Pasquill,  i f the ground if  for  i s used  1972).  P a s q u i l l uses the well known form of the Gaussian plume model for  h i s computations,  assuming  complete  reflection  on the  ground :  *(x,y,0) u  1 =  Q  exp 710*  o*  y  where <r « x z  q  z  (6.11)  exp 2cr  2<r  and or /or = constant (independent of distance). y  z  With the shape of the i s o p l e t h given by (6.11), a c r i t e r i o n f o r the source area i n terms of the particle-source analogy i s  - 72 defined : the area  bounded by the %  /2 i s o p l e t h i s chosen,  max  i.e. the area i n which the concentration from an elementary unit point  source  is  greater  than  one  concentration from such a source, D i f f e r e n t i a t i o n of (6.11) with gives the expression f o r Y  half  of  the  maximum  as sensed at the same height. respect  to x (and with  y = 0)  '•  max X  2 <r  /Q  z  max  so that the x  /(TTZ  s  (6.12)  e u <r ) y  /2 i s o p l e t h i s given by max  r 1  =  exp  1 -  • exp •  2ar  z  2  •>  s  2<r  (6.13)  2  z  (note that since both or and or are functions of x, (6. 13) i s y z two-dimensional and symmetric about the x-axis). Pasquill  (1972)  tabulates  the dimensions  of t h i s  region  r e l a t i v e to the receptor l o c a t i o n f o r a set of height, roughness and area  stability  conditions  estimates  variability.  (see Figure  are the f i r s t  6.4). P a s q u i l l ' s  attempt  to cope  with  source patchy  In view of recent developments i n d i f f u s i o n theory,  there are some points i n h i s approach which may be updated i n a revised version of the same concept.  - 73 6.5  A RECIPROCAL PLUME MODEL TO ESTIMATE THE SOURCE AREA  As mentioned, P a s q u i l l  (1972) assumes Gaussian d i f f u s i o n i n  both the horizontal and the v e r t i c a l d i r e c t i o n s f o r h i s source area computations. (1957) r e c i p r o c a l  This assumption  allows him  to use  Smith's  theorem, which states that the concentration  at ground l e v e l downwind of a point- or line-source at height z s  is  identical  height  to the concentration at the same x and  z , due  to an  exactly  similar  source  at  y but at  ground  level  s  (Smith,  1957). Smith's proof of t h i s theorem r e l i e s h e a v i l y on  the assumption that the d i s t r i b u t i o n of v e l o c i t y f l u c t u a t i o n s i n any d i r e c t i o n i s symmetrical,  which indeed i t i s assumed to be  i n the Gaussian model. In h i s review paper on updating applied diffusion  models  Weil  (1985)  diffusion  i s acceptable  states  i n the  that,  horizontal,  c l e a r l y not Gaussian i n the v e r t i c a l .  The  while real  Gaussian  plumes  are  implication i s that,  with skewed v e r t i c a l d i f f u s i o n , the symmetry assumption  i n Smith  (1957) i s i n v a l i d and the concept of r e c i p r o c i t y , as employed by P a s q u i l l (1972), should be modified or replaced. It w i l l be shown i n the following that a s l i g h t l y concept problem,  of r e c i p r o c i t y regardless  characteristics.  is valid of  the  different  i n the context of the present functional  In p a r t i c u l a r ,  i t will  form not  assume that an elevated source at height z  be  of  diffusion  necessary  to  generates the same s  e f f e c t pattern on the ground as a ground source w i l l produce at height z . As isn the  one  dimensional  case,  the  by  now  familiar  - 74 assumption  of  a  horizontally  homogeneous  windfield  average atmospheric s t r a t i f i c a t i o n has to be applied. Simpson  (1982)  mention  that  surface  changes  of  and  an  Hunt and  a  limited  amplitude have only very weak dynamical e f f e c t s and the zone of influence i s the same f o r a l l such changes. it  i s assumed that horizontal windshear  s p a t i a l and temporal  In the present study  i s n e g l i g i b l e over the  scales of concern, and that Monin-Obukhov  s c a l i n g i s applicable as a f i r s t approximation. Consider  a  sensor  of humidity,  t h e i r f l u c t u a t i o n s at x = 0,  temperature,  y = 0 and  wind-speed  at a height z  or  i n the s  surface layer (see Figure 6.5).  An a r b i t r a r y elementary  source  on the ground and i n a sector upwind from the sensor w i l l create a  time  averaged  concentration d i s t r i b u t i o n  whose  maximum  Is  located a c e r t a i n distance d i r e c t l y downwind from t h i s source. If  this  source  direction,  the  i s moved  a  given  distance  concentration d i s t r i b u t i o n  in  will  the  x  shift  or  y  in  a  complementary fashion, due to the h o r i z o n t a l l y homogeneous flow. Accordingly,  the sensor at z  will  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 effect  at the sensor w i l l  away from total  the  effect  decrease,  maximum source  that  location.  Obviously, the  when the source  location  i n any  i s moved  direction.  the sensor experiences i s determined  weighted  contributions  resulting  two-dimensional  of  all  source  sources weight  upstream, distribution  fi(x,y) has i t s maximum at the maximum source location.  by  and  The the the  function  height  mox  wind  P-SOURCE A R E A  F i g u r e 65 . is  :  Schematic c r o s s - s e c t i o n o f a P - c r i t e r i o n s o u r c e a r e a .  d e f i n e d as the  relative effect  r e g i o n i n which any p o i n t  level  of x  source weight d i s t r i b u t i o n of  the  p  at  source causes a  the s e n s o r l o c a t i o n .  fi(x,y)  is equivalent  It  minimum  The c o r r e s p o n d i n g  t o the  plan-projection  z - e f f e c t l e v e l d i s t r i b u t i o n o f a v i r t u a l s o u r c e beneath s s e n s o r (and w i t h a v i r t u a l wind i n the r e v e r s e d i r e c t i o n ) .  the  - 76 Consider furthermore an a r b i t r a r y sensed concentration or e f f e c t effect  level,  Again,  i f the source  location  f o r a source  i n any  c r i t e r i o n ' P' to define a  level x^, as the minimum sensed  to belong  to the P-source  i s moved away from  direction,  the  sensed  the maximum effect  point sources,  equals x^,  whose e f f e c t  level  will  l o c a t i o n of  at the sensor  location  forms a closed curve, which i s the boundary of the  P-source area. This curve i s i n e f f e c t a tracing of the r  source  level  eventually decline to the x\p l e v e l . The geometric all  area.  (on the ground)  i s o p l e t h (on an imaginary plane at l e v e l  z ) derived  P  s  from the source at the maximum source location.  Therefore, the  source area may be obtained by a simple geometric t r a n s l a t i o n of the Xp i s o p l e t h distribution translation  (see Figure 6.5). S i m i l a r l y ,  function  the source  weight  (flcx.y)), with u = Q(x,y) i s a geometric  of the concentration or e f f e c t  level  function at l e v e l z . Mathematically, the P-source  distribution area  outline  s  and the x  p  and  isopleth  (or the source weight  the concentration d i s t r i b u t i o n  distribution  function  at z , respectively) are s  i d e n t i c a l except f o r a change of axes.  In practice,  Q(x,y) may  be obtained by simply reversing the wind d i r e c t i o n , virtual  source  at  the  ground  underneath  the  placing a  sensor,  projecting the v i r t u a l e f f e c t level d i s t r i b u t i o n at z  and  down onto  s  the ground (see Figure 6.5). This concept (1972),  i s s i m i l a r to the reciprocal  but i t i s independent  of Smith's  plume of P a s q u i l l (1957)  reciprocity  theorem, however, and i s not constrained by any p a r t i c u l a r of v e r t i c a l or horizontal d i f f u s i o n .  form  - 77 -  F i g u r e 6.6  :  Integrated effect 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  effect  - 78 Having  established  the  source  o u t l i n e f o r a P-source area, ' P'  which  defines  obvious suggestion  the  weight  distribution  a s u i t a b l e form f o r the  effect  i s that  level  'P'  isopleth  i s the  is  fraction  of  e f f e c t which i s contributed by the P-source area. above,  the  total  effect  (E )  at  the  weighted sum of a l l upwind sources  :  t  sensor  and  the  criterion needed. the  An  total  As mentioned  location i s  the  1  For point sources the summation becomes a double i n t e g r a l of the weight d i s t r i b u t i o n function £2(x,y), and considering the of  each source  relative  to  i t s source  strength,  effect  the  drop  out : + 00  E  =  t  J* J* 0(x,y) dx dy -oo  so that E  +00  (6. 15)  0  i s the volume under the Q(x,y)-function  6.6a). The horizontal area bounded by each w = w  p  w  p  corresponding  fraction  'P'  to x  of  p  before  which  the  i s contained p  + 00  =  Figure  isopleth (i.e.  t r a n s l a t i o n ) represents  surface portion bounded by the w -isopleth  P  (see  underneath  the  (Figure 6.6b)  that Q(x,y)  :  +00  J\fQ(x,y) dx dy / J" J" Q( ,y) dx dy u=w -oo o X  .  (6.16)  P  It  follows  effect  that  which  ' P*  i s the  i s contributed  portion by  the  of  the  total  integrated  P - c r i t e r i o n source  bounded by the weight d i s t r i b u t i o n function i s o p l e t h w = The  P-source area i s therefore defined  area, w. p  i n a s i m i l a r fashion  - 79 to  the F-percentage  seen  e f f e c t i v e f e t c h of Gash (1986) and may/ be  as an extension of i t to two-dimensional  Depending on the kind of plume source  diffusion  model  characteristics,  chosen  however,  inhomogeneity. to describe the  the  source  area  c a l c u l a t i o n s are not limited to neutral conditions. The present work i s aimed at unstable thermal s t r a t i f i c a t i o n and the source area model presented i n the next chapter i s applicable only i n those conditions.  - 80 7.  The Source Area Model  7.1  (SAM)  INTRODUCTION  The concepts of the source area model (SAM) diffusion  characteristics  of  a  ground-level  are based on the point  source.  A  short-range plume model forms the core of the source area model. The  choice  of  importance,  plume  diffusion  model  is  therefore  of  prime  since i t contains a l l the physics and boundary layer  parameterization schemes that w i l l  eventually be  reflected  in  f o r use  is  described, with s p e c i a l focus on adaptions f o r the purposes  at  the source area r e s u l t s . In the  f o l l o w i n g the  hand and  the computer  data from  independent  These test  diffusion  model  implementation.  selected  It i s then  tested with  passive tracer releases at ground  r e s u l t s are presented  i n Appendix B,  level.  together  with  the FORTRAN-77 code of the source area model. Since the source area  itself  is  not  directly  measurable,  possible v a l i d a t i o n of the physical source area model.  this  processes  is  the  determining  only the  - 81 7.2  AN APPLIED DISPERSION MODEL BASED ON METEOROLOGICAL SCALING PARAMETERS  In h i s recommendations f o r applied plume  dispersion  models  f o r ground l e v e l sources, Weil (1985) mentions three classes of models, layer  into  which recent  scaling  Gaussian  developments of convective boundary  are incorporated.  model,  the  probability  These  include  density  the  function  classic (p.d.f.)  approach and 'impingement' models, such as the one by Venkatram (1980). A l l three classes use Gaussian concentration p r o f i l e s i n the l a t e r a l d i r e c t i o n , but only the Gaussian model also assumes a symmetric d i s t r i b u t i o n i n the v e r t i c a l .  Observations of r e a l  plumes make clear, however, that the assumption of homogeneous turbulence and Gaussian d i f f u s i o n i n the v e r t i c a l d i r e c t i o n does not hold  i n the presence of a surface boundary ( P a s q u i l l and  Smith, 1983). Therefore the Gaussian approach has to be r e j e c t e d e s p e c i a l l y f o r the case of ground-level releases. 'Impingement'  models  have been designed  primarily  to cope  with the s p e c i a l features of buoyant plumes (Weil, 1985) and are not s u i t a b l e f o r the present study. This leaves the p.d.f. model class  as the most  appropriate  and up-to-date  applied  plume  modelling approach f o r present purposes.  7.2.1  The Gryning et a i .  The p.d.f. Gryning et it  al.  model  (1987) P.D.F. Model (GHIS)  selected  f o r this  work was presented by  (1987) (hereinafter r e f e r r e d to as GHIS). While  i s easy to use on a routine basis,  i t incorporates  recent  - 82 developments stability  i n the  scaling  parameters. This  of  boundary  layer turbulence  s e c t i o n summarizes the  and  f i n d i n g s of  GHIS and follows the format of that paper c l o s e l y . GHIS divide  the  boundary  regimes depending on  the  layer  into  relative  a  height  number within  of  the  scaling boundary  layer and the atmospheric s t a b i l i t y conditions (see Figure  7.1).  The present a p p l i c a t i o n i s concerned only with unstable  daytime  conditions and  tens  i s limited  metres. From Figure 7.1  to heights  less than a few  t h i s means that surface  of  layer s c a l i n g  applies i n most daytime s i t u a t i o n s . In addition, GHIS point out, based on Holtslag's (1984) study  on  the  ' P r a i r i e Grass'  data,  that surface layer predictions compare well with f r e e convection layer r e s u l t s at small distances. As mentioned, spread assumed to  be  i n the  Gaussian and  only In conditions where u/w,  l a t e r a l d i r e c t i o n may  s a f e l y be  axial  diffusion  becomes  Important  s 1.2  ( W i l l i s and Deardorff,  1976)  ( i . e . at windspeeds below about 1 m-s , as a rule-of-thumb f o r -1  most  situations).  separated  It  follows  into a cross-wind  that  the  plume  model  may  be  integrated concentration (CIC) model  of the v e r t i c a l spread to which a Gaussian component i s applied to account f o r the l a t e r a l d i s t r i b u t i o n .  7.2.1.1  V e r t i c a l Dispersion  Nieuwstadt  and  van Ulden  (1978)  dispersion from a ground level source adequately  show  that  the  vertical  i n the surface  layer i s  described by K-models, using Monin-Obukhov s i m i l a r i t y  theory to account f o r the e f f e c t s of s t a b i l i t y on  the  vertical  - 83 -  UNSTABLE  NEUTRAL 01MENSI0NLESS  STABLE  STABILITY h/'L  Figure 7.1 : The s c a l i n g 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 i s 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 p a r t i c l e s , 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)  =  H  It  i s interesting  reverse  analogy  *  to note that  to  describe  .  (7.1)  H  the present study assumes the  a  heat  plume  based  designed f o r a p a r t i c l e or gas plume, so that  on  a  model  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 p r o f i l e s as i n (7.1) together with the f a m i l i a r diabatic numerical  log-linear  wind  solutions  profile  of  the  have  often  crosswind  been  integrated  used  for  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 P a s q u i l l and Smith (1983, p. 96) as CIC  0  as x,z  CIC  oo  at x = z = 0  K -a(CIC)/az  -+  0  as z  oo  0,  x>0  H  and when the p r o f i l e s of wind and d i f f u s i v i t y simple  power laws, an approximate  analytical  are replaced  by  s o l u t i o n becomes  possible and i s expressed by van Ulden (1978) as CIC/Q  with  =  (A/zu)-exp  j-  (B-z/z) Js  (7.3)  - 85 oo  z  =  z-CIC(z) dz  o  (7.4)  CIC(z) dz 0  as the weighted mean height o f the p a r t i c l e s that have t r a v e l l e d a distance x and  (7.5)  being  the mean  horizontal  velocity  o f the p a r t i c l e s .  exponent 's* i n (7.3) i s termed the shape factor  The  and i s c l o s e l y  r e l a t e d to the exponents of the power law p r o f i l e s o f wind-speed and d i f f u s i v i t y (van Ulden, 1978). The x-dependency o f CIC/Q i n (7.3) i s contained  i n t h i s x-dependent exponent as well as i n z.  B  from  and A  follow  a  substitution  of (7.4) into (7.3)  (Nieuwstadt and van Ulden, 1978) as A  =  B  =  s-r(2/s)/[r(l/s)] r(2/s)/  r(i/s)  s  (7.6)  s  (7.7)  where I* i s the gamma function. CIC/Q i s thus obtained mean plume height  from the  z, the shape parameter s, the mean transport  v e l o c i t y u and the functions A and B. GHIS summarize equations for  the c a l c u l a t i o n or approximation o f these q u a n t i t i e s i n  t h e i r appendix. A r e l a t i o n between t r a v e l distance, roughness length, Obukhov  length  and the mean plume height  Monin-  i s approximated by  van Ulden (1978) and may be written f o r a ground source and unstable conditions :  - 86 x  =  (z/k )-[ln(cz/z ) - i/r (cz/L)]-[l - pa z / ( 4 L ) ] ~ . 2  0  1  M  The c o e f f i c i e n t s p and c depend on s but are rather to  variations  p = 1.55  (7.8)  1/2  of  s.  van Ulden  for practical  (1978)  suggests  insensitive a  value  applications and Gryning et al.  of  (1983)  propose c = 0.4 f o r unstable conditions. After f i t t i n g the power law of wind speed to the d i a b a t i c form of the l o g - l i n e a r p r o 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 f a c t o r  s is  determined by Gryning et al. (1983) as  s  1 - a c-z/(2L)  1  =  1 - a c-z/L l  (1 - a c - z / L ) ~ * + -— ln(c-z/z ) - </• (c-z/L) o M x /  (7.9)  f o r unstable conditions. The  constants a  and  a  1  are r e l a t e d  to the choice of  the  2  non-dimensional p r o f i l e functions f o r heat and momentum <p and <p .  In the present a p p l i c a t i o n the functions suggested by Dyer  M  and  Bradley  (1982)  are  used  with  a,^ = 14  and  a  g  = 28.  S e n s i t i v i t y tests conducted here show that differences between this  and  the Dyer  (1974) version of a  i  = a  g  = 16  are  minimal  (see Appendix B). F i n a l l y , the mean horizontal plume v e l o c i t y u i s approximated by van Ulden (1978) f o r L < 0 u  =  u / k - [ l n ( c - z / z ) - 0 (c-z/D] . o  M  In practice, equations (7.8),(7.9) and  (7.10)  (7.10) are used as a  - 87 t r i p l e t i n an i n i t i a l value i t e r a t i o n to obtain z, s and u f o r a given set of x, z  and L. With these values, A i n (7.6) and B i n  o  (7.7) are known and CIC/Q may be obtained from (7.3). The  shape  factor,  s,  varies  with  stability  and  downwind  distance. In most r e a l s i t u a t i o n s t h i s v a r i a t i o n seems to range between values of 1 and 2 at short range and unstable conditions (Gryning  et  Therefore  al.,  the  1983). Gaussian  When s = 2, model  is a  (7.3)  special  becomes  Gaussian.  case  a  of  p.d.f.  model. In the present a p p l i c a t i o n s was prevented from exceeding a  value  of  numerical  2  (by  exponential  damping- f o r s > 1.9)  to  avoid  i n s t a b i l i t i e s i n the model.  P a s q u i l l and Smith (1983) present an equivalent  approximation  f o r the x-s r e l a t i o n taking a s l i g h t l y d i f f e r e n t route.  7.2.1.2  Lateral Dispersion  Lateral  d i s p e r s i o n i s caused  wind-shear.  In  intensities  and  unstable over  a  becomes i n s i g n i f i c a n t the  cross-wind  by  turbulence  conditions  short  range  with  only,  and h o r i z o n t a l  high the  latter  (Pasquill and Smith, 1983).  spread  of  the  plume,  turbulence  <r , may  process  Consequently,  be  related  to  y  turbulence  via  the  standard  deviation  of  lateral  wind  f l u c t u a t i o n s , cr^, f o l l o w i n g Taylor (1921). From experimental the  lateral  distribution. time-averaged  profile Once  observations i t i s concluded of  a  plume c l o s e l y  the  concentration  appropriate at  any  by GHIS that  resembles CIC  point  is may  a  Gaussian  known, therefore  the be  - 88 obtained from  CIC(x.z) %<x,y,z)  =  where x(x,y,z)  (7.11)  exp  V2ir ar  i s the normalized  2<r  concentration  ( i . e . conc./Q).  Taylor's formula may be written as  ar  y  =  cr t-f •  (t/T )  y  (7.12)  y  following GHIS, where t i s the t r a v e l  time (= x/u),  and  T  is y  the Lagrangian  time scale f o r l a t e r a l dispersion, so that f  a f u n c t i o n of the non-dimensional travel time t/T . In a  is  review  y  of various suggestions  f o r the form of the f  function,  Irwin  (1983) found the best r e s u l t s with a version by Draxler (1976) :  f The  choice  resulting f  of y  estimates.  T  y  =  l/[l+(t/2T )  1/2  y  obviously  has  a  ]  (7.13)  significant  effect  on  the  and introduces considerable uncertainty into the cr  y  For  very  short  distances  and  travel  times  f y  approaches u n i t y and a l i n e a r model i n place of (7.12) appears to be a reasonable s i m p l i f i c a t i o n : cr  y  This  linear  approximation  *  ar • x/u  (7.14)  v  was  also  proposed  by  Pasquill  and  Smith (1983) f o r short distances from the source. Comparisons of the r e s u l t s given by t h i s model with the data from 35 runs from the  Hanford-30 data  series  (see  Appendix B)  unstable showed  - 89 good agreement. Since the concept a  scaling  parameter  of a Lagrangian  becomes questionable  over  time s c a l e as highly  complex  surfaces (and close to the roughness elements) t h i s l i n e a r model which does not r e l y on estimates  of a T  has  been adopted f o r  y  the present work. <r^ i s not estimated  u s u a l l y a v a i l a b l e on a routine basis.  It may  be  from the standard f l u c t u a t i o n s of wind d i r e c t i o n or , <P  following Panofsky and Button (1984), as  <r  «  or -u  If <r^ cannot be  obtained,  the mixing height, z ^  (7.15)  If)  V  an a l t e r n a t i v e parameterization v i a  i s proposed by GHIS. For ground sources : Z  (o-/u) V  In summary,  2  *  0.35-  (7.16)  *  the  k-L p.d.f.-plume model  surface  layer  provides  a concentration estimate  range  of  the  independent observations  scaling  source.  comparison . The  *  conditions  B  of  model  required  o  input  v  GHIS ( f o r unstable,  ground-level  f o r any  Appendix the  by  and  and e i t h e r or , or  includes u , L, z  ->2/3  i  <p  gives  point the  with  1  within close  results tracer  information or z .  sources)  for  of  an  diffusion the  model  - 90 7.3  IMPLEMENTATION OF THE  DISPERSION SUB-MODEL IN THE  SOURCE  AREA MODEL  Chapter 6 shows how estimate the weighted  a r e c i p r o c a l plume model may source area of a sensor at a  be used to level  z . s  This r e c i p r o c a l plume i s i d e n t i c a l to the v i r t u a l plume from a v i r t u a l source on the ground beneath the sensor, dispersed.by a virtual  wind  i n the opposite d i r e c t i o n to the r e a l  source area f o r a given s i g n i f i c a n c e area bounded by a c e r t a i n  level  wind.  The  i s defined as  the  i s o p l e t h of t h i s v i r t u a l  plume.  In  order not to confuse the r e a l plume with the v i r t u a l plume, the 'concentration' of the v i r t u a l plume was or  source weight,  termed 'effect  level'  <*>, where each value i s scaled by the maximum  along the centerline. The source weight,  w,  of a point (-x,-y,0) i n respect to a  sensor at point (0,0,z ) i s i d e n t i c a l to the concentration, %' , s  at  a point (+x,+y,z ), scaled by the maximum concentration on s  the c e n t e r l i n e ( i . e . on the l i n e 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 )  s ' max  In  the  following,  an  axis  reversal  .  (7.17)  s  i s assumed,  so  that  negative s i g n f o r the x and y p o s i t i o n and the primes of %  the m a  y  be l e f t out. The concentration f i e l d of a continuous point source on the ground  is  equations  defined (7.3),  for  any  (7.6)-(7.11)  level and  in  the  (7.14).  surface  layer  Consequently,  by the  - 91 normalized point  c o n c e n t r a t i o n %(x,y,z)  (x,y, z ) ,  once  the s i t e  may be determined  conditions Z  f o r any  and d and the  q  meteorological s c a l i n g parameters L, u and cr are known. In p r a c t i c e numerical  i n s t a b i l i t i e s and overflow c o n d i t i o n s  l i m i t the distance i n the x - d i r e c t i o n f o r which x i s n u m e r i c a l l y defined t o regions w i t h x £ x  , and x  min  b i s e c t i o n search r o u t i n e . Subsequently, are  obtained  by a s i m i l a r  i s determined  using a  min  %  and i t s x - l o c a t i o n  max  'regula f a l s i '  search  along the  centerline. In order t o c a l c u l a t e the l o c a t i o n o f a given i s o p l e t h the GHIS-model equations are solved n u m e r i c a l l y f o r any one member o f the  triplet  (%,x,y) when the other  Because o f the l a t e r a l have  two elements are given.  symmetry of d i f f u s i o n o n l y p o s i t i v e y  t o be considered.  Nested  combinations  o f forward  and  backward stepping, b i s e c t i o n and secant algorithms are used f o r t h i s purpose. With  these  numerical  tools  the e f f e c t  level  field  may be  i n t e g r a t e d portion-wise, along i s o p l e t h s o f decreasing f r a c t i o n s of x  • The i n t e g r a t i o n  i s performed by a double  Simpson's  max  r o u t i n e . The i n t e g r a l over the whole e f f e c t f i e l d ( i . e . down t o the zero e f f e c t l e v e l ) i s approximated  by a simple second order  e x t r a p o l a t i o n , i n which the curvature of the l a s t three nodes i s preserved.  The  integrated  bands  are then  translated  into  f r a c t i o n s o f the t o t a l i n t e g r a t e d e f f e c t and serve as nodes f o r a cubic s p l i n e i n t e r p o l a t i o n t o map rounded values o f i n t e g r a t e d effect  f r a c t i o n t o an appropriate e f f e c t  level  i s o p l e t h . For  each rounded i n t e g r a t e d e f f e c t f r a c t i o n a set of  characteristic  F i g u r e 7 . 2 : The is  defined  by  source the  characteristic  set  area of  dimensions  of  its  a:  d i s t a n c e t o downwind edge o f the isopleth d i s t a n c e from the downwind edge to the p o i n t where t h e i s o p l e t h i s widest d i s t a n c e from 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 t h e upwind edge  b:  c:  d:  bounding  maximum w i d t h isopleth  a+b+c: a In  isopleth.  of  the  maximum  fetch  : minimum  fetch  addition,  distance  to  the the  source  Vocation  by t h e  source  axial maximum is  area  given model.  SOURCE AREA MODEL - RESULTS  e -  Figure 7 . 3 :  1  SAM Calculations for Jul.Day. 212 at 12:30 L A X at SUNSET 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  Example plot of source area model r e s u l t s  - 94 source area dimensions i s determined  from these isopleths. These  dimensions include (see Figure 7.2),  a : the distance from the  sensor to the downwind c e n t e r l i n e l i m i t ,  b : the distance from  the downwind c e n t e r l i n e l i m i t to the distance with the maximum width, c : the distance from that point to the upwind c e n t e r l i n e l i m i t and d : the maximum half-width of the isopleth. The area bounded  by  each  of  these  isopleths  i s evaluated  Simpson's integration algorithm. This area, dimensions f o r each rounded x-distance  of the maximum  integral  source  a  the c h a r a c t e r i s t i c  effect  location  using  fraction  and the  i n respect  to the  sensor, form the p r i n c i p a l r e s u l t s f o r a given source area model run. As an option, these r e s u l t s may be presented graphically, i n a three dimensional plot, such as i n Figure 7.3.  7.4  THE FORTRAN-77 CODE OF THE SOURCE AREA MODEL  The  FORTRAN-77 code of the source  area model c o n s i s t s of a  set of 23 nested subroutines and functions, adding 2100  up to some  l i n e s (including comments). The model i s independent of any  library  routines (other than  the i n t r 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  calls  the  shell  - 95 subroutine, the model may  be run i n a loop with imported  input  variables, or i n t e r a c t i v e l y , where the user i s prompted f o r the model input. The e n t i r e code i s presented i n Appendix C.  7.5  A STATISTICAL VERSION OF THE NUMERICAL MODEL (mini-SAM)  As mentioned above, a SAM run i s quite time consuming and be  too  tedious  necessary.  The  in situations results  may  where only a rough estimate  of 715  runs  on  a  PC  and  on  the  is UBC  mainframe computer, with a wide range of input values, have been used  to  approximate  the  isopleths by a polynomial  chararacteristic  dimensions  of  the  fit.  The r a t i o of the dimensions  of the various integrated e f f e c t  f r a c t i o n isopleths i n 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,  of  one  are known. The  not exceed  i f the c h a r a c t e r i s t i c dimensions  errors of these approximations  a few percent. Table 7.1  of i s o p l e t h dimensions  normally  do  shows the standardized set  i n r e l a t i o n to the 0.9-isopleth together  with the expected standard errors. Each equal  area  between  portion  two  ( i . e . 10  sequential %)  of  the  isopleths total  represents  integrated  an  effect,  although t h e i r s i z e s vary by more than an order of magnitude. Therefore,  the  relative  source  depends on  i t s p o s i t i o n r e l a t i v e to the set of isopleths.  source weight per area, w, may  weight  of  an  elementary  then be written i n terms  of  area The the  Table 7. 1 the  : Standardized set of isopleth dimensions r e l a t i v e tp  0.9 isopleth.  The source  weight  per area,  w ,  i s given  p  r e l a t i v e to w int .eff 0. 10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90  (see equation (7.20)).  Ap +/ - % 0.0126 0.0311 O.0571 O.0931 0.1421 0.2107 0.3147 0.5047 1 .0  7.3 6.4 5.6 5.0 4.6 4.3 3.7 2.5  a +/ - 'A 2.1082 1.7706 1.5719 1.4347 1.3314 1.2472 1.1716 1.0942 1 .0  2.0 1 .4 1.2 1 .0 0.9 0.8 0.7 0.6  b  +/•  0.0787 0.1275 0.1817 0.2420 0.3107 0.3955 O.5016 0.6663 1 .0  1  4  4 4 4 4 4 4 4 4  3 6 5 6 2 3 3 3  +/ - %  d +/•  0.1068 9 0 0. 1728 8 .6 0.2372 8 5 0.3062 7 8 0.3819 7 5 0.4651 7 1 0.5711 6 0 0.7207 5 1 1.0  0.1357 0.2053 0.2700 0.3359 0.4060 0.4849 0.5821 0.7242 1 .0  C  „  1  4  3 1 2 5 20 17 14 13 1 1 0 8  V*o.i 1 .00 0.68 0.48 0.35 0.26 0. 18 0. 12 0.066 0.025  - 97 source  weight  distribution  f u n c t i o n Q(x,y) (and normalized by  the t o t a l e f f e c t E ) as  x+Ax y+Ay v  =  J x  /  J £l(x,y) dx dy / ( A x - A y E ) y /  .  t  (7.18)  For the average source weight per area between i s o p l e t h s P and (P-0.1), w , follows p  v  so that the w  p  p  =  0. 1 / [A - A ] ' p p-o.r L  ,  (7. 19)  may be normalized with the average source weight  per area f o r the smallest isopleth, W  , as  Q  A o.i w/v ^ °-  1  A  p  is  m  [A - A  1  P  (7.20)  ]  P-0.1  J  i s the area bounded by the P-isopleth. This r a t i o , a measure f o r the importance  of a unit  area  wp/w  Q  »  x  of a given  surface inhomogeneity depending on i t s p o s i t i o n r e l a t i v e to the sensor.  For example,  a  disturbance  somewhere  0.9-isopleth and the 0.8 i s o p l e t h i s of l i t t l e  between the  consequence to  the measurement, compared to a similar" disturbance within the 0. 