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Parameterization of solar irradiation under clear skies Mächler, Meinrad A. 1983

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PARAMETERIZATION  OF SOLAR IRRADIATION.UNDER CLEAR SKIES  by  Meinrad A. Machler Diplom Maschineningenieur, E i d g . Techn. Hochschule Z u r i c h , 1977  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in the Department of Mechanical E n g i n e e r i n g 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  March 1983  (c) Meinrad Machler, 1983  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by  the head o f  representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  1  written  ABSTRACT  T h i s study compares 5 e x i s t i n g i n s o l a t i o n models l e v e l s of complexity. The model by Leckner integration and by  model,  Hoyt  A.S.H.R.A.E  are is a  simple  seasonal  models  well  model  as  will  increase  on be  received  model  of  atmospheric  aerosols  instead,  several  improvements  to  or to make the a p p l i c a t i o n e a s i e r w i l l be  notably f o r the d e t e r m i n a t i o n of a e r o s o l a t t e n u a t i o n .  Simplification  resulting  the  & Hay  aspect of the s c a t t e r e d r a d i a t i o n . No new  proposed;  the accuracy  presented,  between  the  and  Davies  model. The emphasis of the  comparison was kept on the a t t e n u a t i o n by as  three  represents a s p e c t r a l  the models by B i r d & Hulstrom, parameterization  on  has  horizontal in  been  achieved  ground  easy-to-handle  by  using  correlation  v i s i b i l i t y and a e r o s o l a t t e n u a t i o n , equations.  While  only minor changes i t was necessary  A.S.H.R.A.E. a l g o r i t h m  the  substantially.  most  models  to r e s t r u c t u r e the  i i TABLE OF CONTENTS  Abstract  i  L i s t of Tables  iv  L i s t of F i g u r e s  v  Acknowledgements  vi  Nomenclature and G l o s s a r y I  Introduction  1  1.1  O u t l i n e of the Atmospheric A t t e n u a t i o n  3  1.2  Rayleigh Scattering  9  1.3  Ozone Absorption  12  1.4  Gaseous Absorption  15  1.5  Water Vapor Absorption  18  1.6  Aerosol Attenuation  20  I. 6.1 II  III  vii  The Aspect of V i s i b i l i t y  25  Treatment of D i r e c t R a d i a t i o n  28  II. 1  The S p e c t r a l Model of Leckner  32  11.2  The Model of Davies and Hay  34  11.3  The Model of Hoyt  38  11.4  The Model of B i r d  11.5  The A.S.H.R.A.E. Model  (and Hulstrom)  43 46  Treatment of D i f f u s e R a d i a t i o n  58  111.1  General Approach f o r S i n g l e S c a t t e r i n g  60  111.2  Multiple Scattering  61  111.3  D i f f u s e R a d i a t i o n i n the Models  69  111.3.1  Model of Leckner  69  111.3.2  Model of Davies and Hay  72  111.3.3  Model of Hoyt  74  iii 111.3.4  Model of B i r d and Hulstrom  76  111.3.5  A.S.H.R.A.E.  78  Model  IV  Concluding Remarks  85  V  Further Work  86  VI  References  87  VII Appendix  90  iv L i s t of Tables  Table I  Hoyt's v a l u e s f o r R a y l e i g h  Table II  Hoyt's values f o r a e r o s o l  scattering scattering  Table I I I S o l a r spectrum and a b s o r p t i o n c o e f f i c i e n t s Table IV  A.S.H.R.A.E. c o e f f i c i e n t s  V  L i s t of F i g u r e s  Fig.  1 ; The E x t r a t e r r e s t r i a l  Fig.  2; The Transmittance  due t o R a y l e i g h S c a t t e r i n g  1 1  Fig.  3; The Transmittance  due t o Ozone A b s o r p t i o n  14  Fig.  4; The Transmittance  due t o Absorption by  17  due t o Water Vapor  19  S o l a r Spectrum  8  Mixed Gases Fig.  5; The Transmittance  Fig.  6; O p t i c a l P r o p e r t i e s of Atmospheric A e r o s o l  23  Fig.  7; The Transmittance  24  Fig.  8; Performance of 5 Models, D i r e c t R a d i a t i o n I  51  Fig.  9; Performance of 5 Models, D i r e c t R a d i a t i o n II  52  Fig.  10; Performance of 5 Models, D i r e c t R a d i a t i o n I I I  53  Fig.  11; Performance of 5 Models, D i r e c t R a d i a t i o n IV  54  Fig.  12; Transmittance  due to Water Vapor  55  Fig.  13; Transmittance  due to A e r o s o l  Fig.  14; Performance of the o l d and new ASHRAE Model  57  Fig.  15; M u l t i p l e A t t e n u a t i o n  64  Fig.  16; M u l t i p l e S c a t t e r i n g P a t t e r n  66  Fig.  17; Davies and Hay Layer  79  Fig.  18; Performance of 5 Models, D i f f u s e R a d i a t i o n I  80  Fig.  19; Performance of 5 Models, D i f f u s e R a d i a t i o n II  81  due to A e r o s o l  Absorption  Attenuation  Absorption  Attenuation  Pattern  Model  56  F i g . 20; Performance of 5 Models, D i f f u s e R a d i a t i o n I I I  82  F i g . 21 ; Performance of 5 Models, D i f f u s e R a d i a t i o n IV  83  F i g . 22; Performance of ASHRAE Models, D i f f u s e R a d i a t i o n  84  ACKNOWLEDGEMENT  I would l i k e t o express my s i n c e r e thanks and to  my  supervisor,  appreciation  Dr. Muhammad Iqbal f o r h i s help and advice  throughout the p r o j e c t . I a l s o would l i k e t o  thank  Douglas  T.  B r i n e f o r many i n s p i r i n g d i s c u s s i o n s .  Dr.  John  Hay's comments on the f i r s t d r a f t of t h i s t h e s i s have  been very u s e f u l i n preparing the f i n a l document. The support  of the N a t u r a l Sciences and E n g i n e e r i n g  of Canada i s g r a t e f u l l y acknowledged.  financial  Research C o u n c i l  vi i Nomenclature  A  A.S.H.R.A.E. c o e f f i c i e n t  BA  A e r o s o l forward s c a t t e r i n g r a t i o  AMS  A i r mass  B  A.S.H.R.A.E. c o e f f i c i e n t  C  A.S.H.R.A.E. c o e f f i c i e n t  C toH  F a c t o r s as d e f i n e d  DS  Double s c a t t e r e d  FR  Forward s c a t t e r i n g  I  unsubscripted: t o t a l irradiance;  [W/m«m]  subscripted:  [W/m*m«Mm]  i n text  diffuse radiation  [W/m«m]  ratio  spectral  irradiance  Idif  Diffuse  irradiance  Irr  Direct  K  Davies' and Hay's t u r b i d i t y parameter;  irradiance  Optical  [W/m«m]  [W/m»m] [W/m-m]  depth  M  M u l t i p l i c a t o r i n A.S.H.R.A.E. a l g o r i t h m  N  Molecule  P  Pressure  PR  Attenuation p r o b a b i l i t y  SC  Solar  SH  Scale height  SS  Single  scattered  diffuse radiation  [W/m«m]  TS  T r i p l e scattered  diffuse radiation  [W/m«m]  UG  Concentrated atmospheric gas  [cm]  UO  Concentrated atmospheric ozone  [cm]  UW  P r e c i p i t a b l e water  [cm]  density [kPa]  constant  [W/m-m] [km]  vi i i VIS  Visibility  WO  R a t i o of  f  Hoyt's R a y l e i g h s c a t t e r i n g  g  Hoyt's a e r o s o l s c a t t e r i n g parameter  k  Absorption  ra  Atmospheric  rg  R e f l e c t i v i t y of the ground  s  Pathlength  a  Absorptance  a  Angstrom wavelength exponent  0  Angstrom p a r t i c l e number d e n s i t y  X  Wavelength  T 6  Values  [km]  (AS)/(AA+AS) parameter  coefficient  [1/cm]  albedo (albedo) [km,cm]  [nm,Mm]  Transmittance Zenith angle  [deg]  i n Brackets  (A)  Aerosol  (AA)  Aerosol absorption  (Abs)  Absorption  (AS)  Aerosol  (C)  Carbon d i o x i d e  (G)  Gas  scattering  (0)  Ozone  (OX)  Oxygen  (R)  Rayleigh  (Uv)  U l t r a v i o l e t part of the  scattering spectrum  ix (Vi)  V i s i b l e part of the spectrum  (Sc)  Scattering  (tot)  total  (1)..  General o p t i c a l p r o p e r t i e s  ..(4)  Subscripts  h  on h o r i z o n t a l  surface  n  a t normal  0  E x t r a t e r r e s t r i a l value  X  s p e c t r a l value  incidence  Glossary  Airmass:  The r a t i o of the a c t u a l p a t h l e n g t h versus the s h o r t e s t  possible  pathlength  of  the r a d i a t i o n through the e a r t h ' s  atmosphere. (See page 6/7 and eq. (1-6).  A.S.H.R.A.E:  American S o c i e t y  of  Heating,  Refrigeration  and  number;  the r e a l part  isa  A i r c o n d i t i o n i n g Engineers.  Index  of  refraction:  measure f o r the indicates  the  A  complex  optical  difraction,  dielectric  f u n c t i o n of the wavelength.  properties.  the  imaginary  This  index  part is a  X  Insolation:  The t o t a l s o l a r r a d i a t i o n which reaches the ground.  P r e c i p i t a b l e water:  The  height  of  the  completely  condensed  water i n a column from ground to the edge of the atmosphere (mostly i n [cm]).  S c a t t e r i n g phase f u n c t i o n : gives  the  relative  A f u n c t i o n i n t h r e e dimensions which distribution  of  radiation  after  scattering.  Solar constant: earth  in  The amount of r a d i a t i o n which would the  absence  the  of an atmosphere at mean sun-earth  d i s t a n c e . The exact amount of the s o l a r constant of d i s c u s s i o n s i n the s c i e n t i f i c  Turbidity:  reach  i s subject  community.  'Turbid' a i r r e f e r s to a t t e n u a t i o n of  radiation  in  the atmosphere from sources other absorbing gases.  Zenith  angle:  The  angle between the normal to the ground and  the p o s i t i o n of the sun. The z e n i t h a l t i t u d e add up to 90 degrees.  angle  and  the  solar  1 I.  Introduction  The received has  popular  interest  an a p p r e c i a b l e boost i n the  brought  the  aspects  attendance, i t must be of  in s o l a r r a d i a t i o n and  research  The  century;  subject  research  century  ago  on (for  of s o l a r i r r a d i a t i o n f i n d a wide  i n such f i e l d s  - c a l c u l a t i o n of c o o l i n g  as:  loads f o r a i r  conditioners  agriculture  - performance of s o l a r  - material  a  [1]).  range of a p p l i c a t i o n s  - h e a t i n g of  this  to a broader  infact,  done more than a  methods of p r e d i c t i o n  - f o r e s t r y and  irradiation  s o l a r r a d i a t i o n was  this  e x t i n c t i o n of r a d i a t i o n was  l a s t decade. Although  solar  s a i d that  throughout  example: Lambert  of  i t s research  cells  buildings  deterioration  under  sunlight  - thermal power generation The as  s o l a r r a d i a t i o n reaches the  direct  scattering  (beam) and  radiation  reflection.  or  surface  as  While  diffuse the  s p e c i a l i n t e r e s t to such a p p l i c a t i o n s diffuse  radiation  i n s o l a t i o n and  can  contributes not  amount  of  as  beam  earth  either  radiation  after  radiation  i s of  focusing  devices,  a c o n s i d e r a b l e percentage of  the the  be n e g l e c t e d i n most c a s e s .  While i t i s s a t i s f a c t o r y for total  of the  irradiation  wavelengths, other a p p l i c a t i o n s  many over call  purposes the for  to  whole spectral  know  the  spectrum  of  values  of  2 diffuse  and  direct  irradiation  -  notably  in  the  f i e l d of  photovoltaics. Often, conditions sky  the is  knowledge sufficient  conditions.  of  irradiation  under  cloudless  because peak loads occur under c l e a r  Therefore  this  study  will  be  limited  to  i r r a d i a t i o n under c l e a r s k i e s . Five  models  of  d i f f e r e n t complexity  w i l l be the b a s i s of  t h i s study which has the goal to s i m p l i f y the use and extend a p p l i c a b i l i t y of v a r i o u s p a r a m e t e r i z a t i o n will  be  described  improvement  of  the  briefly,  followed  respective  model.  models. by  Every  suggestions  Fundamentals  the  model f o r an  of  the  s c a t t e r i n g and a b s o r p t i o n of s o l a r r a d i a t i o n through the e a r t h ' s atmosphere are b r i e f l y d i s c u s s e d i n the next  chapter.  3 1.1  O u t l i n e of the Atmospheric A t t e n u a t i o n  The  extinction  of  radiation  constant over the spectrum; t h i s fact  that  blue and  the  in  the  becomes  atmosphere  evident  i s not  through  the  d i f f u s e r a d i a t i o n of the c l e a n c l o u d l e s s sky i s  turns red f o r very h i g h z e n i t h a n g l e s , i . e the sun  c l o s e to the h o r i z o n . Before an attempt can  be  made  to  very model  broadband a t t e n u a t i o n , a short d e s c r i p t i o n of the s p e c t r a l  (as a  function  seems  of  wavelengths  X)  attenuation  phenomena  necessary. Solar  radiation  is  not  emitted  spectrum. I t has a maximum at around 480 end  of  the  over  the  nm which i s at the  blue  v i s i b l e spectrum. F i g . 1 shows the s o l a r  a r r i v i n g above the e a r t h ' s after  homogenously  p a s s i n g through  atmosphere,  as  passing  as  radiation  the atmosphere. A l s o shown i n t h i s  i s a graph of the r a d i a t i o n which reaches after  well  radiation  through  a  the  earth's  figure surface  t y p i c a l atmosphere. I t becomes q u i t e  c l e a r that i n some bands of the spectrum the a t t e n u a t i o n i s very strong and The the  i n others i t i s r a t h e r weak.  e x t r a t e r r e s t r i a l r a d i a t i o n reaches  atmosphere  without  100  km  above  sea  level)  of  virtually  any a t t e n u a t i o n , simply because the outer space p r o v i d e s  an almost surface  (approx.  