1-isopleth,  and w^/W  Q  ^ indicates how much less  important i t  i s . The standardized set of source weights per area, r e l a t i v e to the 0.1 i s o p l e t h i s also contained i n Table 7.1. The  0.9-isopleth  approximated  characteristic  have  been  i n terms of z , z /z , u . L and o* by f i t t i n g a s  set  dimensions  of orthogonal  polynomials  s  0  *  v  to the model r e s u l t s ,  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 f o r dimension X  (X = a, b, c, d, x 1  , A) i s given max  i  by :  n In (X )  =  t  n  l  E a  ^ k=0  •(lnz) + k  Uk  s  2  V a  u  k=0  n  •(ln(z /s ) ) + k  12k  s  0  n Ea  i 3 k  .(lnu.)  Ea  k +  k=0  U k  .(ln-L)  (7.21)  1  k=0  n E a u  k=0  • ( l n <r ) lBk  ; ( i = 1,2  v  In t h i s way each c h a r a c t e r i s t i c dimension one l i n e a r equation of f i v e independent variables  are constant  6).  i s approximated  by  variables. The f i r s t two  f o r any given s i t e ,  which reduces the  number of variables f o r a s p e c i f i c s i t e to three meteorological controls. Table  7.2  summarizes  the regression constants  a  (as i n ijk  6  equation (7.21)) and Table 7.3 contains the summary of the polynomial f i t . of  the comparison  polynomial  statistics  Figures 7.4 - 7.9 show the s c a t t e r p l o t s  between  source  area  model  runs  and the  approximations.  Though the number of terms necessary to approximate a l l the dimensions  i s large, the a c c e s s i b i l i t y of source area estimates  i s increased considerably by t h i s  statistical  version  of the  ln i  tai i  00  2. 6652154  11  - S. 3899568  '?  1 .SB07017  13  -  -22.573248  In  21  0. 32944544 0. 257286a3E-01  i  - 2.23S5605  17.244166 -  4.9954973  -  4.0627402  31  0. 12238973  0.18670063 0.25348158E -01  - 0.360962S7E -01  0.10104497  -  33  -  0. 79245576E-01 —  41  1. 7965164  42  -  0.61049839 - 0.26358443E -01  -  -  34  0. 25708377  -20. 997688  1.3945269  -  -  - 0.64599710E- 02 -  - 0.97617692 0.38747429E- 01  0.26010320  1.778B597  0.27445103  - 0.72653457E -01  -  0.11555328E -01  - 0.21603852E -01  •  0. 120770B6E-01  44  —  51  0. 78914032E-01  0.19311151  0.15864185  32  0. 53996396E-01  0.11940725  0.9109B499E -01  33  - 0. 32373654E-01  -  - 0.59892080E -01  -  4. 7530946  constant Int:) •  1. 3733871  ln'd > • ln'<z I  0. 10B24610  ln'd  «  -  23.235270 -  In  •  )  (z / z ) • o  6. 7472282  In* (z / z ) » o  0. B243109B  In'(z  In" (z / z )  •  /z ) o  -  0. 35580032E -01  0. 10103001  -  0. 63979019  In tu_)  0. 12667125  -  0. 69395441E -01  In t >  a  O  1  u >  —  -  -  0. 39B53203  In'  (uJ  —  -  -  0. 10672003  In  <uj  3. B322710  ln <-L)  0. 4904 2522  In*(-L)  0. 20778019E -01  In t-L)  1. 9027159  0.582B7161E- 01 -  0. 25730721 0. 116B5460E -01 —  0.99485475  6. 2979254  variable  0. 25351517E -01  0.82672876  0.17S39108E -02  - 0.69132516E -01  -  0. 5B4269B5  - 0.21545915E -01  43  13. 830045  16. 326991  0. 14192713  - 0.74067156E -01 2. 1733248  -45. 205787  0. 10049848 —  0.497B679S  0.23067548  In (A)  1. 4336980 0. 86714119  0.10884310 -  13.980797  —  0. 320B1366E-01  -  1.2944537  - 0.43476245E -01  24  -  -  0.51065259  0. 17321196E-02  32  In (xm)  Id) 1 4 1  4.9746758  23  -  ln  lc)  i*  -21.003838  0.83391462  -  —  -  (b) 1 2 1  0. 19333715-.  14  22  ln  i  -  -  —  -  4  1  In't-L)  (cr )  0. 1962273B  1. 2280201  ln  0. 20817542  0. 13440652  In (a J  -  0. 74461739E -01  -  0. 37152093E -01  - 0. B6402650E -01  2  In'  (a ) V  54  —  -  —  —  Table 7.2 : Summary of mini-SAM polynomial f i t c o e f f i c i e n t s . M u l t i p l y coefficient with variable and add v e r t i c a l l y , to obtain the c h a r a c t e r i s t i c dimensions for the 0.9-source area (see eq. (7.21)).  —  -  In (CT > 4  V  -  100 -  100  -  40  30  40  50  60  SAM Isoplech-dimension  Figure 7 . 4 : dimension a  ' a'  70  00  (m)  mini-SAM v a l i d a t i o n s c a t t e r - p l o t  3000 F  2500  2000  1500  1000  100  1000  1500  SAM Isopleth-dimension  Figure 7 . 5 :  2000  2500  ' b ' (m)  Same as Figure 7 . 4 for dimension b  - 101 -  2000  1500  -  1000  500  400  000  1200  SAM Isopleth-dimenston  Figure 7.6  :  1600  2000  ' c ' (m)  Same as Figure 7.4 for dimension c  3000 h  2500 H  .2000  .1500  h  ;1000 -  500  0  1000  2000  SAM I s o p l e t h - d i m e n s i o n  Figure 7.7  :  3000  4000  ' d ' (m)  Same as Figure 7.4 for dimension d  - 102 300 -  -  103 -  0  Table 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  a n R  715 2  d  715  715  • 715  d  c  b  x  m  715  0.9 715 A  • 96  • 92  • 94  • 93  .96  • 93  • 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  8k. 8  93.7  8.3*1  420.9  15.8  50. k  1.08  162.4  83-3  79.0  8.27  388.3  RMSE„ „• tot RMSE sys RMSE unsys un i t s  3.03 0.18 3.02 m  120. 2k. 6 118. m  m  m  m  10- m. 5  - 104 numerical source area model (mini-SAM).  A  programmable  c a l c u l a t o r may be used to compute the dimension equations perform with  (7.21).  A SHARP PC-1403 pocket  this calculation  Table  7.1,  i n approximately  estimates  of the source  pocket  estimates  from  computer i s able to 30 seconds. area  Together  dimensions  at  different significance  l e v e l s and the r e l a t i v e importance  specific  within the source area may be evaluated  quickly.  inhomogeneity  of a  - 105 -  PART III : MEASUREMENTS AND RESULTS  8.  8.1  Evaluation of S p a t i a l V a r i a b i l i t y  v i a Spectral  Analysis  INTRODUCTION  A method to quantify the representativeness of a measurement and the homogeneity  of a dataset was presented i n Chapter 5. It  involves the computation of a two-dimensional Fourier transform of  a discrete  dataset. The variance  spectrum  i s obtained by  taking the squared modulus of the transform and a q u a n t i t a t i v e measure of representativeness i s a v a i l a b l e from the integrated variance d i s t r i b u t i o n curve i n the r a d i a l 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  i n Chapter 3 and the data a c q u i s i t i o n was  described i n Chapter 4. The  discrete  application  Fourier  (Moore,  transform (DFT) package  1984) i s most e f f i c i e n t  used  i n this  when datasets with  x- and y-dimensions of integral powers of 2 are used. For t h i s reason,  the DFT  was  performed  on quadratic  sub-arrays with  dimensions up to 512 x 512 data-points. The two-dimensional spectra were obtained i n a s i m i l a r 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, to remove the large  i n order  magnitude spike at zero wavenumber which  - 106 represents the mean data value. This array was then  multiplied  by a c i r c u l a r f i l t e r of i d e n t i c a l size, having a value of unity over most of the domain and a cosine taper at the f r i n g e s . This taper gradually reduces the values to zero i n a cosine fashion across  an annulus with  Values  at a l l greater r a d i i  shape  similar  to a f l y i n g  "Frisbee-fliter" frequency  a width of 10% of i t s outer  noise  will  be zero. This f i l t e r  disk  and was  therefore  radius. takes a  named the  (see Figure 8.1). It i s a p p l i e d to avoid high introduced by the edge d i s c o n t i n u i t i e s  of the  dataset (Justice, 1981). The tapered data array could be p l o t t e d i n a perspective view using a DISSPLA p l o t t i n g routine (see below). A f t e r 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 v i s u a l i z a t i o n of the two-dimensional variance spectrum by a perspective view plot can be used to assess the d i r e c t i o n a l i t y or degree of isotropy of the data and i l l u s t r a t e s the dominant spatial  scales  within  e x i s t ) are expected  the dataset.  i s given  scales  (if  any  to show up as peaks i n the variance spectrum  (see following sections). F i n a l l y , (which  Dominant  the variance spectrum array  i n Cartesian co-ordinates  domain) can be transformed  f o r the wavenumber  into the wavelength domain and plane  polar co-ordinates as described i n Chapter 5. Both the spectrum and  the wavelength dependent  integrated variance  may be p l o t t e d f o r individual sectors  or  distribution  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 F r i s b e e - f i I t e r  - 108 sectors. It was suggested i n Chapter 5 that t h i s cumulative  relative  variance d i s t r i b u t i o n  integrated  i s equivalent  measure f o r the representativeness of a c i r c u l a r with a diameter  equal  to the wavelength  or  to a  block-sample  of the d i s t r i b u t i o n  curve. The  spectral  power components are subject  uncertainty r e l a t i n g the  to two kinds of  to the magnitudes and the wavenumbers of  i n d i v i d u a l components. The l a t t e r type of uncertainty i s due  to the d i s c r e t e nature of the data, so that the s p e c t r a l values are  given only at the natural  wavenumbers  (2/N, 3/N  N/N;  N = number of data points i n one dimension). The power of any one  component (e.g. at wavenumber  the  power of a wavenumber band ( i - i / 2 ) / N £ k > ( i + 1 / 2 ) ,  is  the wavenumber,  (Priestley,  rather  i/N) must be interpreted as  than  1981; pp. 429ff).  of  one  specific  I f the spectral  where k  wavenumber  components are  p l o t t e d against wavelength (A), t h i s 'spectral smearing' appears to  be more  wavelength  dominant  at longer wavelengths  resolution  decreases  because  as  d i s t r i b u t i o n of power (or variance) within  the given  The  spectral  a wavenumber band i s  unknown. The usual procedure to handle the question of the amplitude uncertainty i n spectral analysis involves the composition of an average spectrum from several the  problem  can  be  independent r e a l i z a t i o n s ,  treated  statistically.  In  so that  the present  context, however, s i n g l e r e a l i z a t i o n s are treated as i n d i v i d u a l cases  and  thus  a  statistical  treatment  of  the  amplitude  uncertainty i s meaningless. Consequently, the i n t e r p r e t a t i o n of  - 109 i n d i v i d u a l spikes i n any  given spectrum can be v a l i d  the s p e c i f i c sample from which the spectrum was The  d i s c u s s i o n of  the  spectra  in  the  only f o r  obtained.  following  and  the  i n f e r r e d correspondence of individual spikes or groups of spikes with geometric  patterns i n the data should be  interpreted with  the above i n mind.  8.2  SPATIAL VARIANCE OF SURFACE TEMPERATURE  Figure 8.2 temperature  shows the square blocks within the daytime surface data  domain  transformations.  The  i s shown i n Figure  that  were  selected  for  the  area used from the nighttime  Fourier  temperatures  8.3.  The dominant street layout i n the study area i s perpendicular In the E/W  and N/S  d i r e c t i o n s , except  f o r a small area i n the  north-eastern corner of the data domain, NE of Kingsway, where the  pattern  is  approximately  diagonal  d i r e c t i o n s . Area (0) (Figure 8.2) in  this  region  directions  to  to compare other  'inhomogeneities'  the  regions.  was  the  cardinal  e x p l i c i t l y chosen to l i e  effect Area  to  of (1)  a  change  contains  i n street few  large  ( i . e . larger than block s i z e non-residential  areas), with the exception of Kensington Park and Tecumseh Park. Area  (2),  on  the  'inhomogeneities',  other namely  hand,  is  Memorial  characterized Park  and  data (Figure 8.3)  is  almost  large  Mountain  Cemetery. Area (3) i s a high density r e s i d e n t i a l with no major ' inhomogeneities' . The  by  View  neighbourhood  square block of the night  equivalent  to  area  (1)  of  the  11  • ©  : Sunset Tower;  ©  : Mountain  ©  : Gordon Park  ©  :  ;  View  Cemetery;  'Hot-Crossed-Buns  1  ;  ©  : Memorial  ©  : L a n g a r a Community  Figure -2 : Daytime temperature sub-domains selected Fourier-transforms. Area (0): 128x128, areas (1), (2): 512x512, (3): 256x256 pixels. Areas (1) and (2) have a p a r t i a l overlap. 8  Park;  for area  ©  : Kensington College  Park  '  I  Li  i  (D  : Sunset Tower;  ©  : Mountain  ©  : Gordon Park  ©  :  Figure -3 : Fourier-transform 8  ;  Nighttime  1  View C e m e t e r y ;  Hot-Crossed-Buns  temperature  1  ;  : Memorial  ©  : L a n g a r a Community  sub-domain  selected  Park;  (R)  ©  for  : Kensington College  Park  i:-;vvv  - 112 daytime temperatures ( i t i s merely s h i f t e d to the north a l i t t l e b i t , due to a d i f f e r e n t f l i g h t - p a t h at night). The surface temperature data are best viewed as f a l s e colour images  on VZIP.  Figure  8.4  shows  the r a d i a t i v e  temperature  d i s t r i b u t i o n of the central section of the daytime data domain. The  major  landmarks  i n the area  are c l e a r l y  recognizable :  Kingsway cuts diagonally through the north eastern corner and the f i r s t Street.  major  north/south street  The f a l s e  colour  from the r i g h t  coding  maps  is Victoria  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  i s expected due to t h e i r low albedo values. They show up as red, corresponding to r a d i a t i o n temperatures around 40 °C. Dry grass areas of lawns, parks and cemeteries (Mountain View Cemetery i s shown near the l e f t edge of Figure 8.4) appear with colours of red,  green and yellow corresponding to r a d i a t i o n  temperatures  o  between 40 and 50  C. Irrigated grass i s of course considerably  cooler. The two large grey rectangles i n the upper r i g h t centre of Figure 8.4 are i r r i g a t e d soccer f i e l d s The  hottest  the  prominent  surface temperatures correspond to roofs,  b u i l d i n g roofs) roofs  i n Kensington Park.  feature  of the "Hot Crossed  i n the top right  of commercial  buildings  quadrant  Buns"  such as  (apartment  of Figure 8.4, the  along V i c t o r i a  Street  and the  dotted rows of r e s i d e n t i a l house roofs. These surfaces are coded as  white,  corresponding to temperatures  above  50 °C.  Sensor  o  s a t u r a t i o n i s reached at about 60 show saturation  can  be  traced  C. The very few p i x e l s that to  hot  a i r outlets  of a i r  F i g u r e 8.4 : study area  False  colour  image  of  daytime  surface  temperature  in  the  - 114 conditioners or other commercial a c t i v i t y . C l e a r l y Figure 8.4 contains enough s p a t i a l information to used as the base f o r a d e t a i l e d c i t y map.  In other  be  words the  d i s t r i b u t i o n of r a d i a t i v e surface temperatures r e f l e c t s both the regularities  and  the  breaks  to  the  dominant objects and surface types found i n a suburban area.  The  spatial  scales of the r e g u l a r i t i e s seem to be governed by  the  street  spacing,  which  in regularities,  in  turn  determines  relating  the  spacing  of  house-rows, rows of boulevard trees and a l l e y s . Further, i n more or  less  because  regular they  intervals  serve  V i c t o r i a Street on edge of Figure 8.4 north  as the  the  major right  s t r e e t s are avenues  and  or  wider  the  usual  thoroughfares  (e.g.  Fraser Street near the  or, perpendicular to these, 41  of Memorial Park and  than  "Hot  Crossed  left  Avenue just  Buns").  Property  l o t - s i z e s on the other hand determine the house spacing  along  the i n d i v i d u a l s t r e e t s and avenues. These f i n d i n g s are  not  They become important  s u r p r i s i n g and  though,  i f the  might  seem  analysis of  the  trivial. spatial  temperature d i s t r i b u t i o n i s generalized, so that the r e s u l t s and conclusions  become  configuration  at  Section 8.4).  The  scales  of  independent  the  surface  time  of  of the  argument here temperature  the  specific  remote sensing  i s that  variation  the are  temperature flight  dominant reflected  (see  spatial in  the  (more e a s i l y accessible) d i s t r i b u t i o n of permanent structures i n the  same area  and  are  therefore  similar  at  causal r e l a t i o n s h i p i s obviously the other way  a l l times.  (The  around, but that  i s unimportant at the moment). This hypothesis w i l l be  examined  - 115 -  1  large c i t y - b l o c k  or 'super-block'  850 m  2  along-row-street  spacing ( a l l e y spacing)  100 m  3  across-row-street  4  house-row ( s t r e e t / a l l e y )  5  inter-house  210 m  spacing  s p a c i n g ( a l o n g row)  Figure 8.5 : Dominant s p a t i a l c i t y - b l o c k system  50 m  spacing  scales  11-15 m i n suburban  - 116 by comparing the daytime temperature d i s t r i b u t i o n  (Figure  8.4)  with an e n t i r e l y d i f f e r e n t night-time c o n f i g u r a t i o n (see below). In summary, i t  appears  that  the  surface temperature i n Figure 8.4  spatial  distribution  of  has a dominant component of a  highly regular pattern and a minor component of breaks i n t h i s r e g u l a r i t y caused by parks,  school grounds etc. It i s expected  that  the  spacing  will  show up  wavelength of the as  peaks  i n the  of  these  variance  regular features  spectrum.  The  linear  dimensions of these regular spacings were estimated from Figure 8.4  and from a topographic map  schematically avenues,  i n Figure 8.5.  which  approximately could  be  are  The  typically  850 m,  termed  of the area. They are summarized spacing  wider  than  a  ' superblock'.  With  no  streets,  length)  is  about  park  or  contains 32  the house-rows ( i . e . the  100 m  on  s t r e e t - t o - a l l e y spacing about 50 m.  average,  or is  p a i r s of them school blocks,  i n a four by eight array, as shown i n Figure 8.5.  spacing of the s t r e e t s along block  normal  and the area bounded by two  i n t e r r u p t i n g the pattern such a superblock arranged  of major s t r e e t s  which  The  shorter  makes  the  With two rows of houses per  block t h i s gives a 50 m house-row spacing. The street spacing i n the  d i r e c t i o n of  210 m.  Street  the  widths  longer block vary  dimension amounts to explains  the  apparent inconsistency of 8 short blocks adding up to 850 m.  The  number of houses per row and  15  with  an  considerably,  which  about  i n a block varies t y p i c a l l y between 10  average spacing  of  11  to  15 m.  Oke  (personal  communication) measured an inter-house spacing of about 23 m i n the  same area as  a weighted mean of along-row and  across-row  house spacing. A s i m i l a r estimate, given the preceeding  spacings  Figure 8.6 : Same s t r e e t s and houses  as  Figure  8.4;  special  colour  coding  to  show  - 118 of rows, and houses along rows, i s found here. These values are order o f magnitude estimates  and claim no high accuracy.  expected, however, that the 50 m, 100 m or 200 m scales identified  as peaks  i n the variance  spectra.  It i s may be  Figure  8.6  i l l u s t r a t e s the dominant way i n which these scales are r e f l e c t e d in  the surface  temperatures.  In t h i s  image  selective  false  colour coding shows only small bands o f r a d i a t i o n temperature, corresponding  to the common street  temperature  and the roof  temperatures. Figure  8.7  temperature  shows  a  perspective  ' topography' of  view  the square  length o f about 0.4 km, a f t e r  area  of  the surface  (0) with  a side  i t has been m u l t i p l i e d with the  " F r i s b e e - f l i t e r " . The two main street axes (and accordingly the house-row  axes)  are NNE/SSW  and WNW/ESE.  The "ridge" and  "canyon" sequence created by t h i s street pattern corresponds to "waves"  propagating  perpendicularly  to  the  street-axis  d i r e c t i o n , i.e. i n WNW/ESE and NNE/SSW d i r e c t i o n s r e s p e c t i v e l y . The  corresponding  two-dimensional variance spectrum o f area  (0) i s shown i n Figure 8.8. clearly visible, sharper  The b i - b i r e c t i o n a l i t y of the data i s  with two branches of s p e c t r a l ridges, a major  one i n the SSW d i r e c t i o n and a minor but wider one i n  the ESE d i r e c t i o n .  Because of symmetry r e l a t i v e to the o r i g i n ,  only h a l f the spectrum i s shown. After transforming the spectrum into polar co-ordinates, the angular  variance  distribution  relative  to the t o t a l  computed. It i s displayed as a rose diagram with Figure 8.9.  was  16 sectors i n  Because of the d i s c r e t e width of the sectors the two  SPATIAL VARIANCE ANALYSIS OF S U R F A C E T E M P E R A T U R E  SPATIAL VARIANCE ANALYSIS OF S U R F A C E T E M P E R A T U R E  Plot of surface temp, variance spectrum in wavenumber domain  Figure 8.8  :  Temperature variance spectrum, daytime,  area  (0)  SPATIAL 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  w  \  I CO I  Total Variance = 110.79  Rose Diagram of Angular Variance-% Distribution  K Directional d i s t r i b u t i o n of surface temperature daytime, area (0) F i  u r e  8  9  :  variance:  - 122 main axes do not appear to be orthogonal Figure 8.8  400 m)  spectral  i s too  variance  small  to  this  daytime  surface  (with  compute a  distribution,  s t r e e t - p a t t e r n i n determining of  the NW/SE arm  seems to be d i s t r i b u t e d on two sectors i n Figure  The domain of t h i s p a r t i c u l a r transform only  and  a  the  temperature  of  integrated  importance  the anisotropy and  8.9.  sidelength  meaningful  in  of  the  directionality  distribution  is  clearly  indicated. In area (1) (see Figure 8.2) E/W  and  N/S  s t r e e t s and avenues run i n the  d i r e c t i o n s respectively. The  transform  of a  small  subsection of t h i s area (400 m diameter) i l l u s t r a t e s t h i s change i n d i r e c t i o n a l i t y , from that of area (0) very n i c e l y (see Figure 8.10). When the domain f o r a transform  i s larger, the d i s c r e t e  spectrum becomes very dense, spikes and and a g r i d - p l o t such as Figure 8.10  peaks become  sharper,  i s incapable of showing a l l  the d e t a i l : only the highest peaks are distinguishable. Figure 8. 11 shows the equivalent to Figure 8.10 area (1). The is  lost  plotting  f o r the whole domain of  impressive structure of the two  in this grid.  plot  The  due  rose  to  the  diagram  lack of for  the  variance branches resolution in  whole  of  area  the (1)  (Figure 8.12), however, i l l u s t r a t e s that t h i s structure i s s t i l l conserved.  Apart  from the  two  c a r d i n a l axes, corresponding  the s t r e e t d i r e c t i o n s i n t h i s case,  to  the area would q u a l i f y  as  i s o t r o p i c , but as expected, the influence of the s t r e e t g r i d  on  the anisotropy i s very strong. Note that the t o t a l variances of areas  (0) and  (1) compare quite well, i f one considers that  (1)  i s sixteen times larger than (0) (the standard deviations d i f f e r  SPATIAL VARIANCE ANALYSIS OF S U R F A C E T E M P E R A T U R E  Plot of surface temp, variance spectrum in wavenumber domain ELgure 8.10 : Same as Figure 8.8 p i x e l sub-section  : daytime,  area (1),  128x128  SPATIAL VARIANCE ANALYSIS O F S U R F A C E T E M P E R A T U R E  Plot of surface temp, variance spectrum in wavenumber domain  F g iu r e 81.1  :  S a m e as  F g iu r e 81.0  :  w h o e l a r e a  -  125 -  SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE  W  I po.o  r IS.  i 13.0  1 E 30.0  Total Variance =  129.43  Rose 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 t r i b u t i o n  F 1  g u r e 8.12  : Same as Figure 8.9  : daytime, area  (1)  I" •0 0 2  io-» rov.UfielA  X (in  km)  Figure 8.13 : Temperature variance spectrum ( r a d i a l ) : daytime, area (1)  Figure 8.14 : Normalized integrated r a d i a l variance spectrum of surface temperature :daytime, area (1)  - 126 by only 0.85 °C. ). In polar co-ordinates against  radial  the variance spectrum may be p l o t t e d  wavelength by i n t e g r a t i n g over  a l l angles. The  r a d i a l variance spectrum of the daytime surface  temperature i n  area (1) i s given i n Figure 8.13. The spectral variance density has  been divided by the wavelength to account  visual  weighting  values  have  of the spectrum  been  scaled  f o r the proper  i n a logarithmic plot. The  by the maximum  occurring  variance  density, since absolute magnitudes are unimportant here. Due to a l i m i t i n the number of spikes that can be drawn, the smallest ones had to be omitted. integral  with  respect  around the o r i g i n , linear,  radial  Note  that  to angle.  this  plot  i s not a  real  It i s merely a "sweep" 360°  so that a l l the spikes are aligned on one  axis.  In contrast,  the integrated  variance  spectrum plot (below) involves true integrations. The  i n t e r e s t i n g features i n Figure 8. 13 are the three  peaks. The dominant one corresponds 25 m  and  the  two  minor  ones  to  high  to a wavelength of about around  20 m  and  50 m  respectively. Following from the discussion of Figure 8.5, the 50 m peak can e a s i l y be i d e n t i f i e d as the scale of the house row and  street/alley  spacing.  Figure 8.4 showed that  both  surfaces and house roofs are r e l a t i v e l y warm. Therefore suspected  that  the regular  25 m spacing  of house  street it  is  rows and  s t r e e t s or a l l e y s i s r e f l e c t e d i n the c l u s t e r of high peaks at the 25 m wavelength i n Figure 8.13. The o r i g i n of the minor peak at  about  inter-house  20 m  i s less  spacing  along  clear.  It might  the rows.  correspond  20 m i s rather  to the long f o r  - 127 that,  but  could  be  a t t r i b u t e d to  'spectral  smearing'  (see  above). Figure 8.14 shows the integrated r a d i a l variance spectrum f o r area  (1). In normalized  form, the integrated s p e c t r a l variance  i s equivalent to the measure f o r specific  wavelength,  as  representativeness,  described  in  Chapter  5.  increases very q u i c k l y at short wavelengths and about  200 m,  means that average,  where R  i s already  a circular  more  than  block  90 %  well  sample  R,  of  a  Here,  R  l e v e l s o f f at  above 0.9.  This  result  of 200 m diameter  i s , on  representative  of  the  temperature  d i s t r i b u t i o n i n area (1). In addition, the very strong curvature away from the diagonal across the plot of homogeneity contribution wavelengths wavelength  (compare to  is  minimal,  components.  nighttime  Chapter 5 and Figure  the. t o t a l  are discussed below, and  with  indicates a high degree  variance  compared  Further  to  made  by  that  by  5.1) : the the the  shorter  implications of these f i n d i n g s  a f t e r examining more examples  temperature  longer  distributions  and  of daytime  the  roughness  element d i s t r i b u t i o n . The integrated r a d i a l variance spectrum i s broken up into 16 component sectors of width 22.5° f o r symmetry reasons) and  (only 8 sectors are relevant,  i s displayed i n Figure 8. 15a-h.  influence of each sector on the anglewise integrated (Figure 8.14)  The  variance  i s equivalent to the r e l a t i v e c o n t r i b u t i o n of that  sector to the t o t a l variance, as i n Figure 8. 12. It i s apparent from Figure  8.15a-h that  small. In general i t  is  the v a r i a t i o n observed  that  among sectors  i s very  the  of  R-values  the  - 128 -  D r r C C R A T t OR J . D U L V * R I A M C E S P t C T R U U Of S U R f t k C T T C U P C I U T U R C  ,  5  wrtCRmo  •uouLv.RUMCE-sPr.cTRUw or s  U R  r * c r TEMPERATURE  IMTICRATCD RASUL VARUNCr-SPECTBUU Or SURT1CE m i P E R A T U R t  <  M T I C I U I T D RADIAL VARIAHCE-SPECTRUU  or SURPACE  TWERA™..  o.« c) Stcter o.o  o.2  o . 4  .» - 20Z.S*tt  o.a o.o i.o i.a Vaw«l«n0i't X f i n J k m J  t.2S*  d) (SS*/NNE)  1 . 4 j.a  Stelor 0.0  0.2  0 . 4  0 A  . • - rdO.O**M.25*  1.0  O A  1.2  *awl*n iK X fin kmj  (S/H)  1 . 4 i.fl  9  Figure 8.15 : S e c t o r i a l break-up o f normalized integrated variance spectrum of surface temperature : daytime, area (1)  - 129 -  1NTECRATED RADIAL VARIANCE-SPECTRUM Of SURrACE TEUPERATURE  INTECRATEO RADIAL VARIANCE-SPECTRUM O r SURrACE TEMPERATURE  (Figure 8.15,  continued)  [NTECRATED RADIAL V A R I A N C E - S P E C T R U M O f  SURFACE TEMPERATURE  INTEGRATED RADIAL VARIANCE-SPECTRUM Or SURTACI TEMPERATURE  - 130 dominant sectors r i s e  quickest, whereas minor sectors contain  some large scale variance components, but they are r e l a t i v e l y i n s i g n i f i c a n t f o r the spectrum as a whole. A possible reason f o r this  effect  i s the more  or less  irregular  d i s t r i b u t i o n of  inhomogeneities such as small parks which may contribute to the variance i n any d i r e c t i o n , according to t h e i r p o s i t i o n r e l a t i v e to each other. In area (2) the N/S components o f both the 25 m and the 50 m wavelengths are much more d i s t i n c t than i n area (1) (see Figure 8.16). This i s to be expected since the comparison o f areas (1) and  (2) i n Figure 8.2 shows that the s t r e e e t / a l l e y - and s t r e e t /  house-row-spacing i n N/S d i r e c t i o n i s much more dominant i n area (2) than i n area (1). Figure 8.2 shows that area scale  inhomogeneities  (2) contains a s e r i e s  i n NW/SE d i r e c t i o n ,  Cemetery, John O l i v e r School  o f large  namely Mountain View  and Memorial Park.  In f a c t ,  area  (2) was s p e c i f i c a l l y chosen because o f these features. Although i t cannot e a s i l y be determined from Figure 8.16, the large peaks at very low wavenumber l i e i n the NW/SE sector. This conclusion is  confirmed  by the s e c t o r i a l  variance spectrum  (see below).  break-up  o f the integrated  Curiously, the double  peak near  wavenumber 40 km" , corresponding to 20 m and 24 m wavelengths, 1  already  observed  i n area  (1)  i s also  present  i n the  E/W-direction i n Figure 8.16, but i s not i d e n t i f i a b l e i n the N/S direction. The shape of the rose diagram f o r area (2) (Figure 8.17) i s as expected and compares c l o s e l y with the rose plot of area (1).  SPATIAL VARIANCE ANALYSIS OF S U R F A C E TEMPERATURE  Plot of surface temp, variance spectrum in wavenumber domain  Figure 8.16  : Same as Figure 8.8 : daytime, area (2)  -  SPATIAL VARIANCE  132  ANALYSIS OF  SURFACE  TEMPERATURE  N poc  /'  o  wi  i  30.0 \  r^£%S S C J '  I5.01—^<q  1  ' IS.0  1E 30.0  ***. 9 ***>-  *  s Total Variance =  Rose Diagram of Angular Variance-/!  M9.56  Distribution  F i g u r e 8.17 : Same as F i g u r e 8 . 9 : d a y t i m e , a r e a  10-*  IO'  1  ravUngtn X (in km)  10°  F i g u r e 8.18 : Same as F i g u r e 8 . 1 3 : d a y t i m e , a r e a (2)  0.0  0.2  0.4  O.fl  (2)  0.8  1.0  1.2  1.4  TawUngth \ (in km)  I.a  F i g u r e 8 . 1 9 : Same as F i g u r e 8 . 1 4 : d a y t i m e , a r e a (2)  - 133 INTEGRATED RADIAL VARIANCE - SPECTRUM Or SURFACE T E M P E R A T U R E  INTEGRATED RADIAL VARIANCE-SPECTRUM Or SURrACE TEMPERATURE  : o.«  a)  0.0  0.2  0.4  0.4 0.8 1.0 1.2 favtwngth X (in km)  1.4  1.0  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  SURFACE  TEUPERATURE  d) Sielor .• - 160.0'tt 1.25* (S/N) o.o  o.z  0.4  06  a.a  favtltnglh  Figure 8.20  : Same as Figure 8.15  : daytime, area (2)  10 X (in  km)  1.2  1.4  1.6  - 134  0-0  0.2  0.4  0.0  0.0  1.0  1.2  1.4  l.fl  r*v*ltnQtn k (\n km)  (Figure 8.20, continued)  0.0  0.2  0.4  0.6  06  10 1.2 JTavWtnftA  1.4  1.0  X (in km)  - 135 Also the r a d i a l variance spectrum (Figure 8.18) (Figure area  8.19) compare quite well  with  their  (1). A l l three peaks or peak c l u s t e r s  and the R-curve equivalents from  identified  i n the  r a d i a l variance spectrum f o r area (1) are again present and even more dominant  (Figure 8.18).  Representativeness  (Figure 8.19)  reaches a value o f about 0.9 at 0.2 km wavelength and l e v e l s o f f towards longer wavelengths, as seen before. The two large steps at approximately 750 m and 1200 m may be a t t r i b u t e d to the large scale inhomogeneities  i n t h i s area. Figures 8.20a-h support the  general impression gained from the equivalent f o r area (1). The long wavelength contributions i n the non-street d i r e c t i o n s are stronger than i n area (1). This i s e s p e c i a l l y true f o r the SE/NW and  the SSE/NNW sectors  (Figure 8.20e,f),  which confirms the  i n t e r p r e t a t i o n that the s e r i e s o f peaks i n the NW/SE d i r e c t i o n i n area  (2) i s responsible f o r large long-wave peaks i n that  sector (and, as i t turns out, the adjacent sector). While the dominant street/house row/alley sequence occurs i n the E/W d i r e c t i o n i n area (1), the street block arrangement i s turned  by 90 degrees  i n area  (3), and the dominant  sequence  occurs i n the N/S d i r e c t i o n (Figure 8.2). The very d i s t i n c t peak at wavenumber 40 km N/S that  -1  i n the N/S d i r e c t i o n (Figure 8.21)  bias o f the rose diagram (Figure 8.22)  are c l e a r proofs o f  observation. Note that the domain of area  quarter  o f the s i z e  of areas  (1) and  and the  (3) i s only a  (2). The regular  street-block structure i n t h i s area (3) i s almost undisturbed by large scale inhomogeneities  (Figure 8.2) and accordingly, the  low wavenumber peaks i n Figure 8.21 are comparatively small.  SPATIAL VARIANCE ANALYSIS OF S U R F A C E T E M P E R A T U R E  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 =  154.14  R o s e D i a g r a m o f Angular Variance-% D i s t r i b u t i o n  Figure 8.22  10*  1  : Same as Figure 8.9  10"  Watttlrngth X {in km)  Figure 8.23 : Same as Figure 8.13 : daytime, area (3)  10°  : daytime, area (3)  0.0  0.1  0.2  0.3  0.4  0.S 0.6 o.? o a Vawtmgtn X (ir\ tmj  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  IH TEC RATED RADIAL VAJIUHCE-SPCCTnUU Of SURFACE TEUPERATURE  0.7  WavtlingtS A (in km)  rNTTGRATED RADIAL VARIANCE-SPECTRUM Of SURrACE TEUPERATURE  INTEGRA TED RADIAL VARIANCE-SPECTHUU Of 5URTACC TEMPERATURE  c) SieUr  0.2  Figure 8.25  0.3  0.4  -  0.5  Z0t.i*tlt.2S*  0.0  (SST/NNE)  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)  : Same as Figure 8.15 : daytime, area (3)  -  0.0  0.1  0.2  0.3 0.4 0.S r t v . l n i f f A A f o , km)  0.0  O.T  0.0  INTEGRATED I U 0 U L VARIANCE-SPECTRUM Or SUREACE TEMPERATURE  139  0.0  0.1  0.2  0.3 0.4 O.J f a i . l « n ; I A A f i n km)  continued)  0.7  0 »  INTEGRA TCP RADIAL VARIANCE-SPECTRUM Of SURf ACE TEMPERATURE  0.3 0.4 0.5 rovUnfltx X (tn km)  (Figure 8.25,  00  a  0.0  - 140 Figures similarity  8.23  and  with  8.24  both  are  area  a  familiar  (1)  and  sight  area  by  (2)  now.  is  The  striking  e s p e c i a l l y f o r the R-curve (note that the maximum wavelength i s only h a l f of the ones f o r areas (1) and  (2)). The peaks at 20,  25 and 50 m observed i n areas (1) and (2) are also present here, however, they are supplemented by two a d d i t i o n a l peaks at about 15 m and 35 m. Since area (3) i s smaller than areas (1) and (2), small inhomogeneities or i r r e g u l a r i t i e s i n the spacing s t r u c t u r e of s t r e e t s and  houses can cause higher peaks i n the variance  spectrum than f o r the larger areas. The nature of these peaks i s uncertain. towards  It might  the  15 + 35 = 50)  be  streets or  traceable to or  might  towards  reflect  a  shift  the  the  of  alleys  spacing  integrated variance  into  compares well with the previous r e s u l t s  the  houses  (note  between  trees etc. , but t h i s i s not of prime importance break-up of the  the  that  garages,  here. Also the various sectors  (see Figure 8.25).  The  large long wavelength contributions i n the non-cardinal sectors are  apparent  rather  than  significant  since  these  sectors  contribute l i t t l e to the t o t a l (compare with the rose diagram i n Figure 8.22). This  strong  consistency  between  the  spectral  variance  d i s t r i b u t i o n s between d i f f e r e n t areas, and even d i f f e r e n t domain sizes,  indicates  temperature  an  overall  distribution  homogeneity  i n the data domain.  to  the  It also allows  claim f o r v a l i d i t y of the r e s u l t s f o r more than one configuration (1),  (since  (2)  and  (3)  similar).  The  results  the  are  temperature  different, for  the  spatial  particular  configurations of  but  daytime  the case  results can  a  are be  areas very  further  - 141 generalized  when they  are  compared  with  the  results  for  the  nighttime temperature d i s t r i b u t i o n . Figure  8.26  shows a  sub-section  of  the  nighttime  surface  temperature d i s t r i b u t i o n i n a perspective view a f t e r i t has been m u l t i p l i e d by the F r i s b e e - f i l t e r . 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 f a r the warmest  and roof-tops are coldest, due  surface-type  to t h e i r large sky view f a c t o r ,  which f a c i l i t a t e s r a d i a t i v e cooling, as i l l u s t r a t e d by the VZIP f a l s e colour image i n Figure 8.27  (see also Roth, Oke and Emery,  1988). The  variance  distribution  spectrum  (Figure  of  the  8.28)  nighttime  reflects  temperature  this  diminished  v a r i a b i l i t y : the sharp peaks are considerably smaller than the highest  ones f o r areas  automatic  scaling  exaggerates the  of  (1) and the  (2) of the daytime cases  spectrum  peaks i n the  nighttime  daytime configurations the 40 km were  the  most  important  power  the  daytime  relatively  warm  "topography", vegetation  both  components,  and  with in  street represent  "troughs"  between  (see  case).  in  this  this  street/house/alley  and  "ridges" of  in  cooler,  Figure  spacing)  i n the nighttime  surfaces  plot  Whereas i n the  wavenumbers (= 25 m  -1  spacing appears to be almost n e g l i g i b l e In  axis  (the  8.4)  street/house/alley spacing corresponds to the  house the often so  roofs  case. are  temperature irrigated, that  the  distance between  "wavecrests". At night, the s i t u a t i o n i s d i f f e r e n t : s t r e e t s are warmest, vegetation i s intermediate and house-roofs are  coldest  SPATIAL VARIANCE ANALYSIS OF SURFACE TEMPERATURE  Plot of surface temperature  Elgure 8.26  data in s p a c e d o m a i n  : Perspective view of nighttime temperature 'topography*  s a t ! * -  CO  HWIMIiflitHHil *•  2f2  *t  MtfL 4Mf M « • " • ' • K M Ml/  «M  1Mb.  4  I  *M *».  F i g u r e 8.27 : False colour the s t u d y a r e a  image o f n i g h t t i m e  surface  temperature i n  SPATIAL V A R I A N C E ANALYSIS O F S U R F A C E T E M P E R A T U R E  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  '  30.0  9  li-o-  \  ^  S>  ~~*P.O  30.0  q  s  Total Variance =  Rose Diagram ofAngular Variance-%  Figure 8.29  10-'  21.05  Distribution  : Same as Figure 8.9 : nighttime  I0V a v . L n o l n X fin km) 1  10°  Figure 8.30 : Same as Figure 8.13 : nighttime  0.0  0.2  Oi  0« 0.8 1.0 ' o v t U n f f l A X f i n km)  1.2  l.«  1.0  Figure 8.31 : Same as Figure 8.14 : nighttime  - 146 -  W I TECRATED RADA IL VARIAHCE-SPECIRUM or SURrACE TEMPERATURE  INTEGRATED RADIAL VARIANCE-spEcrnuw  or swrxci  TEMPERATURE  a)  00  0.2  04 . fivtltnfflh 00 . 00 . \ (intO . km) 12 .  00 OS tO . 12 .  |.4  **av«Unf(/t k (in km)  N ITEGRATED RADA IL VARA INCES-PECTRUM Of SURrACC TEUPERATURC  H ITECRATE0 RACA IL VARA INCES-PECTRUM Of SURFACE TEMPERATURE  c)  00 . 08 .  0.4  Figure 8.32  *<8 00 . S0 .el»r .•10 .- !0t.fm.2S' 12 . 1.4 (SSf/NNE)  r a w i n g * * A (in km)  : Same as Figure 8.15  nighttime  IHTCCRATXO RADIAL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE  0.0  0.2  0.4  0.0  0.6  1.0  Wavttnotn X (in km)  1.2  1.4  t.8  INTEGRATED RADIAL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE  147 1.0,  INTEGRATED RADUL VARIANCE-SPECTRUM OF SURFACE TEUPERATURE ,  0.0  0.2  0.4  O.a  0.8  10  Stet»r - 112.5* kt t .23* (ESt/WHW) o.2  o.4  o.a  o.a  1.0  1.2  1.4  rawtingth * (in km)  (Figure 8.32, continued)  1.4  16  INTEGRATED RADIAL VARIANCE-SPECTRUM OF SURFACE TEMPERATURE  g)  oo  1.2  WaviitnftK A (in km)  i.a J'av*Itna(A X (in  km)  - 148 (Figure  8.27).  As  a  result  the  "wavecrest"  to  "wavecrest"  distance corresponds to the s t r e e t / a l l e y spacing and i s twice as long as the street/house/alley spacing. Accordingly the 20 km  1  wavenumber peaks (= 50 m wavelength) are dominant at night, as i l l u s t r a t e d i n Figure 8.28. The peak i n the centre, at very low wavenumbers seems to be s i m i l a r to the one i n Figure 8.11 f o r area  (1) of the day case,  but i t i s much lower. The  nighttime  rose diagram (Figure 8.29) i s almost i d e n t i c a l to Figure  8.12 :  the s t r e e t d i r e c t i o n s dominate. Note that the t o t a l variance has decreased  by more than a f a c t o r of 6 between the daytime and the  nighttime temperature d i s t r i b u t i o n s . In correspondence with the perspective view of the variance spectrum,  the r a d i a l  plot  i n Figure  8.30  shows  a  dominant  c l u s t e r of peaks at about 50 m wavelength, but not at 25 m. The r e l a t i v e l y greater importance of larger scales i s also apparent in  the integrated spectrum (Figure 8.31) : an R of 0.9 i s only  reached at 0.4 km. relation  to  the  However, t h i s  magnitude  of  effect  the  should  total  temperature d i s t r i b u t i o n i s generally f l a t ,  be viewed i n  variance.  the e r r o r r e s u l t i n g  from an i n s u f f i c i e n t l y representative sample i s s t i l l to  be very small. More important  of well  over  0.9  during  differences are large) i s s t i l l with the much decreased The  break-up  expected  here i s to note that a c i r c u l a r  block sample with 200 m diameter R-value  I f the  (which was shown to have an the day,  when  temperature  about 80 % representative even  t o t a l variance at night.  of the t o t a l  variance  into  sectors  (Figure  8.32a-h) shows s i m i l a r c h a r a c t e r i s t i c s to the daytime cases. The  - 149 large s c a l e contributions to the variance are most apparent i n the non-street axis d i r e c t i o n s . In summary the spectral d i s t r i b u t i o n of surface variance seems to be configurations  and  nighttime case, is  i n general  temperature  consistent f o r several d i f f e r e n t  the  few  differences when compared  are of minor importance. The r e l a t i v e l y homogeneous and  daytime to  the  spatial structure  the dominant  spatial  scales can e a s i l y be c o r r e l a t e d with the spacing of permanent structures.  8.3  SPATIAL VARIANCE OF ROUGHNESS ELEMENTS  The  area  covered  by  the  roughness  Chapter 4) i s shown i n Figure 8.33.  element  inventory  (see  Since the p i x e l dimensions  of the d i g i t i z a t i o n have a non-unity r a t i o , the three 512 x 512 pixel (in  blocks used f o r the Fourier transform  space).  lines  and  They are labelled  inventory area. The VZIP  image  are  Figures 4.3 and  indicated i n Figure according  to  their  are not  8.33  by  position  quadratic the  dashed  within  the  heights of the surface cover elements i n a  assigned  colours,  was  already  presented  in  4.4.  A small s e c t i o n of the area i s shown i n a perspective view plot  i n Figure 8.34.  The  distinct  street-canyon  morphology i s  c l e a r l y v i s i b l e and over-emphasized by the approximately height  exaggeration  i n t h i s plot.  twofold  It i s therefore no s u r p r i s e  that the house-row spacing of 50 m forms the strongest component i n the variance spectrum shown i n  Figure  8.35.  In  fact,  the  (D  : Sunset  Tower;  ©  : Gordon Park  ;  (G)  : Mountain  ©  :  View C e m e t e r y ;  'Hot-Crossed-Buns  1  ;  (0)  : Memorial  Park;  0  : L a n g a r a Community  (R)  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.  : Kensington College  Park  SPATIAL V A R I A N C E A N A L Y S I S O F R O U G H N E S S E L E M E N T S  Figure 8.34 : Perspective view of roughness element d i s t r i b u t i o n (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 V A R I A N C E ANALYSIS O F R O U G H N E S S E L E M E N T S  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  1 E 30.0  ToUl Variance =  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 - %  Figure 8.36 : D i r e c t i o n a l variance : area SE  62.20  Dislribulion  d i s t r i b u t i o n of  roughness  0.4 o.s r a v « U n f t \ X (in tm)  Figure 8.37 : Roughness variance spectrum ( r a d i a l ) :' area SE  Figure 8.38 : Normalized integrated r a d i a l variance spectrum of roughness elements : area SE  - 154 INTEGRATED RAOUL VARIANCE-SPECTRUM Of ROUGHNESS  0.4  ELEMENTS  INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHHESS  ELEMENTS  Or ROUGHNESS ELEMENTS  INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHKnSS  ELEMENTS  O.fl  Wavlmgth k (in km)  IWTECRATED RADIAL VARIANCE-SPECTRUM  VavWrntflA  A (in  km)  Figure 8.39 : Sectorial break-up of normalized i n t e g r a t e d variance spectrum o f roughness elements : area SE  - 155 -  p-rac RATED  RAOIAL  vARiANCE-sprcnunj  or  ROUGHNESS ELCUEHTS INTEGRATED RAOIAL VARIANCE-SPECTRUU Or ROUCHNESS  »"av(mgth K (\n km)  ( F i g u r e 8.39, c o n t i n u e d )  0.4  o.a  Ifavtlmgth \ (in km)  EUUCNT5  SPATIAL V A R I A N C E A N A L Y S I S O F R O U G H N E S S E L E M E N T S  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 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 t r i b u t i o n  Figure 8.41  : Same as Figure 8.36  : area SW  ifowUnglh \ (in ttn)  Figure 8.42 : Same as Figure 8.37 : area SW  Figure 8.43 : Same as Figure 8.38 : area SW  -  0 0  0.2  0.4  O.A  158  -  OA  0.0  Tavcttngth X (in km)  INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUGHNESS ELEMENTS  : Same as Figure  0.4  0.0 o.a ITawlmgth \ (in km)  INTEGRATED RADIAL VARIANCE-SPECTRUM Of ROUCHNESS ELEMENTS  0.4 o.a f<t\nt*ngth X (in km)  Figure 8.44  0.2  favtUngtK X (in km) 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 fin km)  INTEGRATED RADIAL VARIANCE-SPECTRUM Or ROUCKKESS ELEMENTS INTEGRATED RADIAL VARIANCE-SPECTRUM  or  ROUCHNESS ELEMENTS  h) Sictor r*vltngth X (in km)  (Figure 8.44,  continued)  oo  o.2  oa.  o.4 '«vfJ«n«tA  _ 90.0*iJ I Si* (£/*)  A (in  km)  SPATIAL V A R I A N C E A N A L Y S I S O F R O U G H N E S S E L E M E N T S  E i g u r e 8.45 : Same as Figure 8.35 : area NE  - 161 -  SPATIAL VARIANCE ANALYSIS OF ROUGHNESS ELEMENTS r•J  . - --o«•» •'  9  „ jai -  /  w  i — r ^ ~ ^^ C T j  30.0  li^O—•  X^J  v**  i  >,.o  E  30.q  o c Total Variance = 57.97  Rose Diagram of Angular Variance-?! Distribution  Figure 8.46  : Same as Figure 8.36  Figure 8.47 : Same as Figure 8.37 : area NE  : area NE  Figure 8.48 : Same as Figure 8.38 : area NE  WTECRATEQ RADIAL VARIAMCE-SPECTRUV  or ROUCKHESS  162  IKTECRATED RADIAL VARIANCE-SPECTRUM Or ROUCKNESS  ELCUEHTS  ELCUEHTS  a) s.eiw . • -  o.<  i47.s'in.ts'  (rsr/ri/t)  o.e  0.2  Jawtmgtti X (in km)  INTEGRATED RADIAL VARIANCE-SPECTRUM Or ROUCHHCSS ELEMENTS  ;  0.4 ' a v a l t n g t v X (In  0.0 km)  INTEGRATED RADIAL VARIANCE-SPECTRUM o r ROUCHNESS  ELEMENTS  d) Sictor 0.4  - 130.0* tit.25' (S/N) 0.0  r*vii*tigt\ \ (in km)  Figure 8.49  : Same as  Figure  8.39  area NE  0.0  - 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  INTEGRATED RADIAL VARIANCE-SPECTRUU Of ROUGHNESS ELEMENTS  g) Stetor  - ttZ.S'tlt.ZS' (ESC/WHW)  WavltnQtn A (in km) ( F i g u r e 8.49,  continued)  r*v*l»netn X (in km)  - 164 r e s u l t s of the s p a t i a l variance analysis of roughness elements are  so  similar  nighttime  to  the  surface  results  of  temperature  both  the  analyses,  daytime  that  r e s u l t s need to be discussed only b r i e f l y . The Figures 8.35  to 8.49  the  and  the  roughness  various p l o t s i n  are p e r f e c t l y analogous to those  f o r the  surface temperature analysis. They mostly speak f o r themselves and are included here f o r completeness. Summarizing i t may three sub-sections NE,  SE and SW,  be mentioned  that  the  results  of the roughness inventory domain,  are the dominant f a c t o r s determining the house-row spacing  dominant  spatial  significant  scale might  (about  strong  found  i s the  the  only  h a l f the  where the street/house/alley  length of the  house-row spacing)  superior, due to the e f f e c t described i n Section  area  be  area  d i f f e r e n c e to the s p a t i a l structure of the daytime  spacing  (see  s t r e e t axes  the anisotropy of the  scale. This s p a t i a l  temperature d i s t r i b u t i o n ,  curve  labelled  (or the s t r e e t / a l l e y spacing)  surface  The  the  are very s i m i l a r , both to each other and to t h e i r  analogous i n the surface temperature analysis. The  and  for  curvature  e.g.  Figure  for  representativeness  the  of  the  8.38)  confirms  surface  measure,  at wavelengths of only 0.2  integrated  R,  the  reaches a  8.2.  variance  spectrum  homogeneity of  temperatures. value  is  Similarly, well  km (see e.g. Figure 8.38).  over  the the 0.9  - 165 8.4  ESTIMATING SPATIAL REPRESENTATIVENESS  The preceding discussion of r e s u l t s from the s p a t i a l  variance  analysis of surface temperature and roughness elements indicates clearly  that  governed types.  by  the the  dominant spacing  In p a r t i c u l a r  variance  curves  generalized  of  scales  permanent  of  f o r both temperature and of  the  variance  structures  the consistency of the  description  meteorological  spatial  or  integrated  are  surface radial  roughness suggests  spatial  structure  surface parameters i n terms of the  a of  distribution  of permanent structures and surface cover patterns. It was  stated i n Chapter 3 that the s p a t i a l  variability  of  sensible heat f l u x and the representativeness of measurements of t h i s f l u x may of  surface  Therefore, surface  be examined by analysing the s p a t i a l  temperature  and  the  dominant  roughness  given the f i n d i n g that the s p a t i a l  temperature  and  roughness  distribution  elements  elements.  distributions are  similar,  of it  follows that the common scale of representativeness f o r surface temperatures  and  roughness  elements  is  also  the  scale  of  representativeness f o r a sensible heat f l u x measurement. Choosing an  a r b i t r a r y R-value of 0.9  as  a criterion for a  s u f f i c i e n t l y representative block sample,  i t also follows from  the  block  preceding  analyses  that  a  circular  sample  diameter of 200 m w i l l meet t h i s c r i t e r i o n i n most cases  with  a  i n the  present study area. 200 m happens to be four times the length of the s i n g l e most dominant s p a t i a l scale i d e n t i f i e d : the average spacing between house-rows across the s t r e e t s or a l l e y s and  the  - 166 spacing of the street and  a l l e y pairs.  Alternately, i t i s  two  times the spacing of the s t r e e t - p a i r forming the long sides of a block.  Due  to  the  anisotropy  towards the street axes), involves an  area  of  the  area  i t i s important  across  the  house-rows,  (bias of  variance  that a block since  d i r e c t i o n that the variance i s concentrated  sample  i t i s i n that  (as r e f l e c t e d i n the  rose diagrams of the previous sections). This precaution  i s of  course t r i v i a l i n the case of a c i r c u l a r block sample, since the dimension i s 200 m i n a l l d i r e c t i o n s . It i s not non-circular  (e.g.  elliptical)  sample.  The  trivial  for a  source  area  considerations i n Chapter 6 show that the shape of the  "sample"  of a turbulent f l u x measurement i s only r a r e l y c i r c u l a r . conservative,  i t may  To  be  therefore be recommended that the smallest  a x i a l dimension of a block sample should be at least four times the house-row spacing, c r i t e r i o n of R -£  i n order to meet the  0.9.  This scale of representativeness spacing)  is valid,  on  average,  where the measurements that are were taken. with  It may  (four times the  i n suburban  house-row  Vancouver,  the basis f o r t h i s  of  similar  c l a s s R2 of Auer (1978)). whether these  character  (e.g.  It i s very d i f f i c u l t f i n d i n g s may  be  the  B.C.,  conclusion  e a s i l y be extended, however, to other  suburban areas  t h i s point,  representativeness  cities  land-use  to speculate at  used f o r non  North  American c i t i e s or f o r complex surfaces i n general. Further work i n t h i s area i s p o t e n t i a l l y very i n t e r e s t i n g f o r a wide research community. The  measure  of  representativeness,  R,  Chapter 5 to minimize the dependence of t h i s  was  formulated  in  representativeness  - 167 on a s p e c i f i c r e a l i z a t i o n of the sample. Some contingency upon the configuration of the sample remains,  however. For example,  i f the sample i s located over a large-scale inhomogeneity (e.g. a large park), the r e s u l t whole area.  i s probably not representative of the  I f i t i s possible to monitor  the l o c a t i o n of the  sample, such unrepresentative cases can e a s i l y be i d e n t i f i e d . The  combination  of  the  source  area  model  presented  in  Chapters 6 and 7 and the representativeness estimate developed here  serves as an objective  particular  sensible  heat  c r i t e r i o n to accept  flux  measurement  or r e j e c t  as being  a  spatially  representative. In the next chapter p r a c t i c a l examples are used to show how the s i z e of the source area may a f f e c t representativeness of a turbulent f l u x  measurement.  the s p a t i a l  - 168 -  9.  Simultaneous Eddy C o r r e l a t i o n Measurements a t Two S i t e s : F i v e Configurations  9. 1  INTENTIONS  Consider flux  at  two eddy c o r r e l a t i o n measurements of sensible heat  two  different  approximately  locations  i n the  same  area  and at  the same height i n the surface layer. Source area  c a l c u l a t i o n s f o r the two sensors, following the method described i n Chapter 6, are expected to be s i m i l a r as f a r as t h e i r and  orientation  concerned. sensors,  However,  the  possible  to  the  respective  due to the d i f f e r e n t  be d i f f e r e n t .  area that  have  Therefore,  considerable  the  two  sensor  are  l o c a t i o n of the two  the actual l o c a t i o n of the two source  ground w i l l in  relative  size  areas  on the  i f the surface conditions  spatial  measurements  variability, will  have  i t is  different  magnitudes. It  i s suggested  i n Chapters  5  and  8  that  the  spatial  representativeness of measurements depends on the s i z e of t h e i r source  areas : the larger the two source  areas are, the higher  the s p a t i a l representativeness of the measurements and therefore the smaller  the difference between the two measurements, i . e .  the smaller the s i t e uncertainty. I f the source areas are small, and  the  measurements  are not  representative,  the  deviation  between the two measurements w i l l not n e c e s s a r i l y be large, but i t w i l l be very v a r i a b l e from case to case. The two measurements  - 169 might,  by  chance,  be  very s i m i l a r ,  depending  on  the  type  of  surface elements that the two source areas include. Over a large number of r e a l i z a t i o n s i t i s therefore expected that the spread i n the differences between the two  measurements i s wider when  the source area i s small and  i t becomes smaller as  source area increases.  that  the  It follows that the s i t e uncertainty of  the measurement i s inversely dependent on the s i z e of the source area. A large number of eddy c o r r e l a t i o n measurements of sensible heat  f l u x density  (Q ) at paired  sites  were conducted  i n the  H  summer  of  1986  to  examine  the  above  hypothesis.  These  measurements and the r e s u l t i n g r e l a t i o n s h i p between the range of the observed f l u x differences and the source area s i z e  (using  the 0.9-level source areas as computed by the source area model described i n Chapter 7) are discussed i n the following sections. In  Chapter  determined  8  for  measurements distribution  in of  a  spatial  the the  source study  scale  of  area  representativeness  of  area,  sensible  based  surface c h a r a c t e r i s t i c s .  "on  This  heat the  scale  was flux  spatial and  the  concept of representativeness are developed i n terms of d i s c r e t e block samples,  where the elements within the sample a l l have the  same non-zero weight and those outside The  source  area  of  a  time  have  averaged  none. turbulent  flux  measurement, however, i s not a block sample i n the above sense. Its source weight d i s t r i b u t i o n function reaches a maximum at a s i n g l e point and then f a l l s o f f asymptotically on a l l sides, as described  in  Chapters  6  and  7  (see  e.g.  Figure  7.3).  In  - 170 accordance with t h i s notion, the source area i t s e l f  i s defined  in  effect i t  terms of  the  fraction  of  the  total  integrated  contains : the source area f o r each integrated e f f e c t l e v e l contains the source areas of a l l 6). Obviously, i t Is d i f f i c u l t  fraction  lower l e v e l s (see Chapter  to decide which l e v e l of e f f e c t  f r a c t i o n i s relevant f o r estimates of representativeness. Each measurement  "samples"  ( i n the  course  period) over a very large area and measurement  would  Inevitably be  of  an  averaging  the source area f o r every  considered representative, i f  only a high enough integrated e f f e c t l e v e l i s chosen. the  source  always  area f o r an  infinitely  demonstrates,  integrated  effect  level  (Note that  of  unity  large i n theory. ) The source weight  however,  that  time  points f a r away from  is  function  the maximum  source l o c a t i o n have p r a c t i c a l l y 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 s i z e can be demonstrated, r e l a t i o n s h i p may  help to decide how  such a  large the 0.9-level source  area needs to be f o r the s i t e uncertainty of the measurements to become n e g l i g i b l e is  anticipated  have dimensions  (compared to the instrument uncertainty). It  that  this  critical  0.9-level source  area  will  that are considerably larger than the scale of  representativeness  as  determined  in  Chapter  8,  because  it  includes a large portion with almost no weight, f o r the reasons described above. It i s suggested that the appropriate l e v e l of the source area can  be  found  by  matching  the  smallest a x i a l  dimension of the various source area levels with the scale of  - 171 representativeness  and  at  -  the  same  time  meeting  requirements of the 0.9-level of t h i s source In  the  following  instrumentation  and  sections the  data  the  the  size  sites,  the  area.  measurement  processing  involved  will  be  outlined. A f i e l d comparison of the two eddy c o r r e l a t i o n systems i s used to determine the instrument helps  to evaluate  between the  uncertainty, which i n turn  the s i t e - r e l a t e d portion of the d i f f e r e n c e s  measured Q  values  at  the  two  sensors.  After  the  area dimensions have been computed using  the  H  0.9-level source  meteorological conditions of each hourly Q -observation above 10 H 2  W/m  , a method i s suggested to evaluate  the contingency  of  the  s i t e uncertainty on the source area size. 9.2  THE SITES  During  late July, a l l of August and  e a r l y September,  1986,  simultaneous sensible heat f l u x density measurements were taken using two  s i m i l a r eddy-correlation systems (see below). One  set  was  fixed  the  held  at  the  Sunset  meteorological  tower,  while  other set was  mounted on a mobile tower. During the measurement  period  mobile  this  locations (see Figure The  Sunset  tower  tower  was  operated  at  five  different  9.1). site  served  as  the  main  site  and  all  meteorological data necessary f o r the SAM-runs were c o l l e c t e d at that s i t e  (see below). The  Section 1.3,  Sunset tower s i t e  was  here a b r i e f d e s c r i p t i o n of the f i v e  described i n mobile  sites  - 172 -  S 1 2 3 k 5  : : : : : :  Sunset Tower Culloden Site Argyle Site Waver ley Site Memorial East Site Memorial West Site  Figure 9.1  :  Measurement s i t e s  SCALE  500 m  1  km  Figure  9.3  : Argyle  site  ( l o o k i n g W)  Figure  9-4  :  Waverley  site  (looking  N)  F i g u r e 9.7 : Memorial ( l o o k i n g NE)  West  site  Figure  9.8  : Memorial  West  site  ( l o o k i n g W)  - 177 i s given. With the mean daytime winds expected to be governed by the westerly sea-breeze 1986),  sites  (1)  (Hay and Oke,  Culloden,  chosen to be spread out and  tower  by  a  and  distance  respectively  conditions,  i n the  order  m distant.  of  i n order  inhomogeneity  on  to  the  (3)  direction  Waverley were  Memorial examine  sensible  from  km,  the  whereas  downwind  during  flux.  and  sea-breeze  influence of  heat  9.1  (4) Memorial East  l i e just  Park  the  1  These  (Figures 9.5-9.8) are separated by approximately 1 km  Faulkner,  (see Figures  of  Sites  (5) Memorial West were selected to  upwind  Steyn and  Argyle are both separated  Waverley i s only about 400 and  Argyle  i n the N/S  9.2-9.4). Culloden and  Sunset  (2)  1973;  this  major  two  sites  500 m,  and l i e  (West) and 780 m (East) away from Sunset.  With a u t h o r i z a t i o n from Vancouver City, Engineering Dept., i t was possible to park the mobile t o w e r - t r a i l e r i n any regular car parking space and operate i t f o r a period of up to three or four days.  A  pick-up  camper-unit  data-loggers  and  operator  night.  at  instruments  provided  and  Residents'  served  mistrust. Culloden  through The  most  Street,  total  who  comment  complained  to  the  to  open  came from that  the  the  f o r the  odd-looking  i n q u i s i t i v e n e s s and  indifference  notable  for  as a s h e l t e r  reactions  "spying device" ranged from interest, admiration  security  even  suspicion a  resident  tower  and of  "attracts  mosquitoes and bugs" ( s i c ! ) . In general, however, there were no difficulties for  with such potential problems as residents' claims  parking space or vandalism.  each s i t e i s given i n Appendix D.  A more d e t a i l e d d e s c r i p t i o n of  - 178 -  9.3  EQUIPMENT  The  mobile  INSTRUMENTATION AND DATA  tower was the key piece of equipment  f o r this  study. Because of the limited space a v a i l a b l e  at the side of  residential  impeding  traffic,  streets  and along  parks  without  i t was necessary to operate the mobile  normal  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 i s a pneumatic Ltd.  system  of nine telescopic sections by Hilomast  (UK). It i s extendable to a height of 28 m, driven by an  e l e c t r i c air-compressor that can be run from a 12 V c a r battery. It has a maximum head-load capacity of 14 kg and i s designed to withstand wind-speeds up to 12.2 m-s  -1  without  guys.  