the outer l i m i t s  complete vacuum. On to  the  earth's  i t s way  the  apparent  s u r f a c e the quantum may  e i t h e r s c a t t e r i n g or a b s o r p t i o n p a r t i c l e s o c c u r s . The  from  solar  be subject to  (or both) when i n t e r f e r e n c e with  p r o b a b i l i t y of i n t e r a c t i o n grows with  the  4 increasing  density  of  suspended  matter.  d e n s i t y the p r o b a b i l i t y of i n t e r a c t i o n  At a given  particle  i n c r e a s e s with i n c r e a s i n g  pathlength. The  probability  transmittance  and  for  must  non-interference  be  a  fraction  called  of u n i t y . There i s no  interference in  vacuum,  unity.  medium i s t o t a l l y opaque and thus the mean free  If  the  therefore  is  the  transmittance  p a t h l e n g t h zero, the t r a n s m i t t a n c e becomes most  intermediate  amount  of  cases  directly  we expect  transmitted  zero  as  becomes  well.  For  an e x p o n e n t i a l decay of the  radiation  at  a  particular  wavelength. The  main  strongly  in  absorption") mainly The  selected and  of  a t t e n u a t i o n a r e gases, which  parts  of  the  spectrum  absorb  ("bands  of  a l s o s c a t t e r , and p a r t i c l e s i n the a i r , which  s c a t t e r , but have (weaker) a b s o r p t i v e p r o p e r t i e s as w e l l .  term  quantum  sources  "absorption" energy  is  describes  converted  energy) and the quantum ceases  to  the  interaction  heat  Under exists in  the a  the  (or some other form of  to e x i s t .  "Scattering"  change i n d i r e c t i o n of the quantum through c o r p u s c l e without  where  means  a  an i n t e r a c t i o n with a  any l o s s of the quantum energy. assumption  homogenuous  that medium,  only the  one  type of a t t e n u a t o r  transmittance  for  any  p a r t i c u l a r wavelength can be w r i t t e n as: r(1)  X  = exp-{K(1) } X  The term ' r' stands  (1-1)  f o r t r a n s m i t t a n c e and 'K' f o r the o p t i c a l  depth. The i r r a d i a t i o n can now be w r i t t e n a s :  5 Irr  nX  another combination be  = I  • exp-{K(l) } X  OnX  attenuators  can  (1-2)  be added to t h i s atmosphere.  of the i n f l u e n c e s of two  d e s c r i b e d by way  (or more)  attenuators  The can  of m u l t i p l i c a t i o n of the t r a n s m i t t a n c e s f o r  the v a r i o u s a t t e n u a t o r s : r ( t o t ) = T(1) -T(2) -r(3) • X X X X  (1-3)  H t o t ) = e x p - ( K d ) + K(2) + K(3) +...} X X X X  (1-4)  or  In atmospheric a p p l i c a t i o n s those K's  represent the  optical  depth of atmospheric c o n s t i t u e n t s , such as ozone, dry a i r , water vapour, a e r o s o l s or mixed gases. T h i s leads to of  a  transfer  direct  formulation  equation which d e s c r i b e s the amount of incoming  radiation: Irr  nX  Naturally and  the  = I  OnX  one  the s c a t t e r i n g  -exp-{K(l) + K(2) + K(3) +...} X X X  (1-5)  would l i k e to d e f i n e e x a c t l y the of  solar  radiation  at  every  absorption wavelength.  R e s t r i c t i o n s i n the r e s o l u t i o n of the measuring equipment and i n the  p r a c t i c a b i l i t y of data h a n d l i n g make i t necessary  to  devide  the spectrum i n a f i n i t e number of small s p e c t r a l bands. More than 98% of the s o l a r 290  nm  and  4000 nm.  For reasons  radiation  is  emitted  between  to be e x p l a i n e d l a t e r , most of  the r a d i a t i o n o u t s i d e the above l i m i t s does not  reach the  earth  6 and  i s therefore  of no i n t e r e s t . T h i s study uses the s p e c t r a l  d i v i s i o n of Thekaekara  [2] which d i v i d e s the spectrum  into  144  p a r t s w i t h i n the range of 290 nm to 4000 nm as o u t l i n e d i n Table III.  The  bandwidths  at the longer end narrower  close  intervals  absorption  to  would  absorption  spectrum  was  allow  the  based World  but  the  a  spectrum  with  use of many sets of  in  adopted  are  by  -  available  coefficients proposed  4000 nm  not  coefficients  Thekaekara's  spectrum,  a r e not as small as d e s i r a b l e - e s p e c i a l l y  the  literature.  by  NASA  and many sets of  on  i t . Recently,  Radiation  Center  a  new  has been  adopted by the World M e t e o r o l o g i c a l O r g a n i z a t i o n [ 3 4 ] . To  understand  the  transfer  equation,  the  mathematical  h a n d l i n g of the s p e c t r a l a t t e n u a t i o n f o r the v a r i o u s atmospheric c o n s t i t u t e n t s has to be understood. In the absence of a complete and  comprehensive  theoretical spectral aerosol  attenuation,  a b s o r p t i o n and  water  set of  measured  transmittances  data  this  f o r dry  ozone  absorption,  vapor  absorption  study  uses  a i r scattering,  mixed atmospheric gas as  criteria  f o r any  p a r a m e t e r i z a t i o n . These v a l u e s were taken from Leckner [ 3 ] . It  is  important t o determine the p a t h l e n g t h of r a d i a t i o n .  The s h o r t e s t p a t h l e n g t h p o s s i b l e f o r the s o l a r r a d i a t i o n  i s with  the sun i n the z e n i t h . Commonly the p a t h l e n g t h i s expressed  in  non-dimensional  form of the 'airmass'. At sea l e v e l , an airmass  of u n i t y i s the  shortest  zenith  angles  higher  pathlength  then  zero  (zenith  the  airmass  angle  6=0); at  i n c r e a s e s . The  s i m p l e s t mathematical f o r m u l a t i o n to determine the airmass the f o l l o w i n g  assumptions:  - there i s no c u r v a t u r e of the e a r t h  uses  7 - the index of r e f r a c t i o n  of the a i r i s equal to u n i t y .  Thus the airmass takes on the form:  AMS  Other  = l/cos(0)=sec(0)  f o r m u l a t i o n s that e l i m i n a t e  are i n t r o d u c e d i n l a t e r  (1-6)  the  restrictive  assumptions  s e c t i o n s whenever they are used.  9 1.2  Transmittance  Due  To  Dry  Air  Scattering  (Rayleigh  scattering)  Approximately 80% of the atmosphere and  nitrogen  and  radiation  -  of  than the  wavelength  the  atmosphere,  Rayleigh. the  The  biatomic  Rayleigh  property found  scattering the  that  the  i s wavelength  transmittance  due  oxygen,  to  the  but  the  molecular that  transmission  than the wavelength  of the r a d i a t i o n becomes an exact f u n c t i o n of the  fourth  of the wavelength and can be put i n t o the f o l l o w i n g  X  is a  dependent. Under the assumption  dry a i r with molecules much smaller  r(R)  largest  i s of second order between 290 and 4000 nm.  s c a t t e r i n g molecules are p e r f e c t spheres,  through  of  founded  next  molecular s c a t t e r e r as w e l l . I t a l s o absorbs r a d i a t i o n absorptive  nitrogen  and molecules are j u s t that - i s w e l l  goes back to the work of Lord  constituent  of  a f f e c t s r a d i a t i o n mainly by s c a t t e r i n g . The theory  of s c a t t e r i n g by p a r t i c l e s much smaller the  consists  power  form:  = exp{K(R)-AMS}  (1-7)  where the o p t i c a l depth K(R) at u n i t y AMS  i s given  as (Penndorf  [4]): 2 K(R)  The  s  term 'N' g i v e s  density  at  refraction.  2  =8-7T.(n -1)-N /  sea  (3«N  o  4 -X)  the molecule d e n s i t y level  and  'n  (1-8)  with  subscript  'N s u b s c r i p t o' the s'  the  index  of  10  Leckner departure  [3] uses an exponent of  of molecules  -4.08  to  allow  from the theory of p e r f e c t  for  spheres  the  (based  on R a y l e i g h ) :  r(R) = e x p { k ( R ) - ( X  -4.08  )-AMS}  (1-9)  X  with:  k(R)  Thus the t r a n s m i t t a n c e due study  represented  to dry a i r s c a t t e r i n g as used i n t h i s Eq.  (1-10). A p l o t  of  T ( R , X ) a g a i n s t X i s shown i n F i g . 2. As we  can  see  2,  t r a n s m i t t a n c e due  by  (1-10)  and  the  is  X i n [jum]  = -0.008735  Eq.  (1-9)  to s c a t t e r i n g by molecules  i n c r e a s i n g X. Thus R a y l e i g h s c a t t e r i n g has in the short end  of the s o l a r  spectrum.  from F i g .  i n c r e a s e s with  i t s strongest  effect  11  F i g u r e 2, The Transmittance due  to R a y l e i g h S c a t t e r i n g  12 1.3  Transmittance  due t o Ozone Absorption  Oxygen atoms not only come i n p a i r s but a l s o i n t r i p l e t s i n form  of ozone. Not only does ozone s c a t t e r  Eq. 1-9),  but i t a l s o absorbs  r a d i a t i o n . Ozone absorbs  in the s h o r t e r wavelengths of mainly  responsible  the  solar  The  following  spectrum  f o r the a b s o r p t i o n  r a d i a t i o n , e x c e s s i v e amounts of such earth.  transmittance  (this i s included i n  of  would  and  the  harm  radiation  ultraviolet  the  life  on  due t o ozone can be d e s c r i b e d by the  equation: r(O) = exp{-k(0) -UO-AMS} X X  The  i t is  (1-11)  term 'UO' denotes the amount of ozone i n a v e r t i c a l column.  Depending on season  and  latitude  the value  of  'UO'  varies  between UO=0.20 [cm] and UO=0.50 [cm] a t NTP. Furthermore, concentration altitude  the  shows  with  a  a  peak  vertical distinct at  profile  of  concentration  around  22 km  m i d l a t i t u d e s . The peak e l e v a t i o n decreases  above  the  at  ozone  very  high  sea l e v e l f o r  s l i g h t l y towards  the  poles. Ozone a b s o r p t i o n c o e f f i c i e n t s k(O) taken  from  Vigroux  Leckner  used i n t h i s study were  [3] who i n turn took the c o e f f i c i e n t s from  [ 5 ] . These c o e f f i c i e n t s a r e reproduced  i n Table I I I . The  t r a n s m i t t a n c e of ozone as a f u n c t i o n of wavelength i s shown Fig. values  3.  As  we  smaller  in  see, the t r a n s m i t t a n c e approaches zero f o r Xthan  300 nm.  Below  290 nm  no  radiation  is  13 transmitted  due  to  the t o t a l a b s o r p t i o n  by ozone. T h i s i s the  reason why no p o r t i o n of r a d i a t i o n below 290 nm account as mentioned above i n S e c t i o n  1.1.  is  taken  into  14  1.0 2.0 Wavelength (micrometers)  Figure  3, T r a n s m i t t a n c e  due t o Ozone  Absorption  3.0  4.0  15 1.4  Absorption  Among  all  ozone has a direction  the dry a i r gases ( i . e . e x c l u d i n g water  distinct and  ozone was  by Atmospheric Mixed Gases  it  concentration  i s due  profile  to t h i s reason  in  the  vapor), vertical  that the a b s o r p t i o n  t r e a t e d s e p a r a t e l y i n S e c t i o n 1.3.  All  the  remaining  gases (such as oxygen and carbon d i o x i d e , e t c . ) are more or homogenously  distributed  in  the  c o n c e n t r a t i o n does not vary g r e a t l y .  atmosphere In  this  by  and  study  less their  they  are  referred  to as "mixed gases". Among these mixed gases, the main  absorbers  of s o l a r r a d i a t i o n between 290  d i o x i d e and  occurs  4  shows  the  monochromatic  of  the  visible  of an oxygen band at 760  threshold  transmittance  by atmospheric mixed gases. Most  outside  exception  4000 nm are  of  human  perception.  of  the  the  The  the transmittance  which  is  right  due  f o r an  to mixed gases would  T ( G ) = exp{-K (G)-AMS) X X  (1-12)  that the a t t e n u a t i o n by mixed gases occurs  narrow bands with steep f l a n k s . The is  much  the  i r r a d i a t i o n w i t h i n those  coefficients  at  f i r s t expectation  be something l i k e the f o l l o w i n g :  too  absorption  nm  probably  shows  set of wavelength  mixed  i n t e r v a l s are c o n s t a n t .  gases  in  intervals  crude f o r the assumption that the a t t e n u a t i o n  for  to  the  to formulate  Fig.4  due  p a r t of the spectrum with  equation  But  carbon  oxygen.  Fig. absorption  and  and  Absorption  are t h e r e f o r e averaged over  the  i n t e r v a l s . There are many approaches to model the a b s o r p t i o n  by  16 mixed atmospheric gases (Fowle [ 6 ] , Howard [ 7 ] ) . T h i s study uses the  approach  of  Leckner  [3]  who  in  p u b l i c a t i o n s of Yamamoto [ 8 ] , McClatchey  part based h i s work on [9] and Elterman [10].  The shape of a b s o r p t i o n bands can be c l a s s i f i e d . Each c l a s s of a b s o r p t i o n band t h e o r e t i c a l l y c a l l s  for a different  function  to d e s c r i b e the e x t i n c t i o n , as o u t l i n e d by Goody [11] and T i w a r i [12].  W i t h i n the s p e c t r a l range of 290 t o 4000 nm however, i t i s  sufficiently  accurate  to  use  one  function  t r a n s m i t t a n c e due t o the a b s o r p t i o n by mixed  only;  atmospheric  the gases  can then be d e s c r i b e d by the f o l l o w i n g e q u a t i o n :  / T(G)  X  = exp-  1.41-  K(G)  • AMS  I 0.45 \(1+118.3-K(G) -AMS) X  \ / / '  (1-13)  The c o e f f i c i e n t s K(G) used with t h i s equation to produce F i g . where taken from Leckner [3] and are reproduced i n Table I I I .  