The mast  has an o v e r a l l weight of 120 kg and i s mounted on a road t r a i l e r equipped  with  extendable  stabilyzer  posts.  In the r e t r a c t e d  p o s i t i o n the mast has a length of 4.56 m and can be pivoted into the horizontal f o r road transport. These s p e c i f i c a t i o n s make t h i s tower t r u l y mobile. After some p r a c t i c e i t was possible to convert the tower from i t s transport position  into  an operational  minutes,  complete  meteorological tower  with an eddy c o r r e l a t i o n  system  within  30  and a net  radiometer mounted on top. The salvage of the instruments from sudden strong winds or r a i n took about 3 minutes. All  meteorological  instruments  previously been described by Roth  used  i n this  (1988) and Cleugh  the present s e c t i o n i s heavily based on t h e i r work.  study  have  (1988) and  - 179 Sensible heat flux, or rather the kinematic heat f l u x was  measured on both the mobile and  identical  eddy-correlation  anemometer  and  a  the Sunset  systems  fine-wire  consisting  thermocouple  towers of  (w9), by  a  probe  two  sonic  (Campbell  S c i e n t i f i c CA-27T). The o r i g i n a l version of t h i s s p e c i f i c sonic anemometer was thermocouple  introduced by Campbell  i s a welded  and Unsworth (1979).  chromel/constantan  The  junction of wires  with a diameter of 12.7 fim. This type of thermocouple has a very large Seebeck e f f e c t of 58.7 fiV- ° C  _1  at 0 °C (Fritschen and  Gay,  1979). The junction i s 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  s i g n a l s are amplified i n a electronics-box a f t e r only about  1.5  m  wire-distance from  datalogger.  This  the  sensor  and  electronics-box  then  passed  contains  on  a  to  the  reference  temperature probe with a large enough thermal mass to make the temperature d r i f t component of the (w8)-covariance n e g l i g i b l e . Two  different  data-loggers  had  to  be  used  for  the  two  eddy-correlation sets. At the Sunset s i t e the voltages from the amplifier  were sampled  and  accumulated  over  a  period  of  15  minutes, f o r which the covariance was computed and stored i n a Campbell  Scientific  CR-21X data-logger. These  eddy c o r r e l a t i o n  Q -values were stored on cassette tape as 30 minute averages. H  The data-logger at the mobile tower was a Campbell CR-5  logger equipped with a K18  Scientific  eddy-correlation module.  Here,  the covariance was obtained f o r a period of 5 minutes and then averaged over 15 minutes,  i n which form the data were stored on  - 180 cassette tape and also printed on paper. The c o n t r i b u t i o n to the covariance by eddies with a time-scale longer than 5 minutes i s missed by t h i s data-logger. The w'8'-cospectral analysis by Roth (1988) shows that a non-dimensional (corresponding  to  a  5  minute  c u t - o f f frequency of 0.022  averaging  time  computed  with  z = 20 m and u -= 3 m-s" ) could introduce a considerable e r r o r 1  in  the covariance. As  he  points out,  however,  the  cospectral  estimates i n t h i s frequency range are poorly defined and show large f l u c t u a t i o n s . An instrument comparison of the two systems with  5-  and  significant  15  minutes  difference  correlation  intervals  over a wide range  of f l u x  showed values  no (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 f o r a d i f f e r e n t  (Cleugh,  1988).  In Figure 9.10  study  the CA-27T i s mounted on  mobile tower together with a net radiometer, which was  the  used by  Cleugh (1988). Note the e l e c t r o n i c s box on the t i p of the tower. The  input data (u, 0,  <p and <r^) f o r the SAM-runs were a l l  obtained at the Sunset tower. Figure 9.11 a schematic overview of the e n t i r e Sunset was  operated i n summer 1986  1988;  Roth,  used  in this  instrumentation as i t  (see also Cleugh,  1988;  1988 and Steyn and McKendry, 1988). The study  (other  than  described above) included : a Met-1 and  (from Roth,1988) gives  a PT-100 resistance  the  Grimmond, instruments  eddy-correlation  system  cup-anemometer and wind-vane  thermometer. The  data f o r these were  logged by a CR-21X data-logger and stored on cassette tape.  Figure 9 . 9 : Arrangement o f i n s t r u m e n t s on the Sunset tower. L e f t t o r i g h t : two n e t r a d i o m e t e r s , Krypton hygrometer, sonic anemometer/thermocouple.  Bi  Figure 9. 1 0 : Arrangement of instruments on the mobile tower. Net radiometer (left) and sonic anemometer/thermocouple (right).  - 182 -  Is O  >  NE  P  — C  SW •Q-  o 4  —  .5  V  22.0  (30.51  19.0 18.4  (27.5) (26.9)  13.9  (22.4)  12.1  (20.6)  8.9  (17.4)  Figure 9.11 : Schematic of the Sunset tower and instrumentation. Instruments used i n t h i s 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  Q H  at  Sunset  Culloden (solid) for the period from JD 212-214.  (dashed) and  - 184 From the casette tapes the data were t r a n s f e r r e d to the. UBCmainframe computer.  minute values  were  averaged to give hourly means, determined every h a l f hour.  Data  f o r cases  when Q  The  15  minute and  30  at the Sunset s i t e f e l l  below 10 W-m  _2  were  H  excluded and not used f o r processing. The r e s u l t i n g time periods with v a l i d and complete measurements, the data and some of the derived values  (eg. L, u^, etc.) are summarized i n Appendix D.  This appendix also includes r e s u l t s from the corresponding  SAM  runs described below. Although principal  the  daily  interest  variations  of t h i s study,  themselves  Figure 9.12  are  not  illustrates  the the  general c h a r a c t e r i s t i c s using the data from the Culloden s i t e i n comparison  to  the  concurrently evaluated  Q  data  from  Sunset  H  (dashed l i n e s ) . 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 a f t e r sunset,  i s at i t s peak. Shortly  convection i s once again damped so much that the  sensible heat f l u x becomes very small. At  first  sight  i t i s questionable  between the Culloden-Q  whether  and the Sunset-Q H  r e a l d i f f e r e n c e s due source  areas)  instrument range  of  addresses  or  flux  i n Figure 9.12  whether  the  locations ( i . e . d i f f e r e n t  differences r e s u l t  The  instrument  comparison  conditions,  described  in  t h i s question.  reflect  H  to the d i f f e r e n t  uncertainty.  the d i f f e r e n c e s  the  mainly  from  for a  wide  next  section,  - 185 9.4  INSTRUMENT COMPARISON  Between JD 255 (September 12) and JD 269 (October the  same two eddy c o r r e l a t i o n  field  systems  that  6), 1986  were used  i n the  programme were mounted on the Sunset tower side-by-side,  with a spacing of 15 cm between the two sonic paths, interference. The comparison o f 184 hourly Q  to avoid  measurements with  H  the two systems showed remarkable agreement over a wide range o f values, e s p e c i a l l y i f the much wider s c a t t e r encountered during the i n t e r - s i t e f i e l d programme i s considered. the  scatterplot  of  the instrument  Figure 9.13 shows  comparison  during  the  c a l i b r a t i o n period alone and Figure 9.14 shows the c a l i b r a t i o n data  (diamonds) together  with  the values  from  the i n t e r - s i t e  f i e l d programme (stars). The RMSD between the two sensors  from  the i n t e r - s i t e measurements i s more than four times as large as that  during  Impression  the c a l i b r a t i o n  gained  by Figure  period.  9. 14  Further,  the  indicates c l e a r l y  visual  that the  diamonds and s t a r s belong to two d i f f e r e n t data sets, where the amount of s c a t t e r and bias i s governed by d i f f e r e n t phenomena. The  principal  axis analysis and the break-up of the RMSD into  i t s systematic calibration  and unsystematic  period  (Figure  parts (see Appendix E) f o r the  9.13) confirms  the  unsystematic  nature o f the differences between the two sensors. Therefore, i t is  suggested  that  the increase  i n scatter  between the  c a l i b r a t i o n period and the measurement period r e f l e c t s effect  of  the  differences  Furthermore i t i s suggested  among  that  the  a real  individual  the strength  of t h i s  sites. site  -  186 -  350  12.7 (W m-2) 3.2 (W m-2) 12.3 (W m-2) 50  100  150  200  250  300  350  Q_H-MobMe (W m ) -2  Figure 9.13  Scatterplot of Q  instrument comparison  H  350  * : intersite * : ca1i brat ion  300  250 CM  '  E  200  +j  150  OJ  (/> c  CO 100 I  50  12.9 (W m-2) 45-9 (W m"2) 0.97 0.86  ' RMSDcaHb RMSDi tersite ^calib R intersi te n  R  -50 -50  50  100  ISO  200  250  300  350  Q -Mobile (W m ) -2  H  Figure 9. 14 : Scatterplot of Q &  Hmob  vs. CJ Hsun  - 187 e f f e c t may be measured by the absolute Sunset  and the mobile Q  difference  between the  values, normalized by the RMSD of the  H  calibration  period (which appears  to be f a i r l y  constant f o r Q H  values above 10 W-m  Q' H D  , see Figure 9.13):  (9.1)  I F F  This measure may be interpreted as a signal-to-noise r a t i o f o r the s i t e - e f f e c t  of the sensible heat f l u x  measurements.  It  is  t h i s 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,  i n order to  assess  heat  the  representativeness  measurement. A l l the Q„  of  a  sensible  flux  f o r the measurements of the f i e l d  fiDIFF  programme are included i n the summary i n Appendix D. An a d d i t i o n a l conclusion from the instrument  comparison and  the close agreement between the two sensor systems i s that the error  introduced by the 5 minute  covariance  calculations  averaging  of the CR-5 system  cut-off  for  the  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 t h i s chapter, the s p e c i f i c source area l e v e l , f o r which representativeness estimates become relevant, i s unknown. It has been shown i n Chapter 7 (Table 7.1) that  the dimensions  of  the various  minor  levels  can be  - 188 approximated i f the dimensions of the 0.9 source area l e v e l are known. These 0.9 source complete sets of Q  area dimensions were computed f o r a l l  measurements using the necessary  input data  H  gathered during the f i e l d programme and the mainframe v e r s i o n of SAM  (see Chapter 7 and Appendix C). The r e s u l t s are included i n  the data and r e s u l t s summary i n Appendix D. As  mentioned,  the v a r i a b i l i t y  of  the Sunset/mobile  Q H  differences size.  should  In order  quantify  this  be studied i n r e l a t i o n  to do t h i s ,  a method  v a r i a b i l i t y : that  s c a t t e r of Q  IT  H D I F F  to the source  needs  which  area  to be found to  i s reflected  i n the  from a large number of measurements. In other  words, i f a s c a t t e r p l o t of 0.9 source area s i z e s vs. Qjjjj  IFF  *  s  considered, i t i s the changing spread i n Qjjpjpj. along the source area axis which i s of interest. According to the hypothesis i n Chapter  5, i t i s a n t i c i p a t e d  that  should  be much more  v a r i a b l e with small rather than large source areas. et.al.  Chambers  (1983)  propose  a  method  to examine the  dependence of the spread of an ordinate v a r i a b l e on the a b s c i s s a value  by smoothing  implemented ^ H D I F F  V  S  '  S  absolute  f o r the present O  U  R  C  E  a  r  e  a  -  values purposes  scatterplot  of residuals. as follows.  in  detail  was  F i r s t the  i s smoothed using a LOWESS  routine with f = 2/3 and three robustness a non-parametric  This  i t e r a t i o n s . LOWESS i s  l o c a l l y weighted regression scheme (described  i n Appendix E). The r e s u l t  (Figure 9.15) shows the  s c a t t e r plot and a s o l i d l i n e which i s the varying p o s i t i o n of the median of the Q  IT  d i s t r i b u t i o n within i n t e r v a l s of source  H D I F F  area s i z e . In the next step the absolute  residuals  from  this  -  189 -  Q  0.9-source  area  (in  100  1000  120  m ) 2  140  Figure 9.15 : Scatterplot of ^ H D I F F V S . 0.9-source area s i z e . The LOWESS-curve i s shown as a s o l i d S  2  line. The 6-10 ra area (dashed line) is equivalent to the 3.14-10 area of the 0 . 3 - l e v e l . 4  -o  (0 t_QJCL (/>  ZOO  400 000 000 1000 1200 0.9-source  area  Figure 9.16 : Spread of s i z e (residuals of |Q„  H D I F F  ( i n 1000 H D I F F V S .  m)  1 40  2  0.9-source - LOWESS-curve|)  area  - 190 -  o  zoo  400  aoo  0.9-source  Figure 9.17 period  1000  (in  400  aoo  0.9-source  aoo area  .  M O O 2  1000  (in  1200  1 0 0 0 m -)  : Same as Figure 9.15,  zoo  Figure 9.18 period  aoo area  for  calibration  1200  M O O  1000 m )  : Same, as Figure 9.16,  2  for  calibration  - 191 smooth median curve are evaluated and p l o t t e d against the source area s i z e . Once again a robust LOWESS curve ( f = 2/3) i s p l o t t e d through the r e s i d u a l s  (Figure  9.16).  This  curve  shows the  p o s i t i o n of the median of the residual d i s t r i b u t i o n i n r e l a t i o n to the source area s i z e . Figure 9.16 demonstrates that the spread of Q  quite c l e a r l y  i s wider f o r small source areas than  HDIFF  for  large ones, as was anticipated by a v i s u a l  examination of  the s c a t t e r p l o t i n Figure 9.15. An immediate r e a c t i o n to t h i s r e s u l t might be that there are many more points at the small source area end ( l e f t ) than on the r i g h t hand side of Figure 9.15 and that small source areas are related  to large  heat  fluxes  (see Chapter  7) so that  more  s c a t t e r i n 0 ^ ^ . i n t h i s region i s due to the higher Q -values H  and i s therefore a t r i v i a l r e s u l t . For t h i s reason an i d e n t i c a l analysis  was  performed  f o r the Q  measurements  during the  H  c a l i b r a t i o n period, where the two sensors were placed very close together (see above). Figure 9.17 shows that the range of source area s i z e s during the c a l i b r a t i o n period was s i m i l a r to that i n the i n t e r - s i t e measurement period, but the Qj^  are generally  much smaller : the LOWESS curve stays at a value of about over a wide range of source area sizes, from  equation  (9.1). The very  clear  unity  as i s to be expected  decrease  of the spread  estimator curve towards larger source areas i n Figure 9. 16 i s almost  entirely  absent  i n Figure  9.18, f o r the c a l i b r a t i o n  period. It i s concluded that the e f f e c t of decreasing v a r i a b i l i t y of the  measurements with increasing source area (as a n t i c i p a t e d  - 192 from  theoretical  Figure  9.16.  arguments  This  i n Chapter  result  is  also  5) a  i s demonstrated  confirmation  of  in the  suggestion that the s p a t i a l representativeness of sensible heat flux  measurements  i s s e n s i t i v e to  computed by the source  the  source  area  area model. It i s the f i r s t  size, proof  as that  the source area model describes an e f f e c t which i s r e a l and  may  be observed by measurements. It i s somewhat subjective to decide at what source area s i z e in  Figure  9.16  the  spread  of  might argue that the spread  Q^^.  becomes n e g l i g i b l e .  around the median curve  9. 15 i s about equal to the random spread due uncertainty 9.16  (see Figure 9.17)  i n Figure  to the  instrument  when the LOWESS curve  i n Figure  reaches a value of unity ( i . e . at a 0.9-level source S  size  One  of 6-10  area  2  m . This corresponds to a median  over 2 i n Figure 9.15.  Accordingly, the Q  of  just  v a r i a t i o n s due to the H  s i t e d i f f e r e n c e s (the non-representativeness  component) s t a r t to  be of the same order of magnitude as the random v a r i a t i o n s due to  the  instrument  uncertainty.  In  other  words,  the  differences become undistinguishable from the instrument  site noise,  5  when the 0.9-level source area i s larger than about 6-10  2  m . As  noted already, t h i s area i s not equivalent to the representative sample s i z e defined i n Chapter 8, due  to the continuous  source  weighting function which defines the source area. S  If the 0.9-level source for  area needs to be  at  least  6*10  a s p a t i a l l y representative heat f l u x measurement, how  t h i s area s i z e be r e l a t e d to the scale of representativeness 200 m as determined from the s p a t i a l  structure of the  2  m can of  surface  - 193 c h a r a c t e r i s t i c s found i n Chapter 8 ? The  200 m  result  refers  to  the  diameter  of  4  c i r c u l a r sample area, i . e . an area of 3.14*10  a  discrete,  2  m . It was stated  i n Chapter 8 that the sample area (or source area i n t h i s case) needs  to  have  a  minimum  axial  dimension  of  200 m  in  d i r e c t i o n across the house-rows. For a non-circular source it  may  the area  be postulated that i t has a minimum s i z e of 3.14-10  and that i t s smallest a x i a l dimension  i s 200 m.  4  m  2  Because of the  asymptotic character of the source weight f u n c t i o n a source area that meets these s i z e requirements large enough e f f e c t  level  can always be defined, i f a  i s 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 i n Chapter 7 that  the s i z e - r e l a t i o n s h i p between the various source area l e v e l s i s more or less constant. According to Table 7.1 4  s i z e of 3.14-10  m  the corresponding 5  6-10  i s reached by the 0.3-level source area when 0.9-level source  2  5  area has  4  m  2  source area. The smallest a x i a l dimension  l i m i t on the 0.3-level of the source area i s  most commonly the cross wind dimension,  2d  axis,  the  criteria  for  a  characteristics requirements procedure  7.2).  In  discrete analysis  reflected  of  l i m i t on the 0.9-level source area  i s e n t i r e l y equivalent to a 3.14-10  Figure  a minimum s i z e  2  m .Therefore, a 6-10 m  see  the minimum area  2  this  sample are  by  way,  resulting  matched  the  (d i s only the h a l f  with  observation  representativeness from the  the  surface  source  area  programme.  This  could be seen as equivalent to a c a l i b r a t i o n of  the  source area model : i t shows both that the source area model i s  - 194 in  effect  modelling  appropriate  an  integrated  estimates i s the 0.3  existing effect  phenomenon  level  for  and  the  source  first  point  Figure 9. 15,  area  the source areas f i t into the  study region, three cases have been picked from  The  the  integrated e f f e c t f r a c t i o n .  As an i l l u s t r a t i o n to show how  data i n Figure 9.15  that  the scattered  : (1)  i s well on  the representative side of  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 s i t e , f o r t h i s data point are p l o t t e d onto a  map  of  the  direction)  study  (oriented  i n Figure 9.19.  obtained on JD 222 tower was  area  at 8:00  The  data  (LAT)  i n the  appropriate  for this  at Sunset,  case  (1)  while the  wind were  mobile  located at the Argyle s i t e (see Appendix D).  It i s i n t e r e s t i n g to compare case (1) with case (2) which has a source area of s i m i l a r size, but i s an 'odd point' i n Figure 9.15,  with  a  relatively  large  of  over  five  times  value of case (1). The data f o r case (2) were recorded on JD at  17:30  site. the  (LAT), with the mobile  The Sunset  the 232  tower located at the Waverley  source area outlines are drawn only with respect to site  for clarity  i n Figure  9. 19.  It  i s easy  to  imagine respective p a r a l l e l source area outlines with respect to the mobile  sites  the Sunset  source areas because of the s l i g h t l y  tower).  (although they would be a l i t t l e  larger  higher  than  mobile  li.t:;-L  > 11  .... —~ |li'"Y V"" ia iKnwpDO"'/t"""-.— ,  -M * S TV' ! V~ l/' 1  (D  : Sunset Tower;  ©  : Mountain  ©  : Gordon Park  ©  :  ;  View C e m e t e r y ;  'Hot-Crossed-Buns'  ;  (R)  : Memorial  Park;  ©  : L a n g a r a Community  Figure 9.19 : Source area outlines for three cases described i n the text. Cross-hatched : 0.3-level; s o l i d line : 0.5-level; dashed l i n e : 0.9-level.  (R)  1  ;  : Kensington College  Park  "  - 196 The source area of the measurement i n case misses the large i n t e r s e c t i o n of 49th Ave. reaches  mainly over r e s i d e n t i a l  (1) just  barely  and Knight St. and  neighbourhoods.  The 0.3-level  area i s wide enough to include a minimal a x i a l dimension of =<200 m as required by the representativeness c r i t e r i a .  In contrast, a  large p o r t i o n 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 i s s u f f i c i e n t l y f a r away to play a minor r o l e f o r the measurement at the the Sunset s i t e .  This i s  not the case f o r the mobile tower, which was located at Waverley site  i n case  (2)  (Figure  9.1).  It was  downwind of Memorial Park. A s h i f t the case  therefore c l o s e r  and  (which has to be imagined) of  (2) source area from the Sunset  s i t e to the  Waverley  s i t e would make a considerable portion of the source area l i e over Memorial without  Park,  influence.  whereas the substation The  c h a r a c t e r i s t i c s f o r the two large Q^^pp  different  would be  source  entirely  area  s i t e s i n case (2) may  surface  explain the  excursion noted.  A s i m i l a r (imagined) s h i f t of the source area i n case (1) to the Argyle s i t e  (see Figure 9.1)  shows that  the source  area  c h a r a c t e r i s t i c s are much more s i m i l a r between the Sunset and the mobile s i t e here : hence CL^pp i s expected to be small. A t h i r d example i s given i n case area combines with the The  largest  (3), where a small source  Qjjjjjpp recorded  data i n t h i s case were recorded on JD 221  with the mobile tower at the Waverley i s no  site  (Figure at 13:30  (Appendix D).  9.15). (LAT) There  large park, p l a y f i e l d or parking l o t i n the v i c i n i t y of  - 197 the source area of e i t h e r the Sunset s i t e or ( i f i t i s s h i f t e d ) the  Waverley s i t e  (Figure 9.19), but  the  source  area  smaller than required by the representative c r i t e r i a . the large QJJ^ In  may  summary,  be expected, but i s not it  representativeness  is  (as  suggested  determined  Therefore,  necessary.  that  from  i s much  the  the  scale  of  variability  of  surface c h a r a c t e r i s t i c s i n Chapters 5 and 8) f o r sensible heat f l u x measurements can be r e l a t e d to the s i z e and minimal a x i a l dimension of the 0.3-level source area. Therefore  the  suggested  criteria  for  a  spatially  representative sensible heat f l u x measurement are : the 0.3-level source area needs to be at least  o  circular  area  with  a  diameter  equal  to  as  large  the  as a  scale  of  representat iveness o  the d-dimension of t h i s 0.3-level source area needs to be at least h a l f the scale of representativeness For  the  identified  study as  area  twice  the  the  scale  of  mean spacing  representativeness of  house-rows or  was 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 s p a t i a l representativeness 9.15 pairs  shows quite of  acceptable  the  of sensible heat clearly  Sunset  and  that the  a  flux  large mobile  measurements. number of sites  are  Figure  measurement within  range of each other even though the source  an  area i s  much smaller than stated by the above c r i t e r i a . This s i m i l a r i t y  - 198 i s subject to considerable uncertainty however, as indicated by the wide s c a t t e r on the l e f t this  hand side of Figure 9. 15. It i s  uncertainty which the representativeness  above help to evaluate.  criteria  stated  - 199 -  PART IV : DISCUSSION AND CONCLUSIONS  10.  How  good  are  the  SAM-Estimates  ?  -  Comparison  with  E x i s t i n g Work  A l l numerical model r e s u l t s are questionable, i f they be  supported  by  observations.  Unfortunately  cannot  source  area  estimates are not simple e n t i t i e s that can be v e r i f i e d by d i r e c t measurements. However, can  be  validated  evidence  of  a  i n the present  indirectly  changing  by  source  case,  comparing area  with  the SAM the  estimates  observational  corresponding  SAM  results. The need to develop a model which estimates the source area of a turbulent f l u x measurement o r i g i n a l l y arose because of a concern  about  the  spatial  representativeness  of  such  measurements. The suspicion that the s p a t i a l representativeness of turbulent f l u x measurements i s r e l a t e d  to the s i z e  of  the  source area was formalized i n 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 s e n s i b l e heat flux  measurements  from  two  spatially  separate  sensors  is  inversely dependent on the s i z e of the source area which may estimated by SAM i n Figure 9.16 considered  be  (see Chapter 9). The shape of the LOWESS curve  led to the acceptance of t h i s hypothesis which i s  to be  one  of  the  fundamental  conclusions of  this  - 200 thesis.  The f a c t  that the r e s u l t s from the source  were involved i n t h i s conclusion and that t h e i r is  supported  by observations  i s an i n d i r e c t  area model  interpretation (but  important)  v a l i d a t i o n of SAM, at least i n a q u a l i t a t i v e sense.  Comparison with Observations i n Cabauw (the  Netherlands)  A more d i r e c t v a l i d a t i o n i s possible by applying SAM r e s u l t s to some observations of Beljaars et  al.  (1983). They discuss the  r e s u l t s of some surface-layer turbulence observations i n Cabauw, the Netherlands. These observations were conducted conditions,  due  to the presence  upstream of the instrument vicinity  of the tower  consisted  of short  and b u i l d i n g s  tower. The surface i n the immediate  (within a  grass  of orchards  i n poor f e t c h  with  radius  of 300  an estimated  z  -  500 m  )  « 0.02 m (see  o below).  Turbulent  flux  22.5 m indicated a u  #  measurements at heights  of 3.5 m and  at 22.5 m which was c o n s i s t e n t l y  higher  (by a r a t i o 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). S i m i l a r measurements of sensible heat flux  and  evaporation  did  not  i n t e r p r e t a t i o n given by Beljaars et o  The 22.5 m  level  show al.  this  effect.  The  (1983) i s as follows:  shows the integrated e f f e c t  of a  large  upstream area. The shear s t r e s s measured at that l e v e l includes the form drag on obstacles f a r away and therefore i t i s larger than  the l o c a l  surface drag.  At the 3.5 m height,  the shear  s t r e s s tends towards the surface drag on the smoother near-tower terrain.  - 201 -  UK(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 t r i a n g l e (from Beljaars et.al. , 1983).  Table 10.1 : Input data f o r Cabauw SAM-runs He i g h t : 3 - 5 m / 22 .5 m  1 A1  1 kO  JD  2  o = 0.02  m  160  161  17*1  1831  183II  u.,(0  0. 30  0.21  0.53  0. 56  0.13  0.31  0.21  e  0.53  0.37  0.93  0.98  0.23  0.54  0.37  236  253  212  200  f  ( 2 )  V  36  (20m)  5.1  3-7  9.1  9-6  2.3  5.3  3-7  L  -300  - 50  -800  -170  - 16'  -135  -203  u  (''  b a s e d on zo = 0 .02 m b a s e d on 3(  V  - u •;• 1 .75  - 202 o  Heat and  vapour fluxes are not  the ground temperatures  n e c e s s a r i l y affected, since  and moisture values could be s i m i l a r f o r  both surface types. This problem i s equivalent to the step change i n roughness and Beljaars et al.  described i n Section 6.2;  (1983) f i n d t h e i r  interpretations confirmed by the internal boundary layer growth estimates of Rao et al. The  input  data  (1974).  necessary  cases are given i n Beljaars et  f o r SAM-runs al.  f o r seven  (1983) and are presented i n  Table 10. 1. In order to examine the r e l a t i v e roughness elements, the SAM  importance of the  estimates were computed f o r the  measurement l e v e l s f o r each of the seven cases. that  the  level,  large obstacles have no e f f e c t  so  that the  model  unstable  input value  It was  two  assumed  at e i t h e r measurement  of z  was  that f o r the  o  grass surface (z = 0.02  m)  i n both cases  . If i t can be shown  o  that  a  considerable  portion  of  the  source  area  (with  the  appropriate weighting) f o r the 22.5 m l e v e l includes the rougher orchards  or  buildings,  hypothesis  has  Beljaars et  al.  to  be  but  not  rejected  for and  the the  3.5 m  level  this  interpretations  of  1983 are confirmed by the source area model.  The r e s u l t s f o r three of these cases are presented i n Figures 10.2  to 10.7.  Note the d i f f e r e n t s c a l i n g of the alongwind- and  cross wind-axes i n the various plots : the shapes of the  source  area o u t l i n e s are  strong  distorted.  These p l o t s  indicate  the  s e n s i t i v i t y of the source area to the sensor height. The e f f e c t s of s t a b i l i t y and u^ are secondary,  but s t i l l  strong, e s p e c i a l l y  f o r the 22.5 m level (compare Figures 10.3 and 10.5). Variations  SOURCE AREA MODEL - RESULTS  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  Figure  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 C A B A U W  W i n d et s e n s o r level (sp./dir.) : — / 1 4 8 . 0 ; Zi : n/e ; S V : 0 . 6 3 0 Stability (Zs/L) : - 0 . 0 7 6 ; U ' : 0 . 3 0 0 ; S e n s o r height : 2 2 . 6 A l l data are in SI units  Figure  10.3  : Cabauw S A M - r e s u l t s  ( J D 140,  22.5  m)  SOURCE AREA MODEL - RESULTS  SAM Calculations for Jul.Day. 174 at  — L A T . at CABAUW  W i n d at s e n s o r level (sp./dir.) : — / 4 6 . 0 ; Z i : n / a ; S V : Stability ( Z e / U : - 0 . 2 1 9 ; U : 0 . 1 3 0 ; S e n s o r height : 3 . 6 #  A l l data are in SI units  Figure  10.4  : Cabauw S A M - r e s u l t s  ( J D 174,  3.5  m)  0.230  SOURCE A R E A M O D E L - RESULTS  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  Figure  10.5  : Cabauw S A M - r e s u l t s  ( J D 174,  22.5  m)  SOURCE AREA MODEL - RESULTS  e -  Figure  10.6  1  : Cabauw S A M - 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.) : — / 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 ( J D 161,  3.5  m)  SOURCE A R E A MODEL - RESULTS  SAM Calculations for Jul.Day. 161 at  -  L A T . at CABAUW  W i n d at s e n s o r l e v e l (sp./dir.) : — / 2 5 3 . 0 ; Z i : n/e ; S V : 0 . 8 8 0 Stability ( Z s / L ) : - 0 . 1 3 2 ; U " : 0 . 6 8 0 ; S e n s o r height : 2 2 . 6 A l l data are in SI units  Figure  10.7  : Cabauw S A M - r e s u l t s  ( J D 161,  22.5  m)  - 209 at the lower l e v e l are much smaller. The  undistorted outlines  of the 0.3-level  source areas of these three cases  and 0.5-level  (JD 140, 161 and 1741)  for  both heights are plotted i n the appropriate wind-direction on a map of the Cabauw tower and v i c i n i t y the  wind  direction  was  from  (Figure 10.8). On day 161  an almost  unobstructed  sector.  Judging from the source area outlines i n Figure 10.8 f o r JD 161, it  i s not s u r p r i s i n g that Beljaars et al. (1983) d i d not f i n d a  stress  discrepancy on t h i s  day : the source  sensor  l e v e l s contain a s i m i l a r ,  fairly  areas  f o r both  uniform surface type,  although t h e i r s i z e s d i f f e r by about an order of magnitude. The s i t u a t i o n i s d i f f e r e n t f o r JD 140 and 1741 : the a i r flows over the  obstructed  respective  zone  source  before  area  i t reaches  outlines  the  i n Figure  sensors.  10.8 confirm the  reason f o r the r e s u l t i n g stress discrepancy quite c l e a r l y . the  3.5 m  source  roughness elements map),  areas  are p r a c t i c a l l y  The  unaffected  While  by the  (indicated by dotted and hached areas on the  the corresponding 22.5 m source areas reach f a r into the  obstructed zones (the 0.3-levels are plotted by dashed l i n e s i n Figure  10.8).  The maximum  small s o l i d c i r c l e s :  locations  are indicated  f o r the 3.5 m measurement l e v e l  very close to the tower distinguishable  source  from  by  these are  (between 27 and 36 m) and are hardly  the  mast-marker.  The  maximum  source  locations f o r the 22.5 m level are much further away : f o r 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 t e and surroundings, with observed winddirections and estimated source area outlines f o r JD 140, 174 and 161. Day numbers are indicated at the end of the l i n e s that mark the wind-direction during the different measuring runs. 0 . 3 - and 0 . 5 - l e v e l source area outlines are indicated by dashed and s o l i d l i n e s respectively. The larger sets refer to the 22.5 m, the smaller sets to the 3.5 m sensor l e v e l . Maximum souce locations are indicated by large dots (o) (based on Beljaars e t . a J . , 1983).  - 211 A considerable difference  i n surface character between the  source areas f o r the two measurement l e v e l s on JD 140 is clearly  indicated.  f o r JD 140 and  1741  The  above hypothesis has  and  to be  1741  rejected  and thus the v a l i d i t y of the SAM-estimates  i s supported by these observations from Cabauw.  Estimation of the Growth of the Internal Boundary Layer The  growth of the  estimated  from  the  internal  variation  boundary of  the  layer  (5 « x ) n  maximum source  can  be  location  between the two levels, as follows :  z  = s  b-x  n  ,  (10.1)  m  where b i s a constant of p r o p o r t i o n a l i t y (see also Chapter 6). After taking the log, the r e s u l t i n g sets of two l i n e a r equations f o r the seven cases i n Table 10.1 are solved f o r the exponent n (Table 10.2).(This internal boundary layer growth evaluation i s not dependent on the Cabauw data, they are used here simply f o r convenience). A plot of n versus the s t a b i l i t y parameter (-L) i s given i n Figure 10.9 and includes the corresponding estimates by Rao n  (1975). The general character of the s t a b i l i t y v a r i a t i o n of is similar  although  f o r both  Rao  (1975) and  values  are  consistently  Rao's  the  present  slightly  results, lower.  A  hand-drawn curve through the present r e s u l t s indicates a neutral limit  of  agreement  n = 0.8 with  for  both  the  exponent  theoretical  in  results  f i e l d observations (e.g. Munro and Oke,  (10.1)  which  (Peterson,  1974).  is  1969)  in and  - 212 -  T a b l e 1 0 . 2 : C o m p u t a t i o n o f exponent o f i n t e r n a l boundary l a y e r g r o w t h . n  L  35-62 327-34  0.84  - 300  3.5 22.5  32.60 232.65  0.95  -  3.5 22.5  36.05 348.93 35.10 305.61  0.82  - 800  0.86  - 170  27.27 147.52  1.10  -  JD  s  x  140  3.5 22.5  141 160  z  3.5 22.5 1741 3.5 22.5  161  m  50  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 "  14 " O  -\-  1 3 "  SAM-values  Rao (1975)  O (R)  1 2 '  limit  in Rao (1975)  \  n  1 1  neutral  +  1  \ \  1 0 0 9 0 8-  — ,  10°  10  1 1  10  — 2  —1  10 -L  (R)  t— 3  10  14  ( i n m)  F i g u r e 1 0 . 9 : Exponent o f i n t e r n a l boundary l a y e r g r o w t h v s . - L  - 213 With t h i s r e s u l t as another  -  confirmation of the v a l i d i t y of  the SAM-estimates i t may be concluded general  qualitative  and  that these estimates show  quantitative agreement  presently a v a i l a b l e observations.  with  the  few  - 214  11.  -  The Structure of the Surface Layer over Complex Surfaces  In Chapter 2 rough surface lower  i t was  (e.g.  portion  noted that  tall  vegetation  called  characterized by the  the  or  layer above  buildings)  roughness  involves  sublayer,  the inert ial sublayer,  i s h o r i z o n t a l l y homogeneous and authors  surface  which  a a is  influence of individual surface elements,  and an upper porion c a l l  Several  the  have  where the  flow  Monin-Obukhov s c a l i n g applies.  speculated  on  the  height  of  the  upper  boundary of t h i s roughness sublayer. An e a r l y suggestion by Tennekes (1973) states that  turbulence  sensors should be at a minimum height of 50 - 100 z . In a more o  recent  study  arranged  of  rough  wind  tunnel  surface,  experiments  Raupach  et  al.  over  a  regularly  (1980) estimate  the  height of the roughness sublayer i n terms of the mean height of the roughness elements  (h) and  (D)  This  as  z  = h + 1.5 D.  the  mean inter-element  estimate  applies  to  the  spacing momentum  f l u x i n neutral conditions. Garratt slope  of  (1978 the  temperature Australia.  a,b)  non-dimensional  over He  examines  a  very  the  height  profiles  heterogeneous  interprets the height  v a r i a t i o n s of for  wind-speed  savannah  of the  surface  roughness  the and in  sublayer  (he c a l l s i t t r a n s i t i o n layer) as the l e v e l where such p r o f i l e s converge to the indicate a z  #  heat f l u x and  familiar  Dyer-Businger r e l a t i o n s .  which i s higher show a v a r i a t i o n  f o r the in z  #  His  momentum f l u x with  stability  results than f o r f o r both  - 215 z  fluxes :  is  0  higher  in  near-neutral s t r a t i f i c a t i o n . heat  i s interpreted  to  unstable  conditions  The d i f f e r e n t be  z  the r e s u l t  than  in  f o r momentum and  m  of  the  different  d i s t r i b u t i o n s of momentum sinks and heat sources e s p e c i a l l y f o r surface considered by Garratt (1978a,b). connection of these z  The lies  estimates with the present work  %  i n the i n t e r p r e t a t i o n  sublayer.  As mentioned  of the nature  i n Chapter  of the roughness  2, t h i s  interpretation i s  summarized very well by Raupach and Thorn (1981). roughness sublayer i s three-dimensional  Flow i n the  due to the i n d i v i d u a l  wakes and "heat plumes" of the surface elements.  A sensor close  to these surface elements measures the e f f e c t s of some surface elements  preferentially  inhomogeneities  compared  to  others.  These  become more and more mixed and d i f f u s e d  increasing height, u n t i l  with  they disappear altogether. The height  where horizontal differences become n e g l i g i b l e i s interpreted as the  upper  boundary  of the roughness  sublayer,  above  which  Monin-Obukhov s c a l i n g can be applied. This i n t e r p r e t a t i o n of the nature of the roughness sublayer makes  the height  requirements  for a  sensor  (i.e. z  > z)  s  equivalent suggested sensor  to  the  i n Chapter  i s located  representativeness suggestion having because  spatial  would  representativeness  9. Accordingly, above  z  criteria  stated  be i n agreement  the roughness the source  sublayer  area  size  requirements  i t i s suggested  i f i t s source  m  area  i n section  with  height  Garratt change  f o r a given  *  that a  meets the  9.5.  Such  a  (1978 a,b) i n with  level  stability  varies  with  - 216 stability.  The value of z  w i l l also depend on the spacing and  0  d i s t r i b u t i o n of surface elements as suggested by Raupach et al. (1980). Judging from the v a r i a t i o n s of the s c a t t e r and 9.16  i n Figures  9.15  and the f i n d i n g that the minimum s i z e of the 0.3-level 2  4  source  area  (for  the  Sunset S  (corresponding  to  area)  should  be  3.14'10  m  2  6.5* 10  m  of  representativeness c r i t e r i a ,  the  0.9-level) to  meet  the  i t seems that the majority of the  measurements taken f o r t h i s study have been conducted  within,  rather than above, the roughness sublayer of the study area. This  conclusion  sensible  heat  is  flux  supported  by  measurements at  various mobile s i t e s  the the  comparison Sunset  site  (see Figure 9.14). On average,  of  the  and  the  the Sunset  s i t e over-estimates i n comparison with the sensible heat f l u x at the mobile s i t e s by a considerable amount. The mean values f o r the e n t i r e measurement period are 128.5  W-m  _2  (Sunset) and  _2  99.6 -2  W-m  i n an overestimation of 28.9 W-m  (mobile) r e s u l t i n g  about  25%  in  the  mean  (calculated  as  [2-|Q  - Q Hsun  [Q  + Q Hsun  ])  or |]/  Hmob  '  . Since the mobile measurements involve several  Hmob  d i f f e r e n t s i t e s , the mobile mean i s more s p a t i a l l y averaged than the  Sunset  mean,  and  therefore  is  considered  to  be  more  s p a t i a l l y representative. A  possible  preferential Sunset  tower,  Mainwaring  explanation  influence which  of  for the  consist  this  overestimation  immediate of  the  surroundings  non-vegetated  substation to the NW and a major road  and associated commercial  is  the  of  the  and  dry  intersection  buildings to the SE (see Chapter 9).  - 217  -  Such dry, non-vegetated surfaces may of  the  enthalpy  flux  to the  well s h i f t the p a r t i t i o n i n g  sensible heat side compared  with  other s i t e locations. This  possible overestimation  of  sensible heat  Sunset s i t e and the conclusions from Figures 9.15  flux  at  and 9.16  the  point  to a roughness sublayer height above the suburban surface i n the study area which i s often higher than the sensor height Sunset  tower  so  that  the  measured  values  are  not  on  the  entirely  representative of the larger land-use area. In order to  estimate  the range of conditions when Sunset measurements are expected to be  fully  representative,  the  source area c a l c u l a t i o n s f o r  measurement period were examined i n r e l a t i o n lateral  wind f l u c t u a t i o n s (Figure  cases (0.3-source area > 3.14*10 not  fully  representative  11.1).  to s t a b i l i t y  Fully  the and  representative  m) are p l o t t e d as squares and  cases as  dots.  The  hand-drawn curve  separating the two groups can be interpreted as the c o n d i t i o n i n which z. = z *  the Q  Hsun  at the Sunset tower. This does not  imply that a l l  s  -data above t h i s l i n e are useless. It simply means that *  measurements obtained  i n 'not  fully  J  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 l o c a t i o n of the source area should be monitored. Turbulence spectra obtained indicate  important  smooth t e r r a i n .  discrepancies  with  spectra  It i s therefore suggested that  s t a t i s t i c s at that height scale-lengths  by Roth (1988), however, do  although  may  they  be are  obtained the  over  turbulence  independent of l o c a l not  not  representative  geometry of  the  - 218 -  0 not  fully  representative '*•  •* .•.»»  (z  >  A  z )'. 5  _  . •"•  • Jj> ' *  -50 ^  z* = z  • • * •  Q~0  a  s  0  -100  -150  £  •  -  •/-  fully  • •  • •  ?  a  •  «  -200  •  a  •  •  a  : 0.3-source a r e a <C 3- 1 it 10*  m  2  3- 14 10*  m  2  a : 0.3-source a r e a >  • •  -  <  -a  o  -250  (z...  o  a a  -  representative  o  -300  -  a a  -350 i  1.5  i  2.0  i  2.5  1  3.0  1  3.5  1  4.0  4.5  Figure 11.1 : Representativeness conditions at Sunset tower  5.0  5.5  - 219 larger-scale  surface  communication). top  of  conditions  (Roth,  1988;  personal  I f t h i s i s the case, the i n t e r p r e t a t i o n of the  the roughness  sublayer  given  above  needs  to  be  re-examined f o r complex surfaces, where a wide range of s p a t i a l v a r i a b i l i t y scales co-exist. It seems appropriate to d i f f e r e n t i a t e between two d i f f e r e n t mechanisms  of s p a t i a l  inhomogeneities.  averaging  and d i f f u s i o n  of horizontal  Bluff-bodies create a turbulent wake,  where  mixing i s g r e a t l y enhanced by a process termed "wake d i f f u s i o n " by Thorn et al. (1975) and Raupach (1979), inhomogeneities  so that  horizontal  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  f o r other changes i n  surface character (such as changes i n temperature  or non-bluff  roughness), where s p a t i a l averaging i s therefore expected to be slower. Observations show that at the height of the sensor on the Sunset tower, flow d i s t o r t i o n s due to individual bluff-rough inhomogeneities  do  not a f f e c t  the shape  of  the turbulence  spectra (Roth, 1988). Since these spectra are normalized by the total  flux,  changes  individual  element  comparison  of  i n heat spacing  sensible  however, does show these lack  of a continuum  variance  flux  on a larger  are not r e f l e c t e d  heat  flux  at  of s p a t i a l 8,  than the  i n them.  the d i f f e r e n t  inhomogeneities.  spectra i n Chapter  scale  scales  The  sites,  It follows that the (as indicated  e.g. Figure  by the  8.11) over  very  complex surfaces r e s u l t s i n nested sublayers, each one r e f e r r i n g to  a  different  "roughness  scale  sublayer"  of refers  variability. to  the  Clearly,  sublayer  due  the  term  to the  - 220 bluff-rough processes  variability  scale,  which  i n the atmosphere d i r e c t l y .  inhomogeneities  on  the  other  hand  considering s p a t i a l representativeness.  affects  the  transfer  "Sublayers" due to other are  only  relevant  when  - 221  12  -  Summary of Conclusions  Following the format of the o b j e c t i v e s o u t l i n e d i n Chapter 1, the main conclusions of t h i s t h e s i s are summarized as f o l l o w s . o be  The small s c a l e s p a t i a l v a r i a b i l i t y of s e n s i b l e heat f l u x can evaluated  distributions  by of  an the  analysis spatial  of  surface  the  structure  temperature and  dominant b l u f f - r o u g h s t r u c t u r e s i n the area. that a measure f o r the s p a t i a l  of  It i s  of  the  suggested  representativeness of a  sample i s given by the normalized  the  block  i n t e g r a t e d variance spectrum  of the data i n the wavelength domain. For two dimensional  data,  the spectrum should be transformed  i n t o plane-polar coordinates  and  to  integrated  with  respect  angle,  to  yield  a  representativeness curve r e f e r r i n g to c i r c u l a r samples w i t h the diameter equal to the r a d i a l wavelength of the spectrum. o  In  the  context  representativeness temperature and  of  the  sensible  of  block-samples  heat of  flux, both  the the  the roughness element d i s t r i b u t i o n  spatial surface  need to  be  examined. o in  The the  a n a l y s i s of surface temperatures and roughness elements study  area  indicates  that  a  sensible  heat  flux  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 r e p r e s e n t a t i v e , i f i t includes the averaged c o n t r i b u t i o n s of an area that extends across three  - 222 house-rows.  Further  work w i l l  f i n d i n g can be generalized,  be needed to show whether  so that  this  i t applies to the s p a t i a l  structure of any complex surface. o  A  reverse-plume  model  provides  an  Eulerian  approach  account f o r the surface area of influence of a turbulent measurement  in  suggestion  the  surface  layer,  by P a s q u i l l (1972).  based  A numerical  on  the  source  to  flux  original  area  model  (SAM) to evaluate t h i s area of influence was developed using the p.d.f.-plume  model  of  et.al.  Gryning  (1987)  as  a  basis.  Observations which can be r e l a t e d to changes i n the source area of a sensor or of a set of sensors, give i n d i r e c t support validity  of  the  SAM-estimates  both  to the  qualitatively  and  quantitatively. o  It i s concluded that the SAM-estimates are a useful tool to  evaluate  the  measurement.  spatial  representativeness  Observations  used f o r turbulent  indicate that  fluxes other  of a  turbulent  the model  may  flux  also be  than sensible heat. The model  shows good agreement with t h e o r e t i c a l and observational  studies  of internal boundary layer growth. The  most  connection  original  contribution of t h i s  between the source area estimates  measurements  and  the  representativeness  of  statistical a  given  measure  work  lies  i n the  f o r turbulent of  the  contributing  area.  flux  spatial This  combination of turbulent d i f f u s i o n modelling and two-dimensional s p a t i a l spectrum analysis i s a new method f o r the evaluation of the s p a t i a l measurements  representativeness  of turbulent  surface  layer f l u x  i n an objective and quantitative fashion.  In the  - 223 present scale  case  i t i s applied over  spatial  variability  of  complex  surfaces with  temperature  and  small  roughness.  Micro-meteorologists can use the source area estimates to r e f e r with  some  precision  to  the surface  to  which  their  flux  measurements r e l a t e , rather than pointing vaguely i n the upwind direction  with  some coarse  estimate  of f e t c h  derived from  a  measurement height. It i s f e l t in  this  study  application  that future work r e l a t i n g to some of the notions promise  interesting  of the source averages  area  spatially  weighted  Grimmond,  1988) or experiments  results.  weighting  For example the f u n c t i o n to model  f o r turbulent (e.g. i n a  fluxes wind  (see e.g. tunnel) to  examine the r e l a t i o n s h i p between s p a t i a l representativeness and the source area of turbulence measurements of d i f f e r e n t f l u x e s over complex surfaces.  - 224 -  REFERENCES  Arnfield,  Auer,  A.J.: 1982, 'An Approach to the Estimation of the Surface Radiative Properties and Radiation Budgets of Cities', Phys. Geog. 3, 97-122.  A.: 1978, 'Correlation of Land Use and Cover with Meteorological Anomalies', J. Appl. Meteorol. 17, 636-643.  Baldocchi, D.D. and Hutchison, B.A.: 1987, 'Turbulence i n an Almond Orchard: Vertical Variations i n Turbulent Statistics', Boundary-Layer Meteorol. 40, 127-146. 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Welty, J.R. , Wicks, C E . 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 V a l i d a t i o n of Models', Geography 2, 184-194.  Physical  Young,  G.S. and Pielke, R.A. : 1983, 'Application of T e r r a i n 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 Table  A . l . The  operated  by  the  shown i n a scanner  aircraft,  a  of the f l i g h t s  Falcon-20  Canada Centre  schematic  (MSS)  details  i s given i n  (twin-jet) owned  f o r Remote Sensing  i n Figure A. 1.  Both  the  and  (CCRS),  multi-spectral  and the push-broom imager (MEIS II) were operated  during the d a y - f l i g h t , but only the i n f r a r e d data from the are  used  through  for  this  study.  Figure  A. 2  shows  introduced  by  Zwick  et.al.  MSS  a cross-section  the scan-head of the Daedalus-1260 MSS.  originally  is  This MSS  (1980).  The  was  rotating  scanning mirror projects an instant f i e l d of view (IFOV) of  2.5  mrad over a set of f o l d i n g mirrors to the Hg-Cd-Te detector (the beam i s s p l i t - u p by a d i c h o r i c and  part of i t i s passed  spectrometer  near  f o r the  detection of  infrared  and  to a  visible  l i g h t channels). This infrared detector has a s e n s i t i v i t y window of 8-14  um.  In the present application,  scans/second  with a 73.7  f l i g h t p a t h . The 2.5  the MSS  scanned at  degree swath width orthogonal  50  to the  mrad IFOV i s sampled and d i g i t a l l y recorded  to 1 byte at 716 p i x e l s per scan-line. At the beginning and of each scan-line the  IR-sensor  i s c a l i b r a t e d by two  end  on-board  black-body sources. A d e t a i l e d d e s c r i p t i o n of the data format i s given by Zwick et.al. During  both  the  (1980) and by Edel and McColl day-  measurements were performed College,  and  the  night-flights,  (1981). ground-truth  on the grounds of Langara Community  i n the study area, using a PRT-10 hand-held  thermometer. Figure A.3 shows a plan of the area with  infrared the  four  - 232 Table A . l  : Technical d e t a i l s of the remote sensing  flights  Dates  :  Aug. 25 (JD 237) ; A u g . 26 (JD 238), 1985  T imes  :  -14:15 (LAT)  A i rp1ane  :  Fa Icon-20  Height  :  ~1560 m above Ground  Sensor  :  Daedalus-1260 MSS (see F i g u r e A.2)  Scan-rate  :  50 per second  Ground-speed  :  105 m s "  Pre-processing :  ; - 0 5 : 0 5 (LAT)  (see F i g u r e A . l )  2.1m/scan  1  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. 2. • 3. 4. 5.  Multispectral Scanner (MSS-Daednlus I 260) Metric Camera (RC-101 Pushbroom Visible Imager (MEIS II) Mission Manager's Console (camera controls and navigation data logger) Inertial Navigation System  Figure A.1 :  6. 7. 0. 9.  MEIS Console and Real-time Display for MSS and MEIS MSS Console High-Density Digital Tape Recorder Navigation Data Logger Tape Recorder  Schematic of Falcon-20 a i r c r a f t  - 233 -  SCAN  HEAO CROSS  SECTIONAL ILLUSTRATION  Legend ©  INFRARED DETECTOR "  (§) SCANNING MIRROR  (f) SPECTROMETER  ©  MOTOR  ©  GYRO  (§) ENCOOER  ©  FOLDING MIRRORS  @  PARABOLIC MIRROR  (5) DICHROIC *  Hg-Cd-Te sensor  Figure A.2 :  (8-14  MSS scan-head  u)  - 234 ground t r u t h s i t e s . These four s i t e s included a t y p i c a l range of surface types f o r a suburban area : i r r i g a t e d grass, concrete, a gravel/tar roof and tarmac. VZIP image of the daytime area,  with a  specific  For comparison, radiation  colour  Figure A.4  temperatures  coding to  i s the  of the same  highlight  the  pixel  values of the ground-truth s i t e s . Ignoring  emissivity  measurements deviation  within  of only  errors  each  0.8  °C  for  site  the  showed  moment, an  (the PRT-10 was  the  average  PRT-10 standard  calibrated  over  a  temperature c o n t r o l l e d water surface). Due to the d i f f i c u l t y of i d e n t i f y i n g the exact ground t r u t h s i t e  locations on the VZIP  image, an uncertainty of ± 5 p i x e l values was estimated f o r the assignment of the temperatures to a pixel-value. The Stefan-Boltzmann law of thermal r a d i a t i o n f o r grey bodies can be written  F  =  e-<r-T  ,  4  (A.l)  where F i s s t r i c t l y an integrated radiance over a l l  wavelengths,  but f o r n a t u r a l l y occurring surface temperatures i n t h i s region F may be approximated by the radiance of the 8-14 um window.  A  survey  of  emissivities  natural  surfaces occurring  of  building  wavelength  materials  and  i n the study area as given by  Oke  (1978) and A r n f i e l d (1982) suggests a mean e m i s s i v i t y of  e  *  0.95 ± 0.02  ,  (A.2)  mean  if  the values are weighted according to the r e l a t i v e  f r a c t i o n s estimated by Oke e t . a i .  (1981). This  result  land-use compares  - 235  Figure A.3 : College  Plan of  -  ground truth s i t e s at  Langara Community  - 236 -  Figure  A.4 :  VZIP-image  of  temperatures  at  ground  truth  sites  - 237 well with the mean e m i s s i v i t y f o r R2-land-u.se zones i n snow-free conditions  of e  = 0.944  reported  by  Arnfield  (1982).  The  R2  e m i s s i v i t y error of ± 0.02 translates to a temperature error of ± 2 K, i f the true  temperature i s 295 K. This  with the e m i s s i v i t y error estimates by Garratt et.al.  (1981)  and  Leckie  (1980).  Since  i s i n agreement (1978a), only  Carlson relative  temperatures are of interest here, the PRT-10 temperatures were adopted without correction,  with an assumed e m i s s i v i t y - r e l a t e d  error range of ± 2 K. Following atmospheric attenuation  Leckie  (1980) the e r r o r due to  (or r a d i a t i o n at night) i s assumed to be  accounted f o r by the ground t r u t h values.  The mean values of  these temperatures versus the corresponding p i x e l values of the four  sampled surface  day-flight. pixel  types are p l o t t e d  The error-bars  indicate  i n Figure  A. 5 f o r the  the uncertainties  value and the PRT-10 standard error  i n the  (not the e m i s s i v i t y  error). According to (A.l) the radiance i s proportional  to the fourth  4  power of the temperature.  This  T - r e l a t i o n i s plotted  between  the two i n - f l i g h t c a l i b r a t i o n points on Figure A.5. Compared to the  assumed e m i s s i v i t y error of ± 2 K the d i f f e r e n c e from the  l i n e a r T - r e l a t i o n i s small  f o r the l i m i t e d temperature range of  the present a p p l i c a t i o n . Therefore, the p i x e l value-temperature relation  i s evaluated d i r e c t l y from the ground t r u t h data by  s t r u c t u r a l analysis (see Appendix E) as :  60  50  (0  rad.temp.[oc] = 15.9 + 0.17 pix  2V  R2 = 0.99  CL  2 lil  <  DC  30  irrigated grass concrete gravel/tar roof tarmac  20  in-flight calibration #1 in-flight calibration #2  10  20  (0  60  60  100  120 PIXEL  Figure A.5  Daytime pixel value-temperature  HO VALUE  relation  160  180  200  220  2(0  - 239 -  rad.temp.  [ C]  day  A s i m i l a r analysis  =  15.9 + 0.17  (pixelvalue)  was performed for the  .  nighttime  (A.3)  data,  with  the r e s u l t i n g r e l a t i o n s h i p :  O  rad.temp. *  [ C] = night  L  J  -1.565 + 0.13  (pixelvalue)  (A.4)  - 240 -  APPENDIX  Two  B  :  V a l i d a t i o n of  the  P.D.F.-Plume Model  datasets were a v a i l a b l e to v a l i d a t e the plume model used  f o r the SAM : "Project P r a i r i e Grass" Experiment In t h i s experiment  SC"  2  source at a height of 0.46  was  Data  released from a continuous point  m and van Ulden  values f o r distances of 50,  200  and  800  (1978) reports CIC/Q m.  These values  are  estimated from samplers at 1.5 m height along arcs at i n t e r v a l s of 2° at 50 and 200 m and 1° at 800 m. The roughness length of the area i s taken as 0.008 m. The data set i s supplemented with L and u  values derived by Nieuwstadt  s  CIC/Q  was  modelled  for  34  runs  (1978). in  unstable  conditions  according to equation (7.3), using the Dyer and Bradley (1982) versions of the non-dimensional  p r o f i l e s <f> and M  7).  A scatterplot  <p (see  Chapter  H  ( i n l i n e a r space) of the modelled  versus the  observed CIC/Q i s given i n 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 r e f e r to Appendix E). The values  i n brackets r e s u l t  from  a reduced  data set, where the  three t r i a n g l e s on the f a r right and the g r e a t l y s t a r on the l e f t  In Figure B. 1 are rejected as o u t l i e r s .  four data points account error  over  overestimated  f o r more than h a l f of the systematic  a l l distances. The  s t a t i s t i c s i n Table B. 1 by  These  the  improvement omission  of  of  the  these  validation points  is  - 241 -  so  s  MODELED CIC/Q vs. OBSERVED CIC/Q from PROJECT PRAIRIE-CRASS  at 50 m dsitance and 15. m heg i ht at 200 m dsitance and 15. m heg i ht at 600 m dsitance and 15. m heg i ht  70  U) 60 O C 50  2 40  ? 30 The CC IQ / was modeeld wtih the parameters : p - 15.5 ; c - 04. ; k - 04.0 Fu lx profile relations by Dyer and Brade ly (1982) »„  = (1-28  i/L)-'^  +„ = (1-14 z/i;-'/*  30  CIC/Q (in  40  10~  50 60  s/m )  3  2  80  70  OBSERVED  Figure B.1 : Scatterplot of CIC/Q model v a l i d a t i o n (with ' P r a i r i e Grass' data)  Table B. 1 : S t a t i s t i c s for 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) Stat ist i c  all  di st. 80  n  ( 76 )  50 m 29  ( 26 )  200  m  800  30  ( 29)  21  0.92  (0.96)  0.65  (0.76)  0.13  (0.25)  0.51  d  0.97  (0.99)  0.84  (0.92)  0.58  (0.70)  0.82  0  20.4  (18.8)  43-9  (41.4)  10.7  (11.0)  1.8  P  19-7  (18.6)  40.4  (39.7)  12.1  (11.7)  2.0  5.8  ( 3.7)  8.9  ( 5.6)  3.8  ( 2.8)  0.7  3.4  ('1.6)  6.9  ( 3-D  2.1  (1.8)  0.3  4.7  ( 3.4)  5.5  ( 4.6)  3-2  ( 2.1)  0.6  R  2  RMSE  t o t  RMSE. sys RMSE. "unsys  M O D E L E D C I C / Q ( D y e r k = . 4 l ) v s . ( D y e r ic B r a d l e y  MODELED C I C / Q  k=.40)  ( D y e r ic B r a d l e y v e r s i o n )  k = .41 v s . k = 0.40  at SO m dsitance and 15 . m heg i ht at 200 m dsitance and 15 . m heg i ht al 600 m dsitance and 15 . m heg i ht  The CC IQ / was modeeld wtih the paramee trs p - 15 .5 ; c - 0.4 10  20 30 40 50 60 CIC/Q (in  10-  3  s/m ) 2  [Dyer]  Figure B.2 : Effects of Dyer (1970) p r o f i l e s compared to Dyer and Bradley (1982) profiles on p.d.f.-model results  eo  0  10 20 30 CIC/Q  (in  40 10-  3  s/mSO ) 2  60 70 60 00  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  considerable measurements  omission  uncertainty  is  justified  involved  and the derived u  in  because  both  the  of  the  concentration  and L values which are used to  #  run the model. Figure B.2 shows that 2  ( 0 = 0 = (1-16- z / L )  —1/4  )  H  M  (1982)  versions  result  Figure B.3 i l l u s t r a t e s of  k = 0.41  the  is  use  of  as  opposed  in  slightly  the to  Dyer (1970) the  Dyer  higher  profiles  and Bradley  model-values  and  that the effect of a von K£rm£n constant  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 i s from the "Hanford-30 Series", as  published by Fuquay e t . a l .  release times,  (1964).  These data  include  the  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" f o r arcs at  distances of  200,  800,  output of zinc s u l f i d e  1600  and 3200 m,  the  total  i n grams, and estimates of o-  source  along the  y  sampling arcs  for  each run.  Ten of  these  runs  were made  in  unstable s t r a t i f i c a t i o n .  The conditions during the releases are  summarized  (1984).  by  Draxler  The  zinc  sulfide  tracer  was  released at a height of 1.5 m i n the gently r o l l i n g t e r r a i n with sagebrush  vegetation  of  Washington. The roughness with a displacement From  this  1  2  m height  near  Hanford,  length of the area is given as 0.03 m  height  information  to  of the  1.4  m (Doran and Horst,  required  quantities  1985).  for  the  - 244 model-validation could be obtained i n the following manner. The  Richardson number was  conditions,  following  taken  Golder  equal  (1972),  to  where  z/L in z  unstable  refers  to  the  geometric mean of the heights for which Ri was determined (7-50) ' 0  5  = 18.7  ft  or  5.7  m).  The roughness  (i.e.  wind-speed  was  determined according to  u,  where the i/»(Ri)  =  u(z)  • k  I  ^ln((z-d)/z ) o  «/»(Ri)j  ,  f a m i l i a r integration by Paulson (1970) — 1/4  (with  following  "exposure"  (B.D  was used  for  —  <j> (Ri) = (1-28RD  ).  <r -u was  taken as  equal  to  <p  M  cr^,  -  Panofsky is  and Dutton  defined  concentration along an arc,  as  (1984,  the  p  maximum  i n [g*s*m  ].  159ff).  The peak  time-integrated  Therefore CIC/Q could  be established from Exp. CIC/Q  =  / . • ar • / 2it  ,  (B.2)  Q-t where Exp. i s the peak exposure and t the duration of the tracer release as given by Fuquay e t . a i . In Figure B. 4 the unstable  (1964).  