4  1 7  F i g u r e 4 , Transmittance due  to Absorption by Mixed Gases  18 1.5  A b s o r p t i o n by Water  The  attenuation  behaviour  of  complex than the one of ozone and mixed  atmospheric  water  vapor  is  i s s i m i l a r to the one  f a r more of  gases. Water absorbs i n c e r t a i n bands of the  spectrum with a very steep r i s e of the a b s o r p t i o n at the of  these  bands  the  flanks  which are o f t e n c l o s e to bands of almost  total  transparency. S i m i l a r to the mixed gases t h i s study adopted water  absorption  treatment  of  Leckner  [3]  who  the  o b t a i n e d the  t r a n s m i t t a n c e equation f o r water vapor as:  1  0.2385- UW •  • k(W)  0.45 / -AMS) /  «k(W) X  1  i s of the same form as the t r a n s m i t t a n c e equation f o r  mixed gases - Eq.(I-13) describing  \ (1-14)  (1+20.07 • UW  Eq.(l-14)  • AMS  the  varying  with amount  the of  extension water.  of  The  a b s o r p t i o n c o e f f i c i e n t s used t o produce F i g . 5 w i t h  an set Eq.  element of water (1-14)  are reproduced i n Table I I I ; they were taken from Leckner [ 3 ] . The  total  amount  of the p r e c i p i t a b l e water  column can be estimated from a number of dew  point  temperature  Leckner [ 3 ] ) .  in a v e r t i c a l  observations  such  as  or p a r t i a l p r e s s u r e of water vapor (see  19  JWavelength D  2.0 (micrometers)  F i g u r e 5, Transmittance due to Water A b s o r p t i o n  3.0  20 1.6  Attenuation  The  by Atmospheric  Aerosol  a c c u r a t e mathematical d e s c r i p t i o n of the a t t e n u a t i o n  atmospheric a e r o s o l s i s extremely d i f f i c u l t . T h i s the  v a r i e t y which e x i s t s i n the most important  i n f l u e n c e the determination  - the p a r t i c l e s i z e and  is  shown  by by  parameters which  of a e r o s o l a t t e n u a t i o n :  i t s distribution  - the d i e l e c t r i c p r o p e r t i e s which determine the  absorption  - the shape of a p a r t i c l e - the wavelength - the number of a e r o s o l p a r t i c l e s per u n i t volume - a e r o s o l s both s c a t t e r and Probably the  absorb.  the most d i f f i c u l t  task  p a r t i c l e s i z e d i s t r i b u t i o n and  number  and  the  size  the  determination  suggestions  to  difficulty  i s the c u t - o f f p o i n t : The  the more d i f f i c u l t The  Rayleigh  determine  d i s t r i b u t i o n of a e r o s o l s i n a given  volume of the atmosphere - none of them e n t i r e l y One  satisfactorily.  smaller  the  particle  i t becomes to prove i t s e x i s t e n c e . theory  of  s c a t t e r i n g does not d e s c r i b e  s c a t t e r i n g by a e r o s o l s because the a e r o s o l p a r t i c l e s  within  the  spectrum  developed a theory which t r e a t e d larger  than  the  of the  interest.  Gustav Mie  scattering  by  the  (or most of  them) are s u b s t a n t i a l l y l a r g e r than the wavelength of the radiation  of  i t s o p t i c a l p r o p e r t i e s . Over  the decades there have been numerous the  is  solar [13]  particles  wavelength of the i n c i d e n t r a d i a t i o n but  this  21 theory it  i s a l s o r e s t r i c t e d to s p h e r i c a l  proved  particles.  Nevertheless  to be a v a l u a b l e t o o l f o r the d e s c r i p t i o n of a e r o s o l  s c a t t e r i n g , though a treatment w i t h i n the range  of  engineering  a p p l i c a t i o n s r e q u i r e s some s i m p l i f y i n g assumptions. Under the assumption that the p a r t i c l e diameter distribution  follows  a  power  n e g l i g i b l e or i s nonexistent, form  of  a  rather  simple  law  and  that  (or r a d i u s )  absorption  one can apply Mie's theory relation  refraction  the  index  has to be small or zero which i n t u r n means that  the a e r o s o l s have t o have d i e l e c t r i c p r o p e r t i e s . The water  i n the  of wavelength dependence.  N e g l i g i b l e a b s o r p t i o n means that the complex p a r t of of  is  fact  that  condenses around p a r t i c l e s i n c r e a s e s the a p p l i c a b i l i t y of  t h i s assumption. A s t i l l widely assumptions  was  first  formulation  f o r the  used approach t o implement  published spectral  by  Angstrom  transmittance  [14,15]. due  these His  to aersosol  a t t e n u a t i o n takes on the f o l l o w i n g form: a  r(A) = exp{0 • AMS / (X )} X The c o n s t a n t s at  (I-15)  a and 0 are o b t a i n e d with f i l t e r  two wavelengths. With only two equations  the  result  measurements  to determine a and 0  i s a s e t of two c o n s t a n t s . At a given wavelength the  value of a i s a f u n c t i o n of the p a r t i c l e s i z e . I t becomes  quite  c l e a r , that f o r every wavelength and f o r every c l a s s of p a r t i c l e size  a  unique  a  i s valid.  An average constant  a i s thus an  averaged value over 2 parameters. The p a r t i c l e s i z e  distribution  of the atmospheric a e r o s o l s u s u a l l y f o l l o w s a closely.  Exceptions  from  this  rule  have  power been  law  quite  observed f o r  22 maritime single  a e r o s o l s and source  f o r unusual  accumulations  of dust  l i k e v o l c a n i c e r u p t i o n s or f o r e s t  from  f i r e s of l a r g e  e x t e n s i o n . The average a over the whole spectrum between 290 4000 nm has a value between 0.9 observations  around  and  2.0  with the  most  r a d i i of the a e r o s o l p a r t i c l e s called  and  frequent  a=1.3. Fig.6 shows the r e l a t i o n s h i p of the  t u r b i d i t y parameters at p a r t i c u l a r wavelengths and  Angstrom  a  for  various  (from McCartney [ 1 6 ] ) .  the constant  0 ' P a r t i c l e Number D e n s i t y ' .  T h i s might be m i s l e a d i n g because 0 i s not only a f u n c t i o n of the p a r t i c l e numbers per u n i t volume but even more a f u n c t i o n of the a e r o s o l mass d e n s i t y . For a p p l i c a t i o n s of s p e c t r a l i r r a d i a n c e values i t might of  interest  to  increase  the  number  f i l t e r measurements are taken to constant concern value  l a y s with the of  determine  parameterization  the s p e c t r a l t r a n s m i t t a n c e transmittance  determine  one  of wavelengths at which 0  not  the l a t t e r  value  over  models  as  a  the  exact  i s not as important  as the  the  i t is sufficient  and  whole  spectrum.  respect  to  the  accuracy  t r a n s m i t t a n c e . F i g . 7 shows the  of  the  transmittance  was  broadband due  to  a t t e n u a t i o n as a f u n c t i o n of wavelength f o r average 0=0.1 a of u n i t y .  To  to o b t a i n 0 as a constant  i t seems that Angstrom's c h o i c e of wavelengths in  only  but as a f u n c t i o n of the wavelength. T h i s study's main  average  and  be  a  good  aerosol aerosol and  an  23  TOTAL SCATTERING AND EXTINCTION BY HAZE AEROSOLS  Wav*»!ennfh  gure  6, O p t i c a l  Properties  Wavelength.  (From  (um)  of  Aerosols  McCartney [ 1 6 ] )  as  a  Function  of  24  F i g u r e 7, Transmittance  due  to A e r o s o l  Attenuation  25 The  1.6.1  The  Aspect of V i s i b i l i t y  determination  of the transmittance  due to a e r o s o l s  t u r b i d i t y measurements i s q u i t e a formidable desirable  to  link  the  measureable q u a n t i t y problem  arises  defines  the  difference  with  the  0.98  background,  of  human  times  visibility  maximum  as  apparent  immediately. N e g l e c t i n g  sensitivity  to a f i x e d q u o t i e n t  d e f i n i t e advantage that the  horizontal  The  Linke [17]  a  contrast  invisible at  Rayleigh  and  we o b t a i n :  k(A,0)= (3.912/VIS)-0.01162  of  a  f o r the The  of c o n t r a s t has the  optical  depth  follows  scattering:  [1/km]  s c a t t e r i n g (0.01162 1/km)  black  X=550 nm.  (1-16)  s c a t t e r i n g i s taken i n t o account, the o p t i c a l  for  first  brightness  any amount of molecular  k(A,0)= l n ( l - 0 . 9 8 ) / V I S  If R a y l e i g h  perception  the  be  t o a more e a s i l y  d e f i n i t i o n of v i s i b i l i t y .  the o b j e c t i s c o n s i d e r e d  human eye, which has a linking  of  would  2%: I f the apparent b r i g h t n e s s of an i d e a l  body i s more than diffuse  a e r o s o l transmittance  l i k e the h o r i z o n t a l v i s i b i l i t y .  threshold  of  task. I t  from  depth  [10] has t o be s u b t r a c t e d  [1/km]  (1-17)  A v a r i e t y of authors have e s t a b l i s h e d r e l a t i o n s between the v e r t i c a l a e r o s o l d e n s i t y d i s t r i b u t i o n and  the  aerosol  density  26 near the ground. A very convenient McClatchey  [9] who e s t a b l i s h e d the ' s c a l e h e i g h t ' of an a e r o s o l  distribution. ground  way to do so was p u b l i s h e d by  'Scale h e i g h t '  level  aerosol  is  the  density  to  equivalent obtain  pathlength  the  same  at  aerosol  a t t e n u a t i o n as f o r the v e r t i c a l path through the atmosphere: SH = K ( v e r t i c a l p a t h ) / Mground, h o r i z ) It  is  quite  clear  that  the s c a l e h e i g h t becomes higher  d e c r e a s i n g p o l l u t i o n because the highest  (1-18)  aerosol  pollution  d e n s i t y c l o s e to the ground. Under normal  - no a e r o s o l of one s i n g l e source, f i r e - i t was found less  linearily  [18],  with data  shows  the  circumstances  i.e volcanic eruption, forest  that the s c a l e height SH i n c r e a s e s  with  with  increasing  visibility.  from McClatchey [10], found  more  Buckius  or  and King  the s c a l e height  a  visibility  of  5 km to be SH(5)=1.132 km. These r e s u l t s f i n d support  at  of 23 km to be SH(23)=1.577 km and at a v i s i b i l i t y  from Zuev [ 1 9 ] . With Eqs.  (1-17)  and  (1-18),  0  i n data  can  now  be  w r i t t e n as [ 1 8 ] : a /3.912 0=(.55) • \VIS Although size  X /( 1 .577-1 . 1 32)•(VIS-5) \ .01162 • +1.132  J\  23 - 5  a guess has to be made about the value of the  distribution  exponent  'a', the equation  Eq.  For  (1-19) i n  results fact  values  for  of 'a' d i f f e r e n t  conjunction  with  equal v i s i b i l i t i e s ,  particle  proves very  under usual a e r o s o l d i s t r i b u t i o n s where 'a' i s 1.3.  (1-19)  /  commonly  valid around  from 1.3 however, the use of  Eq.  (1-15)  yields  diverging  which can be a t t r i b u t e d to the  that the maximum s e n s i t i v i t y of the eye  (at  550 nm)  does  27 not  coincide  w i t h the maximum s p e c t r a l i r r a d i a n c e  (at 480 nm).  T h i s study t h e r e f o r e suggests a s l i g h t m o d i f i c a t i o n of 19)  so  Eq.  that equal t r a n s m i t t a n c e s f o r equal v i s i b i l i t i e s  (I-  can be  obtained: 0=(3.912/VIS-.O1162)•{(16.2385+VIS)•(F-G-a)+H}  (1-20)  where F= 2.3575E-02  (l-20a)  G= 9.387E-03  (l-20b)  H= 0.278863  (l-20c)  and  and  These Eqs. (1-19) and (1-20) do not cover  visibilities  i n fog  where the p a r t i c l e s become very b i g . The  link  between  ground  visibility  and a e r o s o l o p t i c a l  depth w i l l be used i n the p a r a m e t e r i z a t i o n models as o u t l i n e d i n the next  chapter.  28 II  Treatment of D i r e c t R a d i a t i o n There are c u r r e n t l y 3 l e v e l s of models to  estimate  (and d i f f u s e ) r a d i a t i o n on c l e a r days. Models which use transmittances  and  irradiances  and  perform  direct spectral  numerical  i n t e g r a t i o n s represent the h i g h e s t l e v e l . T h i s study p r e s e n t s rather  simple  for t h i s  model  of t h i s kind by Leckner  [3] as an example  approach.  The  middle  parameterization the spectrum  level  is  models  which do not perform  but s t i l l  split  represented  by  presents  three  [20], Hoyt [21] and The  models  complex  i n t e g r a t i o n s over  constituents.  This  of t h i s kind from Davies and  from B i r d and Hulstrom  lowest l e v e l of  less  the d e t e r m i n a t i o n of the e x t i n c t i o n  i n t o v a r i o u s p a r t s r e p r e s e n t i n g atmospheric study  a  models  Hay  [22].  simulates  insolation  through  j u s t one equation with airmass as a v a r i a b l e and constants  valid  f o r e n t i r e months not c o n s i d e r i n g any changes of the atmosphere. The  most  widely known such model i s the A.S.H.R.A.E. a l g o r i t h m  [31] which w i l l be presented The b a s i c problem of solution  of  wavelength and  the  modeling  transfer  direct  equation  study.  insolation  (Eq. 1-5)  is  for a p a r t i c u l a r  n  =  4000 r Ii 290 J  OnX  • r(tot)  X  • dX  (II-1)  a model of the top l e v e l the main problem i s the d e f i n i t i o n  of the s p e c t r a l t r a n s m i t t a n c e s f o r the v a r i o u s a t t e n u a t o r s the  the  i t s i n t e g r a t i o n over the whole spectrum:  Irr(tot)  For  i n S e c t i o n II.5 of t h i s  whole spectrum. S e c t i o n s 1.1  through  1.6  have shown one  over way  29 of s o l v i n g t h i s t a s k . Other models - not presented here - d i v i d e the atmosphere i n t o a number of homogenous l a y e r s f o r which a t t e n u a t i n g p r o p e r t i e s are d e f i n e d i n d i v i d u a l l y The  transfer  the  [24,25].  