runs from the  are compared with the model r e s u l t s  Hanford-30  Series  f o r CIC/Q and a summary of  the v a l i d a t i o n s t a t i s t i c s i s given i n Table B.2. Two of the runs included  in  consistently  the at  data all  set  overestimate  distances  by an amount  exhibited i n the rest of the runs. considerable  error  It  involved i n the  inputs (L and u^) here and  that  the  the  observed  values  much larger  than  i s probable that there  determination of overestimation  is  the model does  not  - 245 MODELED C I C / Q  0  5  vs. OBSERVED CIC/Q  10  IS CIC/Q (in  20 IO'  3  from  25 s/m ) 2  HANFORD-30  30  35  SERIES  40 45  OBSERVED  Figure B.4 : Scatterplot of v a l i d a t i o n (with Hanford-30 data)  CIC/Q  model  Table B.2 : S t a t i s t i c s for CIC/Q model v a l i d a t i o n (with Hanford-30 data)  Stat i st ic n  c  200 m  11 d i s t . 35 (  27)  9  ( 7 )  800 m 9  ( 7  )  1600 m  3200 m  8  9 ( 7 )  ( 6 )  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  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)  0.29  ( 0.6) 0 56 (0.72) 0 40 (0.73) 0 85  RMSE  t o t  5. 9  (3 4 ) 11.3 ( 6.6) 2 2  RMSE  s y s  2. 2  (0 6 )  7.4 ( 4.2) 1 4  5. 5  (3 3 )  8.6  RMSE un s y s  ( 5.D 1 7  (0.93)  1 3  (0.57) 0 34  (0.84)  (0.19)  (0.33) 0 7  (0.17) 0 23 (0.03)  1 1  (0.55) 0 24 (0.19)  (0.87)  - 246 -  Table B.3  : S t a t i s t i c s for <r -model v a l i d a t i o n y  (with Hanford-30 data) n  35  R  2  0.74  d  0  P  0.91  140.0  140.2  RMSE  t o t  64.3  RMSE  s y s  39-4  R M S E  unsys  50.8  - 247 necessarily  portray  a  d e f i c i e n c y of  s t a t i s t i c s f o r the reduced  the  model  itself.  data set (without the o u t l i e r s ) are  included i n brackets i n Table B.2. The values of these two are connected responsible  The  runs  by dashed l i n e s i n Figure B.4. These o u t l i e r s are 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  distances.  The  favourably with  concentration  the observations  variation  with  at a l l  distance  is  described quite well by the model, whereas the modeling of CIC/Q at a s p e c i f i c distance i s somewhat less s a t i s f a c t o r y . Note that the o r i g i n a l developed  version of the CIC/Q-model  (van Ulden,  1978)  was  to f i t the P r a i r i e Grass data, but i s independent of  the Hanford-30 Series. The c -observations from the Hanford-30 data y  were  compared  with the model f o r l a t e r a l spread, using the l i n e a r r e l a t i o n s h i p between cr and cr v  The  results  statistics agreement  as described i n Section 7.2.1.2 (<r = or -x/u).  y  are  y  plotted  are given  i n Figure  i n Table  of the CIC/Q  model  B.3. with  B. 5  and  the  v  validation  It i s concluded observations  that the  i s good  over  distances up to a few km, whereas the equivalent f o r the l a t e r a l spread i s at least acceptable f o r the present  purposes.  - 248 -  APPENDIX  C :  SAM - F o r t r a n - 7 7 Code  1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48  C C C C  ' This i s theshortest ' interactively. CALL STOP END  C  p o s s i b l e way t o r u n SAM  SAMSUB(AZS.AD.AZO.AUS,AQH,AZI.ASV.AUST,AL,0)  ================================================================= SUBROUTINE SAMSUBt A Z S , A O , A Z O , A U S , A Q H , A 2 I , A S V , A U S T , A L , I F L G ) ================================================================ S o u r c e A r e a Model =================================================================  C C C C C » > S A M « < C C * This subroutine evaluates thesource region o f a turbulent C * f l u x s e n s o r a t h e i g h t AZS. C " The arguments are as f o l l o w s : C * AZS : s e n s o r h e i g h t AD : displacement height C ' AZO : r o u g h n e s s l e n g t h AUS : w i n d s p e e d a t AZS C * AQH : s e n s , h e a t f l u x AZI : m i x i n g h e i g h t C ' ASV : l a t . w i n d s t d . d e v . AUST: r o u g h n e s s w i n d s p e e d C * AL : Monin-Obukhov l e n g t h IFLG: f l a g C ' C * I f IFLG=1, t h e v a l u e s o f t h e a r g u m e n t s a t e n t r y a r e u s e d C * a s i n p u t t o t h e m o d e l . AZS, AD a n d AZO a r e n e c e s s a r y site 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 i n p u t may b e AUS a n d AQH o r C " AUST a n d A L , a n d ASV o r A Z I . F o r t h e l a t t e r c h o i c e s , t h e C * m o d e l 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 a n d ASV. C * C * E l s e , t h e m o d e l g o e s i n t o i n t e r a c t i v e mode a n d t h e u s e r i s C * prompted f o r a l l input 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 C * D I S S P L A ( 1 0 . 0 ) , i s a v a i l a b l e i n t h i s mode. I f D I S S P L A ( 10.0) i s C * n o t a v a i l a b l e , SUBROUTINE MONET n e e d s t o b e r e p l a c e d f o r C * compilation. C * R e s u l t s a r e w r i t t e n t o UNIT* 2, w a r n i n g s , e r r o r m e s s a g e s a n d 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. I n i n t e r a c t i v e C • mode UNIT# 6 = 'SINK* a n d UNIT# 5 = 'SOURCE*. C 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 t h a t upwind C ' r e g i o n on the g r o u n d which i s d i r e c t l y a f f e c t i n g t h o s e a i r C * p a r c e l s that are encountering t h e s e n s o r by. t u r b u l e n t t r a n s p o r t . C * T h e m o d e l makes u s e o f t h e a n a l o g y b e t w e e n t h e m e c h a n i s m s o f C * d i f f u s i o n o f any s c a l a r a n d t h e plume b e h a v i o u r o f a p a s s i v e C * t r a c e r from a g r o u n d s o u r c e . ( R e c i p r o c a l plume model, s e e C * Chapters 6 and 7 ) . 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 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). C * C * C I M P L I C I T REAL'8 (A-H.L.O-Z) REAL * 8 X C U T ( 2 ) , C F R A C ( 1 0 , 4 ) , E F ( 1 0 , 2 ) , S N 0 ( 0 : 5 0 , 2 ) , A U X ( 11) REAL*8 X L I M t 2 ) , P L E T H ( 9 , 2 , 5 1 ) , P D O T S ( 1 1 0 ) , D U M P ( 2 ) REAL'8 PROFX( 101 , 2 ) , P R O F Y ( 1 0 1 , 2 ) , P C H A R ( 1 0 , 7 )  - 249 -  49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106  C C C  " Specify COMMON  input  constants  ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN.XBMIN  C C=0.4 VK=0.4 C C C  *  I f IFL=1  : not i n t e r a c t i v e  I F ( I F L .NE . 1) THEN C C C  " Give  site  specific  information  : sensor  W R I T E ( 6 , * ) ' G I V E S I T E INFO. : SENSOR WRITE(6,*)'ZSZSZS ZOZOZO DDDDDD' REA0(5,*)ZS,ZO.D  height  ( Z S ) , ZO & D  HEIGHT, ZO, D  ; as i n :  C  c c c c  c c c c c c c c c c c c  ZS=22.5 Z0=.5 D=3.5 W R I T E ( 6 , * ) ' D O YOU HAVE U* AND L ? READ(5,*)JJA IF(JJA.EQ.1)THEN W R I T E ( 6 , ' ) ' G I V E U* AND L : ' READ(5.*)UST,L END I F  (YES=1)'  * R e a d J u l i a n Day, l o c a l a p p a r e n t t i m e , w i n d s p e e d a n d d i r e c t i o n * ( a t sensor h e i g h t ) , s u r f a c e layer temperature ( i n C e l s i u s ) , • 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 . W R I T E ( 6 , * ) ' G I V E J.DAY, L . A . T . . U, WD, Z I , SV, QH, TEMP. :' WRITE!6,*)'DJD TH TM UUUUUU WDWDWD Z I Z I SVSV QHQHQH T T ' READ(5,*)JD,JTH,JTM,US,WD,ZI,SV,QH,TT IF(JJA.EQ.1)GOTO * Calculate CALL *  5  L a n d U* b y a c a l l  t o MONOB  MONOB(US,ZS,D,ZO,TT,QH,L,UST)  I f there  occurred  an e r r o r  I F ( L . E Q . 9 9 9 . ) G O T O 999 5 CONTINUE ELSE ZS=AZS D=AD ZO=AZO US=AUS QH=AQH ZI=AZI SV=ASV UST=AUST L=AL IF(L.GE.0.)THEN  i n MONOB, c u t s h o r t .  - 250 -  107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164  C  C  CALL MONOB (US , ZS , D , ZO, TT , QH , L , UST) END IF END IF IF(SV.EQ.O.>THEN SV=SIGV(L,ZI,UST) END IF CALL MINIM(ZBMIN,XBMIN) WRITE(6,*)'MINIMUM UPWIND DISTANCE  = ',XBMIN  c c c c c c c c  * Set up an a r r a y w i t h a u x i l i a r y data that may be c a r r i e d over t o * t h e p l o t - s u b r o u t i n e MONET . A r r a y AUXI10) c o n t a i n s the f o l l o w i n g * information : JD, L.A.T., Wind ( s p . / d i r . ) , Z i . S V . Z S / L (stab.),U*,ZS & FLAG " i f FLAG=0, the run r e f e r s t o the SUNSET s i t e , * i f FLAG=1, the run r e f e r s t o a MOBILE s i t e .  c c c  * W r i t e header f o r output d a t a - f i l e (#2)  c c c c c c c c  AUX(1)=DFL0AT(JD) AUX(2)=DFL0AT(JTH) AUX(3)=DFL0AT(JTM) AUX(4)=US AUX(5)=WD AUX(6)=ZI AUX(7)=ZS/L AUX(8)=UST AUX(9)=ZS IF(IFLG.NE.1)THEN WRITE(6,*)'WHICH SITE ? (0 = SUNSET; 1 = MOBILE)' READ(5,*)AUX(10) ELSE AUX(10)=0. END IF AUX(11)=SV  WRITE( 2,*)'RESULTS FROM S A M - CALCULATIONS' WRITE( 2 , ' ) ' =-==-==—=—==-==-===-===-=====— • WRITE(2. 101)JD,JTH,JTM WRITE12,122)QH,TT,SV WRITE(2,121)US,WD,L,UST,ZI 101 FORMAT!//.'DATA FOR : JD*,I4.' ; a t ' .12.' : ' . 12.' L.A.T.'./) 122 FORMAT('SURFACE HEAT FLUX : '.F6.2,' ; AIR TEMP. : '.F5.1. #' SV ='.F5.2) 121 FORMAT('WINO (SPEED/DIR.) : '.F5.2,'/ '.F5.1,' L ='.F8.2. #' U* = ',F5.3,' Z i ='.F5.0) WRITE(2,128) 128 FORMAT(/,'(ALL DATA ARE IN SI-UNITS)'.///)  * F i n d the p o i n t o f maximum e f f e c t CALL CMAXUMX ,CMX) * S p e c i f y the number of i n t e g r a t i o n s t e p s f o r the e f f e c t - f i e l d * s e c t i o n s ( 1/2 number i n X and Y d i r e c t i o n ) .  - 251 -  165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222  C C C C C C  WRITE(6,")'GIVE HALF # OF INTEGRATION STEPS IN X DIRECTION :' READ(5 , * )N WRITE(6,*)'GIVE HALF # OF INTEGRATION STEPS IN Y DIRECTION :' READC5,* )M N=10 M=25  C C C C C  *  C C C  • I f an e r r o r o c c u r r e d i n ADDUP, w r i t e  C C C C C C C C C  C C  c c 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 s e c t i o n - w i s e 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 f o r • further d e t a i l s CALL ADDUP(N,M,CMX,XMX,CTOT,CFRAC) t o #3 and c u t s h o r t  IF(CTOT.LE.O.)THEN WRITE(3,*)'ADDUP WAS UNABLE TO INTEGRATE THE EFFECT FIELD' WRITE(3,")'CTOT = ',CTOT GOTO 999 END IF • Use a s p l i n e i n t e r p o l a t i o n t o e v a l u a t e the i s o p l e t h s that • c o n t a i n d e c r e a s i n g f r a c t i o n s o f the whole i n t e g r a t e d e f f e c t • f i e l d (eg. 0.95, 0.90, 0.80, e t c . ) . NOTE : In CFRAC t h e • f r a c t i o n a l e f f e c t l e v e l s a r e g i v e n i n % , the f r a c t i o n a l • i n t e g r a l s in decimals. *  F i r s t s e t up the nodes f o r the s p l i n e , w r i t e  WRITE(3,33) 33 FORMAT!//,'NODES FOR SPLINE : CUM. INTEGR. J=0. DO 503 1=1.9 IF(CFRAC(I,3).EQ.0.)GOTO 77 J=J+ 1 SN0(J,2)=CFRAC(I,1) SN0(J,1)=CFRAC(I,4) 77 CONTINUE 503 CONTINUE SNO(0,1)=0. SN0(0.2)=100. SN0(10,1)=1. SN0(10,2)=0. NM=J+1 DO 366 J=0,NM WRITE(3,32)SN0(J,1),SN0(J,2) 32 FORMAT(19X.F12.5.F12.2) 366 CONTINUE  them t o comment f CONC. %',/)  *  Set the d e s i r e d i n t e g r a l f r a c t i o n s . EF (EF f o r e f f e c t ) . • 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 %) * W r i t e subheader f o r comment listing. 189  WRITE(3, 189) FORMAT(/,'INTEGRAL FRACTION OF TOTAL EFFECT VS. ISOPLETH %') WRITE(3 •)'-==-==-==-=====-===-========-===-==-====-==—=='  - 252 -  223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280  WRITE(3,')' WRITE(3,')'  (int.frac.)  (cone.  % o f max.)'  '  C X=0.95 J=0 NP=NM+1 DO 504 1=10,1.-1 I F ( X . G T . S N O ( N M , 1 ) ) G 0 T 0 88 j=j+1 C CALL  SPLYNE(SN0,NP,X,Y)  C EF(I,1)=X EF(I.2)=Y C C C  • Write  results  into  comment-file  WRITE(3,1881X.Y FORMAT(F20.5,F20.2) CONTINUE X=0.1'DFLOAT(I-1) 504 CONTINUE NJ=J 188 88  C C C C C C C C  c c c c c c c c c c c c  *  Now  t h e a r r a y EF c o n t a i n s round v a l u e s o f t h e i n t e g r a l fraction t h e a s s o c i a t e d e f f e c t l e v e l i n % o f t h e maximum. * The n e x t step i sto f i n d the location of the isopleths f o r these * effect l e v e l s . T h i s i s d o n e b y i n v o k i n g s u b r o u t i n e CONC • w i t h t h e FLAG=2. S o t h a t X a n d t h e e f f e c t l e v e l a r e t h e i n p u t and Y(+) t h e o u t p u t . T h e maximum r a n g e o f X f o r e a c h • e f f e c t l e v e l c a n b e f o u n d w i t h CONC a n d FLAG=1. • The r e s u l t s w i l l b e p u t 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 :  • versus  *  • •  PLETH([number],[x/y].[case  »])  ; dimension  (10,2,1000)  • The  l i n e number ( i - v a l u e ) i n E F ( i , j ) corresponds to the [number]-va1ue i n PLETH. * In o r d e r t o 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 , * PLETH c o n t a i n s t h e x/y p a i r s f o r both pos. and neg. y ' s . * PDOTS c o n t a i n s o n l y h a l f t h e i s o p l e t h and i s used f o r t h e • c a l c u l a t i o n o f t h e a r e a bounded by each i s o p l e t h by c a l l i n g on • s u b r o u t i n e SIMPS. WRITE(2,179) 179 FORMAT(//.' # ' ) WRITE(2.')' WRITE(2,181) 181 FORMAT('S A M - RESULTS #DIMENSIONS)') WRITE(2.182) 182 #====--==—=-' ) WRITE(2,*) ' i n t . e f f . # d xm' WRITE(2,*)' Y0=0. NJM=NJ-1 -  SUMMARY  area  (ISOPLETH CHARACTERISTIC  a  b  c  - 253  281 282 283 284 285 286 287 288 . 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338  -  XMIN=100. XMAX=0. DO 505 1=1,NJM C C C  •  Chose  the  isopleth,  write  choice  CP=EF(I,2)'CMX/100. W R I T E ( 3 , * ) ' I S O P L . #',I.'  =  into  comments-file  ',EF(I,2)  C  c c  * Look  CALL  c  c  c c c  123  into  comments-file  C0NC(CMX,XMX,XC,XLIM,YO,CP,1)  far X =  ',F9.2)  DX=(XH-XL)/100. PLETH(I,1.1)=XL PLETH(I,2,1)=0. PLETH(1,1,51)=XL PLETH(I,2,51)=0. PD0TS(1)=0. JJ=5 '  Calculate DO  the  location  of the  isopleths  506 J=2,101 XP=XL+DX * J CALL  CONC(CMX,XMX,XP,XCM,YP.CP,2)  IF(J.EQ.JJ)THEN JJJ=((JJ-1)/4)+1 P L E T H d . 1,JJJ)=XP PLETH(I,2,JJJ)=YP  c  K=52-JJJ P L E T H d , 1,K)=XP PLETHd,2,K)=YPM0.-1.) JJ=JJ+4 END I F IF(YP.GT.PDOTS(J-1))THEN YYMX=YP XYMX=XP ENDIF * Put  506  c c c  result  XL=XLIM(1) XH=XLIM(2) IF(XL.LT.XMIN)XMIN=XL IF(XH.GT.XMAX)XMAX=XH WRITE(3,123)XL.XH FORMAT('near X = ',F8.2,'  c c  c c c  f o r X - r e a c h and w r i t e  ' •  t h e +-ve  YP's  into  PDOTSfJ)=YP CONTINUE WRITE!3,*)'YYMX Compute t h e a r e a over the X-range  =  PDOT f o r u s e  ',YYMX,'  AT  b o u n d e d by t h e s e f r o m PDOTS( 1) t o  i n the area  X =  calculation  '.XYMX  isopleths; PD0TSM01)  i.e.  integrate  -  339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396  C C C C C C C C  C C  c c  c c  c c  -  • u s i n g s u b r o u t i n e SIMPS, and m u l t i p l y by 2. • A r r a y FF c o n t a i n s the i n t . f r a c . and a s s o c i a t e d area CALL SIMPS(POOTS,XL.XH.101,FP) *  S p e c i f y the c h a r a c t e r i s t i c dimensions o f the i s o p l e t h s and put • them i n t o a r r a y PCHAR(10,7) : ( # ; i n t . e f f . , a r e a . a , b , c , d , x m ) • Write r e s u l t s into r e s u l t s - f i l e PCHAR(I,1)=EF(1,1) PCHAR(I,2)=FP'2. PCHAR(I,3)=XL PCHAR(I.4)=XYMX-XL PCHAR(I,5)=XH-XYMX PCHAR(I,6)=YYMX PCHAR(I,7)=XMX WRITE(2,199)(PCHAR(I,J),J=1,7) 199 FORMAT(F9.2,F12.1.4F10.3.F9.2) 505 i CONTINUE • C a l c u l a t e a n o r m a l i z e d p r o f i l e i n X - d i r e c t i o n and put the X.CONC ft v a l u e s i n t o a r r a y PROFX, t o be used f o r p l o t t i n g purposes. Y0=0. PDX=DABS(XMAX-XMIN)/100. XXP=XMIN 00 509 1=1,101 CALL CONC(CMX,XMX.XXP,DUMP.YO,CPX.0) PR0FX(I.1)=XXP PR0FX(I,2)=CPX/CMX  c c c c c  254  509  XXP=XXP+PDX CONTINUE  »  C a l c u l a t e a n o r m a l i z e d p r o f i l e i n Y - d i r e c t i o n at the XMX, • put i n t o a r r a y PROFY to be used f o r p l o t t i n g . PDY=PCHAR(NJM,6)/50. YYP=0. DO 510 J=51.101 CALL CONC(CMX,XMX,XMX,DUMP.YYP,CYP,0) PR0FY(J,1)=YYP PROFY(J,2)=CYP/CMX K=102-J YYPN=0.-YYP PROFY(K,1)=YYPN PR0FY(K,2)=CYP/CMX  YYP=YYP+PDY 510 CONTINUE IF(IFLG.EQ.1)GOTO 999 WRITE(6,*)'DO YOU WANT A PLOT ? READ(5,*)IESNO IF(IESNO.NE.1)G0T0 999 D i s p l a y the r e s u l t s g r a p h i c a l l y  (YES = 1 / NO = 2 ) '  by c a l l i n g on s u b r o u t i n e  - 255 -  397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454  C C C C C  *  MONET(PLETH,AUX,PROFX,PROFY) CALL MONET(PLETH,AUX,PROFX,PROFY)  999 CONTINUE RETURN END SUBROUTINE MONOB(UB,Z,D,ZO,TT,QH,L,USTR)  C  c c c c c c c c c c c c c c c c c c c c cc c  This subroutine computes the Monin-Obukhov length (L) • i t e r a t i v e l y via log-profile U* correction : • • U* = UB(z) k /[1n((z-d)/zo)-PSY((z-d)/L)+PSY(zo/D] • • PSY = 1n[((1+x*2)/2)((1+x)/2)*2]-2arctan<x)+(PI/2) • a x = (1-28(z-d)/L) .25 = PHY -1 (unstable) *  A  * *  ft  L = -RHO Cp TB U**3/(k g QH)  • In this application the constants take the following values • related to the SUNSET s i t e and for unstable conditions : • • k = 0.4 RHO = 1.204 ft • Cp = 1010  ft Equations (1) to (4) appear in the subroutine. ft *  c c  (1)  ft  * *  (2)  *  (3)  *  (4)  *  a  • •  a •  ft  «  •  ft  • •  IMPLICIT REAL'S (A-H.L.O-Z) REAL * 8 UB . WD. TT , QH, QHM. OL 1 . 0L2 . US. XZ . XO . PSYZ , P'SYO . D . OLD , QRD ft Specify the constants RH0=1.204 CP=1010. VK=0.4 G=9.81  c c ft I n i t i a l US=0. 1 c • Convert c c  A  ft  value of U* is 0.1  '  to absolute temperature * IF(TT.LT.150.) THEN TT=TT+273. END IF • Exclude cases where QH <= 10. W/mi2 * IF(QH.LE.10.)G0T0 305 • Equation (4) ( » f i r s t estimate) * 0L1 = ( - RHO*CP)*TT*US* * 3./(QH*VK*G) N=0 * Begin U" iteration ' 300I CONTINUE • Equation (3) for (z-d) and zo * XZ=( 1 -28. * (Z-D)/OLD * "0.25 X0=(1-28.'Z0/0L1)"'0.25  - 256 -  455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512  C C C  • Equation (2) for (z-d) and zo * PSYZ=DL0G((( 1 + X Z " 2 . )/2. ) * (( 1 + XZ)/2. ) * *2. -2.) *0ATAN(XZ)+1 PSY0=0L0G(((1 + X0* *2.)/2.)*((1 + X0)/2.)* *2.)-2. •DATAN(X0)+1 * Equation (1) ( » U* correction) • US=UB * VK/(DLOG({Z-D)/ZO)-PSYZ+PSYO) tl Equation (4) ( » second estimate) * 0L2=(-RH0*CP) TT*US**3./(QH*VK*G) * Evaluate improvement of estimate * DD=0L1-0L2 0LD=0ABS(D0) 0L1=0L2 N=N+ 1 • Maximum iteration steps set to 300 * IF(OLD.GT.0.0001.AND.N.LT.300)GOTO 300 ,  C  C C  IF(N.EQ.300)THEN WRITE(3,')'ITERATION FOR L UNSUCCESSFUL AFTER 300 STEPS' L=999. GOTO 306 END IF  C C  c c c c c c c  c c c  L=0L2 USTR=US GOTO 306 305 WRITE(3,205) 205 FORMAT ( T u n is *** STABLE OR QH < 10 W/M2 •* • ' ) 306 CONTINUE . RETURN END SUBROUTINE MINIM(ZBMN,XBMN) • This subroutine determines the minimum upwind distance, which an effect-level calculation is defined.  *  IMPLICIT REAL*8 (A-H.L.O-Z) COMMON ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN,XBMIN TOL=1.E-5 ZBA=D+1.E-8 PHYA=PHYMIN(ZBA,L) PSYA=PAUL(PHYA) QA=DL0G(C'(ZBA-D)/ZO)-PSYA • Find i n t e r v a l l where QB changes sign. ZBB=ZS IK=0 401 CONTINUE PHYB=PHYMIN(ZBB,L) PSYB=PAUL(PHYB) QB=DL0G(C *(ZBB-0)/ZO)-PSYB PROD=QA'QB IF(PROD.GE.O.)THEN ZBB=ZBB+1 . IK=IK+1  for  - 257 -  513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 566 569 570  C C C  C  c c c  c  IF(IK.GT.500)GOTO GOTO 401 ENDIF " Bisection begins  DO 500 1=1,100 ZBP=ZBA+(ZBB-ZBA)/2. PHYP=PHYMIN(ZBP,U PSYP=PAUL(PHYP) QP=DL0G(C *(ZBP-D)/ZO)-PSYP DIFF=DABS((ZBB-ZBA)/(2.*ZBB)) IF(QP.EQ.O..OR.DIFF.LT.TOD THEN ZBMN=ZBP IF(ZBMN.L T.ZO)ZBMN=ZO PSYQ=PAUL(PHYMIN(ZBMN,L)) QQ=DLOG(C*(ZBMN-D)/ZO)-PSYQ XBMN=XZB(ZBMN,PSYQ,L) IF(XBMN.LT.ZO)XBMN=ZO GOTO 402 ENDIF PR=QA'QP IF(PR.GT.O.)THEN QA=QP ZBA=ZBP ENDIF IF(PR.LT.O.)THEN QB=QP ZBB=ZBP ENDIF 500 CONTINUE WRITE (3,*)'MINIM : NOT FOUND AFTER 100 ITERATIONS' WRITE (3.*)'REL. DIFF IN ZB : ' . D I F F , ' QP = ',QP 402 CONTINUE ' Double check ZBMIN and XBMIN. CALL CICZ(ZMIN.ZS.UB,S,XBMN.CIC,0) ZBMN=ZMIN PSYP=PAUL(PHYMIN(ZBMN,L)) XBMN=XZB(ZBMN,PSYP.L) 999 CONTINUE RETURN END  c c c c c c c c c  999  SUBROUTINE CMAX(XMX.CMX) * ' * * * •  This subroutine calculates the maximum CONC/Q at level ZS for a surface point-source, together with the downwind location (X) of that point. The sensor level ZS, and surface geometry parameters ZO and D are treated as constants, residing in common memory space with the main program (also wind at sensor level (US), U*. ZI & L ) . IMPLICIT REAL'8 (A-H.L.O-Z) COMMON ZS.ZO,D,C,VK,US,UST.SV.L,ZBMIN,XBMIN  - 258 -  571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628  C C C C C C C C C CC CC CC CC CC CC C C C  C=0.4 VK=0.4 ZS=22.5 Z0=.5 0=3.5 *  WRITEO,')' WRITE(3,')'THE MAXIMUM CONC(ZS) IS REACHED WITH THE DATA : ' WRITEO,') ' - - ' WRITEO, * ) ' ' WRITEO, *)' X ZB ZS S CONC(ZS)' WRITEO,')' ' *  c c c c c c c c c c c c c c c c c c c c c c c C C C C C  Write subheader into comment-file  I n i t i a l guess for CMX is at ZB = 0.6 ZS ZB1=ZS'0.6 IF(ZB1.LT.ZBMIN)ZB1=ZBMIN  • compute CIC(ZS) with ZB = ZB1 • Adjust for lateral diffusion with LAT and Y=0  Y=0. CALL CICZ(ZB1.ZS.UB1,S1,X1,C1,1) C1=C1'LAT(Y,SV,.UB1 ,X1) *  The search for a maximum in CMX begins. The method of "REGULA  • FALSI" is used with a steplength of DAB in ZB.  DAB=ZB1 ZB2=ZB1+DAB * *  Compute CIC(ZS) with ZB = ZB2 Adjust for lateral diffusion with LAT and Y=0 CALL CICZ(ZB2,ZS,UB2,S2,X2,C2,1) C2=C2*LAT(Y,SV,UB2,X2)  * * * *  This is the beginning of the algorythm, an imitated "WHILE" loop via IF-THEN & GOTO. Two values (ZB1.C1) and (ZB2.C2) are evaluated . Based on the comparison of C1 and C2 it is decided in which direction the search is to continue. Every time the search is changing d i r e c t i o n , the steplength DAB is halved. LOOP 402 : Main bisection routine.  402 CONTINUE DAB=DAB/2 *  The search is terminated when DAB < 1E-10 IF(DAB.GE.1E- 10)THEN  • LOOP 400 : While C2 > C1 the search is continued to the right  - 259 -  629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686  C  400  CONTINUE IF(C2.GT.C1)THEN ZB1=ZB2 C1=C2 ZB2=ZB1+0AB CALL CICZ(ZB2,ZS,UB2.S2.X2.C2,1) C2=C2*LAT(Y,SV,UB2,X2)  C C C  • In case of an "overshoot" skip out of LP-400, back to bisection  C C C  *  IFIC2.LE.C1)G0T0 402 GOTO 400 ENDIF  401  C  LOOP 401 : While C2 <= C1 the search is continued to the  left.  CONTINUE IF(C2.LE.C1)THEN ZB1=ZB2-DAB CALL CICZ(ZB1,ZS.UB1,S1,X1,C1, 1) C1=C1'LAT(Y,SV,UB1,X1)  C C C  • In case of an "overshoot" skip out of LP-401, back to bisection  C C C C  • End of "REGULA FALSI" algorythm. * Print last estimates with last steplengths as uncertainties.  IF(C2.GT.C1)G0T0 402 ZB2=ZB1 C2=C1 GOTO 401 ENDIF ENDIF  EC=DABS(C2-C1) EX=DABS(X2-X1) XMX=X2 ZB=ZB2 S=S2 CMX=C2  C CC CC CC CC  WRITEO,')' ' WRITEO,')'THE REMAINING UNCERTAINTIES ARE WRITEO, *)'E-CIC : ' . E C , ' ; E-X : ',EX WRITEO,')' ' RETURN END  C  c c c c c c c  SUBROUTINE CICZ(ZB.Z,UB,S,X,CIC,IFLAG) Distance which plume has travelled to reach ZB is X , computed from GHIS (A1a). • If IFLAG=0. X is input and ZB is output, if IFLAG=1 , ZB is input • and X is output. • For that, compute PSY(C(ZB-D)/L) and give ZO, 0, K, P, C, using *  - 260 -  687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744  C * C * C " C * C * C *  PSYM = PSYM(PHY -1) from Paulson, 1970 PHY*-1 = XM = (1 - 28(ZB-D) .25 ; K = 0.-4 (Dyer&Bradley. (using the correlations for heat d i f f u s i o n ) . P = 1.55 ; C = 0.4 (GHIS.1987; 1983) A  A  IMPLICIT REAL*8 (A-H.L.O-Z) COMMON ZS.ZO.D,C.VK,US,UST,SV,L,ZBMIN,XBMIN IF(IFLAG.EQ.1) THEN  C C C C  * Want X for Z * compute PHY -1 or XM  C C C  * compute PSYM using the Paulson(1970) formula  C C C  ' compute X from GHIS (A1a)  C C C  * compute S from GHIS (A2a)  C C C  * compute UB from GHIS (A3a)  C C C  * compute A from GHIS (A4)  C C - C  * compute B from GHIS (A5)  C C C C  C  A  XM=PHYMIN(ZB.L)  PSYM=PAUL(XM)  X=XZB(ZB,PSYM.L) IF(X.LT.XBMIN)THEN X=XBMIN ENDIF ENDIF IF(IFLAG.EQ.O) THEN IF(X.LT.XBMIN)THEN X=XBMIN ENDIF ZB=Z0FX(X,L) ENDIF  S=SPAR(ZB.PSYM,L)  UB=UBAR(UST.ZB.PSYM.L)  A=AGAM(S)  B=BGAM(S) * compute CIC at Z when the mean plume height  is ZB  CIC=CHYQ(A.B.ZB,UB,Z.S)'1.E5 IF(X.LE.O)THEN WRITEO, *)'CICZ : X <= 0. ENDIF RETURN  IFLAG = ',IFLAG  1982)  - 261 -  V 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802  C C C C C C C  C C C C C C C C  C C C C C C C  c  c c  END FUNCTION PAUL(XX) ' This function computes the PaulsonC 1970) PSY * for a given XX for unstable conditions. REAL'8 XX,PAUL PAUL=0L0G( ( 1 + XX"2. ) * < 1+ XX) " 2/8) -2*DATAN( XX ) + 1 . 5708 PAUL=XX-1. RETURN END FUNCTION XZB(ZB.PSYM.LD) * • * "  This function is equivalent to equation (A1a) in GHIS, with P=1.55, C=0.4 (GHIS,1983). A1 = 14 XZB is the distance the plume has to travel to reach height ZB.  IMPLICIT REAL'8 (A-H.L.O-Z) COMMON ZS,ZO.D,C.VK,US,UST,SV,L,ZBMIN,XBMIN ZZ=ZB-0 V=VK"2. XZB=(ZZ/V)*(DL0G(C'ZZ/Z0)-PSYM)'( 1 -(5.425'ZZ/L)) " ( -.5) RETURN END FUNCTION SPAR(ZB.PSYM.LO) • * * *  This function is equivalent to equation (A2a) in GHIS, with C=0.4 (GHIS,1983), A1=14, A2=28 SPAR is the shape parameter for the exponent of the v e r t i c a l concentration p r o f i l e .  IMPLICIT REAL * 8 (A-H.L.O-Z) COMMON ZS,ZO,D,C,VK.US,UST,SV,L.ZBMIN,XBMIN ZZ=ZB-D PSYM=PAUL(PHYMIN(ZB,L)) SP1=(1-7.'C*ZZ/L)/(1-14.*C'ZZ/L) SP2=(1-28.'C'ZZ/D " (-.25)/(0LOG(C'ZZ/ZO)-PSYM) SPAR=SP1+SP2 IF(SPAR.GT.1.9)THEN SPAR=1.9+(1.-DEXP(-1.0*DABS(SPAR-1.9)))/10. ENDIF IFtSPAR.LT.O.)THEN WRITE(3,*)'SPAR : '.SPAR.' QQ=DL0G(C'ZZ/Z0)-PSYM WRITE(3,*)'QQ=',QQ ENDIF RETURN END  SP1=',SP1,'  SP2=',SP2  - 262 -  803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860  C C C C C C C  C C C C C C  C C C C C C  C C C C C C  C  C  , FUNCTION UBAR(USTD.ZB.PSYM.LD) * This function is equivalent to equation (A3a) in GHIS, * with C=0.4 * UBAR is the mean travel-speed of the plume IMPLICIT REAL*8 (A-H.L.O-Z) COMMON ZS,ZO.D.C.VK,US,UST,SV.L.ZBMIN,XBMIN ZZ=ZB-0 UBAR=UST/VK*(DLOG(C'ZZ/ZO)-PSYM) RETURN END FUNCTION AGAM(S) * This function is equivalent to equation (A4) in GHIS * AGAM is a function of the shapefactor S IMPLICIT REAL*8 (A-H.L.O-Z) S1=1/S S2=2/S AGAM=S'DGAMMA(S2)/(DGAMMA(S1))* *2 RETURN END ; FUNCTION BGAM(S) * This function is equivalent to equation (A5) in GHIS * BGAM is a function of the shapefactor S IMPLICIT REAL*8 (A-H.L.O-Z) S1=1/S S2=2/S BGAM=OGAMMA(S2)/DGAMMA(S1) RETURN END FUNCTION DGAMMA(X) * This function approximates the Gamma-function according to * equation (A6) in GHIS. IMPLICIT REAL * 8 (A-H.L.O-Z) REAL*8 B(8) DATA 8( 1 )/-.577191652/,B(2)/.988205891/,B(3)/-. 897056937/, » B(4)/.918206857/,B(5)/-.756704078/.B(6)/.482199394/, # B(7)/-.193527818/,B(8)/.035868343/  DG=1. DO 500 1=1.8 DG=DG+((X-1.)**I)'B(I) 500 CONTINUE  - 263 -  861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918  DGAMMA=DG  C  RETURN END  C C  FUNCTION CHYQ(A.B,ZB,UB,Z,S)  C  * This function is equivalent to equation (3) in GHIS * CHYQ is the crosswind integrated concentration divided • by the source strength.  c c c c  IMPLICIT REAL*8 (A-H,L,0-Z) COMMON ZS.ZO.D.C,VK.US,UST,SV,L,ZBMIN,XBMIN ZZ1=ZB-D ZZ2=Z-D CHYQ=A/(ZZ1*UB)*DEXP(-(B*ZZ2/ZZ1)* *S) RETURN END  c c c c c c  FUNCTION SIGV(L.ZI.UST) • This function computes the V-standard deviation • SIGV (or sigma-V) according to eq.(18a) of GHIS. IMPLICIT REAL*8 (A-H.L.O-Z) SIGV=0.8*UST*((0.-ZI)/L)"*0.333 RETURN END  c c c c c c c c c c c c c c c c c c c c c  FUNCTION LATfY.SV.UB.X) This function computes the factor that the CIC w i l l have to be multiplied with to obtain a concentration value with • consideration of lateral spread and other than on the center• line. * The linear relationship between Sigma-V and Sigma-Y as * suggested by Pasquill & Smith (1983) for the diffusion very * close to the source is adopted here. * SV is the V-standard deviation e.g. from FUNCTION SIGV *  *  IMPLICIT REAL*8 (A-H.L.O-Z) Compute the traveltime T = X/UB T=X/UB *  Compute the lateral diffusion parameter SIGY SIGY=SV*T  *  Compute LAT EX=(Y/SIGY)**2*(-0.5) IF(EX.LT.- 180.)THEN LAT=0.  - 264 -  919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976  GOTO 44 ENDIF LAT=DEXP(EX)/(SIGY•2.5066) 44 CONTINUE RETURN END  C C  FUNCTION ZOFX(X.LO)  C  c c c c c c  • • • • •  IMPLICIT REAL'8 (A-H.L.O-Z) COMMON ZS.ZO.D.C.VK.US,UST.SV.L.ZBMIN,XBMIN JK=1  c  ccc c  c  c  This function i t e r a t i v e l y determines the ZB value for a given X in Eq. (A1a) of GHIS, using a secant root-finding routine. ZOFX is equal to ZB. It c a l l s on functions PAUL and XZB  IF(X.LT.XBMIN)THEN WRITEO, *)'ZOFX : X < XBMIN' ZOFX=ZBMIN GOTO 402 ENDIF  T0L=1.E-5 ZB0=X/5. IF(ZBO.LT.ZBMIN)ZB0=ZBMIN ZB1=ZB0+1 XMO=PHYMIN(ZBO,L) PSYMO=PAUL(XMO) Q0=XZB(ZB0,PSYM0,L)-X 1=2 401 CONTINUE IFM.LE.50) THEN XM1=PHYMIN(ZB1,L) PSYM1=PAUL(XM1) Q1=XZB(ZB1.PSYM1,L) -X IF(01.EQ.QO)THEN WRITEO, •)'ZOFX : DIVIDE BY 0 ; X = ' , X WRITEO, *)'ZB1 : ',ZB1 Z0FX=ZB1 ENDIF ZB=ZB 1 - Q1 * ( ZB 1 - ZBO) / (Q1 - 00) IF(ZB.LT.ZBMIN)THEN ZB=ZBMIN IF(JK.GT.1)THEN WRITEO, *)'ZOFX : TRIED TO SHOOT BELOW ZBMIN ' , JK, ' TIMES' ENDIF JK=JK+1 ENDIF DIFF=DABS(ZB-ZB1) IF(DIFF.LT.TOL) THEN ZOFX=ZB GOTO 402  - 265 -  977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034  C  C C C C C C C C  ENOIF 1=1+1 ZB0=ZB1 Q0=Q1 ZB1=ZB GOTO 401 ENDIF WRITE(3,*)'ZOFX : FAILED AFTER 50 ITERATIONS; OIFF =',DIFF 402 CONTINUE RETURN END FUNCTION PHYMIN(Z,LD) This function is equivalent to the reciprocal of a • empirical f l u x - p r o f i l e relation . • In this case for momentum flux as suggested by Dyer & * Bradley, 1982. *  IMPLICIT REAL'S (A-H.L.O-Z) COMMON ZS.ZO.D,C,VK,US.UST,SV,L,ZBMIN,XBMIN ZMIN=D+1.E-9 IF(Z.LT.ZMIN)THEN WRITE(3,*) ' WARNING (Z-D)= M Z - D ) . ' ; IS SET TO +1.E-9 ' Z=ZMIN ENDIF PHYMIN=(1.-28.'C*(Z-DJ/L) " .25  c  c c c c c c c c c c G  RETURN END SUBROUTINE CONC(CMX,XMX,XC.XCM,YC,CC,IFLAG) This subroutine computes the concentration of a surface point source at coordinate location (XC.YC.ZS). or any • third of the t r i p l e t (XC.YC.CC). when the other two are * given and IFLAG set accordingly (see below). If the X -locat ion • for a certain pair (CC.YC) is wanted, the result resides in the * array XCM, with XCM(1) being closest and XCM(2) farthest from * the source. * The maximum concentration CMX at (XMX.O.ZS) is imported as a * restraint for the secant search. * *  •* C •- • IFLAG=0 : XC.YC = Input; CC = Output IFLAG=1 : CC.YC = Input; XCM(2)= Output c ft c * IFLAG=2 : CC.XC = Input; YC(+) = Output  c ft cc c  c  IMPLICIT REAL * 8 (A-H.L.O-Z) REAL'S XCM(2) COMMON ZS,ZO,D,C,VK,US,UST,SV,L,ZBMIN,XBMIN SV=SIGV(L,ZI,UST) IF(IFLAG.EQ.0)THEN CALL CICZfZB.ZS.UB.S.XC.CIC.O) CC=CIC*LAT(YC,SV.UB,XC) FF=DABS(XMX-XC)  - 266 -  1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092  C  C C C C C  C C  c  c c c  c  CR=DABS(CC-CMX)/CMX IF(CC.GT.CMX.AND.CR.LE.0.01)CC=CMX ENDIF IF(IFLAG.EQ.1.AND.CC.LT.CMX)THEN T0LX=1.E-3 DO 500 J=1,2 • Search for XCM-points by going on either side of N=2 XP0=XMX CQ0=CMX-CC IF(J.EQ.I) XP1=XBMIN IF(J.EQ.2) XP1=XMX+20. * Bisection CH=CMX-CC IF(CC.EQ.CMX)THEN XCM(J)=XMX GOTO 402 ENDIF • Look for  IF(J.EQ.2)THEN DXX=50. DO 551 1=1,300 XP1=XP0+DXX CALL CICZ(ZB,ZS,UB.S,XP1.CQC.O) CQ1=CQC*LAT(YC.SV,UB,XP1)-CC IF(CQ1 . EQ.OTHEN XCM(J)=XP1 GOTO 402 ENDIF  c c c c c  c c c  i n t e r v a l ! where CQ changes sign  CQP=CQ0*CQ1 IF(CQP.LT.O.)THEN • Found the  interval 1. Start Bisection now GOTO 451 ENDIF IF(I.EQ.300)THEN WRITEO,')'OUTER REACH IS > 15000 M' XCM(J)=15000. GOTO 999 ENDIF  * Update everything CQ0=CQ1 XP0=XP1  -  1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 11.18 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1 132 1 133 1134 1135 1136 1137 1 138 1139 1 140 1 141 1142 1143 1 144 1145 1 146 1147 1148 1 149 1150  551 C C C  451  267 -  CONTINUE ENDIF CONTINUE  * Bisection begins 452  NBB=1 CONTINUE IF(NBB.LT.250)THEN PB=XP0+(XP1- XP01/2. RAT=DABS(XPO -XP1)/DABS(XPO) CALL CICZ(ZB.ZS.UB,S,PB,CPB,0) FPB=CPB'LAT(YC,SV,UB,PB)-CC IF(FPB:GT.CH)FPB=CH IFCFPB.EQ.O. .OR.RAT.LT.TOLX)THEN XCM(J)=PB  C C C  * Found XCM(J)  C C C  * Iteration begins (secant method)  C C C  GOTO 402 ENDIF NBB=NBB+1 PR0D=CQ0'FPB IF(PROD.GT.O .)THEN XPO=PB CQO=FPBENDIF IF(PROD.LT.O .)THEN XP1=PB CQ1=FPB ENDIF GOTO 452 ENDIF  401  CONTINUE CALL CICZ(ZB,ZS,UB,S,XP1,CQC,0) C01=CQC*LAT(YC,SV,UB,XP1)-CC IFIN.LE.50)THEN CCC=CQ1-CQO IF(CCC.EQ.O. ) THEN WRITE(3. ') 'CONC(I) : DIVIDE BY WRITE(3.')'XP1 : '.XP1 ENDIF P=XP1 -CQ1*(XP1-XPO)/(CCC) DXP=DABS(P-XP1) IF(DXP.LT.TOLX)THEN XCM(J)=P GOTO 402 ENDIF  * Update everything N=N+1 XP0=XP1 CQ0=CQ1 IF(P.LT.XBMIN)P=XBMIN  - 268 -  1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1 182 1183 1184 1185 1186 1 187 1188 1189 1190 1 191 1192 1193 1 194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208  C  C  C C C  IF(P.LT.XMM)P=XMM XP1=P GOTO 401 ENDIF WRITEO, *) ' SECANT SEARCH IN CONCU) FAILED AFTER 50 STEPS' WRITEO. ' ) 'DIFF =' .DXP 402 CONTINUE 500 CONTINUE ENDIF T0LY=1.E-2 IF(IFLAG.EQ.2.AND.CC.LE.CMX)THEN IF(CC.EQ.CMX)THEN YC=0. GOTO 412 ENDIF * Search for YC N=2 YP0=0.  