equation f o r a p a r t i c u l a r wavelength or w i t h  enough accuracy f o r a very narrow band i s given below: T = r(R) T ( A ) -T(W) T ( G ) «T(0) X X X X X X However,  i t i s mathematically  (II-2)  not sound t o formulate the  broad-  band t r a n s m i t t a n c e as f o l l o w s : T = r(R)«T(A)•T(W)-T(G)'T(O) Despite  this  the middle  obvious  mathematical  l e v e l presented  of the s p e c t r a l l y  (II-3) i n c o n s i s t e n c y a l l models of  i n t h i s study  f o r c e the d e t e r m i n a t i o n  i n t e g r a t e d i r r a d i a t i o n values i n t o the form of  Eq.(II-3) or s i m i l a r e x p r e s s i o n s . E q . ( l I - 3 ) i s t h e r e f o r e the  'broadband t r a n s f e r equation', although there i s an inherent  error  in t h i s formulation. Because  despite  the  the  appropriate  'broadband  obvious to  assess  the e r r o r  To what extent can a over  the  attenuator.  whole  t r a n s f e r equation' i s widely  mathematical  The problem can be s p l i t  1.)  called  i n t o two  it  seems  inherent i n t h i s f o r m u l a t i o n . separate q u e s t i o n s :  "broadband  spectrum,  deficiency,  used  be  transmittance", established  for  averaged a single  30 2.)  To what  extent  attenuators  be  can  the  used  "broadband  averaged transmittance  assumed  ideal  occurs over following  condition  for  in conjunction  with  multiplicatively  each other and with the s o l a r The  transmittances"  constant.  for a  where  single  only  one  the whole spectrum - i s now  attenuator  -  an  kind of a t t e n t u a t i o n  d e f i n e d by means of  the  equation:  T(1) =  4000 r {I • T(1) • dX 290J OnX X  (II-4)  4000 r I • dX 290 OnX J  where: 4000 f Ii 290^ It  is  OnX  • dX = SC  possible  (II-5)  to i n t e g r a t e t h i s equation  r e s p e c t i v e T as a f u n c t i o n of the airmass  and  and d e f i n e  the  the d e n s i t y of  the  a t t e n u a t o r . T h i s f u n c t i o n - which w i l l most l i k e l y closed  form  accuracy  -  can  then  be  parameterized  as a f u n c t i o n of the airmass  attenuating  medium.  All  models  of  and  the  with  not be  in  a  any d e s i r e d  density  of  the  the middle l e v e l use  this  approach. The easier  second q u e s t i o n understanding  a t t e n u a t o r s . The  Eq.  (II-4)  the  so  example  readily will  be  answered:  For  an  r e s t r i c t e d to  two  e f f e c t of three or more a t t e n u a t o r s can then  e x t r a p o l a t e d . For one by  i s not  a t t e n u a t o r the t r a n s m i t t a n c e  above. For two  is  be  described  a t t e n u a t o r s the c o r r e c t equation  31 becomes: 4000 r r(tot) =  It  290 J  11  OnX  T(1)  •  T ( 2 ) • dX  •  X  X  (II-6)  SC  i s now q u i t e obvious,  t h a t the product  two d i f f e r e n t a t t e n u a t o r s i s not the same as the  integral  the product one  of  a product  Despite  this  Eq.(lI-6)  f u n c t i o n s degenerates  mathematical  verdict  functions,  over  very  most p a r a m e t e r i z a t i o n rather  small  l a r g e p a r t s of the spectrum only one of the  a t t e n u a t o r s i s dominant or complete one a t t e n u a t o r , where another However,  unless  to a constant.  models use t h i s approach. The e r r o r s i n v o l v e d a r e because  because  of two f u n c t i o n s i s not the same as  of two i n t e g r a l s of the same two  (or both) of those  of Eq.(II-4) f o r  the  extinction  occurs  through  would be q u i t e s t r o n g .  user of p a r a m e t e r i z a t i o n models i s c a u t i o n e d  not to use s i n g l e elements of the models because the performance of a s i n g l e  transmittance  might  be  less  no  attempt  accurate  then  the  product. On  the  lowest  t r a n s f e r equation; as  a  level  to s o l v e the  i n s t e a d , a simple power law with the  v a r i a b l e i s adopted. The next  models i n d e t a i l .  i s made  airmass  sections w i l l outline  these  32 I1.1  The  S p e c t r a l Model by  Leckner  T h i s study presents the s o l a r [3]  as  an  numerical  example  the  models  of  i n t e g r a t i o n methods are used to  insolation Leckner's  for  i n s o l a t i o n model  incident  on  the  model where presented  earth's  by  the top l e v e l , where calculate  the  s u r f a c e . The  in parts  Leckner  in  the  solar  elements of  sections  1.2  through 1 . 6 . The  basic  equation of Leckner's  model of s o l a r  insolation  has the f o l l o w i n g form: 4000 r Irr = I I - T ( R ) -T-(A) «r(W) - T ( G ) - T ( O ) -dX n 290 J OnX X X X X X  (II-7)  The v a r i o u s s p e c t r a l t r a n s m i t t a n c e s are c a l c u l a t e d with respective  expressions discussed e a r l i e r  2 to 6. The performance of Leckner's of  water  vapor,  ozone and  (at the end of Chapter comparison.  turbidity  i n chapter  model f o r  I I ) . A l s o shown are  ( 1 - 1 9 ) . The  the z e n i t h angle; unless otherwise ( 1 - 6 ) was  to  predict  other  amounts  models  noted  assumed.  as  a  Turbidity  h o r i z o n t a l a x i s shows  f o r a p a r t i c u l a r model,  used to o b t a i n the airmass. Leckner's  model  tends  s l i g h t l y higher values then the other models except  f o r c o n d i t i o n s of high  turbidity.  While the model of Leckner small  various  The v e r t i c a l a x i s shows the i r r a d i a n c e i n watts per  were o b t a i n e d with Eq.  Eq.  I, s e c t i o n s  i s shown i n F i g ' s . 8 to 11  square meter. Mean sun to e a r t h d i s t a n c e was values  the  calculator,  exceeds the p o s s i b i l i t i e s  of  a  i t i s n e v e r t h e l e s s easy to use on a computer.  33 Because of the good models)  combined  time the Leckner  performance with  (compared  to  more  elaborate  a t o l e r a b l e amount of r e q u i r e d  model has been chosen as  a  standard  computer for  development of components f o r the other models presented  the  in this  study. If  f u r t h e r s t u d i e s should show some s i g n i f i c a n t  systematic  d e v i a t i o n s of Leckner's model from r e a l i t y , c o r r e c t i o n s  on  the  elements of the other models should be easy to perform. The  next  sections  present  these  models  improve t h e i r a p p l i c a b i l i t y and/or  and  some changes suggested  practicability.  to  34 II.2  The Model by Davies and Hay  Davies and Hay irradiation  on  [20] present  horizontal  a  model  surfaces.  to  broadband  Chapter vapor  transmittances"  "transfer  mixed gases absorb  model  equation  as o u t l i n e d at the beginning of  II (Eq. II-3) with a s l i g h t m o d i f i c a t i o n . Because and  solar  The b a s i s f o r t h i s  (from now on r e f e r r e d to as Model A) i s the for  calculate  water  i n p a r t s of the spectrum where no  ozone a b s o r p t i o n occurs, the t r a n s m i t t a n c e s f o r water vapor (and gases) and ozone are not m u l t i p l i e d and the  following  approach  i s used i n s t e a d : I r r = SC • cos(f5) . [ (0) • r (R)-a(W) ] • r (A) h  (II-8)  T  The  absorptance  transmittance  due  to  ozone  and  the  are given as f o l l o w s ( L a c i s and Hansen  the ozone a b s o r p t i o n band i n the u l t r a  resulting [ 2 6 ] ) . For  violet:  0.02118 • UO a(0,Uv) =  and  1.082  • UO  .805 (1 + 138.6-UO)  +  0.0658 • UO 3 1 + O03.6-UO)  (II-9b)  the a b s o r p t i o n due to ozone as the sum of the above: a(0) = a(0,Uv) + a(0,Vi)  and  (II-9a)  f o r the band i n the v i s i b l e p a r t of the spectrum: a(0,Vi) =  and  :— -4 2 1 + 0.042 • UO + 3.23-10 • (UO)  finally  f o r the t r a n s m i t t a n c e due to ozone:  (ll-9c)  35 T(0) = 1 - a(0) The  combined  atmospheric  (II-9)  absorptance  gases  by  i n Model  water  vapor  and  mixed  A was taken from L a c i s and Hansen  [26] as 2.9 • UW a(W) =  (11-10)  .635 (1+141.5-UW)  + 5.925-UW  T h i s f u n c t i o n approaches a s y m p t o t i c a l l y the value 0.49 f o r l a r g e airmasses.  The  transmittance  r=(1-a)  will  t h e r e f o r e never be  l e s s than 0.5. The  t r a n s m i t t a n c e due to s c a t t e r i n g by  scattering) published  was a  presented  polynomial  transmittance  dry  a i r (Rayleigh  i n t a b u l a t e d form . The authors a l s o  expression  to  f i t  the  tabulated  data:  2 3 4 r(R)=.972-.08262AMS+.00933AMS -.00095AMS +.0000437AMS (11-11) The  above  expression  fits  the  table  well within  reasonable  l i m i t s of the airmass. Davies  and  establishing  a  Hay  circumvented  the  difficult  task  p a r a m e t e r i z a t i o n f o r the a e r o s o l t r a n s m i t t a n c e .  Instead, f o l l o w i n g a suggestion by Houghton [27], they the simple  of  employed  relationship: AMS T(A)  No procedure  (11-12)  = K  was given t o determine -K- and the  user  was  left  with the suggestion t o use K = 0.95 "as a g l o b a l average v a l u e " . This  value  would  be  matched  by  "clean  Atlantic  a n t i c y l o n i c days a value of K = 0.88 was suggested;  a i r " . For  the  authors  36 f u r t h e r recommend " l o c a l c a l i b r a t i o n a g a i n s t measurement". It  is  and Hay  obvious  that the weak p o i n t of the model of  f o r any a p p l i c a t i o n  i s the u n c e r t a i n t y i n v o l v e d  Davies in  the  d e t e r m i n a t i o n of the a e r o s o l t r a n s m i t t a n c e . T h i s study t h e r e f o r e attempts  to  improve Davies' and Hay's model by r e p l a c i n g t h e i r  t r a n s m i t t a n c e equation  for aerosols.  I n t e g r a t e d values were obtained from the model  of  Leckner  [3] i n form of the f o l l o w i n g equation: 4000r  J  290r(A) =  4000 U r I 290 J  As was  shown i n chapter  transmittance  for  an  -a OnX  exp{-/3-X  }«dX (11-13)  OnX  ••dX <  I I , Eqs. II-4  to  II-6,  the  broadband  a t t e n u a t o r with inhomogenous a t t e n u a t i o n  c h a r a c t e r i s t i c s over the spectrum can never be represented by  a  power f u n c t i o n . As  a  simplifying  step,  the  fiction  of a power law  upheld to o b t a i n a f o r m u l a t i o n which i s easy to handle, t h i s approach n e c e s s a r i l y means a l o s s airmasses. the  of  accuracy  at  was  although higher  Because of the s i g n i f i c a n c e of peak r a d i a t i o n v a l u e s ,  constants  of  the  f o l l o w i n g Eq.(11-14) were chosen to f i t  Eq.(11-13) at low airmasses  (AMS  = 1 and AMS  T ( A ) = C + D • exp(-E • 0 • AMS)  = 2): (11-14)  where: C  = (0.12445 • a - 0.0162) ,  (ll-14a)  37 D  = (1.003 - 0.125-a)  (II-14b)  E  = (1.089 • a + 0.5123)  (II-14c)  and  The Eq.(11-14) makes the assumption measured  values  of  is  (11-14).  measurement a n e c e s s i t y . To  a c c u r a t e v a l u e s of 0 without  suggested The  a g a i n s t other  to  0. The d e t e r m i n a t i o n of 0 - as o u t l i n e d i n  s e c t i o n 1.6.1 - makes f i l t e r reasonably  that the user has access  t h a t Eq. (1-20) be used performance  filter  in  obtain  measurements, i t  conjunction  with  Eq.  of Model A i s shown i n F i g s . 8 t o 11  parameterizations.  The  correspond w e l l with other models except  results for high  of  this  model  turbidity.  38 II .3  The  The Model by Hoyt  model  by  Hoyt  (from  now  on c a l l e d Model B) uses a  d e r i v a t i v e of the broadband t r a n s f e r equation. Hoyt o b t a i n s  the  d i r e c t p a r t of the s o l a r r a d i a t i o n a s : r(AS) He d e f i n e s f i v e carbon  different  absorptances  r(R) a  for  (11-15) water,  d i o x i d e , oxygen and a e r o s o l and two t r a n s m i t t a n c e s due to  scattering  by  particles  (aerosol)  and by dry a i r ( R a y l e i g h ) .  T h i s Eq. (11-15) i s b a s i c a l l y the 'broadband t r a n s f e r as  ozone,  o u t l i n e d i n chapter  equation  has  been  equation'  II "Treatment of D i r e c t R a d i a t i o n " . T h i s  modified  by  Hoyt  by  substituting  the  T by r=(1-a) and by then n e g l e c t i n g a l l terms of  transmittances second or higher  order:  Based on: T  =T  • T  2  • T  3  he s u b s t i t u t e s T = ( 1 - a )•(1-a )•(1-a ) 1 2 3 which i s r =1 a -  -a -a +a »a +a «a +a -a -a 1 2 3 1 2 1 3 2 3  «a -a 1 2 3  and i s s i m p l i f i e d to T  =1~a  1  -a -a 2 3  (11-16)  39 The  influence  of t h i s s u b s t i t u t i o n and  f o r small absorptances but absorption delivers  may  become  results  5%  simplification  i s small  i n c r e a s e s with higher airmasses where  quite  large.  smaller  Hoyt's  than i f he had  "broadband t r a n s f e r equation"  with  the  equation used the  respective  easily original  absorption  values.  Hoyt's absorptance values are t h e r e f o r e a d j u s t e d to h i s  equation  and cannot be used p r o p e r l y with other models  airmasses.  The  at  high  absorptances f o r Model B are  f o r water: -4 a(W)  = 0.110-(6.31•10  0.3 +UW)  -0.0121  (II-17)  f o r ozone: a(0)  = 0.045-(8.34-10  -4  +UO)  0.38  -3.1-10  -4  (11-18)  f o r carbon d i o x i d e : 0.26 a(C)  = 0.00235.(0.0129 + UC)  -4 -7.5-10  (11-19)  f o r oxygen: -3 a(OX)= 7.5-10  0.875 - AMS  (11-20)  for a e r o s o l : AMS a(AA)=  (l-wo)-{g(/3)}  (11-21 )  U n f o r t u n a t e l y , Hoyt d i d not g i v e c l o s e d form the  calculation  absorption  by  of  the  aerosol.  two  scattering  Instead,  he  r e l a t i o n s h i p to o b t a i n the t r a n s m i t t a n c e s  formulas  components  used due  the  and  for the  following  to s c a t t e r i n g :  40 t r a n s m i t t a n c e due t o R a y l e i g h s c a t t e r i n g : AMS = {f(AMS)}  T(R)  (11-22)  t r a n s m i t t a n c e due to a e r o s o l  scattering:  AMS r(AS)= The  two  (g(0)}  (11-23)  f u n c t i o n s f(AMS) and g(j3) were g i v e n i n t a b u l a t e d  only without a p a r a m e t e r i z a t i o n f u n c t i o n  AMS  f(AMS)  0.0 0.5 1.0 1 .5 2.0 2.5 3.0 3.5 4.0  Table table  0.0 0.909 0.917 0.921 0.925 0.929 0.932 0.935 0.937  (see below). While  0  g(/3)  0.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.24 0.28 0.32  1 .000 0.972 0.945 0.919 0.894 0.870 0.846 0.824 0.802 0.780 0.758 0.714 0.670 0.626  I  form the  Table II  f o r g(/3) extends  to f a i r l y  u s e f u l n e s s of the t a b l e f o r  f(AMS)  extends only to airmass 4 which  l a r g e t u r b i d i t y v a l u e s , the is  restricted  because  it  i s approximately e q u i v a l e n t t o a  z e n i t h angle 8= 75 degrees. Even under e q u a t o r i a l c o n d i t i o n s the table hour  restricts  the use of the model to times of more than one  before sunset or a f t e r s u n r i s e . Although the curve f o r both  functions  f(AMS)  considerable  and  g(0)  extrapolation,  are the  smooth  enough  to  allow  a p p l i c a b i l i t y of the model i s  41 hampered. T h i s study t r i e s t o open model  (graphical  results  this  i n comparison  shown i n F i g . 8 t o 11) t o wider f(AMS)  and  otherwise  g ( 0 ) through  two  AMS=4  and  turbidity  with other models a r e  p a r a m e t e r i z a t i o n formulas which  the range of a p p l i c a t i o n beyond However,  performing  use by r e p l a c i n g the t a b l e s f o r  extend  0=0.32.  well  as  the present  Fig.  11  limits  of  shows, a t very h i g h  the r e s u l t s of t h i s model a r e w e l l below  the  results  of other models. T h i s study p r e s e n t s a p a r a m e t e r i z a t i o n not of f(AMS) but of r(ray)=f(AMS)**AMS:  T(R)=  f o r m u l a t i o n f o r T ( R ) (Eq. 11-24)  The than  (.375566) • exp(-.221185•AMS) + .615958  0.2%  never  (11-24)  deviates  more  from the values given by Hoyt. To estimate the range  of reasonable accuracy f o r e x t r a p o l a t e d v a l u e s beyond AMS=4, the UBC Eq.  Computer Center Software (11-24).  extends around The  The  range  "*OLSF" was used  of  confidence  as f a r as AMS=8. T h i s reduces  the  to  (error  compare  with  l e s s than 2%)  limited  time  range  s u n r i s e / s u n s e t t o l e s s than h a l f of i t s p r e v i o u s v a l u e .  table  f o r g ( 0 ) w i l l be r e p l a c e d by a p a r a m e t e r i z a t i o n of  s i m i l a r c o n s t r u c t i o n as f o l l o w s :  g(0)=1.909267 • exp(-.6670236• 0) - 0.914  This  equation  never y i e l d s a d e v i a t i o n  (11-25)  from Hoyt's t a b l e s  42 of more than 0.32%. As f o r the t r a n s m i t t a n c e f u n c t i o n an attempt of e x t r a p o l a t i o n has been made with *OLSF).  a  powerful  software  (UBC  A g a i n s t t h i s standard, confidence i n Eq. (11-25) can be  maintained  up t o t u r b i d i t y v a l u e s of 0=0.5.  These m o d i f i c a t i o n s of Model B open i t t o atmospheres of high t u r b i d i t y at small s o l a r  applications  altitudes.  of  43  II  Bird direct  The Model by B i r d and Hulstrom  .4  and Hulstrom  published  i n s o l a t i o n and l a t e r  a  extended  model their  [ 2 2 ] to calculate model  to  diffuse  r a d i a t i o n and i n c o r p o r a t e d some minor changes of the d i r e c t p a r t of  the model  presents  to  improve  the performance  the model of B i r d and Hulstrom  as  [23].  This  published  (from now on c a l l e d "Model C") which i s i t s most recent Model  C  its original Irr  h  uses  the "broadband  form; only a constant  = SOcos(t?)-0.9662.  The use of the f a c t o r Hulstrom  used  considered some  t r a n s f e r equation"  a  transmission  past  300  form. almost i n  (11-26)  nm t o  3000  nm.  Because  that only there  is  the above mentioned s p e c t r a l boundaries  the s a i d f a c t o r was i n c o r p o r a t e d . The t r a n s m i s s i o n equations parameterized  [ 2 3 ]  stems from the f a c t that B i r d and  spectrum f o r t h e i r p a r a m e t e r i z a t i o n  the range from  in  f a c t o r i s used:  r ( R ) - T ( A ) «T(W) T ( G ) T ( 0 )  ' 0 . 9 6 6 2 '  study  are  as f o l l o w s : .84  T(R)=  exp{-0.0903-(AMS')  r(0)=  1  -  .1611 • XO  (1  11 . 0 1  •(1+AMS'-(AMS')  +  139.48  (11-27)  - 0 . 3 0 3 5 • XO) (11-28) 2  -0.002715-XO  /(1+0.044-XO+0.0003-XO  )  44  with X O = U O • AMS  (Il-28a)  r(G)=  0.26 exp{-0.0127-(AMS')}  T(W)=  1-2.4959*XW/{(1+79.034-XW)  (11-29) 6828 +6.385-XW}  ( I I - 3 0 )  with XW  = UW  • AMS  (Il-30a)  .873 .7088 .9108 r(A)= exp{-K(A) •(1+K(A)-K(A) )-AMS }  (II~31)  with K(A)=  Bird  and  0.2758«K(A) + 0.35-K(A) X=0.38jum X=0.5jum  Hulstrom  used  Kasten's  [28]  (11-32)  formulation  for  the  airmass as given below i n Eq.(11-33): AMS For  -1.25 = 1/{COS0 + 0.15-(93.885 -0) }  Rayleigh  scattering  and  pressure c o r r e c t e d airmass was  AMS'= AMS Eqs.  (11-33)  parameterization  the  absorption  [P i n kPa]  (ll-33a)  models.  gas  As  are seen  also in  (II-33a)  valid  in  other  Eqs.(11-31) and  (11-32)  above, B i r d and Hulstrom use atmospheric t u r b i d i t y values at wavelengths (380 nm wavelength  are  and  measured  500 by  a  used:  • P/101.3  and  mixed  (11-33)  nm);  turbidity  the U.S.  values  at  two  these  N a t i o n a l Weather S e r v i c e  45 (Flowers et a l . [29]) use  of two  accurate  weighted  f o r some l o c a t i o n s on a r o u t i n e b a s i s . spectral  turbidities  gives  a  p i c t u r e of the atmospheric t u r b i d i t y and  on broadband s o l a r procedure  calls  measurement  at  insolation. for  either  two  a  wavelength  But  for  most  substantial of  the  much  The more  i t s influence  locations amount  visible  of  this filter  spectrum  or  T h i s study attempts to make the Model C more a c c e s s i b l e  and  guesswork.  therefore aerosol data  suggests  to  transmittance.  from  filter  replace In a l l  the  cases  parameterization where  detailed  T(A)  =  Eq.  (11-34)  10.97  h o r i z o n t a l v i s i b i l i t y and 1.6.1.  It  may  be  transmittance:  (11-34)  derived  by  using  the  the t u r b i d i t y as o u t l i n e d  used  to  substitute  t r a n s m i t t a n c e s of models of the  same l e v e l  but  i n the  not  following  - 1 .265  was  f o r the model d e s c r i b e d  the  turbidity  measurements i s not a v a i l a b l e , the  equation a l l o w s the c a l c u l a t i o n of the a e r o s o l  The  for  for  l i n k between in  Section  other  aerosol  (Model A and  following Section  Model II.5.  B)  46 II.5  The A.S.H.R.A.E Model  T h i s model on the lowest [31]. The e x t i n c t i o n over follow  a  power  l e v e l uses a very simple  the  whole  spectrum  and  the  to  A l l attenuative solar  constant  i t s v a r i a t i o n over the year a r e compounded i n t o two s e t s of  12 c o n s t a n t s ,  one s e t f o r every  month.  Only the v a r i a t i o n of the water year  i s assumed  law f u n c t i o n with the airmass.  i n f l u e n c e s a t an airmass of one as w e l l as  algorithm  was  content  over  taken as v a r i a b l e to determine these c o n s t a n t s ,  are reproduced following  vapour  in  App.IV.  The  irradiance  equation  the which  has the  form: I r r = A«cos(0).exp{-B»AMS} h  (11-35)  On a log-normal p l o t t h i s f u n c t i o n transforms  to a straight  line  with an i n t e r c e p t of 'A' f o r AMS = 1/cos(t9) = 1. The major advantage of the A.S.H.R.A.E. easy  algorithm  is its  h a n d l i n g and the major drawback i s i t s i n e l a s t i c i t y  t o any  change of c o n c e n t r a t i o n of atmospheric c o n s t i t u e n t s . Because the i n f l u e n c e of the v a r y i n g c o n c e n t r a t i o n of ozone i s small and the c o n c e n t r a t i o n of gaseous absorbers (except  water  vapour),  the  does  main  not  targets  e l a s t i c i t y of the model are the a b s o r p t i o n  by  change to  very  increase  water  vapor  much the and  the s c a t t e r i n g by a e r o s o l s . It  i s quite  i n t e r e s t i n g t o assess  vapor c o n c e n t r a t i o n on  the  broadband  the i n f l u e n c e of water  transmittance  of  solar  r a d i a t i o n . T h i s has been done by using Leckner's [3] f o r m u l a t i o n  47 for t r a n s m i t t a n c e through 4000/"  /  I  The  UW  • K(W)  . \(1+20.07 • UW  290 J (W)  0.3*  AMS  -K(W)  -AMS)  X  0.45  =  (11-36)  c a l c u l a t i o n s were performed f o r a  precipitable  water  end of Chapter  millimeters  and  multitude  of  values  of  the r e s u l t s are shown i n Fig.12 at the  II.  From Fig.12, of  it  is  quite  obvious  values  that  the  first  few  p r e c i p i t a b l e water have a very strong i n f l u e n c e  on the t r a n s m i t t a n c e while a d i f f e r e n c e higher  •  'expOnX  T  water vapor i n the f o l l o w i n g form:  of  one  millimeter  of water c o n c e n t r a t i o n does not account  at  f o r very  much change i n t r a n s m i t t a n c e . The monthly v a l u e s of 'A' water  vapor  were determined  by using  values f o r the U n i t e d S t a t e s [33],The A.S.H.R.A.E.  Model a l s o  assumed  centimeter  were  that  present  200  dry  with  dust  particles  of the  [30]. Because of h i s use of  also  had  t r a n s m i t t a n c e c a l c u l a t i o n due The  cubic  taken  The from  ' p a r t i c l e s per volume' i n s t e a d  ' a e r o s o l mass c o n c e n t r a t i o n ' the changing atmosphere  per  no v a r i a t i o n over the year.  a l g o r i t h m to e s t a b l i s h the t r a n s m i t t e d r a d i a t i o n was Moon  average  an  water  influence  to condensation  on  the  around the  other parameter with a great spectrum of  the a e r o s o l content. S i m i l a r to Eq.  content  in  aerosol nuclei.  variation  is  (11-36) above, the i n f l u e n c e  48 of  turbidity  on  u s i n g Leckner's following  f o r m u l a t i o n of  form  a e r o s o l s only  amount  range. The  to  obtain  (same as  T(A) =  The  the broadband t r a n s m i t t a n c e was  of  aerosol  an  optical  (ideal)  determined depth  in  transmittance  by the  through  (ii-13):  4000 f -a I I • exp{-/3-X }»dX / OnX 290 J  turbidity  was  v a r i e d by v a r y i n g /3 over a wide  r e s u l t s are shown i n that  the  (11-37)  Fig.13.  becomes  obvious  turbidity  i s much more l i n e a r than  From  transmittance it  is  as the  this a  figure  it  function  of  case  for  water  t r a n s m i t t a n c e . While we can observe a weakened i n c r e a s e of water vapor  absorption  q u i t e obvious on  the  at  higher  water vapor c o n c e n t r a t i o n s , i t i s  that the v a r i a t i o n of the t u r b i d i t y has an  transmittance,  r e g a r d l e s s how  effect  much a e r o s o l there i s i n  the atmosphere. U n f o r t u n a t e l y , the values f o r dust and water c o n c e n t r a t i o n , which  were  represent  used  to  extremely  establish  the  A.S.H.R.A.E  c l e a r s i t u a t i o n s which are r a r e l y  algorithm, observed.  While t h i s might seem to be of advantage f o r the c a l c u l a t i o n peak  insolation  result  values,  f o r monthly averages  from the use of i t .  This fact c a l l s that  over-estimation  of  would  make  f o r a change i n the  the  i n f l u e n c e . T h i s study  varying suggests  A.S.H.R.A.E.  turbidity  algorithm  i t s main parameter of  the use of the ground  visibility  49 as  the  v a r i a b l e i n a m o d i f i e d A.S.H.R.A.E. model. I f a f u r t h e r  increase  of  accounting  accuracy  i s desired,  an  additional  parameter  f o r the v a r i a t i o n of the humidity can be i n c l u d e d . At  any r a t e , as seen from F i g . 