C  C C C  CALL CICZ(2B.ZS.UB.S,XC,CIC,0) CH=CMX-CC K=0 * Search the interval where CQ changes sign, by stepping DO 501 1=1,200 CQO=CIC"LAT(YPO,SV,UB,XC)-CC IF(CQO.GT.CH)CQO=CH IF(CQO.LE.0.)THEN YC=YPO  C  GOTO 412 ENDIF  c  c c c  forward  YP1=YP0+50. CQ1=CIC'LAT(YP1,SV,UB.XC)-CC IF(CQ1.GT.CH)CQ1=CH CQP=CQ0*CQ1 IF(CQP.LT.O.)THEN ' FOUND THE INTERVAL. STARTING BISECTION NOW.  100  GOTO 421 ENDIF IF(I.EQ.200)THEN WRITEO,')'C0NC(2) STEPPED 200 TIMES, BUT NO BANANAS' YC=2.E+5 GOTO 999 ENDIF IF(CQ1.EQ.CQO) THEN K=K+1 IF(K.GT.IO) THEN WRITEO,')'C0NC(2) FAILED TO FIND YC WRITEO, 100) YPO, CQO, YP1 , CQO FORMAT('YPO/CQO = '.2F10.5,' ; YP1/CQ1 = '.2F10.5)  - 269 -  1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266  C C C  C C C  YC=2.E+5 WRITEO,*)'C0NC(2): GOTO 999 ENDIF ENDIF ' Update everything  501  CQ0=CQ1 YP0=YP1 CONTINUE  * Iteration begins (bisection) 421 422  CONTINUE NBB=1 CONTINUE IF(NBB.LT.50)THEN YPP=YP0 IF(YPO.EQ.O.)YPP=YP1 PB=YP0+(YP1-YP0)/2. RAT=DABS(YP0-YP1)/DABS(YPP) FPB=CIC'LAT(PB,SV,UB,XC)-CC IF(FPB.GT.CH)FPB=CH IF(FPB.EQ.O..OR.RAT.LT.TOLY)THEN YC=PB  C C C  * Found YC after bisection  C C C  • 20 Bisections did not do it  C C C  FORWARDSTEPPING UNSUCCESSFUL'  GOTO 412 ENDIF NBB=NBB+1 PR0D=CQ0*FPB IF(PROD.GT.0.)THEN YPO=PB CQ0=FPB ENDIF IF(PROD.LT.O.)THEN YP1=PB CQ1=FPB ENDIF GOTO 422 ENDIF  411  . Try SECANT now.  CONTINUE  * Iteration begins (secant) IF(N.LE.50)THEN 0C=CQ1-CQ0 CPR=CQTCQO IF(DC.EQ.O..AND.CPR.GT.0.) THEN IF(CQ1.LT.O.)YP1=YP1 -100. IF(CQ1.GT.O.)YP1=YP1+100. N=N+1 IF (N.GT.50)THEN  - 270 -  1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324  C C C  C  C C C C C C C C C C C C C C  c  WRITE(3.*)'C0NC(2) ERROR : THE CURVE IS FLAT HERE' YC=1.E+5 GOTO 999 ENDIF GOTO 411 ENDIF P=YP1-CQ1*(YP1-YP0)/DC IF(P.LT.O.)P=YP1+10. DYP=DABS(P-YP1)/DABS(P) IF(DYP.LT.TOLY)THEN YC=P GOTO 412 ENDIF *  Update everything  N=N+1 YP0=YP1 CQ0=CQ1 YP1=P CQ1=CIC*LAT(YP1,SV,UB,XC) IFCCQ1.GT.CH)CQ1=CH GOTO 411 ENDIF WRITE(3,')'SECANT SEARCH IN CONC2) FAILED AFTER 100 STEPS' WRITE(3,*)'DIFF =',DYP 412 CONTINUE ENDIF IF(CC.GT.CMX)THEN WRITE(3.')'THE SPECIFIED CONC. IS GREATER THAN CMAX' WRITE(3,*)'BY ',CR*100.,' %' ENDIF 999 CONTINUE IFCCC.LT.O.)THEN WRITE(3,*)'CONC',IFLAG,' CC= ',CC ENDIF RETURN END SUBROUTINE  ADDUP(N.M,CMX,XMX.CTOT,CFRAC)  ft This subroutine defines a number of effect levels as %-fractions  • of the maximum. It then integrates the 'volume' under the * effect level surface bounded by the previously specified isoft pleths. The integral over the whole surface (down to zero effect ft level) is approximated by an extrapolation to zero by preserving ft the curvature of the last three nodes. ft The integrated bands are then translated into fractions of the ft t o t a l . A cubic natural spline interpolation is used to map round ft values integrated effect to a effect level. • The results are carried over to the main routine in array CFRAC. IMPLICIT REAL*8 (A-H.L.O-Z) REAL*8 CFRAC(10,4),XCM(2)  ft Set s t r a t i f i c a t i o n limits in effect level dimension ft CFRAC(i,1) contains fractions of CMX in 10% increments  (i.e.  - 271 -  1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382  C C C  C C C C  * the lower limit of a band; e.g. 70 -> (70,80]) * CFRAC(i,2) contains the associated effect level FRAC=10. DO 501 1=9,1,-1 CFRAC(I.1)=FRAC CFRAC(I,2)=CMX * FRAC * 0.01 * Update everything FRAC=(11 -1)* 10 501 CONTINUE C=CMX D=CFRAC(1,2) YC=0.  C  CALL CONC(CMX,XMX,XC,XCM,YC,D,1)  C  C C C  AX=XCM(1) BX=XCM(2) CA=BX CB=AX " Start the loop to integrate the bands DO 502 1=1,8 IP=I+ 1  C  CALL DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT)  C  C  C C C  C C  values  756  IF(XINT.LE.O.)THEN WRITE(3,*)'SOMETHING IN DU8SIM WENT VERY WRONG' WRITE(3,*)'XINT = '.XINT,' 1= ',1 IFtXINT.EQ.O.)THEN XINT=1. GOTO 756 ENDIF GOTO 999 CONTINUE ENDIF CFRAC(I,3)=XINT  * Update everything C=D 0=CFRAC(IP,2) CALL CONC(CMX,XMX,XC,XCM.YC,D,1) CA=AX CB=BX AX=XCM(1) BX=XCM(2) 502 CONTINUE CALL DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT) IF(XINT.LE.0)THEN WRITE(3,*)'ERROR  IN THE FINAL ENTRY TO OUBSIM '  - 272 -  1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440  C C C  C  c c c c c c c c c  c c c  GOTO 999 ENDIF CFRAC(9,3)=XINT *  Evaluate CTOT by summation of the CFRAC(i,3)'s  CTOT=0. DO 503 1=1,9 CT0T=CT0T+CFRAC(I,3) 503 CONTINUE • Extrapolate CTOT to zero effect curvature of CTOT-growth.  *  level, by conserving the  CTOT=CTOT+(CFRAC(9,3)** 2/CFRAC(8,3)) WRITEO, *)'TOTAL VOLUME approx. : '.CTOT Evaluate the cumulative fraction of the total volume contributed by each concentration band, e.g. more than 60 cone, contributes XX % of the total volume under the cone. • surface. Put results into CFRACH.4) * * *  1  V0L=0. DO 504 J=1,9 V0L=V0L + CFRAC( J , 3) CFRAC(J,4)=V0L/CT0T 504 CONTINUE • Write CFRAC to comments f i l e  (#3)  CC WRITEO, 101) CC101 FORMAT(/,'PRINTOUT OF CFRAC',/) CC WRITE(3,*)'cone.% cone.val. CC DO 505 1=1,9 CC WRITEO, 100) (CFRAC (I, J) ,J=1 ,4) CC100 FORMAT(4F15.5) CC505 CONTINUE C C CTOT and CFRAC are output of subroutine  c  C  C C C  c c c c c  int.frac.  cum  GOTO 998 999 CONTINUE CT0T=-100. 998 CONTINUE RETURN END SUBROUTINE SPLYNE(SNO,N,X,Y) • • • •  This subroutine uses a. natural cubic spline algorithm to compute the interpolated Y for any given X within the range defined by the nodes SNO(i.j). SPLYNE s a t i s f i e s the boundary conditions Y''(SN0(1,0))= Y''(SN0(1,N-1)) = 0. * N is the number of nodes. • (Ref: Burden & Faires.1985; Algorithm 3.4) IMPLICIT REAL'8 (A-H.L.O-Z)  - 273 -  1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498  C  C  C C C  C  C.  C  C C C  REAL * 8 SNO(0:50,2),H(0:50),A(0:50),L(0:50),Z(0:50) REAL'8 C(0:50),B(0:50),D(0:50),M(0:50) NN=N-1 N1=NN-1 DO 501 1=0,N1 IP=I+1 H(I)=SN0(IP,1)-SN0(I,1) 501 CONTINUE DO 502 1=1.NI IP=I+1 IM=I-1 A(I)=(SNO(IP,2)*H(IM)-SN0(1,2)*(SNO(IP,1)-SNO(IM,1)) # +SN0(IM,2)*H(I))*3./(H(IM)*H(I)) 502 CONTINUE * Solve tridiagonal linear system by Crout reduction method. L(0)=1. M(0)=0. Z(0)=0. DO 503 1=1,N1 IP=I+1 IM=I-1 L(I)=2.*(SNOIIP,1)-SN0(IM,1))-H(IM)'M(IM) M(I)=H(I)/L(I) Z(I)=(A(I)-H(IM)*Z(IM))/L(I) 503 CONTINUE L(NN)=1. Z(NN)=0. C(NN)=0. 00 504 J=N1 .0,-1 JP=J+1 C(J)=Z(J)-M<J)"C(JP) B(J)=(SNO(JP,2)-SN0(J.2))/H(J)-H(J)*(C(JP)+2*C(J))/3. D(J)=(C(JP)-C(J))/(3*H(J)> 504 CONTINUE * Evaluate interpolated value Y for given X DO 505 1=0,N1 IP=I+1 IF(X .GE.SNO( 1,1) . AND.X.LT.SNOdP, 1) ) THEN DX=X-SNO(I.1) Y=SN0(I,2)+8(I)*0X+C(I)*DX**2+D(I) DX**3 GOTO 400 ENDIF 505 CONTINUE ,  C C C C  * If node-range not found, give error message WRITEO,')'ERROR IN SPLYNE : GIVEN X LIES OUTSIDE NODES - RANGE' 400 CONTINUE  - 274 -  1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556  RETURN END C C  c c c c c c c c c c c  c c c  c c c c c c c c c c c c  SUBROUTINE SIMPS(DOTS,A.B.NPT.F) • • • • • •  This subroutine uses Simpson's Composite Algorithm to approximate the integral under a curve described by the nodes in DOTS. A and B are the lower and upper integration limits (values on ordinate, DOTS has only abscisse-values). NPT is the number of points (must be an odd number, to produce an even number of subintervals) . » F is the approximated definite integral. • (Ref : Burden & Faires, 1985; Algorithm 4.1) IMPLICIT REAL'8 (A-H.L.O-Z) REAL'S D0TS(110) M2=NPT-1 M=M2/2 H=(B-A)/DBLE(FL0AT(M2)) XI0=00TS(1)+D0TS(NPT) XI1=0. XI2=0. M21=M2-1 DO 500 1=1,M21 IP=I+1 U=DABS(DBLE(FLOAT(NINT(DBLE(FLOAT(I))/2.)))-DBLE(FLOAT(I))/2. IF(U.EQ.O.)XI2=XI2+D0TS(IP) IF(U.NE.O.)XI1=XI1+D0TS(IP) 500 CONTINUE F=H*(XI0+2. 'XI2+4. ' X I D / 3 . RETURN END SUBROUTINE DUBSIM(CMX,XMX,AX,BX,CA,CB,C,D,N,M,XINT) • This subroutine uses a double Simpson's Composite Algorithm to approximate the integral under a surface. • For reference see Burden & Faires, 1985. IMPLICIT REAL'8 (A-H.L.O-Z) REAL"8 XXM(2) • Step 1 : H=(BX-AX)/(2'N) • Step 2 : XJ1=0. XJ2=0. XJ3=0.  - 275 -  1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614  C C C C C  • Step 3 : 00 501 I=0,2*N • (composite Simpson's method for fixed x) X=AX+I*H CC=C  C  CALL C0NC(CMX,XMX,X,XXM,DX,D,2) IF(DX.EQ.2.E+5)G0T0 999 IF(X.LT.CA.OR.X.GT.CB)THEN CX=0.  C  CALL CONC(CMX,XMX,X,XXM,CX.CC.O)  C  GOTO 402 ENDIF  C 402  c c  c  c c c c  CALL C0NC(CMX,XMX,X,XXM,CX,CC,2) IF(CX.EQ.2.E+5)GOTO 999 CONTINUE HX=(DX-CX)/(2*M) XK1=D+CC XK2=0. XK3=0. DO 502 J=1,2*M-1 Y=CX+J*HX  CALL CONCfCMX.XMX.X.XXM.Y.Z.O) UJ=DABS(DBLE(FLOAT(NINT(DBLE(FLOAT (J))/2.)))-DBLE(FLOAT(J))/2.) IF(UJ.EQ.O.)XK2=XK2+Z IF(UJ.NE.O.)XK3=XK3+Z XL=(XK1+2*XK2+4*XK3)*HX/3. 502 CONTINUE NN=N'2 IF (I. EQ . 0 . OR . I . EQ . NN) THEN XJ1=XJ1+XL GOTO 401 ENDIF UI=DABS(DBLE(FLOAT(NINT(DBLE(FLOATtI))/2.)))-DBLE(FLOAT(I))/2.) IF(UI.EQ.O.)XJ2=XJ2+XL IFtUI.NE.O.)XJ3=XJ3+XL 401 CONTINUE 501 CONTINUE XINT=(XJ1+2*XJ2+4*XJ3)'H/3. GOTO 998 999 CONTINUE XINT=-100. 998 CONTINUE RETURN END  - 276 -  1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672  C C C C C C C C  C  c c c c c  SUBROUTINE MONET(PLETH,AUX,PROFX,PROFY) a This subroutine uses the OISSPLA language to present the • results of the SANCTUM-mode1 graphically in a 3-D p l o t . * Once the object f i l e is created, it has to be loaded and • run together with the 'DISSPLA l i b r a r y . • Dimension data arrays (level 0) REAL*8 PLETH(9,2.51),AUX(11),PR0FX(101.2) REAL'S PROFY(101,2) REAL*4 PRYX(101),PRYY(101),PRXX(101),PRXY(101) REAL*4 PLEX(201),PLEY(201),XSYM(5),YSYM(5) REAL*4 AX(2),AY(2),CX(2),DX(2),DY(2) COMMON /PAKRAY/ IPKRAYUOO) • I n i t i a l i z e a device (rise up to level  1)  CALL DSPDEV( PLOT') 1  *  Set page size  (XPAGE=11. , YPAGE=8.5) in inches (level  XPAGE=11. YPAGE=8.5  c  CALL CALL CALL CALL  PAGE(XPAGE,YPAGE) HWROT('AUTO') HWSHD SHDCHRf30.,1,0.01,1)  c c c  • I n i t i a l i z e coordinates for physical  c c c  • I n i t i a l i z e dimensions for subplot area  c c  **  origin  XPHY = 0. YPHY = 0.  XAREA =11. YAREA = 8. CALL PHYSOR(XPHY.YPHY) CALL AREA2D(XPAGE,YPAGE) ** Draw the caption (level CALL HEIGHT!.15) CALL SWISSL JD=INT(AUX(1)) JH=INT(AUX(2)) JM=INT(AUX(3) ) US=AUX(4) UDIR=AUX(5) 2I=AUX(6) XI=AUX(7) UST=AUX(8) ZS=AUX(9) FL=AUX(10) SV=AUX(11) XMES=4.  1)  1)  - 277 -  1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730  YMES=1.1 CALL MESSAG('SAM Calculations for Jul.Day. $',100, 1 XMES.YMES) CALL INTNOCJD,'ABUT'.'ABUT') CALL MESSAG(' at $', 100,'ABUT','ABUT') CALL INTN0(JH,'ABUT','ABUT') CALL MESSAG( :$',100,'ABUT','ABUT') IF(JM.EQ.0)CALL MESSAG('00 L . A . T . at $',100,'ABUT', 'ABUT' ) IF(JM.NE.0)THEN CALL INTN0(JM,'ABUT','ABUT') CALL MESSAG(' L . A . T . at $'.100,'ABUT','ABUT') ENDIF IF(FL.EQ.O.) CALL MESSAG('SUNSETS',100,'ABUT','ABUT') IF(FL.EQ.1.) CALL MESSAG('MOBILES',100,'ABUT','ABUT') IF(FL.EQ.2.) CALL MESSAGCCABAUWS',100,'ABUT','ABUT') 1  C  C  CALL HEIGHT!. .12) YMES=YMES-.25 CALL MESSAG('Wind at sensor level ( s p . / d i r . ) : $',100,XMES,YMES) CALL REALN0(US.1,'ABUT','ABUT') CALL MESSAGC / $',100,'ABUT','ABUT') CALL REALN0(UDIR,1,'ABUT','ABUT') IF(ZI.GT.O.)THEN CALL MESSAGC ; Zi : $',100,'ABUT','ABUT') CALL REALN0(ZI, 1 ,'ABUT' ,'ABUT') CALL MESSAG!' ; SV : $'.100,'ABUT','ABUT') CALL REALNO(SV,3,'ABUT'.'ABUT') ELSE CALL MESSAGC ; Zi : n/a ; SV : $',100,'ABUT','ABUT' ) CALL REALN0(SV,3,'ABUT','ABUT') ENDIF YMES=YMES- . 18 CALL MESSAG('Stability (Zs/L) : $',100,XMES,YMES) CALL REALN0(XI.3, 'ABUT', 'ABUT' ) CALL MESSAGC ; U* : $',100,'ABUT','ABUT' ) CALL REALNOCUST,3.'ABUT','ABUT') CALL MESSAG(' ; Sensor height : $',100,'ABUT', 'ABUT') CALL REALNO(ZS,1,'ABUT','ABUT') YMES=YMES-.25 CALL MESSAGCAll data are in SI un i ts$ ' , 100 , XMES . YMES) CALL ENDGR(O)  C C C  ' Increment physical origin and subplot area to f i t  C C C  " Define subplot area (rise up to  C C C  YPHY = YPHY+1 .5 YAREA = YAREA- 1.5 CALL PHYSOR(XPHY,YPHY)  CALL AREA2D(XAREA,YAREA) *'** Draw the t i t l e CALL SWISSB CALL ALNMES(.5,.5) CALL HEIGHT(.20)  level  2)  caption (1.  1)  - 278 -  1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788  C C C  C C C C C  c c  CALL MESSAG 1 ('SOURCE AREA MODEL - RESULTSS',100,XAREA/2..YAREA+.2) CALL RESET('ALNMES') CALL RESET('HEIGHT') * Define the 3D plot volume XV0L=12. YVOL=7.0 ZV0L=7.0 CALL  V0LM3D(XV0L,YV0L,ZV0L)  * Define viewpoint of 3D box CALL VUANGL(-60.,30.,30.) * Define user axis system (window) (rise to level 3) CALL GRAF3D(0.,XVOL,XVOL,0.,YVOL,YVOL,0.,ZVOL,ZVOL)  c c c  **** Define the Y-Z GRIFITI plane (jump back to level 1)  c c c  * Set subplot dimensions (viewport)  c c c  '*** Prepare to draw a message  c c c  • • * • Plot a message  c c c c c c c  CALL GRFITKO. ,0. ,0. ,0. ,YV0L,0. ,0. .YVOL,ZVOL) (level  1, r i s e to 2)  CALL AREA2D(YVOL,ZVOL) CALL GRAF(0.,YVOL.YVOL.0..ZVOL,ZVOL) CALL FRAME  CALL CALL CALL CALL  GRACE!10.) HEIGHT(.22) COMPLX ALNSTY(.5.-5)  MAXLIN = LINEST(IPKRAY.400,80) CALL LINES('Cross-Wind P r o f i l e of Effect Leve 1$',IPKRAY, 1) CALL LINES('at the max. e f f . downwind distances',IPKRAY,2) NLINES = 2 XMSG =3.25 YMSG =6.25 CALL STORY(IPKRAY,NLINES,XMSG,YMSG) CALL RESET('HEIGHT') CALL RESET('ALNSTY') • End the Y-Z plane (level 2, r i s e to 3) ( i f you got this f a r , you might as well read to the end) CALL END3GR(0.) * Prepare to draw the curve : set up the curve plane (back to CALL  GRFITI(0.,0.,0.,0.,YVOL,0.,0.,YVOL,ZVOL)  - 279 -  1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831. 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846  XPHY2=XPHY+.25 YPHY2=YPHY+.70 CALL PHYS0R(XPHY2.YPHY2) XLEN=YV0L-2. YLEN=ZVOL-2.5 CALL AREA2D(XLEN,YLEN) CALL CROSS  C C C  * Set axis  C C C  * S p l i t up PR0FY(101,2) into two ID-arrays  C C C  C  CALL CALL CALL CALL  labels.  DUPLX YAXANGOO. ) YREVTK HEIGHTC.22)  DO 501 1=1,101 PRYX(I)=PR0FY(I,1) PRYY(I)=PROFY(I,2) 501 CONTINUE * Set the min. max and step for PRYX and PRYY CALL RNDLINfPRYX,101.XLEN.XMIN,XSTP,XMAX) CALL RNDLIN(PRYY,101,YLEN,YMIN,YSTP,YMAX) IF(XMAX.LE.200.)THEN XSTP=50. XMIN=-200. XMAX=200. ENDIF YPMIN=XMIN YPMAX=XMAX YPLEN=XLEN YSTP=.2 CALL XINTAX CALL XNAME('Cross-wind Distance (m)$',100) CALL YNAME(' $',100)  C C C  * Define user axis system (window) (level 2, r i s e to 3  C C C  * End the current plane  C C C  ***** Begin X/Z-plane  C  CALL GRAF(XMIN,'SCALE',XMAX,YMIN,YSTP,YMAX) CALL RASPLN(2.) CALL CURVE(PRYX,PRYY,101,0.)  CALL END3GR(0)  CALL PHYSOR(XPHY.YPHY) CALL GRFITI(0..YVOL.O.,XVOL.YVOL,0.,XVOL,YVOL,ZVOL) CALL AREA2D(XVOL,ZVOL) CALL FRAME  - 280  1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904  C C  C C C  * Put  in a message  CALL CALL CALL CALL  GRACE(10.) HEIGHT!.22) COMPLX ALNSTY!.5,.5)  * Plot a message MAXLIN=LINEST(IPKRAY,400,80) CALL LINES!'Along-Wind Centerline Profile of Effect 1 IPKRAY,1) NLINES=1 XMSG=5.5 YMSG=6.25 CALL LSTORY!IPKRAY,NLINES,XMSG,YMSG)  C C C  ' End this plane  C C C  ' Prepare to draw the curve  C C C  * S p l i t up PROFX!101,2) into two  C C C C C  -  CALL EN03GR(0)  CALL GRFITI(0.,YVOL,0.,XVOL,YVOL,0.,XVOL,YVOL,ZVOL) CALL OREL!.5,.25) XLEN=XV0L-1.5 YLEN=ZV0L-2. CALL AREA2D(XLEN,YLEN) CALL CROSS CALL DUPLX CALL YAXANGOO. ) CALL HEIGHT!.22) linear arrays  DO 502 1=1,101 PRXX(I)=PR0FX(I,1) PRXY(I)=PR0FX(I,2) 502 CONTINUE CALL RNDLIN!PRXX,101,XLEN,XMIN,XSTP,XMAX) CALL RNDLIN!PRXY,101,YLEN,YMIN,YSTP,YMAX) IF!XMAX.LE.2000.)THEN XSTP=250. XMIN=0. XMAX=2000. ENDIF XPMIN=XMIN XPMAX=XMAX XPLEN=XLEN YSTP=.2 CALL XINTAX * Label the axes CALL XNAME!'Upwind Distance (m)$',100) CALL YNAME('Re 1. Effect Level$',100) * Define user axis system (level 2,  r i s e to  3)  Levels',  - 281 -  1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962  C  CALL GRAF(XMIN,'SCALE',XMAX,YMIN,YSTP,YMAX) CALL RASPLN(2.) CALL CURVE(PRXX,PRXY,101,0.)  C C C  * End this plane  C C C  •*** Begin floor  G C C  * Put in messages  C C C  * Next Message  C  C C  c  CALL RESET('ALL') CALL END3GR(0)  CALL CALL CALL CALL  PHYSOR(XPHY.YPHY) GRFITI(0.,0.,0.,XV0L,0.,0.,XVOL,YVOL,0.) AREA2D(XVOL.YVOL) FRAME  CALL GRACE(10.) CALL HEIGHT!.25) CALL COMPLX CALL ALNSTY!.5..5) MAXLIN=LINEST(IPKRAY.400.80) CALL LINES! 1 'Source Region Outlines with Various Effect- Integra 1s$', 1 IPKRAY.1) NLINES=1 XMSG=XV0L/2. YMSG=YVOL-.35 CALL LSTORY(IPKRAY,NLINES.XMSG.YMSG)  CALL HEIGHT I.22) CALL SIMPLX CALL ALNSTY!.5,.5) MAXLIN=LINEST(IPKRAY,400.80) CALL LINES!'Isopleths are numbered 1 - 9 from centers' , 1 IPKRAY. 1) CALL LINES! 1 '(see inset and separate table for dimensions and area)$' 1 IPKRAY,2) NLINES=2 XMSG=XVOL/2. YMSG=.5 CALL LSTORY(IPKRAY,NLINES,XMSG,YMSG) * End this plane CALL END3GR(0)  c c • Prepare to draw the curves c  CALL GRFITI(0..0..0.,XV0L,0.,0.,XVOL.YVOL,0.) CALL OREL(.70,.15) XLEN=XPLEN  - 282 -  1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020  YLEN=YPLEN CALL AREA20(XLEN,YLEN) CALL CROSS CALL DUPLX CALL YAXANGOO. ) CALL XREVTK CALL HEIGHT(.22) CALL RASPLN(2.)  C C ' Set up data arrays C DO 503 1=9,1,-1 DO 504 J=1,51 PLEX(J)=PLETH(I,1,J) PLEY(J)=PLETH(I,2,J) 504 CONTINUE IF(I.EQ.9)THEN C C * Label the axes C CALL XINTAX CALL YINTAX CALL XNAME(' $',100) CALL YNAMEC $',100) CALL GRAFUPMIN, 'SCALE' , XPMAX , YPMIN , 'SCALE' .YPMAX) ENDIF CALL CURVE(PLEX,PLEY,201,0. ) 503 CONTINUE C CALL END3GR(0) C CALL ENDGR(O) C C ***• Begin inset C CALL PHYSOR(XPHY,YPHY) CALL OREL(-.3,-1.5) C C * set inset dimensions and frame it C XINS=3. YINS=2. CALL AREA2D(XINS,YINS) CALL FRAME C C * Put in a t i t l e for the inset C CALL SWISSM CALL HEIGHT(.15) CALL ALNMESf.5,.5) CALL MESSAG 1 ('Dimensions of IsoplethsS',100,XINS/2,YINS-.25) CALL ENDGR(0.) C C * Prepare to draw the symbol C CALL OREL(0.5,0.3) 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 DATA DATA DATA DATA DATA DATA  XSYM/125.,300..550.,300.,125./ YSYM/O.,40.,0.,-40.,0./ AX/O.,125./ AY/0.,0./ CX/300.,550./ DX/300.,300./ DY/O.,40./  CALL RNDLIN(XSYM 5,XLEN,XMIN,XSTP,XMAX) CALL RNDLIN(YSYM 5,YLEN,YMIN,YSTP,YMAX) CALL XNONUM CALL YNONUM CALL XNAME( $',100) CALL YNAME(' $',100) CALL RASPLN( 1 .) CALL GRAF(0.,600.,600.,YMIN,YMAX,YMAX ) CALL CURVE(XSYM,YSYM,5,0.) 1  Put  in the dimension indicators etc.  CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL  THKCRV(0.03) CURVE(AX,AY,2,0) CURVE(CX,AY,2,0) CURVE(DX.DY,2,0) HEIGHT(0.1) COMPLX RLMESS('a$' , 100,75. 10. ) RLMESSf'b$',100,212. , 10. ) RLMESSf'c$ 100,425.,10.) RLMESS('d$',100,320.,30.) 1  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  (1)  D : Data and Results Summary  : Culloden S i t e The tower was situated  37  i n front of the s i x t h house south of  Avenue on the eastern side of Culloden Street  t h  9.1  and 9.2).  started August 2 small  This  on J u l y  31,  (JD 214)  absolute  was 1986  at  Q  the  first  (JD 212)  19:35 LAT.  values  at  site at  occupied.  Figures  Operations  5:35 LAT and ended on  Due to  night  (see  it  the  consistently  was  decided  that  very the  H  measurements  could be  interrupted during  subsequently  occupied.  Weather conditions  the  night  at  sites  were generally sunny  and warm, with l i g h t but sometimes gusty winds i n late afternoon (see data summary i n Appendix D). (2)  : Argyle S i t e Figure 9.3 shows the mobile tower on the east side of Argyle rd  Street,  just  adjacent 9.1).  to  south the  of  53  south-west  Avenue.  This  corner of  site  was  Gordon Park  (see  directly Figure  David Thompson Secondary School with a b u i l d i n g height of  about 20 m was approximately 50 m to the east of the tower. Data c o l l e c t i o n started at 5:05 LAT on August 9 (JD 221) and ended  at  19:50  measurements negative,  were  LAT on  August  interrupted as  commonly between 19:30  11  (JD  soon as  223). the  At  night,  Q^-values became  LAT and 20:00 LAT, and were  r e s t a r t e d i n the morning between 4:30 LAT and 5:30 LAT (see summary i n Appendix D).  the  data  - 285 Weather conditions were more v a r i a b l e than  at the  Culloden  s i t e with some wind from southerly d i r e c t i o n s and changing cloud cover. (3) : Waverley S i t e Measurements s t a r t e d at 12:10 ended at visual  19:35  LAT  LAT  on August 20  impression  of  the  on August 18  (JD 232).  site,  which  (JD 230)  Figure 9.4 was  9.1).  The  with  clear  second  first  and  the t h i r d day  skies or sparse  day  (JD  231)  was  generally  d i r e c t i o n s ) with variable, fast the  afternoon  instrument started  of  JD  alignment  to  sag  231  and  light  more  windy  15:00  and  LAT)  Figure  very  hot  winds.  The  (W  one  Shut-down  there  element  during  of  the  a  the  to  NW  moving medium-high clouds.  (ca.  problems  slightly.  clouds  in  (see  were very s i m i l a r ,  high  gives  located  north-west corner of the U-shaped Waverley Street  and  In  were  some  the  tower  nights  were  s i m i l a r to the Argyle s i t e (see data summary i n Appendix D). (4) : Memorial East S i t e This s i t e  was  located just  east  of Memorial Park, on  Ross  rd Street just north of 43 the west and park  is  park-trees 100-150  north-west of that  fairly  playgrounds.  m  Avenue (see Figures 9.1  In  open, the  with  and 9.5).  To  location the character of  a  baseball-diamond  SW-direction,  however,  a  and  group  of  the  other high  (up to about 40 m high) i s located at a distance of from  the  mobile  tower  site  (see  Figure  9.6).  Measurements s t a r t e d on August 24 (JD 236) at 10:05  LAT i n l i g h t  winds from southerly d i r e c t i o n s . The  irrigated  park was  not  on  - 286 this f i r s t  day.  The second day  (JD 237)  was  very hot and sunny ,the park  was  i r r i g a t e d and a s l i g h t sea breeze was s e t t i n g i n towards midday. Unfortunately there were data logging problems f o r the data at the main tower, so that source performed  f o r most  occurred on JD 238, 13:45  of  the  area c a l c u l a t i o n s could not  day  (see  below).  be  S i m i l a r problems  on which data c o l l e c t i o n was  terminated  at  LAT.  (5) : Memorial West S i t e Figure 9.7  shows t h i s s i t e looking north-east. On the western  side of Prince Albert Street  (see Figure 9.1)  the  characterized by high density r e s i d e n t i a l housing 9.8.  Measurements s t a r t e d on August 31  (JD 243)  terrain is  as  i n Figure  at 6:00  LAT  in  cloudy and cool conditions. Even though the weather cleared i n the  afternoon,  the  following  day  (JD  244)  was  again  characterized by a complete low s t r a t u s cloud cover. Due  to the  onset of l i g h t rain, the measurements were interrupted e a r l y i n the morning. Data c o l l e c t i o n resumed on JD 246 in  calm  and  generally  sunny  (September 3) at 6:30  conditions,  however,  the  logging problems at the Sunset s i t e continued u n t i l 16:30 that day,  so that only the  LAT data  LAT of  last few data-points could be used  f o r SAM-runs (see data summary below). The measurement period ended on September 4 (JD 247) at LAT again i n and sunny conditions and only l i g h t winds.  14:00  - 287 -  Data and Results Summary  The nomenclature of the various data columns i s as follows  JDD  : J u l i a n day o f 1986  HH: MM  : Local  apparent  time  (LAT) of the measurement.  r e f e r s to hourly averages ending at t h i s time. TT  : Mean air-temperature  UB  : Mean wind-speed at the sensor l e v e l i n [m/s]  W-DIR  : Mean wind-direction i n degrees (90 = E)  SV  : Equivalent to <r i n [m/s]  U*  : Equivalent to u  L  : Monin-Obukhov length i n [m]  QH-SUN  : Sensible heat f l u x at the Sunset s i t e i n [W/m ]  QH-MOB  : Sensible heat f l u x at the mobile s i t e s i n [W/m ]  #  i n [K]  i n [m/s]  2  2  (see Section 9.2) QH-DIFF  : Equivalent to Qj^jpy. [nondimensional ]  AR  : Area of 0.9-isopleth i n [1000 m ]  A - XM  : Dimensions a - xm of 0.9-isopleth i n [m].  2  JOO HH: MM 212 2)2 212 2(2 212 212 212 212 212 212 2 12 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 213 214 214 214 214 214  6: 30 7: 0 7: 30 B: 0 8: 30 9: 0 9: 30 10: 0 10: 30 11: 0 11: 30 12: 0 12: 30 13: 0 13: 30 14 : 0 14 : 30 15: 0 15:30 IG: 0 16: 30 17: 0 17: 30 18: 0 18: 30 19: 0 19: 30 20: 0 7: 0 7: 30 8: 0 8: 30 9: 0 9: 30 10: 0 10: 30 11: 0 11: 30 12: 0 12: 30 13: 0 13: 30 14 : 0 14: 30 15: 0 15: 30 16: 0 16: 30 17: 0 17: 30 18: 0 18: 30 21: 30 6: 30 7: 0 7: 30 8: 0 8: 30  TT 288 .0 288 .2 288 .6 289. . 1 289 .6 290. . 1 290 6 291 .2 291 .7 292. . 2 292. 6 293. . 1 294. . 1 295 0 295. 0 295. 0 295. .3 295. 6 295. 6 295. 6 295. 3 294. 9 295. 0 295. . 1 294. .8 294. .4 293. .8 293. 1 290. 5 291, . 1 291. 6 292. 2 292. 8 293. 4 293. .9 294. 4 295. 0 295. 3 295. 6 296. 3 296. 9 297. 1 297 ..3 297. 6 298. 0 298. 3 298. .5 298 . 9 299. 2 299. 4 299. 5 298. 9 293. 3 291. 8 292. 3 292. 4 292 . 6 293. 5  UB  W-D1R  1 .46 1..49 1 .67 1 .85 1 .74 1 .62 1 .72 1 .82 2 oa 2 .35 2 .28 2 .21 2 . 16 2.11 2. 67 3. .22 3 .36 3 .49 3. .47 3 .46 3 .41 3 .37 3. .20 3 .04 2 .76 2 .48 2. . 14 1 81 1. .22 1..50 1 .78 2 .20 2. 61 2 .54 2. .46 2 85 3. 24 3 .37 3 .50 3. 31 3. 12 3. .37 3 .63 3. .51 3. .39 3. 22 3. 05 3. .09 3 . 14 3. .02 2. 90 2. 47 2. 21 1. 33 1. 01 1 .13 1. 25 1. 25  122. 141. 151. 161. 171. 182. 192. 202. 212. 221 . 238. 256. 246. 236. 247. 259. 255. 252. 252. 252. 253. 254. 264. 275. 277. 279. 283. 287. 206. 208. 211. 229. 246. 244. 241 . 260. 279. 263. 247. 245. 244 . 246. 249. 244. 239. 245. 2S2. 264. 276. 278. 279. 270. 142. 321 . 288. 293. 297 . 310.  SV 0 .63 0 .42 0 58 0 .77 0 .76 0 .75 0 78 0 .81 1. .00 1. .20 1 .25 1. .30 1 .40 1 .51 1. .49 1 .31 1. .20 1. 08 1. .03 0. 97 1 .03 1. . 10 1. 00 0. 91 o, .90 0. 87 1. .04 1. .11 0. 35 0, .50 0 67 0. 83 1. .00 1 06 1. . 11 1. 53 2. .02 1. .57 1 09 1. 10 1. . 10 1. . 12 1. . 12 1. 08 1. ,05 1. 01 0. 97 1 .00 1. .02 1. . 19 1, .35 1, 10 0. 59 0. 69 0. 84 0. 78 0. 68 0. 87  U*  L  0 23 -25, .73 0. 24 -24. .54 0. .27 -23, .16 0 .30 -24, .92 0 28 -23, .83 0. .26 -25. .26 0. .28 -23, 86 0 .31 -17, . 19 0. 34 -22, .90 0. 36 -34. . 19 0. 36 -25 43 0. .37 -19, .93 0 36 -18 .89 0, .36 -18 70 0. 43 -25. ,70 0. .46 -36, .74 0. .48 -49, .67 0. 50 -63. .76 0. 49 -54. ,91 0. 48 -65. .49 0. 48 -61, ,00 0. 47 -62. . 15 0. 45 -62. .22 0. 42 -62. 91 0. .38 -73. .84 0. 34 -73, 64 0. 30 -56. ,48 0. 25 -74. 31 0. 18 -35. 68 0. .24 -24, .45 0. .29 -24. .24 0. 33 -39, ,42 0. 38 -48. .93 0. 38 -35 .39 0. 39 -28, ,89 0. 43 -37. .69 0. 47 -44. 99 0 49 -46. .76 0. SO -48. .73 0. 48 -46, 31 0. 46 -40, .35 0. 50 -42. . 11 0. .52 -51, . 19 0. 51 -46, . 16 0. SO -42, .71 0. 47 -46. 28 0. .44 -46, .44 0. 44 -53. . 13 0. 42 -82 .98 0. 41 -82, .95 0. 39 - B l .65 0. 31 -160. .82 0. 28 -168, .66 0. 19 -45, .20 0. 16 -31 ,.67 0 18 -30 .64 0. .22 -16 .67 0. 22 -16 . 11  OH-SUN  QH-MOB OH-DIFF  43. .95 50. 25 77 ..40 94. .20 84. . 10 62. .20 81. 35 160. .45 153, 75 121 . 35 173 to 229 .55 233. .80 222 00 275. 85 283 40 208. . 10 208. .50 198. 90 153. ,9S 162. .55 152. .65 130. .55 110. 25 66. ,00 48 ,00 44. .SS 18. .35 IS 85 52 .05 88 .30 80 .75 98. 80 146 .20 180. .05 188. .95 214 85 228, .95 241 .95 220. .65 226. . IS 268 .00 253 .05 265, .65 268. .05 204. .70 173 .25 148 .80 84. . to 74, .90 67 ..80 17 . 15 11. .35 14, .60 10 .80 IS 90 54 ,80 SB .00  13. .75 31. . 13 55. .00 80 .25 73 .63 74. SO 84. 63 122. .75 IBS .25 182. SO 153. .25 210 .75 2SB. .00 256 .88 248. .50 228. . 13 225 .25 204. .38 174. .50 147 .00 156. ,00 144. . 13 104. .63 92 .50 61. OO 27, . 13 10 .00 -2 .50 22, .25 46 63 70 .88 95, .25 117. ,25 139 . 13 148 .25 144. .63 178 .38 196 .88 223 .75 223. .88 194. .63 207 .25 205 .50 169. .SO 167, .75 188 .75 146 . 13 102 .25 109 .75 91 . 13 43 .25 26 .25 -4 .25 6 .00 6 .88 12 .25 26 .50 40 .75  AR  2. 38 225 49 1. 51 136. 67 t. 76 158. .43 1. 10 203.81 0. 82 205. 61 233. .05 0. 97 0. 26 211. ,61 2. 97 137. .09 2. 72 213. .76 4. 82 369. .45 1. 56 281. .59 1. 48 218. .24 1. 91 226. 93 2. 75 242. .42 2. 15 283. .22 4. 35 325. .88 1. 35 399. . 14 0. 32 369. OS 1. ,92 366. .09 0. SS 407. ,07 0. 52 408 .07 452. 90 0. 67 2. 04 430. .00 1. 40 421. SS 0. 39 522 .68 1. 64 567. 36 2. 72 618. .69 1 .64 984. 88 0. SO 225. .23 0. 43 161. .37 1 37 178 .09 1. , 14 326 .33 1. 45 416. 16 0 56 319. .88 2. .50 263. .53 3. 49 437 .73 2. ,87 627 .89 2. .53 485, .97 1. .43 343. .26 0. 25 343. .79 2. .48 317. .42 4. 78 309. .53 3. .74 354 03 7, 57 317, . 17 7. 90 293 .71 1. 26 322 .76 2. . 13 331 .34 3. ,67 384 .65 2. .02 585 .68 1. ,28 701 .44 1. .93 826 .75 0. .72 1314. .59 1. 23 798 .05 0 68 531 .87 0. .31 537. .09 0 29 425 .58 2, .23 156 .21 1. .36 191 .34  A 13. .72 13. .42 13 .04 13. .52 13. .23 13. 60 13. ,23 11. . 11 12. ,96 IS ,51 13. 65 12 .06 II 73 11 ..65 13. 72 15, .94 17. .68 18. , 12 18. ,23 19. , 14 18. 79 IB. 88 18 .88 18. 94 19 .74 19. .70 18 38 19. .76 IS. .77 13 40 13 .34 16 .33 17. ,59 15. .72 14. .47 16. 08 17 . 12 17 34 17, .57 17 .28 16. .47 16. 73 17. 84 17 .27 16 .81 17 .28 17, .31 IB OS 20 .26 20 25 20 . 18 22 .65 22. .81 17 . IS 15. 04 14 .85 10. .92 to .70  B 348 .42 337 . IB 307 . 16 334 .40 324 .65 325 . 10 331 .43 257 .42 310 .44 402. .57 338 .95 297 .55 275 .67 268 .75 334 .28 405 .35 492 .88 513 . 15 509 . 14 5S6. .06 568 .87 553 . 16 531 .86 577 .03 622 .42 612 .29 506 .11 577 .42 429 .46 329 .45 315 .95 451 .52 479 .95 403. . 13 360 .80 404 .38 458 .06 468 . 19 507 .88 465 .37 431. .65 451 .05 491 .70 501 .72 472 . 19 483 .98 456, .47 500 .25 619 .45 619 .83 640 .02 799 .94 794 . 18 495. 60 398 . 18 388 .77 246 .81 249 .24  C 296. 80 287 .23 295, . 12 296 .55 287. .90 312. .35 282 ., 33 228 .28 286 56 371 .61 300 .57 243 .45 244 ..46 248 .08 308. .56 405 .35 473 .55 493 03 509. . 14 556. .06 504 .47 531. .46 553. 56 511. .71 551 .96 565 .20 526. .77 600. .99 365 .84 292 . 16 303. .56 400. .40 479. .95 387. .32 333 .04 420. .89 458 .06 468. . 19 450 .38 465. .37 431 .65 433 .36 491 .70 427 .39 418 .74 446 .75 475. . 10 500. .25 619 45 619 .83 590. .79 799 .94 826. .59 422. . 18 339. . 19 331 .. 17 227 ..82 212, .32  0 224 .22 140. 23 167. .97 207 .03 214. 84 233. .59 221 . 09 180. .47 228 91 305. .47 282 .03 25B. 59 278. 91 299. 22 282 ..03 257 .03 264 84 235 . 16 230. .47 23S. 16 244. .53 267. ,97 253. 91 249. .22 286. .72 310. . 16 382. 81 S3S. 94 182 03 166. .41 183. ,59 246. 09 277. .34 258 .59 242. .97 339. .06 439. 06 332. .81 230. .47 236. 72 235. 16 224. .22 230. .47 219. .53 211. .72 222. 66 227 .34 246 .09 303. .91 364 . 06 432 81 529. .69 317. . 19 373. .44 467 . 19 379. .69 210 . 16 264. .84  XM 47 .02 45. .84 44 .41 46. .22 45. . 12 46. .56 4S. 15 37, .24 44. . 14 54. , 14 46. .73 40. .76 39. .47 39. .23 46. .99 55 .94 63 .40 65 .31 65. .82 69 .94 68 .30 68 .73 68 .76 69. .01 72 .64 72 .58 66. .49 72. .78 55. .21 45. .75 45. .54 57 .70 63 ,04 55 .00 49 .91 56. .58 60. .98 61 .93 62 .94 61 .69 58. .28 59. .34 64 . 13 61. .61 59, .70 61 ..67 61 .76 65 03 75 . 17 75 . 16 74 .82 87 .38 88 . 