13, an e r r o r of l e s s than  4% r e s u l t s  from a permanent water content of 1.5 cm. With  a  best  f i t method versus other models d e s c r i b e d i n  t h i s study, the i n c i d e n t r a d i a t i o n can be put i n t h i s form:  (\AMS °'/) 5  I r r = SC • T ( A ) • M(W) n  • (0.775)  (11-38)  with: r(A)=  / 0.85^ (-0.57) \AMS / (1-1.3-VIS )  (11-39)  and with:  . / 0.27\ \AMS) ) M(W) = (1.0223-0.0149-UW)  (11-40)  In most cases, M(W) can be s e t t o u n i t y which i s i t s exact  value  f o r UW=1.5 cm. The Eq.(11-39) above f o r the t r a n s m i t t a n c e due t o aerosol  i s very  Hulstrom  model. I t has , however, been t a i l o r e d t o f i t i n t o  context  of  Eq.  similar  (11-38)  t o the one presented  together  t h e r e f o r e not be used with Eq.(11-39)  other  i n the B i r d and the  with Eq. (11-40) and should parameterization  i s r e p l a c e d by Eq. (11-34),  models.  If  the term (0.775) i n ( I I -  38) has to be r e p l a c e d by (0.745). T h i s may c r e a t e e r r o r s of  up  to 2%. The  Eq.  (11-38)  also  does  not allow a v a r i a t i o n of the  ozone a b s o r p t i o n or the r a y l e i g h s c a t t e r i n g . An amount of 0.3 cm of c o n c e n t r a t e d ozone i n  the  a i r was  assumed  and  that  the  50 simulation  takes  place  elevations different  at  sea  level.  Any  adjustment  for  from sea l e v e l has t o be made v i a the  AMS  relation. In  the event of p r e c i p i t a b l e water d i f f e r i n g s t r o n g l y from  UW=1.5 cm, i t i s suggested that c o r r e c t i o n s a r e made referring  to  f o r equation  the  (11-38).  range  changes  algorithm,  i n c r e a s e d amount of c a l c u l a t i o n s t o  Performance of Eq.  performance of Eq. model  t o the A.S.H.R.A.E  which  of a p p l i c a t i o n to t u r b i d a i r , a r e d e f i n i t e l y  worth the s l i g h t l y model.  by  F i g u r e 12 or by using Eq. ( 1 1 - 4 0 ) as a m u l t i p l i e r  The suggested changes extend  either  (11-38)  i s shown i n F i g .  (11-38)  versus  the  existing  8  use the  t o 11; the  A.S.H.R.A.E.  i s shown i n F i g . 14. These graphs show that the suggested make  the  parameterization  A.S.H.R.A.E.  values. Figure  model  comparable  with  other  14 shows that the v a l u e s of the  o l d A.S.H.R.A.E. model a r e l i n k e d t o extremely low t u r b i d i t i e s .  51  1 30  1  1 40  r  1  50  "ZENJTH ANGLE  F i g u r e 8, Performance of f i v e  models.  52  F i g u r e 9, Performance of f i v e  models.  53  Figure  10,  Performance of f i v e  models.  54  visibility  5 km  ozone c o n t e n t  0 . 3 1 cm  water content  2 . 9 3 cm  CM <  —10 o Q O S -  A  A  A  +  t  t  X  X  X  fl Model B Model C flSHRRE(new)  •»  *  »  LecknEr  1  i  ffl 0  o~1 0  Figure  1  JO  ffl Model  1 23  1  r 33  1  "I 4  ZENJ^H  .  1 1 , Performance of f i v e  1 T _.._?0  -  63  &NGLE  models.  70  B3  55  Figure  12,  T r a n s m i t t a n c e due  to water  vapor  absorption.  Figure  1 3 , Transmittance due  to a e r o s o l  57  58 III  The  All amount that  Treatment of D i f f u s e  the presented models  Radiation  contain  of d i f f u s e r a d i a t i o n . D i f f u s e reaches  the  earth's  surface  (possibly:  multiple)  scattering  distinction  i s made between s c a t t e r e d  ("sky  radiation")  and  from  the  back to the  the  earth  earth.  "multiply  There i s no way  sky  closed  conditions  from  after Often the  a sky  reflected radiation" - reflected then s c a t t e r e d  backward to  two  components  of measurement. T h i s d i s t i n c t i o n  i s more d i f f i c u l t  than the  treatment  of  renders  the  s c a t t e r i n g phase f u n c t i o n  formula mathematical treatment of s i n g l e  practically  and  that of m u l t i p l e  scattering  s c a t t e r i n g becomes  impossible;  inhomogenous  nature of the  the d e t e r m i n a t i o n of the  aerosol  d i s t r i b u t i o n makes  s c a t t e r i n g versus a b s o r p t i o n r a t i o  difficult; reflectivity  i s v a r i a b l e and Simplifying the  but  reflection.  radiation  radiation  because:  extremely d i f f i c u l t  the  directly  to d i s t i n g u i s h these  the complexity of the  very  i s the  the  the d e t e r m i n a t i o n of the d i f f u s e r a d i a t i o n under  the d i r e c t r a d i a t i o n  the  not  calculate  for the convenience of the mathematical treatment.  Generally cloudless  to  radiation  and/or  sky and  of the d i f f u s e r a d i a t i o n by way i s purely  ways  of the  e a r t h ' s surface  i t s determination  models are g e n e r a l l y  be  shown and  "albedo")  difficult. necessary to c a l c u l a t e  d i f f u s e r a d i a t i o n . In the next s e c t i o n s  simplifications will  (called  the  extent of  what r e s u l t s can  be  the  these  expected  59 in  return.  60 III.1  General Approach f o r S i n g l e  Scattering  Most models make use of the s i m p l i f y i n g assumption that a l l s c a t t e r i n g occurs once o n l y . Secondary or m u l t i p l e neglected. This scattering  -  scattering i s  s i m p l i f i c a t i o n makes the c a l c u l a t i o n of R a y l e i g h scattering  by  particles  much  smaller than the  wavelength of the r a d i a t i o n - very easy: The phase f u n c t i o n Rayleigh  scattering  is  symmetrical,  provided  the f i c t i o n of  p e r f e c t l y s p h e r i c a l p a r t i c l e s i s upheld. T h i s means that the  scattered The  the  radiation  i s scattered  treatment of Mie ( a e r o s o l )  Rayleigh  scattering  To determine assumptions  are  the  amount  of  50%  of  forward and 50% backward. scattering  because, the  symmetrical but shows a strong b i a s  for  phase  i s not as easy as function  i s not  i n the forward d i r e c t i o n . diffuse  radiation,  further  neccessary, i . e . r e g a r d i n g the sequence of the  s c a t t e r i n g . They w i l l be i n t r o d u c e d with the r e s p e c t i v e  models.  61 III.2  Multiple  Scattering  None of the models presented i n t h i s study task  to  multiple models  like  the  Model [25] -  one  not  amount  of  of s c a t t e r e d  discussed  in  this  study  computation  time  c a l c u l a t e the d i f f u s e  irradiation  on  scattering.  radiation.  Some  L a c i s and Hansen [26] or the Lowtran  of  multiple  the  c a l c u l a t e the d i f f u s e r a d i a t i o n under c o n s i d e r a t i o n of s c a t t e r i n g or a b s o r p t i o n  excessive  approaches  These  models  -  because  use  the  algorithms  basis  divide  of the  the  of  to  possible  inhomogenous  atmosphere i n t o many l a y e r s (as many as 50) which a r e homogenous in themselves. These models o b t a i n a high degree of accuracy but demand an excessive One  cannot  airmasses  and  computational  expect low  a  large  concentrations  effort. gain  in  scattered  a term of second order - becomes very small scattered  radiation.  But  i n c l u d e the e f f e c t s of m u l t i p l e at  low  diffuse  solar  which  radiation  i f there  i t proves  s c a t t e r i n g f o r very  i s multiply  i s not very  t o be worthwile t o  a l t i t u d e . Under these c o n d i t i o n s  irradiation  f o r low  of s c a t t e r i n g matter i n the  atmosphere because the amount of m u l t i p l y  much  accuracy  turbid a i r  the f r a c t i o n of  scattered  increases  drastically. To  reduce the e r r o r s at l a r g e z e n i t h angles and to put the  c a l c u l a t i o n of d i f f u s e r a d i a t i o n more i n tune with the reality, include  this  physical  study presents a comparatively easy procedure to  the e f f e c t s of double and  triple  scattering  into  the  62  spectral if  Model of Leckner  so d e s i r e d ) . The  (and  i n t o the p a r a m e t e r i z a t i o n  models  f o l l o w i n g assumptions form the b a s i s of t h i s  work: The  single scattered  longer The  r a d i a t i o n has  a p a t h l e n g t h that  is  50%  than the d i r e c t r a d i a t i o n .  twice s c a t t e r e d  r a d i a t i o n f o l l o w s a path that  is  three  times the p a t h l e n g t h of the d i r e c t r a d i a t i o n Radiation  which i s s c a t t e r e d more than twice f o l l o w s a path  5 times as long as the path of the d i r e c t r a d i a t i o n The  s c a t t e r i n g model of homogenous l a y e r s w i l l be  by a model i n which s c a t t e r i n g according The  to the  above  f u n c t i o n of  assumptions  probabilities  are  defined  transmittance.  are the r e s u l t of a  semi-empirical  approach; i t e r a t i o n s were performed to o b t a i n a best other models which i n c o r p o r a t e  replaced  multiple  f i t against  scattering.  At a p a r t i c u l a r wavelength, the d i r e c t r a d i a t i o n i s d e f i n e d by the  transfer  extraterrestrial amount that absorbed  equation  (1-5).  irradiation  is~depleted  in  r a d i a t i o n and  the  to the e a r t h or back to the which  was  scattered  and  The  and the  difference  the  beam  atmosphere;  between  radiation it  space. I t  also  absorbed  the  i t scattered  includes  afterwards.  radiation  Because  the  assumption of a l a y e r e d atmosphere  was  the  question  arises  r a d i a t i o n i n t o the  scattered  p a r t and  how  to d i v i d e the depleted  the absorbed p a r t . T h i s  p r o b a b i l i t y of a t t e n u a t i o n for  a  quantum  is  done  dropped,  i s the  includes  s c a t t e r e d r a d i a t i o n , be  the  by  establishing  f o r a s i n g l e quantum. The  to be absorbed or s c a t t e r e d  the  probability  ( p o s s i b l y more than  63 once)  is: P R = ( 1 - T ( R ) « T ( A ) «r(W) - T ( G ) «T(0) ) X  The  probability  d e f i n e d by the scattering  X  X  quotient the  of  the  sum  optical  of  the  (II1-1)  X  f o r a quantum t o undergo R a y l e i g h  over  attenuating  X  depth  optical  scattering i s for  Rayleigh  depths  for a l l  processes:  PR(R) = PR.log(T(R)  ) / log(r(tot) X  )  (III-2)  X  where: r ( t o t ) =T(R) -r(A) -r(W) - T ( G ) - T ( 0 ) X  The  probability  X  for  occurence of R a y l e i g h  a  X  X  single  X  quantum  scattering  (III-3)  X  to  undergo  only  one  (and not more) becomes:  PR(R,1) = P R ( R ) . r ( t o t )  (III-4) X  Accordingly, and  the p r o b a b i l i t y  then one or more f u r t h e r  f o r a p a r t i c l e to undergo a t t e n u a t i n g processes  PR(R,2+)= P R ( R ) • ( 1 - T ( t o t )  )  Rayleigh  becomes: (III-5)  X  which i s : 2 PR(R,2+)= PR • l o g ( r ( R )  )/ l o g ( T ( t o t ) ) X  The  Figure  a nonlayered  15 i l l u s t r a t e s the process atmosphere.  (III-6)  X  of m u l t i p l e s c a t t e r i n g i n  64  beam=TaTrTm a t t e n u a t e d : 1-TaTrTm of whichI  absorbed  Rayleigh  x(1-TaTrTm)  aerosol  scat,  scat,  z(1-TaTrTm)  y(1-TaTrTm)  s c a t t e r e d and passed: z(1-TaTrTm)TaTrTm further  y(1-TaTrTm)TaTrTm  attenuated: y(1-TaTrTm)  ^  z(1-TaTrTm) I of which  I\  of which 2  i"  2  xy(1-TaTrTm)  y  xz(1-TaTrTm)  yz(1-TaTrTm)  further  (1-TaTrTm)  z  I  (1-TaTrTm)  j zy(1-TaTrTm)  2 2 (y+z) (1-TaTrTm) •TaTrTm  attenuated: (y+z)  (1-TaTrTm)  \  x(y+z)  2  iX  of I which 2  (1-TaTrTm) 2 y(y+z)  r (1-TaTrTm)  Explanation: PR(Abs)= x= log(Tm)/log(TaTrTm) PR(R) = y= l o g ( T r ) / l o g ( T a T r T m ) PR(A) = z= log(Ta)/log(TaTrTm)  z(y+z)  (1-TaTrTm)  Ta=r(AS) Tr=r(R) Tm= r ( A A ) • r ( W ) • T ( G ) • T ( O )  For b r e v i t y , no m u l t i p l i c a t o r was used on t h i s Figure  15, M u l t i p l e A t t e n u a t i o n  Pattern  page  65 Because scattered  50%  of  forward  the  but  Rayleigh  an  scattered  amount  larger  scattered radiation i s scattered  forward,  have  the  to  be  made  to  simplify  radiation  than 50% further  of the  is Mie  assumptions  incorporation  of  multiple  scattering: The  r a d i a t i o n which i s s c a t t e r e d  i n t o the h a l f s p h e r e  the e a r t h with a plane p a r a l l e l to the e a r t h ' s considered  The  In  the  curvature  Fig.  i n t o the  16,  term  respective ratio, [20].  atmosphere i s  the  neglected.  approach to i n c l u d e  multiple  'FR'  of the  scattered  radiation  can  as:  FR=  The  is  s p e c t r a l model w i l l be g r a p h i c a l l y o u t l i n e d .  forward s c a t t e r i n g r a t i o  be d e f i n e d  surface  scattered'.  of e a r t h and  following  scattering The  'forward  facing  0.5'Idif(R)+BA-Idif(A)  'Idif  denotes  s c a t t e r e r and  i.e.  (III-7)  Idif(R)+Idif(A)  the  BA  the i s the  diffuse aerosol  radiation forward  from  the  scattering  f r a c t i o n of I d i f ( A ) which i s s c a t t e r e d  forward  66  SS*  Figure  16, M u l t i p l e  DS*  Scattering  TS*  Pattern  67  The the  total  single,  'sky d i f f u s e  radiation'  the double and the t r i p l e  then becomes the sum of  scattered  radiation:  I d i f ( S ) =SS+DS+TS X  (III-8)  with: SS=I  DS=I  OhX  OhX  •{(1-r(Sc)')-r(Abs)'-FR.T(SC) " }  .