11 61 .09 52 .22 51 .39 36 .53 35 .74  J214 OOK9:H::;M MIT U B WD -R I i . OS 0.U21* 1,.24 324. 0 294, .4 sv  214 214 214 214 214 214 214 2 14 2 14 2 14 214 214 214 214 214 214 214 214 214 214 221 221 221 221 221 221 221 221 221 221 221 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222  9: :30 10:: 0 10::30 11 : 0 11 :30 : 12:: 0 12::30 13:: 0 13::30 14 : 0 14 ::30 15:: 0 15: 30 16:: 0 16: 30 17;: 0 17 ::30 18:: 0 18::30 19:: 0 10::30 11:: 0 I I : ;30 12:: 0 12::30 13:: 0 13::30 14:: 0 14 : 30 IS:: 0 IS::30 6: :30 7: 0 7 ; 30 8: : 0 8: ;30 9: : 0 9: 30 10:: 0 10: 30 11 : 0 1 I:30 : 12:: 0 12: 30 13:: 0 13::30 14:: 0 14:;30 IS:: 0 IS: 30 16:: 0 16::30 17:: 0 17 ::30 18:: 0 IB::30 19:: 0  295, .3 296 .2 295. .9 295. .6 296 .0 296 .4 297. . 1 297 .8 298 .2 298 .6 298 .7 298 a 298 .4 298 . 1 298 .4 298. 6 298. .8 299. .0 298 .6 298. .2 296. .9 297, .4 297, 8 298, .2 298. 8 299. .3 299. .6 299. .8 300. . 1 300. ,4 300. 7 287 .7 287 , 4 287. . 4 287 ,.4 287. .6 287. 9 288. 0 288. . 1 288. 4 288. .6 288, .8 288. 9 288. .9 289. 0 289. ,2 289, 4 289. .6 289. .7 289. 8 289. .9 289. .9 289. .9 289 .9 289. 9 290, 0 290. . 1  1. 67 2. 10 2..71 3,,32 3 .00 2..69 2..94 3.. 19 3. 12 3. 05 3. 10 3.. 14 3. 26 3. 38 3..34 3. ,29 3,.09 2..88 2. 78 2. 67 2..54 2. 66 2 67 2. 67 2. 77 2. 67 3. 25 3 .64 3. 31 2..97 2. 98 4. 54 4. 45 4 .24 4, 02 4. 03 4. 03 3. 72 3. 42 3. 36 3. 30 3. ,50 3, 70 3. 61 3. 52 3. s o 3. 48 3..36 3. 25 2..79 2..34 2. 64 2. 94 3. 00 3. 06 2. 94 2..82  262. 201. 215. 230. 235. 240. 230. 220. 225. 230. 234. 237. 235. 233. 255. 277. 266. 254. 260. 265. 208. 212. 219. 226. 226. 226. 219. 211. 214. 218. 220. 125. 129. 140. 152. 156. 160. 156. 152. 145. 139. 147. 156. 157. 158. 150. 143. 143. 144. 147 . ISO. 148. 146. 144 . 142. 135. 127 .  i . . 11 i . .03 i . . 17 i . .24 i , . 10 0, .97 1 00 1. 02 0 99 0. .96 0. 94 0. 93 0. .98 I..03 1.. 17 1.,30 1.,03 0, .79 0. 90 0 .99 0 77 0 .79 0. 86 0. .92 1..03 1,. 15 1,. 19 1..20 1.. 12 1..04 0 .97 1.. 1 1 1.. 13 1.. 16 1.. 17 1.. 13 1..09 1,. 14 1.. 17 1.. 16 1.. 15 1..25 1. 36 I..24 1 . 13 1 .06 1 .00 1..02 1..03 0 .87 0 .71 0 80 0 .89 0. .78 0 .67 0 .71 0 .74  L O HS -UN 72.OH M -OB 2.0H-D1FF AR 11.A81 43. 30 68 33 297. 36 -19. . 16  0. 28 -20, 85 0. 34 -23. 06 0. 42 -32. 40 0. 47 -55, 55 0. 43 -47, .91 0. 42 - 3 1 . .22 0. 45 -34, 19 0. 48 -38 .56 0. 47 -39, . 13 0. 46 -37, .72 0. 46 -37, .95 0. 46 -46, .62 0. 46 -52. .79 0. 49 -50. .26 0. 48 -48, .27 0. 46 -56. 58 0. 42 - 7 1 . 58 0. 39 -73, .96 0. 37 -97 .38 0. 33 -173. .43 0. .41 -24, .89 0. 42 -26, .91 0. 42 -28 .70 0. 42 -27 ,04 0. 43 -28. .48 0. 45 -28, .07 0. 49 -34 .93 0. 53 -48, .36 0. 48 -44, .73 0. 44 -43, .02 0. 43 -48. .42 0. 55 -281, 31 0. 54 -240. . 14 0. S3 -184 .87 0. .50 -174 .80 0. 52 -117. .76 0. 53 -108 .05 0. 49 -97. .01 0. 47 -67 .71 0. 47 -60 .20 0. .47 -50 .97 0. 50 -49 .09 0. 52 -56 .64 0. 50 -68 .74 0. .48 -77 . 13 0. .47 -77 .70 0. 49 -61 .61 0. 48 -53 .38 0. 47 -49 .28 0. .42 -36 .94 0. 36 -32 . 17 0. 39 -39 .77 0. .44 -39 .30 0. 43 -49 .60 0 42 -68 .79 0. 4 1 -57 .61 0. 39 -69 .05  93. 10 159. 00 204. 10 171. 35 156. 35 211. 40 241. 60 259, 10 237. 60 234. .55 244 .. 15 188. 20 176. , 10 210. .25 215 15 164. . 15 98. ,00 75. 90 46, .95 19..65 250. .95 256. .40 235, .55 258, . 15 267 00 304 318, .75 279 . 10 235 .05 179. .80 153 30 51 .45 58 .60 70 .00 64 .00 107 . 15 120 . 10 109. .05 138 .35 154 .70 185 .65 233 .85 225 .55 159 .80 126 .45 123 . 15 167 .00 184 . 10 186 .80 179 .70 130 . 10 136 .65 192 .05 145 .70 97 .55 110 .90 76 .00  OS  114. 13 125. 38 124. 25 135. 25 156. 25 169. 13 233. 00 264. 13 231. 00 221. 00 166. .38 160. .88 168. 00 • 156. .88 142 00 107. 25 95. 50 67. 00 40. .00 19.,38 167, .50 175, .00 164. .75 154! .63 183. .88 188. .50 165 .75 158. .75 135 .63 135 .25 154 .00 27. .38 29. .75 37 .25 53. .50 70 .38 86 .25 77 .00 107 .38 137 .00 142 .38 150 .88 126 .25 101 .25 88 .75 78 . 13 1 13.38 127 .88 109 .63 112 .25 99 .88 1.14 .00 113 . 13 79 .88 68 .25 60 . 13 32 .50  1. 66 259. 40 2. 65 221 . 72 6. 29 291 . 84 2. 84 464. 66 0. 01 396. 59 3. 33 233. 26 0. 68 246. 38 0. 40 267. 72 0. 52 270. 26 1. 07 257. 58 6. 12 252 . 27 2. . 15 305. .49 0. .64 359. 57 4 .20 . 339. 85 5. .76 379. 10 4. 48 504. .34 0. ,20 529. 62 0. .70 448. 05 0. 55 657, 99 0. 02 1 160.,79 6. .57 149. , 12 6. 41 161. 41 5 58 188. . 18 8. . IS 189. ,03 6. .55 218, 43 9, . 10 229, .91 12 .05 275 .24 9, .48 353. .45 7 .83 340 .34 3. .51 333. .04 0 06 352 82 1 .90 979 .97 2 .27 947 .66 2 .58 874. .02 0 .83 905. .90 2 .90 667 .89 2 .67 596 .69 2 .52 628 .47 2 .44 513 .62 1 .39 464 .54 3 .41 401 .32 6 .53 394 .01 7 .82 466 .99 4 .61 516 .57 2 .97 538 .43 3 .55 518 . 14 4 .22 390 .77 4 .43 360 .64 6 .08 347 .48 5 .31 249 .76 2 .38 205 .42 1 .78 268 .70 6 .21 260 .01 5 . 16 290 .46 2 .31 334 .20 4 .00 314 .88 3 .43 400 . 18  12. 36 13. 01 15. 18 18. 29 17. 47 14. 95 15. 51 16. 22 16. .30 16. 08 16. . 13 17. 32 18. 01 17.,74 17 .52 18. 38 19..57 19 .74 20 .93 22 .87 13 .51 14..02 14 .42 14 OS 14..36 14..28 15 .64 17 .52 17 .08 16 .85 17 .53 23 .95 23. .65 23 .04 22 .89 21 .66 21 .35 20 .92 19 .32 IB .71 17 .81 17 .62 18 .40 19 .39 19 .93 19 .97 18 .83 18 .07 17 .64 15 .97 15 . 13 16 .38 16 .33 17 .66 19 .38 18 .47 19 .39  B 273. .44  290. 67 317, .71 381. .86 543. 12 484 .. 17 373. . 13 402 .57 443. .83 457 .00 413. .03 446. .55 486. 08 499. .80 496. .58 485. .03 527, . 18 579. .90 587. .99 689 .87 869. . 13 328 08 344, .26 352, .40 338. .89 371 .37 368. .27 400 .62 495. .78 457. .23 456. .44 477. . 12 882 .54 865. .35 786. .52 788 .68 730 .28 691. .49 676 . 17 586. .73 533 .82 490 .88 479 .58 527 .58 590. . 12 600 .07 625 .76 528 .71 501 .46 500 .02 407. .36 365. .48 428 .00 415 . 11 503 .35 557 .62 512 .03 616 .67  C 252 .41  268 31 281 .75 366. .89 481, .63 465. . 18 358 .50 371 .61 393 .59 389 .30 4 13 .03 380. .40 448 .69 499. .80 477, . 10 466 .01 506. .51 579 .90 587 .99 636 .80 770 .74 302. 84 317 .78 338. .58 325. .59 316. .35 313 .71 384. .91 457 .65 457. .23 438. .54 477 . 12 995 .21 937 .47 886. .93 654 .41 701 .64 691, .49 649. .66 541 .60 533. .82 490 88 479 .58 506. .90 544 ..73 600. .07 577 .63 550. .29 501 .46 461. .56 407. .36 380. .40 428 .00 432. .05 464 .63 580 . 38 532. .93 525 .31  0 360. .94 296. .09 236 .72 249 .22 291 .41 267 .97 203. .91 203 .91 205. .47 205. .47 199. .22 196. .09 210. . 16 230. .47 224. .22 255 .47 314. .06 292, .97 244. .53 320 .31 457, .81 151. . 17 155 .86 174 22 182. .03 203. 91 216. .41 224, .22 238, 28 238. .28 238. .28 236. .72 335. .94 339. 06 335. .94 354 ..69 300, .78 277 .34 305, .47 292. .97 278 .91 261. .72 263. .28 289. .84 292. .97 288 .28 277. .34 232 03 230 .47 232 . 03 196. .09 175 .78 200 .78 196. .09 192. 97 IBB..28 192. .97 225 .78  XM  39 .81 41 .85 44 .31 52 .79 66 .09 62 .52 51. .86 54 . 14 57 .. IS 57 .52 56. .60 56. .75 61 ..85 64 ..87 63 .69 62. .70 66 .53 71 .95 72 .67 78. .47 88 .52 46. . 19 48. . 14 49. .75 48. .26 49. .55 49, . 19 54 .68 62. .75 60 .84 59. .87 62. .78 94 ..67 92. .88 89. .45 88 .64 82. .11 80 .50 78 .39 70 .70 67. .99 64 .03 63. . 12 66 .55 71 ..04 73. .59 73. .75 68, .53 65 . 14 63 .21 56. .08 52 .61 57 .92 57 ..63 63. .37 71 .05 66 .95 71 . 14  JDD HH: MM 222 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 223 230 230 230 230 230 230 230 230 230 230 230 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231 231  19: 30 8: 30 9: ; 0 9: 30 10: : 0 10:30 tt: 0 11: 30 12: 0 12: 30 13: 0 13: 30 14: 0 14: 30 IS: 0 IS: 30 16: 0 16: 30 17: 0 t7: 30 18: . 0 18: 30 19: 0 19: 30 13: 30 14: 0 14: 30 IS: 0 15: 30 16: 0 16: 30 17: 0 t7: 30 18: 0 18:  6: 7: 7: 8: 8: 9: 9:  10: 10:30  30  30 0 30 0 30 0  30  0  11: 0 11: 30 12: 0 12:  30  13: 0 13: 30 14: 0 14: 15:  30  16: 17: 17:  30  0 IS: 30 16: 0 0 30  TT 289 .9 287 .5 287 .8 288 . t 288 .3 288 .4 288 .6 288 .9 289. .3 289 .9 290. 4 290. .7 291 . 0 291. .4 291 . 7 291 . a 292. . i 292. 0 291. 9 291. 8 291 ..6 291. .4 291. 2 290. .9 296. a 297. 0 297. . 1 297. 3 295. 9 294. 5 294 . 8 295. 0 295. . 1 295. . 1 294. 9 286. t 286. 4 286. 2 286. 0 286. 4 286. 7 287. 8 288. 8 289. 6 290. 5 291. 1 291 . 6 291. 9 292. 3 293. 2 294. 1 294 . 6 295. 2 295. 7 296. 2 296. 4 296. 6 296. 4  UB 3 .03 1 .86 2 .01 2 . 15 2 .30 2 .23 2 . 17 2 . 19 2 .22 2 .39 2 .56 2 .65 2 .74 2 .58 2 .42 2 .36 2 .30 2 .52 2 .75 2 57 2 .39 2 .27 2 . 14 1 .99 2 .00 2 .26 2 .61 2 .96 3 .53 4 .11 3 .57 3 .04 2 .54 2 .04 1 .59 3 .31 3 .43 3 .35 3. 27 3. .04 2 .81 2. 42 2 .04 2 .08 2. . 12 2 73 3. 33 3. .66 3. .99 3 99 3. 99 4. .20 4 .41 4 . 87 5. .33 5. .35 5. 37 5. 01  W-OIR 119.  148. 149. 148. 148. 14S. 143. 141.  139. 147. 155. 156. 158. 152. 147. 154 . 162. tSB. 154. 152. 151.  153. 153. 156. 190. 199. 207. 215. 219. 223. 232.  240. 242.  245. 241. 105. 114. 125. 137. 147 . 158. 167. 176. 202. 228. 243. 259. 253. 248. 260. 273. 286. 299. 299.  300. 294 . 289. 293.  SV 0 79 0 .66 0. 62 0. .69 0. .75 0 80 0 84 0 80 0. 76 0 .87 I. .00 1. 05 1. . 11 1. . 12 1. 12 1. .48 1. .83 1. .40 0. .86 0. 79 0. .73 0. 60 0. .49 0. .50 1. 06 t. 20 1. , 14 1. .02 1. 05 1. 04 t. 09 1. 08 0. 90 0. 72 0. 88 0. 87 0. 98 0. 95 0. 92 0. 85 0. 77 0. 81 0. 80 1. 08 1. 38 1. 41 1. 29 1. 23 1. . t3 1. 23 1. 34 t. 42 1. 50 1. 38 1. 20 1. 19 1. 19 1. 15  U»  L  0 .39 0 .28 0 .30 0 .33 0 .36 0 .34 0 34 0 .34 0 .34 0 .36 0 38 0 .40 0 41 0 39 0 .39 0 .39 0. .37 0 .37 0 39 0 .37 0 .34 0 32 0. 30 0. 27 0 36 0. ,37 0. .40 0 .45 0 51 0. .56 0, 49 0 .43 0. .37 0. 30 0. .23 0, .39 0. 44 0. 45 0. 46 0. 44 0. 41 0. 36 0. 34 0. 36 0 37 0. 43 0. 50 0. S3 0. 56 0. .56 0. .56 0. 58 0. 60 0. 64 0. .68 0. 66 0 65 0. 60  -136 .74 -37 . 14 -41, .23 -30 .74 -30 .92 -32 .26 -30 .98 -32. .23 -31 ,.87 -34 .82 -37 .96 -34. .01 -37 .52 -34, .36 -24. .49 -22 .68 -26. .37 -41 .45 -52. .25 -52 .83 -52 .49 -S4. .07 -60, .27 -69 02 -14, .44 - 2 1 . .14 - 3 t , .00 -33. 48 -50, .54 -79 . 10 -67. 40 -60. 38 -44, .57 -39, . 10 -44 . 07 -364, .20 -121. 05 -78. .90 -61 .41 -44, .33 -43, 03 -36. 68 -20, .37 -16 .62 -16, 04 -26 .89 -36 .96 -47, .50 -62. .34 -58, .25 -57, SO -63 .38 -76 .59 -96. 63 -126. .33 -189 .4 1 -254. 68 -264 . 10  OH-SUN  OH-MOB OH-DIFF  37 .80 52 .40 56 .80 107 .35 130 .35 t i t .60 109 .30 106 .05 112 .43 123 .25 133 .65 174 .43 167 .00 158 .95 218 .50 228 . IS 167 .90 112 .70 104 .90 84 .23 68 .33 56 . 10 40 .30 26 .80 285 .63 227 . 10 195 .65 254 .50 235 .85 198 . IS 162 .00 1 16.75 104 .93 65 .80 26 . IS 14 .60 63 .45 104 .65 137 .55 175 .90 145 .30 1 17.65 172 .00 251 . 10 282 .20 271 .55 307 . IS 283 .50 250 . to 275 .90 281 .85 2B7 .30 256 .45 252 .35 232 .90 140 .70 98 95 76 85  9 .63 40 .25 54 .25 63 .88 71 .50 68 .50 58 .38 47 .63 78 SO too 38 117 .75 124. .75 116 .00 135 . 13 168. .25 137 . 13 101 .75 100. SO 64 ..38 41 .38 35. .75 23. .75 8. .38 6. .75 179 .38 tst .50 139 .25 153 .68 173 . 13 190 .63 172, 50 115 .75 73 . 13 50 .88 IB .88 10 .75 38 .50 52 .63 63 .00 97 .38 125 .25 128. 63 168 .63 203 .50 213 . 13 292. .25 304 .75 259 .75 312 .00 297 .25 224. SO 220 63 199. .00 163 .00 129 .88 101 .38 86 . 13 57 .50  AR  6B3 97 2. .22 0. .96 284. 68 0. .20 279. 19 3. .42 207. 21 4 63 207. ,41 3. 39 245. 26 4. 01 246. 68 4 ..60 244. 78 2. 67 229. .69 1. .80 273. ,84 1. .25 325. 65 3. 91 289. ,38 4. 02 331 ,89 1 88 319 31 3. .96 224. 22 7. , 17 272. .92 5. .21 416 08 0. .96 SIS. 27 3. .19 369. IB 3. 38 359. .76 2 .57 360. 08 2. 55 323. 84 2. .51 307. .70 1. .58 388. SO 8. .37 124. .96 3, .93 215 61 4 .44 284. 80 7. 92 245. .49 4. .94 334. 01 0 .59 432. 74 0 83 458. 60 0. 08 473. 54 2 .31 352 68 t. . 18 309 .29 0 57 549. 21 0 .30 1188. .03 t. .97 696. .95 4. .10 491 .74 5. .71 383. 45 6. . 18 278. .99 t. .58 264 .52 0. 87 268. .54 0 .27 150. .26 3. .75 150, .59 5. .44 179 .74 1..63 281. .46 0 . 19 310. .45 1. .87 357, .67 4 .87 390, . IS 1, .68 401 .22 4, .52 432. .50 5 .25 479. .39 4 .52 568 .60 581. .77 7 .05 a .11 564 .98 3 . 10 729 .69 1 .01 850 .52 1 .52 90S. . tt  A  B  22. 16 772 .01 16. 00 449, .01 16. 61 4S4. .00 14 . 85 376 .23 14. .89 391. .82 15. 15 373 .35 14 ..90 392 .68 IS. 15 380 .21 15. 08 392. .11 15. 61 392 .34 16. . 13 430 .83 15. ,48 393. .58 16. 06 42S .28 15. 54 396 .07 13. ,40 343 .4 1 12. 80 314. .25 13. 88 346. .76 16. 63 447, .31 17. 96 306 .91 18. 02 53B. .95 17. 98 S07, .75 IB.. 13 526. .53 18. .71 544, .51 19. 40 592, .60 to. 01 228 .87 12 .45 299 .29 14. 90 377 .94 ts. 38 381. .91 17. ,77 487 .67 20. .05 594 .41 19. 28 586. .29 IB. 73 566. .29 17 06 456 .02 16. 28 432 .41 17. 00 462. .24 24. .38 1051. .80 21. .76 723 .59 20. 03 606 . 13 IB..62 571. .83 17, 03 445. .71 16. 85 465 .39 IS. 93 404 ,.95 12. 20 280. .49 to. .90 255 .22 10 68 243. .97 14. 01 337 .84 15. 98 423 .25 17 ,,42 491 .55 18. .89 531 .49 18 54 545 . 10 18. ,48 531 .31 18 98 568 .65 19. .90 598 .06 20. 91 648 .86 2t .92 718 .60 23 .09 626 .25 23 76 933 .28 23 84 922 .96  C 74 1 .74 367 .37 419 .07 347 .29 333 .77 373 .35 334 .51 365 .30 347 .72 392 .34 397 .69 376. . 15 395 .33 380 54 280 .97 276. .67 307 SO 429 .77 467 .03 459 . to 487. .84 486. .02 523 . 16 547 .01 194 .96 265 .41 348 .87 381 ..91 487 .67 618 .67 541 . 19 502 . 18 436 .02 415 .43 444 .. 12 832 .73 723 .59 606. . 13 507 . to 463 .90 429 .39 404 .95 269. .49 217 ,4 1 216 .35 324 .59 390 .69 453 .74 553. . 19 503 . 17 510. .48 524 .91 598 .06 675 .34 747 .93 859 .98 896 .68 922 .96  D 291. ,41 224. ,22 205. 47 183, .59 183. 59 210. . 16 217, 97 210. . 16 199. .22 222, .66 252. .34 239. .84 236 .59 263. 28 230. .47 294. 53 407, .81 376. ,56 236. .28 232, 03 232. 03 203. ,47 185. . 16 219. 53 188. 28 244. S3 250. ,78 203. ,47 219. ,53 228. 91 261. ,72 28S. , 16 247. .66 233. 59 389. ,06 389. 06 310. . 16 260. . 16 228, .91 196. .09 189. .84 211. .72 174. .22 203. 91 249. 22 271. 09 244 .,53 242. 97 230. .47 246 09 266. .4 1 282. .03 305. .47 282. 03 247 ..66 278. .91 300. 78 317. . 19  XM 84. .74 56. 21 s a . 82 51 .47 51. 61 32. 68 51 .66 52. 66 52. 37 54. 60 56. 76 54. 01 56. .47 54 ..26 45. ,79 43 .90 47 .63 58. 95 64. ,63 64 .89 64. 74 65. 45 68. .01 71. . 13 33. .25 42. . 18 51. .68 S3 ,61 63. 82 74 . 14 70, 59 68. 06 60. .75 57, SO 60. ,47 97 . 18 82 .61 74, .09 68 .45 60 .63 59 .88 55 .90 41 .28 36 .46 35. .64 48 . 12 56. .09 62 .31 68 .80 67 .21 66. .61 69 . 18 73 .44 78 .31 63 .37 89 .79 93 .57 93 .98  JDD  HH: MM  IT  231 231 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 232 236 236 236 236 236 236 236 236 236 236 236 236 236 236 236 236 236 237 237 237 237 237 237 237 237 237 237 237 237 246 246 246 246 246  18: 0 18: 30 7: 30 8: 0 8: 30 9: 0 9: 30 10: 0 10: 30 11: 0 11: 30 12: 0 12: 30 13: 0 13: 30 14: 0 14: 30 IS: 0 15:30 16: 0 16: 30 17: 0 17: 30 18: 0 11: 0 11: 30 12: 0 12: 30 13: 0 13: 30 14: 0 14: 30 IS: 0 IS: 30 16: 0 16: 30 17: 0 17: 30 18: 0 18: 30 19: 0 7: 0 7 : 30 8: 0 8: 30 9: 0 9: 30 10: 0 10: 30 11: 0 1 1:30 12 : 0 12: 30 17: 0 17: 30 18: 0 18: 30 19: 0  296 .2 295 .6 290 .6 290 .9 292. .2 293 .5 293 .9 294 .2 294 6 294 .9 295 .5 296 . 1 296 .6 297. . 1 297 6 298. .2 298. .7 299. .2 299. S 299. .8 299. .8 299. .8 299. .6 299. .5 292. 6 292. 7 292. .8 293. .4 293. .9 294. 3 294. .7 295. .0 295. .3 295. 6 296. 0 295. 3 294. 5 294. 0 293. 5 293. 4 293. 3 290. 2 290. 7 291. 2 291 . 9 292. 5 293. 3 294. 1 294 . 4 294. 6 294. 9 295. 2 296. 6 296. 7 296. 1 295. 4 294 . 6 293. 8  UB 4 .64 4 .29 1 . 15 1 .56 1 .45 1 .34 1 .82 2 . 29 2 .50 2 .70 2 .86 3 .01 3 .33 3 .66 3 .84 4 .01 4 . 18 4 .35 4 .25 4 . 14 4 .37 4 .60 5 .05 5 .50 3 .50 3 .45 3 .40 3 .24 3 .07 2 .80 2 S3 2 .47 2 .42 2 .28 2. 13 2 .89 3 .65 3 .28 2 .91 2. .48 2 .05 0 .92 1. . 18 1. 45 1..48 1..51 1 .42 . 1. 34 1 .91 2 ..49 2. .94 3. .39 2. 90 1. 95 2. 35 2. 75 2. 49 2. 22  W-DIR 298. 303. 316. 319. 275. 232. 223. 214. 224 . 234. 257 . 279. 285. 291 . 296. 301. 294. 287. 294. 301. 300. 299. 304. 309. 164. 161. 158. 157. 156. 157. 158. 158. 158. 183. 209. 233. 257. 260. 263. 281. 300. 316. 252. 188. 173. 158. 196. 234. 225. 215. 232. 248. 284 . 272. 279. 2B5. 290. 29S.  SV 1.. to 1..38 0 .87 0 .80 1.. 11 1 .36 1 .29 0 .92 0 .93 0. .92 1 . 11 1 31 1..39 1 .47 1 .36 1 .23 1..38 1..54 1..35 1.. IB 1,. 17 1 . 14 t, . IS 1 . 13 t. . 12 1..11 1, . It 1..02 0 93 1 .25 1 .49 1..26 1. 04 1.. 14 1. 21 1 .21 0. .99 0 .84 0. .70 0 .70 0. 66 0. 71 0 68 0. 55 0 76 0. 97 0. 90 0. 83 1. 06 1. 21 1. 40 1. 58 0. 68 1. 47 1. 42 1. 26 0. 97 0. 70  U»  L  0 .56 0 .51 0 .20 0 .26 0 .25 0 .25 0 .31 0 .36 0 .39 0 .41 0 .44 0 .46 0 .49 0 53 0 .53 0 .•55 0 56 0 .57 0 .56 0 SS 0 .58 0 .59 0. .61 0 63 0 SO 0. SO 0. .49 0 48 0 47 0. .44 0 .41 0. .41 0. 40 0. 37 0 .35 0, .43 0. 50 0. .44 0. 39 0. .34 0 28 0 16 0. 20 0 23 0. .24 0. 25 0. 26 0. .26 0. 32 0. 40 0. 45 0, 50 0. 46 0. 32 0. 34 0. 37 0. 33 0. 30  -287 .03 -348. .40 -17 .26 -19 .23 -16 .81 -12 .31 -19 . 14 -28 86 -30 .05 -32 .76 -30 .52 -34 .61 -41. .62 -49. . 12 -63 . 13 -68 . 11 -85 .35 -105, .34 -96 .91 -86 .21 -91 .55 -119 ,65 -237 .83 -642 .79 -50, .72 -47 .SS -48 .33 -42 .57 -33 38 -28, .75 -22 .67 -20 .08 -21 .54 -22 .33 -20 .68 -38 .92 -77. . 19 -89 .21 -86 .89 -69, .22 -65, 01 -16, 78 -17 .77 -26 .04 -21, .03 - IB,.91 - 1 1 . ,74 -8 .80 -18, 34 -26. .48 -31 ,.33 -41 .78 -27 .90 -22, . 13 -44. 50 -87 .20 -96 .80 -88. 89  OH-SUN  OH-MOB OH-DIFF  55 34 40 84 84 109 136 145 178 197 262 253 262 274 223 228 190 162 169 183 198 162 90 36 226 237 222 231 282 267 283 318 269 213 196 188 143 85 61 52 32 21 41 42 63 79 140 189 169 213 273 274 313 135 83 52 33 26  32. . 13 12. 63 21 .63 29 50 40. .88 SO 63 97. .25 148 38 164 .. 13 191 .38 237. .75 219 .25 208. .25 248. 88 241 .75 225. . 13 212. .50. 175 63 133. . 13 102. .00 92 . 13 68 25 36 .50 9 .00 216. . 13 233 SO 237 .00 275 .00 297. . 13 242. .50 219 . 13 212 ,88 169 .88 129 63 123 63 113 .00 87 .38 50 .38 19. .63 2, . 13 - 0 , . 13 1 . 13 13. .38 28 .75 55 .63 75 00 61 . 88 81 . 13 131 .38 153. .75 166. SO 174 .00 213 . 13 94. .00 61 , 38 38 .50 21 .50 0 .88  .20 .60 . IS .75 .35 .70 .95 .60 .55 .90 .00 .80 .60 .95 .05 .95 .65 .35 .35 .40 .90 .65 .35 .65 . 15 .70 .30 .50 .00 .80 .85 .85 .35 .35 .00 .40 .50 .40 .70 . 10 . 10 .45 .45 .75 .20 .30 .75 .35 .35 .50 . 10 .20 .70 .80 .60 . 15 .55 .55  AR  957. . IB 1 .82 . 1. ,73 1423. . 11 1. 46 229. 05 4. 35 183 63 3. .42 225. 92 4 65 187. .95 3. . 13 246. .92 0. .22 236 93 1.. 14 230. .55 0 51 237 .37 1 .91 247 .31 2 .72 320 .07 4 ..28 385. 38 2 .05 438. 38 1..47 SOI .97 0 .30 464. .88 1,.72 606. .72 1..05 772 .27 2. .85 652. .50 6, .41 532. .76 8. .41 524. .38 7. .43 600. .71 4..24 8S2. .56 2. . 18 1122 .85 0 .79 363. .91 0 .33 340 .80 1. . 16 353 .93 3. .43 29S. .34 1 . 19 214 .09 1..99 262. .40 S. . 10 261 .07 8 .34 192 .30 7 .83 176 .31 6. .59 217 .56 5 .70 223 68 5, .94 360 .48 4 ,.42 455. .25 2. .76 484 ..70 3, 31 448 . 14 3. .94 433. . 14 2, .54 472. . 16 1 .60 225. .71 2 .21 184 .77 1. . 10 198 .03 0 .60 209. .94 0 . 34 227, .72 6. .21 112 . 10 8 .52 69 . 17 2 .99 186, .90 4 .71 256, .03 8. .39 314. .42 7 89 431 .93 7 92 131 .93 3 .29 321 ..04 1..75 606. .30 1. 08 850 .96 0. 95 792 58 2 .02 592. .07  A  B  C  0  24 . 00 960. . 13 922 .47 329. .69 24. 32 965. 36 1004 .76 467. . 19 I I . . 14 262 .95 223 .99 300 .78 11. 84 279 .56 247 .91 222 .66 10. 98 252. .46 223 .88 302. .34 9. 02 200 35 170 .66 323 .44 II .81 288 .99 236 .45 300. .78 14 46 354 .25 340 .36 217 .97 14 ,,71 363. .40 349 . 15 207 03 IS. ,25 399. .01 353 .84 202 .34 14. .82 359 .54 359 .54 219. .53 IS. .58 390. .33 390 .33 261. .72 16. ,67 4S5. .71 420 .66 282. 03 17 61 499. 59 461 . 16 292. 97 IB..95 545. .97 545 .97 294, .53 19. .34 566. .57 566 .57 263 .28 20. 38 652 . 19 602 .02 311, .72 21. .25 726 . 15 643: .94 364 ..06 20. .92 676 .56 650 .03 317, . 19 20. .42 617. .33 642 .52 271 .09 20 68 647. .61 647 .61 260. . 16 21. ,71 706. 36 735 . 19 267. .97 23 .62 899, .81 899 .81 305. .47 25 .02 1069 .00 1112 .63 332 81 17. .80 478. .33 497 .85 238. .28 17. .43 452 .99 232. .03 490. .74 17, .52 486. .24 467. . 18 238. .28 16 .80 453. . 15 435 .38 213. .28 15 36 374 .05 389. .32 178. .91 14. .43 374. .00 318. .59 242. ,97 12. .90 314 .03 278 .48 282. .03 12. . 12 282. .60 260 .86 225 .78 12. .57 308. .49 262 .79 . 197. .66 12. 81 298. 85 287. . 13 236. 72 12 31 283. .01 271 .91 257. 03 16. 26 422. .71 422 .71 272. .66 19. .92 601. .26 601 .26 242. 97 20. .59 ' 664..34 613. .24 244. 53 20. .46 619 .30 644 ..58 227 ..34 19. .42 604. .44 536. .01 244. .53 19 . 10 576. .21 531 .88 274. 22 10 .96 252. .44 223 .86 302. .34 11. .33 278. .43 218 .77 23». 28 13. .81 349. .79 297, .97 196. 09 12 .41 298. .42 264 ,.63 238. 28 t l . 72 276. 51 245. ,21 278. 91 8. .74 192. . 15 163. .68 200. 78 7. .07 149. .82 122 .58 161 . 72 11 53 279. 99 229. 08 235. 16 13. .91 334. ,71 321. .58 249. 22 14. .98 373. .36 358 .,72 274. 22 16. .69 430. .61 448 .. IB 314. 06 14 .25 359. .52 318. 82 124 . 61 12. 75 296. 98 285. 33 351. 56 17. 04 464. 73 446. .51 426. 56 432. 61 20 48 632 75 632 .75 20. 91 675. 36 648. 87 385. 94 20. .56 638. .39 638 . 39 297. 66  XM 94. .88 96. ,78 37, .34 39. .90 36 .72 29 .74 39 .79 49 .88 SO .90 53 .07 51 .29 54 .44 59 .05 63 . 13 69 .09 70 .83 75 .76 60 .01 78 .37 75 .97 77 .22 82 .40 92 .76 101 . 18 63. .91 62 .34 62 .74 59. .61 S3 .54 49. .79 43 .89 40 .94 42 .64 43 S2 41 .65 57, .38 73 .61 76. .68 76 . 14 71 . 19 69, .77 36. .68 38. .02 47, .32 42. .06 39 .50 28. .73 22 .94 38 .77 47. ,73 51. .94 59. 15 49. .04 43. 30 60. 71 76 .21 78. 35 76 63  ODD HH; MM 247 247 247 247 247 247 247 247 247 247 247 247 247 247 247  7: 0 7;:30 a:: 0 8: 30 9:: 0 9: 30 10:: 0 10::30 1 1: : 0 11: 30 12:: 0 12: 30 13: 0 13: 30 14:: 0  TT 289. 6 289. 9 290. 1 290. 8 291 .5 291. 7 291. 9 292. 5 293. 1 293. 7 294 . 3 294 . 6 295. 0 295. 3 295. 5  ua 1. 31 1. 82 2 . 33 2 . 13 1. 93 2. 11 2. 29 2. 19 2. 09 1 93 . 1 77 . 1. 98 2. 18 2. 50 2. 83  W-01R 148. 170. 193. 201. 210. 214. 217. 193. 169. 178. 1B7. 225. 264. 248. 232.  sv 0. 28 0 . 50 0 . 78 0 . 90 0. 98 1. 14 1. 31 1. 21 1. 13 1. 29 1. 42 1. 25 1. 01 1. 07 1. 11  U« 0. 20 0 . 27 0 . 33 0 . 32 0. 31 0 . 33 0 . 36 0. 36 0 . 34 0 . 33 0. 32 0. 34 0 . 36 0. 41 0 . 43  L -33 .27 -39 .48 -60 .62 -38, 64 -24 31 -28, .26 -26 .79 -21 .72 -23.. 14 -18 83 -13 .35 - I B 01 -19 ,89 -23 53 -33 .50  0H-!SUN  OH-MOB QH-DIFF  21 .70 45 .25 51 .40 75 . 10 112 .00 116 .60 161 .65 195 .20 154 .30 167 .40 222 .55 194 .30 222 .45 259 .30 220 .95  - 1 , ,63 13..50 37,.75 57 .63 72 .75 69 .88 85 .63 121 . 13 147 .38 175 .25 203 .00 191 . 13 148.. 13 204 .25 250 .25  1..84 2..50 1..08 1..38 3 .09 3..68 5..99 5..83 0 .55 0 .62 1..54 0 .25 s..85 4 .34 2 .31  AR 150..90 240 .03 446..20 355 55 244 81 313..95 312 86 230..03 245 . 19 227. 63 170 .02 202..90 174 .29 196 .03 279..64  A 15 .34 16 .35 18.,75 16 .23 13..36 14..32 13 .98 12 .63 13 03 11..70 9..52 11..42 12 06 13 . 14 15 .39  B 395. 90 425. 58 545.,35 4 19..51 329., 19 356. 52 351..26 298..97 312. 97 280. 21 210. 50 270..93 275. 89 310..70 381,.91  C 365 .45 425 .58 523 .97 419 .51 291 .93 329.. to 311 .50 275 .97 288 .90 238 .70 186 .67 230 .79 265 .07 298 .51 381 .91  0 126..95 180. 47 267. 97 271..09 252. 34 292.,97 302..34 255.,47 260.. 16 280. 47 272..66 258..59 205. 47 205..47 233. 59  XM 53. 45 57. 74 6 8 . 15 57. 20 45. 61 4 9 . 36 48. 02 42. 84 44. 39 39. 40 31 . 50 38. 34 4 0 . 71 44 . 81 53. 63  - 293 -  APPENDIX  E :  Statistical Indices and Methods  E.1 MODEL VALIDATION STATISTICS  For a d e t a i l e d discussion of the s t a t i s t i c a l validate  models,  refer  observed  values  are  to  Willmott  denoted  as  (1981).  (0 , 1  predictions  as  (P^  P ).  indices used to  In the  0  0 )  2  following, and  model  n  The observed  and  predicted  means are 0 and P respectively. Coefficient of Determination : R R•2 .i s the square of the Pearson's Product-Moment C o r r e l a t i o n Coefficient : it n  R  2  * ( I °i* i] P  ~  n 2  = n- £ o  2  (E. 1)  - (n-0)  5  n- [ P  1=1  R  describes  *°*P  1=1  2  - (n-P)  2  1=1  the proportion of the t o t a l  variance explained by  the model. Standard Error  (Standard Difference)  : RMSE, (RMSD)  The root-mean-square error is defined as -•0.5  1 RMSE  =  — n  y (p - o f L i 1=1  l  (E.2)  - 294 If  P and 0 are variables  with comparable u n c e r t a i n t i e s ,  index i s r e f e r r e d to as the root-mean-square difference Systematic and Unsystematic Error : RMSE  sys  this  (RMSD).  , RMSE  unsys  o.s  — V (p - o y  RMSE  sys  n  L  i=i  i  (E.3)  r  o.s RMSE  where P  y (p  -  unsys  n  L l i=i  (E.4)  i  i s the ordinary least-squares estimate of P .  Note that RMSE = RMSE 2  2  Index of Agreement :  + RMSE  2  sys  is  (Willmott,  unsys  d  This d e s c r i p t i v e s t a t i s t i c variate  - py  accurately  reflects  estimated  by  the degree the  to which the  simulated  variate  1981) :  d  =  i=i  1 [ciP  i  - oi  (E.5) + io  i  -  oir  1 =1  Thus,  d specifies  the degree  to which the observed  deviations  about 0 correspond, both i n sign and magnitude, to the predicted deviations about 0, where 0 is considered to be e r r o r - f r e e .  - 295 E.2  STRUCTURAL ANALYSIS / PRINCIPAL AXIS  Ordinary  least-squares  regression  analysis  should  r e s t r i c t e d to p r e d i c t i v e s i t u a t i o n s . For comparisons with or among f i t t e d analysis  lines,  the r e l a t e d technique termed  or s t r u c t u r a l analysis  be  theory  functional  should be used (Mark and Church,  1977). I f ordinary regression analysis has been performed, the slope of the l i n e a r functional r e l a t i o n may be obtained from  [b / R - X] + / [ b 2  b f  2  r  2b  =  V  2  (E.6)  / R - X] + 4X-b 2  r  2  :  i  r  where b  r  i s the slope of the regression l i n e and X i s the r a t i o 2  of  the  error-variances  2  X = E /E . y  The  assumption  X = VARCJO/VARCX) y i e l d s the reduced major axis the equation  for b  f  that  x  becomes the principal  and i f X = 1,  axis  s o l u t i o n (Mark  and Church, 1977).  E.3  LOWESS  LOWESS stands (Cleveland, (1983).  for locally  ueighted  scatter-plot  smoothing  1979). This method i s summarized by Chambers  Here,  the steps  followed  to obtain  a  LOWESS  et.al. curve  through a s c a t t e r - p l o t of n (x/y)-pairs are : 1. Sort the data to increasing x ( x ^ i = l , n ) . 2.  Select  " f " as the f r a c t i o n of points  "neighbourhood",  so that  "q" i s that  used  in a  number  weighting  of points :  -  2 9 6  -  q = nearest integer of (f-n). 3 . Determine the q nearest neighbours to 4.  d = |x - x i  5.  1  (including x^,  |  q-est  neighbour  Assign a weight t ^ j  to each neighbour (k = l,q) according  to 1  'i(k)  -  -  3  x1 -x i(k) d l  W  J  6. Perform a weighted linear regression on the q neighbours of x. i  7 . LOWESS estimator of y : y = a + b -x . J  M  8 . Perform steps 3 - 7 9.  i  l  i  for each x^  Decide on the number of rejection (e.g.  l  "robust iterations" for outlier  3).  1 0 . Compute the absolute residuals : r ^ |y 1 1 . Determine the median r  I , (i = l,n)  (by sorting r ^ : median (r^ = m.  1 2 . Compute a robustness weight for each point : if r i £ 6-m if r i < 6-m  wi =  0  wi =  1  1 3 . Go to step 3 and multiply the t w^s. Thus steps 3 - 7 1 4 . Perform steps 9 - 1 3  , (i = l,n)  -  ri 6-m  2  1  's with the appropriate  update the y^s. for the number of robustness iterations  specified. 15.Plot  a smooth line through the  (XJ.YJ)  using a cubic spline  - 297 interpolation. 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 t h e p r e s e n t work i s i n c l u d e d i n the f o l l o w i n g :  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  SUBROUTINE L O W E S S ( X Y Y . N . F , I T E R . X Y T . W ) C C C C C C C C C C C C C C C C C C C C C C C C C 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 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 ( L O c a l l y weighted 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 r e g r e s s i o n m e t h o d d e v e l o p e d by C l e v e l a n d ( 1 9 7 9 ) . arg  1:  arg arg  2: 3:  arg  4:  arg  5:  arg  6:  XYY i s t h e ( 3 . N ) a r r a y w i t h the d a t a - x and - y in c o l u m n s 1 & 2 (and the smoothed y ' s i n c o l u m n 3 on e x i t ) . N i s t h e number o f d a t a p o l n t s . F is the degree of l o c a l i t y f o r the s m o o t h i n g . I.e. the f r a c t i o n of N in each r e g r e s s i o n s t r i p . ITER i s t h e number o f o u t l i e r r o b u s t n e s s iterations 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 option). XYT i s a ( 3 . N ) workarray ( c o n t a i n s the X.Y & weights of the l a s t strip). W i s a (N) w o r k a r r a y ( c o n t a i n s t h e r o b u s t weights).  I f F i s t o o s m a l l . LOWESS b e c o m e s u n s t a b l e . T h i s w i l l 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 t h e F . A n o t e this, 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 unit 6. 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, which 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 'LIBRARY.  IMPLICIT REAL*4 ( A - H . L . P - Z ) REALM XYY(3.N),XYT(3.N),W(N) W R I T E ( 6 , * ) ' > > > > LOWESS' C C C C  C  * F = FRACTION O F POINTS TO BE INCLUDED IN « 10= N * F = NUMBER OF POINTS CORR. TO F IQ=NINT(N*F) IF(10.LT.3)10=3 « SORT XYY T O INCREASING X (COLUMN CALL ISORT(XYY.3.N.1,N.1.3.1)  1)  C 6 10  503 C C C  •  600 C C C  *  CONTINUE 00 503 I=1.N XYY(3.I)=0. XYT(1,I)"O. XYT(2.I)=0. XYT(3.I)=0. W(I)=1. START  ITERATIVE  ROBUSTNESS LOOP  IITR=0 FF=(IO/FLOAT(N)) WRITE(6,•)'N*F = ' . 1 0 . ' F = ' . F F CONTINUE W R I T E ( 6 . * ) 'NUMBER OF ITERATION = ' . I I T R START  LOOP FOR EACH  POINT  LOWESS  STEPS  of on  is  - 298 -  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91v 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116  00 C C C  *  500  1=1.N  DETERMINE 1 0 NEAREST NEIGHBOURS TO  XYYO.I)  X Y T ( 1 , 1 )=XYY( 1 , 1 ) X Y T ( 2 , 1 ) = XYY ( 2 , 1 ) XYT(3.1)=W(I) C  501  C C C C  * *  K1=I+1 K2-I-1 DO 501 K = 2 , I 0 IF(K1.LE.N)THEN D1=ABS(XYY(1,1)-XYY(1.Kl ) ) ELSE D1»ABS(XYY(1.1)-XYY(1.N) ) ENDIF IF(K2.GE.1)THEN D2=ABS(XYY(1,I)-XYY(1.K2)) IF(D2.LE.D1)THEN XYT(1,K)»XYY(1,K2) XYT(2.K)"XYY(2.K2) XYT(3.K)=W(K2) K2=K2-1 ELSE XYT(1,K)»XYY(1,K1) XYT(2.K)=XYY(2,K1) XYT(3,K)=W(K1) K1=K1+1 ENDIF ELSE XYT(1,K)=XYY(1,K1) XYT(2.K)=XYY(2,K1) XYT(3,K)=W(K1) K1=K1+1 ENDIF CONTINUE D = ABS(XYY( 1 , I ) - X Y T ( 1 , 1 0 ) ) IF(D.EO.0)THEN 10=10+1 GOTO 6 1 0 END IF C A L C U L A T E THE WEIGHT FOR EACH X Y T ( K ) FOR THE WEIGHTED REGRESSION .  AND SUMMARY S T A T I S T I C S  SXY=0. SX=0. SY=0 SX2=0. ST=0. DO 502 K = 1 . 1 0 U=(ABS((XYY(1,I)-XYT(1,K))/D))**3 I F ( U . L T . 1 . )THEN T=(1.-U)*»3 ELSE T=0. ENDIF TK=T*XYT(3.K)  -  1 17 1 18 1 19 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160  502  c c c c  c  SXY=SXY+(XYT(1,K)*XYT(2,K)*TK) SX=SX+(XYT(1,K)*TK) SY=SY+(XYT(2,K)*TK) SX2=SX2+((XYT(1,K)**2)*TK) ST=ST+TK CONTINUE  * CALCULATE THE REGRESSION COEFFICIENTS AND THE LOWESS SDIV=(SX2/SY)-SX**2/(ST*SY) IF(SDIV.E0.0..OR.ST.EO.O.)THEN 10=10+1 GOTO 610 ENDIF BW=((SXY/SY)-(SX/ST))/((SX2/SY)-SX**2/(ST»SY)) AW=(SY/ST)-(BW*SX/ST) XYY(3,I)=AW+(BW*XYY(1,I)) 500 CONTINUE IF(IITR.LT.ITER)THEN * COMPUTE MEDIAN OF ABSOLUTE RESIDUALS :RM 510  c c  -  XYT(1,K)=XYT(1.K) XYT(2,K)=XYT(2,K)  C  C c c  299  DO 510 1=1.N W(I)=ABS(XYY(2.I)-XYY(3.I)) CALL ISORT(W,1,N,1,N.1.3,1) M=NINT(I/2.) RM=W(M)  * COMPUTE THE ROBUST WEIGHTS DO 511 1 = 1 ,N U=ABS((XYY(2.I)-XYY(3,I))/(6.*RM)) IF(U.GE.1.)W(I)=0. IF(U.LT. 1 . )W(I) = (1.-U**2)**2 511 CONTINUE IITR=IITR+1 GOTO 600 ENDIF RETURN END  

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