{(1-T(SC)').r(Abs)'*•(1-T(SC) " ) • T ( S C ) " ' } • {FR-FR+(1-FR)•(1-FR)}  TS=I  OhX  •{(l-r(Sc)')-T(Abs) " '•(1-T(SC) " )•(1-T(SC) " ' } • {FR'FR»FR+(1-FR)•(1-FR)'FR.3}  where  the  scattering.  number As  of  outlined  primes  previously  p a t h l e n g t h s of the s c a t t e r e d Finally,  there  is  corresponds  a  in  radiation  this  require  to  the  class  Section, higher  small amount of r a d i a t i o n ,  which i s back  to  earth. This radiation i s : I(MR) =(Irr +SS+DS+TS)•rg•ra X hX  The term ' r g ' stands f o r the r e f l e c t i v i t y of the ground and  longer  airmasses.  r e f l e c t e d from the e a r t h back t o the sky and s c a t t e r e d the  of  (III-9)  (albedo)  r a i s the atmospheric albedo. The atmospheric albedo can  defined as:  be  68 ra=(SS*+DS*+TS*)/lrr  (III-10) OhX  The t o t a l d i f f u s e r a d i a t i o n can now be d e f i n e d as: l(tot)=Irr  This  hX  +Idif(S)+Idif(MR)  semi-empirical  (111-11)  approach to model m u l t i p l e s c a t t e r i n g  was used with the s p e c t r a l t r a n s m i t t a n c e f u n c t i o n s The  results  are  m o d i f i e d " . The spectrally models.  presented  results  integrated  are values  in very  Figures  of  Leckner.  18 to 21 as  "Leckner,  encouraging  close  to  the  and  bring  the  parameterization  69 III.3  Treatment of D i f f u s e R a d i a t i o n with the Models  III.3.1  The  The Model of Leckner  part of Leckner's Model to c a l c u l a t e d i f f u s e  radiation  uses the f o l l o w i n g b a s i c assumptions: 1. )  The  phase  symmetric.  function  f o r Mie  Therefore  i t can  s c a t t e r i n g i s assumed to be be  treated  like  Rayleigh  scattering. 2. )  The  a b s o r p t i o n by a e r o s o l w i l l be n e g l e c t e d and thus i t i s  assumed  that  scattering 3. )  The  the  aerosol  attenuation  of  earth  i s by  means  of  only.  reflectivity  the  is  zero.  No  multiply  r e f l e c t e d r a d i a t i o n has t o be c o n s i d e r e d .  Leckner's  model  defines  the  d i f f e r e n c e between the d i r e c t has  only  been  subjected  scattered  r a d i a t i o n and a f i c t i o u s  to a b s o r p t i o n  a e r o s o l a b s o r p t i o n . Therefore  radiation  the  as  the  beam that  under e x c l u s i o n of any  f o r m u l a t i o n " of  the  diffuse  r a d i a t i o n takes on the f o l l o w i n g form: 4000Y I d i f = O.5«cos(0) • 1(1 - T ( 0 ) T ( G ) »r(W)-dX)-Irr 29tV OnX n  (111-1 2)  where the i r r a d i a n c e f o r normal i n c i d e n c e i s taken from Eq. 7). The value 0.5  because  symmetric.  f o r FR (forward the  phase  s c a t t e r i n g r a t i o ) i s assumed t o  functions  (IIbe  f o r s c a t t e r i n g a r e taken as  70  Eq. values  (ill—12) delivers fairly and  sacrifices  low on  ground  reflectivity:  Most  the accuracy of the a e r o s o l  neither aerosol absorption influence  good r e s u l t s f o r low  nor  under c o n d i t i o n s of  assumptions  have  t u r b i d i t y . The  significant  performance of  Leckner's approach i s shown i n comparison with other Figs.  18 to 21:  Leckner's d i f f u s e v a l u e s  make  scattered radiation;  phase f u n c t i o n low  turbidity  are very  models  low.  to be expected because Leckner's approach does not  in  This  has  account  for  secondary d i f f u s e r a d i a t i o n . For  the  same  assumptions  p o s s i b i l i t y of m u l t i p l e semi-exact  unnoticed presenting radiation  as  and  with the added  [32]  e a r l y as  published 1928.  language (German), t h i s work seems to  in  the  Anglophone  his results.'Berlage  world  and  shows, t h a t  it the  Probably  have  is  a  well  gone worth  spectral diffuse  becomes:  I d i f = 0.5-  the  is defined  Berlage  treatment  4000 f  where  above  scattering,  mathematical  because of the  as  290J  'T'  /I  OnX  -cost?-T(AB) •  i s Linke's  {1-T(R)  T }-dX  ( 1 1 1 - 1 3)  1-1.4-T-ln(r(R))  Triibungsindex  (turbidity  index) which  as: T  T ( R ) T ( A )  =  T(R)  (111 -1  T h e r e f o r e ( 14000r 1 1 - 1 3 ) can be w r i t t e n {as: l-r(R)-r(A)}-dX Idif=.5' I -COS0-T(AB)290  J  OnX  1-1  .4-ln(r(R)-T-(A) )  4)  (III-15)  71  As shown Leckner's  in  section  III.2,  it  d i f f u s e model by implementing  is  possible  the m u l t i p l e  a l g o r i t h m with Leckner's t r a n s m i t t a n c e f u n c t i o n s .  to  change  scattering  72 III.3.2  The  The  Model by Davies and  Hay  b a s i c idea of the model by Davies and  of the d i f f u s e r a d i a t i o n  d i f f u s e r a d i a t i o n caused by R a y l e i g h  2. )  The  d i f f u s e r a d i a t i o n caused by a e r o s o l  3. )  The  d i f f u s e r a d i a t i o n caused by the  earth's  backscattering An  assumption  be  attenuation processes. exact  construction  surface  made  scattering. scattering.  r e f l e c t i o n of a l l r a d i a t i o n  back  to  from the sky to the  must  is a division  i n t o three p a r t s :  1. ) The  from  Hay  sky  and  the  earth.  regarding  Davies and Hay  the  the  sequence  of  use a l a y e r e d model -  of which i s shown i n Fig.17:  no  the the  attenuation  occurs c o n c u r r e n t l y . With t h i s s i m p l i f i c a t i o n the equations  for  the primary d i f f u s e componeats become: Idif(R)=SC-cos(0)•[T(0)•(1-T(R))T(A)-0.5]  (III-17)  Idif(A)=SC-cos{6)•[(T(O)-r(R)-a(W))•(1-T(A))-WO-BA]  (III-18)  and  The  term  'WO'  a t t e n u a t i o n due scattering  denotes to a e r o s o l  the r a t i o between s c a t t e r i n g and (taken as 0.95)  and  'BA'  the  total  forward  r a t i o . Note that in t h i s model by Davies and Hay  d i f f u s e r a d i a t i o n caused by R a y l e i g h to a t t e n u a t i o n  by water vapor and  the  scattering gases!  is  not  the  subject  73 For  the  m u l t i p l y r e f l e c t e d r a d i a t i o n , Davies  and Hay  are u s i n g  the f o l l o w i n g equation: Idif(MR)=(lrr +Idif(R)+Idif(A)•rg•ra/(1-rg•ra) h The  values  0.2  f o r the ground r e f l e c t i v i t y  but can go as high as 0.9 The  'rg' are u s u a l l y around  f o r f r e s h snow.  atmospheric albedo employed by Davies  from L a c i s and Hansen [26] and  (III-19)  i s given  and Hay  was  taken  as:  ra=0.0685+0.17(1-r(A)')-WO  The  value of T ( A ) ' i s c a l c u l a t e d as r(A) f o r a z e n i t h angle 0=57  degrees. The three and  t o t a l r a d i a t i o n can now  parts  of  be determined by adding  the d i f f u s e r a d i a t i o n and the d i r e c t  up  radiation  i t can be brought i n t o the f o l l o w i n g form: I ( t o t ) = I r r +Idif(R)+Idif(A)+Idif(MR) h  The  inclusion  (111-20)  of the m u l t i p l e s c a t t e r i n g a l g o r i t h m as o u t l i n e d  in S e c t i o n I I I . 2 i s p o s s i b l e . With the absence of measured i t can not be c o n s i d e r e d b e n e f i c i a l because of the increased An  the  complexity  without  a sure measure of any  i n c r e a s e d data base might change the f a c t s and  data  substantially improvement. reverse  this  recommendation. The performance i s shown i n F i g s 18 to 21: While high  at small z e n i t h angles, the r e s u l t s come i n low  angles over  60 degrees.  This  fact  suggests  f u n c t i o n of the z e n i t h angle might not be  that  ideal.  for z e n i t h the  chosen  74 III.3.3  The model  The  Model by  Hoyt  model by Hoyt b a s i c a l l y uses the  as  outlined  radiation  in  II.3.  i n t o three p a r t s  difference  as  did  to the p r e v i o u s l y  that  the  scattered  Mie  scattering  radiation  i s subject  Hoyt  elements of the  also  splits  Davies  and  the  Hay.  o u t l i n e d Model A i s the  the d i r e c t r a d i a t i o n . Thereby the  diffuse The  main  assumption  from R a y l e i g h s c a t t e r i n g to the  direct  and  from  same a b s o r p t i v e i n f l u e n c e s equations  to  determine  as the  d i f f u s e r a d i a t i o n become: (111-21) and (111-22)  [ .75.(I-T(AS) ]  The  R a y l e i g h forward s c a t t e r i n g  aerosol  forward s c a t t e r i n g  r a t i o was  r a t i o was  not  taken as  0.5  while  the  given as a f u n c t i o n  but  as a constant BA=0.75.  (MR)  Hoyt's approach to determine the  multiply  radiation  one  resembles somewhat the  / I d i f ( M R ) = ( I r r +Idif(R+A))•rg•  reflected  by Davies and  that  i t has the  one  important d i f f e r e n c e ;  multiply  reflected  Hay:  5  1-[a  h V 1 [.5.(1-T(R)')+(1-T(AS)')..25]  But  diffuse  It  diffuse  is  (111-23)  obviously  radiation  assumed  i s subject  to  75 further  attenuation.  Physically  and  mathematically,  treatment  i s more a c c u r a t e then the one o u t l i n e d  section.  The t o t a l r a d i a t i o n  sky d i f f u s e p a r t  i n the p r e v i o u s  i s the sum of the d i r e c t p a r t , the  and the m u l t i p l y  reflected  radiation:  I ( t o t ) = I r r +Idif(R)+Idif(A)+Idif(MR) h As  f o r the i n c l u s i o n of the m u l t i p l e  The  performance i s shown i n F i g s  t h i s model as w e l l are reasonably c l o s e C.  (111-24)  scattering  comments t o Davies' and Hay's model are too.  this  valid  a l g o r i t h m , the  for  this  model  18 to 21: The r e s u l t s of to the r e s u l t s of  Model  76 III.3.4  The Model by B i r d and Hulstrom  Like  the models  Hulstrom's  model  describes  the  we  divides  previously the  radiation  diffuse  which  dealt  with,  radiation:  reaches  the  s c a t t e r i n g ; the other  part  was  the e a r t h and b a c k s c a t t e r e d  reflected  surface.  The  from  equation  Bird One  part  earth  after  i s made up of d i f f u s e r a d i a t i o n  f o r the  and  that  t o the earth's  'sky d i f f u s e '  part  of the  i r r a d i a n c e i s given a s : Idif  -cos(0)•(0.79)-r(O)T(W)-T(G)«T(A)•  =1 s  On  1 .02  {.5*(1-r(R))+BA-(1-T(AS)}/{1-AMS+(AMS)  where: BA  (111-25)  }  =0.82  and T(AS)=  r(A) / r(AA)  (111-26)  and: 1 .06 r(AA)=1-KS-(1+AMS  The t o t a l  -AMS)•(1-T(A))  r a d i a t i o n , i n c l u d i n g the m u l t i p l y  reflected  (111-27) radiation  becomes thus: I(tot)=  where  (Irr+ Idif(S))/(l-rg-rs) h  (111-28)  77 rs  The  KS  factor  B i r d and  =  0.0685+(1-BA)•(1-T(AS))  is  dependent on the a e r o s o l  Hulstrom used  calculations.  The  a  value  of  size distribution;  KS=0.0933  d i f f u s e p a r t of Model C can  'multiple s c a t t e r i n g ' study;  (111-29)  pattern  as  for  all  their  be changed to  developed  earlier  in  the d i f f e r e n c e in performance, however, i s so small  i t cannot be c o n s i d e r e d  an  therefore  diffuse  to  use  the  improvement.  absence of b e t t e r  more  standards can  for the d i f f u s e p a r t of t h i s  study  this that  suggests  part of Model C unchanged.  The  18 to 2 1 : Model C  was  performance of the model i s shown i n F i g ' s developed with the a i d of  This  the  elaborate  models  be c o n s i d e r e d  study.  and  in  the  "primary standard"  78 III.3.5  The  D i f f u s e R a d i a t i o n with the A.S.H.R.A.E.  Model  A.S.H.R.A.E a l g o r i t h m [31] uses a simple f a c t o r  m u l t i p l y the d i r e c t  'C t o  radiation:  Idif  = C • Irr n  (111-30)  Irr  = A • exp-(B-AMS)  where:  While the  (11-31)  i t i s a c c u r a t e w i t h i n reasonable l i m i t s t o  diffuse  radiation, and  n  r a d i a t i o n as a constant  the c o n s t a n t s proposed  represent  fixed  fraction  'C  of the d i r e c t  by A.S.H.R.A.E. a r e  circumstances.  A.S.H.R.A.E formula it  i s necessary as  well.  Model of  Bird  and  Because  most  low days. clear  the proposed  change of the  (Eqs.II-38/39) allows v a r i a b l e  visibilities,  to  radiation  very  t u r b i d i t y values of u n u s u a l l y c l e a r  T h e r e f o r e an underestimation occurs f o r a l l but the atmospheric  determine  adapt  the  Comparison Hulstrom,  calculation  of  the  diffuse  with other models, notably the lead  to  the  following  simple  parameterization: Idif  = I r r -(3/VIS+0.1) n  The performance of t h i s equation i s shown i n F i g s .  (111-32)  18 t o 22: The  r e s u l t s are i n very good agreement with the r e s u l t s of Model C  79  Extraterrestrial Irradiance A b s o r p t i o n by Ozone Rayleigh Scattering A b s o r p t i o n by Water vapor Attenuation aerosol  by  Downward s c a t t e r e d component  Attenuation aerosol  by  Downward s c a t t e r e d componend Diffuse radiation (aerosol)  Figure  D i r e c t beam  Diffuse radiation (Rayleigh)  17; Davies and Hay Layer Model; Two Stream Approximation of D i f f u s e R a d i a t i o n  gure  18;  Performance  of  5 Models,  Diffuse  Radiation  81  Figure  19; Performance of 5 Models, D i f f u s e R a d i a t i o n  82  gure 20; Performance of 5 Models, D i f f u s e  Radiation  83  Figure  21; Performance of 5 Models, D i f f u s e R a d i a t i o n  84  F i g u r e 22;  The  ASHRAE Models, D i f f u s e R a d i a t i o n  85 IV  Concluding Remarks  I t was the goal of t h i s study t o put together some wealth  of the  of data and achievements i n the f i e l d of c l e a r sky s o l a r  r a d i a t i o n t o the user with an e n g i n e e r i n g background. The use of the h o r i z o n t a l m e t e o r o l o g i c a l main  parameter  range - the v i s i b i l i t y  for t u r b i d i t y  calculations  l i g h t of t h i s g o a l : L o c a l c o n d i t i o n s the c o r r e l a t i o n between v i s i b i l i t y This and  deficiency  a v a i l a b i l i t y of  r e s u l t s with only The The  slightly  method  The applied  without general  which  of  i n mind:  atmospheric  outlining  a  applications.  on  promises  A  much  better  complexity.  algorithm  to d i f f u s e r a d i a t i o n .  for' d i f f u s e  radiation  time.  proposed treatment of m u l t i p l e  influence  influence  and t u r b i d i t y .  i s a main c o n t r i b u t o r  t h i s f o r the f i r s t  sciences  has t o be seen i n  have a strong  increased  proposed new A.S.H.R.A.E  incorporates  as the  i s outweighted, however, by the s i m p l i c i t y this  turbidity  -  tool  i s offered  components  mathematical  s c a t t e r i n g a l s o has the  on  to  multiple  approach  that  assess  the  scattering disallowed  86  V  Further Work  At  present,  visibilities would  be  from  to high v i s i b i l i t i e s  most  correlation locations,  extrapolations  are used  and  d e s i r a b l e to have an e x t e n s i v e data base on  the  conditions simple  and  and  seasons.  this  turbidity  for  would  be  to conduct, although  in  the best  r e s e a r c h to c o n t a c t simultaneous all  attenuators,  notably  the  availability  measurements  a e r o s o l s and of  of  radiation  measurements of the e f f e c t s ozone,  are  essential.  i n t e r e s t of the s o l a r  water,  gases as w e l l as simultaneous  various  Studies i n t h i s f i e l d  advanced data p r o c e s s i n g c a p a c i t i e s i s c o n s i d e r e d It  range It  visibility  in  medium  study.  between  comparatively  low  of  various  quantities  with  p o s s i b l e high c o r r e l a t i o n such as the m e t e o r o l o g i c a l range. Such studies  would  work that has of  this  allow  to  v e r i f y many e x i s t i n g models and  other  been based on e x t e n s i v e s i m u l a t i o n , i n c l u d i n g much  study.  F u r t h e r work i s a l s o recommended i n the radiation,  both  broadband  e x i s t i n g models s t i l l  and  field  of  diffuse  spectral diffuse radiation.  show wide d i s c r e p a n c i e s and measured  i s sketchy at b e s t .  t  The data  87 References [I]  Lambert, J.H.;  [2]  Thekaekara, M.P.; "Solar energy outside atmosphere", Solar Energy 14, 109-127 (1973).  [3]  Leckner, B.; "The s p e c t r a l d i s t r i b u t i o n of s o l a r radiation at the e a r t h ' s s u r f a c e - elements of a model", S o l a r Energy 20, 143-150, (1978).  [4]  Penndorf, R.; "Tables of the r e f r a c t i v e index f o r standard a i r and the R a y l e i g h s c a t t e r i n g c o e f f i c i e n t " , J . Opt. Soc. of Am. 47, 176, (1957)  [5]  Vigroux, E.; "Contribution a 1'etude experimentale de l ' a b s o r p t i o n de l'ozone", Annals de Physique 8, 709-762 (1953).  [6]  Fowle, F.E.; "The transparency A s t r o p h y s i c s J o u r n a l V o l . 42, (1915)  [7]  Howard, J.N. et a l . ; " I n f r a r e d t r a n s m i s s i o n of synthetic atmospheres", J . O p t i c a l Soc. of Am., Vol.46, (1956)  [8]  Yamamoto, G.; "Direct a b s o r p t i o n of s o l a r r a d i a t i o n by atmospheric water vapor,.carbon dioxide and oxygen", J . Atmos. Soc. 19, 182, (1962)  [9]  "Photometrie",  (1760).  of  McClatchey, R.A. et a l t . ; " O p t i c a l /atmosphere", AFCRL-72-0497, (1972)  the  aqueous  properties  earth's  vapor",  of  the  [10] Elterman, L.; " R e l a t i o n s h i p between v e r t i c a l attenuation and s u r f a c e m e t e o r o l o g i c a l range", J . A p p l . O p t i c s , V o l . 9 , I804ff, (1970) [ I I ] Goody, R.M.; "Atmospheric Press, (1964).  Radiation I",  Oxford:  [12] T i w a r i , S.N.; "Models f o r i n f r a r e d atmospheric Advances i n Geophysics, V o l . 20, 1-85, (1977) [13] Mie, G.; " B e i t r a g e zur Optik t r u b e r Physik, Vol.25, 377ff, (1908)  Medien",  Clarendon radiation",  Annalen  der  [14] Angstrom, A.; "On the t r a n s m i s s i o n of sun r a d i a t i o n and on dust i n the atmosphere", G e o g r a f i s k Annaler 2, 156-166), (1929)  88 [15].Angstrom, radiation [16] McCartney;  A.; "On the atmospheric t r a n s m i s s i o n of sun I I " , G e o g r a f i s k Annaler 3, 130-159, (1930) " O p t i c s of the atmosphere", (1976)  [17] L i n k e , F; "Handbuch der Geophysik", V o l . 9, (1941)  621ff,  Berlin  [18] Buckius, R.O. and King, R.; " D i r e c t s o l a r t r a n s m i t t a n c e f o r a c l e a r sky", S o l a r Energy Vol.22, 297-301, (1979) [19] Zuev, V.E.; "Atmospheric the IR", (1970)  transparency i n the v i s i b l e and  [20] Davies, J.A. and Hay, J . E . ; " C a l c u l a t i o n of the s o l a r r a d i a t i o n i n c i d e n t on a h o r i z o n t a l s u r f a c e " , Proceedings of the F i r s t Canadian S o l a r R a d i a t i o n Data Workshop, Ed. T.Won and J.E.Hay, (1980) [21] Hoyt, D.V.; "A model f o r the c a l c u l a t i o n of s o l a r i n s o l a t i o n " , S o l a r Energy, Vol.21, 27-35, (1978)  global  [22] B i r d , R.E. and Hulstrom, R.L.; " D i r e c t i n s o l a t i o n models", S o l a r Energy Research I n s t i t u t e TR-335-344, (1980) [23] B i r d , R.E. and Hulstrom, R.L.; "A s i m p l i f i e d c l e a r sky model f o r d i r e c t and d i f f u s e insolation on horizontal surfaces", Solar Energy Research Institute TR-642-761, (1981) [24] Selby, J . E . et a l . ; "Atmospheric t r a n s m i t t a n c e from 0.25/xm to 28.5 Mm: Computer code LOWTRAN 3", AFCRL-75-0255; (1975) [25] Selby, J . E . et a l . ; "Atmospheric t r a n s m i t t a n c e / r a d i a n c e : Computer code LOWTRAN 4", AFGL-78-0053; (1978) [26] L a c i s , A.L. and Hansen, J.E.; "A p a r a m e t e r i z a t i o n f o r the absorption of s o l a r r a d i a t i o n i n the e a r t h ' s atmosphere", J . Atmospheric S c i e n c e , Vol.31, 118-133, (1974) [27] Houghton, H.G.; "On the heat balance of the northern hemisphere", J . Meteorology, Vol.11, 1-9, (1954) [28] Kasten, F.; "A new t a b l e and approximation formula f o r the relative o p t i c a l a i r mass", A r c h i v fur Meteorologie, Geophysik und B i o k l i m a t o l o g i e , V o l . 12, 206-233 (1966). [29] Flowers, E.C. e t a l . ; "Atmospheric turbidity over the U n i t e d S t a t e s 1961-1966", J . A p p l . Meteor., V o l . 8 , 955, (1969) [30] Moon, P.; "Proposed standard s o l a r r a d i a t i o n curves f o r e n g i n e e r i n g use", J o u r n a l of the F r a n k l i n I n s t i t u t e 230, 583-617, (1940)  89 [31] A.S.H.R.A.E.; Handbook of Fundamentals, (1972) [32] Berlage, H.P.; "Zur Theorie der Beleuchtung einer horizontalen Flache durch Tageslicht", Meteorologische Z e i t s c h r i f t 45-5, 174-180, (1928) [33] T h r e l k e l d and Jordan; " D i r e c t S o l a r R a d i a t i o n A v a i l a b l e on C l e a r Days", A.S.H.R.A.E. Trans., V o l . 14, 45-56, (1958) [34] E i g h t h S e s s i o n of The Commission Methods of O b s e r v a t i o n ; WMO (1981)  for  Instruments  and  90 Table I I I ; spectrum, a b s o r p t i o n Col Col Col Col Col Col Col  1: 2: 3: 4: 5: 6: 7:  coefficients  I n t e r v a l number AX i n [nm] Center of wavelength i n t e r v a l , [jum] Ozone a b s o r p t i o n c o e f f i c i e n t s f o r E q . ( I - 1 l ) Water a b s o r p t i o n c o e f f i c i e n t s f o r Eq.(I-14) Mixed gases a b s o r p t i o n c o e f f i c i e n t s f o r E q . ( l - 1 3 ) F r a c t i o n of S o l a r Constant w i t h i n i n t e r v a l , [W/(m-m-nm)]  1  2  1 2 3 4 5 6 7 8 9 10 1 1 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  5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0  3 0.290 0.295 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.355 0.360 0.365 0.370 0.375 0.380 0.385 0.390 0.395 0.400 0.405 0.410 0.415 0.420 0.425 0.430 0.435 0.440 0.445 0.450 0.455 0.460 0.465 0.470 0.475 0.480 0.485 0.490 0.495 0.500  4  5  6  38.00 20.00 10.00 4.800 2.700 1 .350 0.8 0.38 0. 160 0.075 0.04 0.019 0.007 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.003 0.003 0.004 0.006 0.008 0.009 0.012 0.014 0.017 0.021 0.025 0.03  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0i0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  7 482.0 584.0 514.0 605.0 689.0 764.0 830.0 975.0 1059.0 1081 .0 1074.0 1069.0 1093.0 1083.0 1068.0 1132.0 1181.0 1157.0 1120.0 1098.0 1098.0 1189.0  1429.0 1644.0 1751 .0 1774.0 1747.0 1693.0 1639.0 1663.0 1810.0 1922.0 2006.0 2057.0 2066.0 2048.0 2033.0 2044.0 2074.0 1976.0 1950.0 1960.0 1942.0  91 Table I I I , cont. 1 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 7.1 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94  2 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 7.5 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0  3 0.505 0.510 0.515 0.520 0.525 0.530 0.535 0.540 0.545 0.550 0.555 0.560 0.565 0.570 0.575 0.580 0.585 0.590 0.595 0.600 0.605 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90  4 0.035 0.04 0.045 0.048 0.057 0.063 0.07 0.075 0.08 0.085 0.095 0.103 0.11 0.12 0. 122 0.12 0.118 0.115 0.12 0. 125 0.13 0.12 0. 105 0.090 0.079 0.067 0.057 0.048 0.036 0.028 0.023 0.018 0.014 0.011 0.01 0.009 0.007 0.004 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.160E-01 0.240E-01 0.125E-01 0.100E+01 0.870E+00 0.610E-01 0.100E-02 0.100E-04 0.100E-04 0.600E-03 0.175E-01 0.360E-01 0.330E+00 0.153E+01 0.660E+00 0.155E+00 0.300E-02 0.100E-04 0.100E-04 0.280E-02 0.630E-01 0.210E+01  6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.300E+01 0.210E+00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  7 1920.0 1882.0 1833.0 1833.0 1852.0 1842.0 1818.0 1783.0 1754.0 1725.0 1720.0 1695.0 1705.0 1712.0 1719.0 1715.0 1712.0 1700.0 1682.0 1666.0 1647.0 1625.0 1602.0 1570.0 1544.0 1511.0 1486.0 1456.0 1427.0 1402.0 1369.0 1344.0 1314.0 1290.0 1260.0 1235.0 1211.0 1185.0 1159.0 1134.0 1109.0 1085.0 1060.0 1036.0 1013.0 990.0 968.0 947.0 926.0 908.0 891 .0  92 Table I I I , 1 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 1 1 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  2 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 30. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 50. 0 75. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0  3 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 .00 1 .05 1 .10 1.15 1 .20 1 .25 1 .30 1 .35 1 .40 1 .45 1 .50 1 .55 1 .60 1 .65 1 .70 1 .75 1 .80 1 .85 1 .90 1 .95 2.00 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0  cont. 4 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0  5 0. 160E+01 0. 125E+01 0. 270E+02 0. 380E+02 0. 41OE+02 0. 260E+02 0. 310E+01 0. 148E+01 0. 125E+00 0. 250E-02 0. 1OOE-04 0. 320E+01 0. 230E+02 0. 160E-01 0. 180E-03 0. 290E+01 0. 200E+03 0. 110E+04 0. 150E+03 0. 150E+02 0. 170E-02 0. 100E-04 . 0.100E-01 0. 510E+00 0. 400E+01 0. 130E+03 0. 220E+04 0. 140E+04 0. 160E+03 0. 290E+01 0. 220E+00 0. 330E+00 0. 590E+00 0. 203E+02 0. 31OE+03 0. 150E+05 0. 220E+05 0. 800E+04 0. 650E+03 0. 240E+03 0. 230E+03 0. 100E+03 0. 120E+03 0. 195E+02 0. 360E+01 0. 310E+01 0. 250E+01 0. 140E+01 0. 170E+00 0. 450E-02  6 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 0. 730E-02 0. 400E-03 0. 11OE-03 0. 100E-04 0. 640E-01 0. 630E-03 0. 100E-01 0. 640E-01 0. 145E-02 0. 100E-04 0. 100E-04 0. 100E-04 0. 145E-03 0. 710E-02 0. 200E+01 0. 300E+01 0. 240E+00 0. 380E-03 0. 110E-02 0. 170E-03 0. 140E-03 0. 660E-03 0. 100E+03 0. 150E+03 0. 130E+00 0. 950E-02 0. 100E-02 0.800E+00 0. 190E+01 0. 130E+01 0. 750E-01 0. 100E-01 0. 195E-02 0. 400E-02 0. 290E+00 0. 250E-01  7 880. 0 869. 0 858. 0 847. 0 837. 0 820. 0 803. 0 785. 0 767. 0 748. 0 668. 0 593. 0 535. 0 485. 0 438. 0 397. 0 358. 0 337. 0 312. 0 288. 0 267. 0 245. 0 223. 0 202. 0 180. 0 159. 0 1 42.0 126. 0 1 14.0 103. 0 90. 0 79. 0 69. 0 62. 0 55. 0 48. 0 43. 0 39. 0 35. 0 31. 0 26. 0 22. 6 19. 2 16. 6 14. 6 13. 5 12. 3 1 11 . 10. 3 9. 5  93 Table IV C o e f f i c i e n t s A,B,C f o r A.S.H.R.A.E a l g o r i t h m [31] A January: February March April May June July August September October November December  1230 .0 1215 .0 1 186.0 1 136.0 1 104.0 1088 .0 1085 .0 1 107.0 1151 .0 1 192.0 1221 .0 1233 .0  B  C 0. 142 0. 144 0. 156 0. 180 0. 196 0.205 0.207 0. 201 0. 177 0. 160 0. 149 0. 1 42  0.058 0. 060 0.071 0.097. 0. 121 0. 134 0. 136 0. 122 0. 092 0. 073 0.063 0.057  

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