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UBC Theses and Dissertations

Diffuse and global solar spectral irradiance under cloudless skies: a simple model Brine, Douglas Toby 1982

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DIFFUSE AND GLOBAL SOLAR SPECTRAL IRRADIANCE UNDER CLOUDLESS SKIES - A SIMPLE MODEL by DOUGLAS TOBY BRINE B.A.Sc., The U n i v e r s i t y o f W a t e r l o o , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department o f M e c h a n i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March 1982 © ,D. Toby B r i n e , 1982. In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of 7V\ JL^A^S^J £SM^^J>J?A^^ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Cx^sCJl J fcX i i ABSTRACT A s i m p l e e m p i r i c a l model t o c a l c u l a t e s o l a r s p e c t r a l d i f f u s e and g l o b a l i r r a d i a n c e u nder c l o u d l e s s s k i e s was i n v e s t i g a t e d . T h i s f o r m u l a t i o n t a k e s i n t o a c c o u n t a b s o r p t i o n o f r a d i a t i o n by m o l e c u l e s s u c h as 0 3 , H 2 0 and t h e u n i f o r m l y - m i x e d a b s o r b i n g g a s e s C 0 2 and 0 2 . A t t e n u a t i o n by R a y l e i g h - s c a t t e r i n g and a e r o s o l e x t i n c t i o n a r e i n c l u d e d . A e r o s o l a t t e n u a t i o n i s c a l c u l a t e d t h r o u g h A n g s t r o m ' s t u r b i d i t y p a r a m e t e r s a and fi. The d i f f u s e r a d i a t i o n i s assumed t o be composed o f t h r e e p a r t s : (1) R a y l e i g h - s c a t t e r e d d i f f u s e i r r a d i a n c e ; (2) a e r o s o l - s c a t t e r e d d i f f u s e i r r a d i a n c e ; and (3) i r r a d i a n c e a r i s i n g o u t o f m u l t i p l e r e f l e c t i o n s between t h e a t m o s p h e r e and t h e g r o u n d . The g l o b a l i r r a d i a n c e i s t h e sum of t h e s e t h r e e components o f d i f f u s e i r r a d i a n c e p l u s t h e d i r e c t i r r a d i a n c e . The i n p u t p a r a m e t e r s i n c l u d e an e x t r a t e r r e s t r i a l s p e c t r u m , z e n i t h a n g l e ©, t u r b i d i t y c o e f f i c i e n t fi, w a v e l e n g t h e x p o n e n t o , g r o u n d a l b e d o p , water v a p o r c o n t e n t and ozone c o n t e n t . The model i s shown t o y i e l d v e r y good r e s u l t s up t o a i r mass two when compared t o a c c u r a t e t h e o r e t i c a l c a l c u l a t i o n s . No c o m p a r i s o n s w i t h m easured s p e c t r a a r e p r e s e n t e d b e c a u s e o f a l a c k o f a c c u r a t e s p e c i f i c a t i o n s o f t h e i n p u t p a r a m e t e r s . R e s u l t s a r e p r e s e n t e d t o show t h e e f f e c t of v a r i a t i o n o f c e r t a i n o f t h e i n p u t p a r a m e t e r s . i i i TABLE OF CONTENTS A b s t r a c t i i T a b l e Of C o n t e n t s i i i L i s t Of F i g u r e s i v L i s t Of T a b l e s v N o m e n c l a t u r e v i G l o s s a r y v i i i A c k n owledgements x I I n t r o d u c t i o n 1 II P a s t F o r m u l a t i o n s 5 I I I M a t h e m a t i c a l F o r m u l a t i o n 8 M o d e l o f D i r e c t I r r a d i a n c e 8 M o d e l o f D i f f u s e I r r a d i a n c e 9 IV M e t h o d o l o g y and V a l i d a t i o n 14 V R e s u l t s 32 VI C o n c l u s i o n s 41 R e f e r e n c e s 42 A p p e n d i x I S o l a r S p e c t r u m u s e d i n M o d e l C o m p a r i s o n s 44 A p p e n d i x I I S o l a r S p e c t r u m u s e d by Dave 47 A p p e n d i x I I I E x p l a n a t i o n o f W a v e l e n g t h D i s c r e p e n c y ... 49 A p p e n d i x IV S p e c t r a l V a l u e s f o r F i g . 1 50 A p p e n d i x V S p e c t r a l V a l u e s f o r F i g . 2 53 A p p e n d i x VI S p e c t r a l V a l u e s f o r Fig.'s 3,4 and 5 56 A p p e n d i x V I I S p e c t r a l V a l u e s f o r Fig.'s 6 and 7 59 A p p e n d i x V I I I S o l a r S p e c t r u m o f P a r a m e t e r C o m p a r i s o n s . 62 A p p e n d i x IX Most R e c e n t S o l a r S p e c t r u m 65 i v L I S T OF FIGURES F i g . 1 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l A . . 15 F i g . 2 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l B . . 17 F i g . 3 C o m p a r i s o n s of D i r e c t : I r r a d i a n c e w i t h Model C .. . 19 F i g . 4 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 21 F i g . 5 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 23 F i g . 6 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 24 F i g . 7 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 26 F i g . 8 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 27 F i g . 9 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 28 F i g . 10 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e w i t h M o d e l C . . 30 F i g . 1 1 C o m p a r i s o n s of D i f f u s e I r r a d i a n c e wi t h M o d e l C . . 31 F i g . 12 E f f e c t o f A i r ] Mass on I r r a d i a n c e . 33 F i g . 1 3 E f f e c t of B e t a on I r r a d i a n c e , m=1 . 35 F i g . 14 E f f e c t o f B e t a on I r r a d i a n c e , m=2 . 37 F i g . 15 E f f e c t o f A l p h a on I r r a d i a n c e ... . 38 F i g . 16 E f f e c t o f Ground A l b e d o on I r r a d i a n c e . 40 V L I S T OF TABLES T a b l e 1 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 1 16 T a b l e 2 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 2 18 T a b l e 3 I n t e g r a t e d D i r e c t I r r a d i a n c e f o r F i g . 3 20 T a b l e 4 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r Fig.'s 4 and 5 ... 22 T a b l e 5 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r Fig.'s 6 and 7 ... 25 T a b l e 6 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r Fig.'s 8 and 9 ... 29 T a b l e 7 I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r Fig.'s 10 and 11 . 29 v i NOMENCLATURE D> d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W irr 2 (f»m)" 1 D &x a e r o s o l - s c a t t e r e d d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W m" 2(»im)" 1 m u l t i p l y - r e f l e c t e d d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W a i " 2 ( c m ) " 1 Dyy R a y l e i g h - s c a t t e r e d d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W m ' M d i n ) " 1 Fa a e r o s o l e f f e c t i v e s c a t t e r i n g r a t i o , d i m e n s i o n l e s s F r R a y l e i g h - f o r w a r d t o t o t a l - s c a t t e r i n g r a t i o , d i m e n s i o n l e s s G\ g l o b a l i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W m"2(>im)"1 Ie\ e x t r a t e r r e s t r i a l n o r m a l i r r a d i a n c e , W m " 2 ( « i n ) " ' 1^ d i r e c t i r r a d i a n c e on a h o r i z o n t a l s u r f a c e , W m~2(»im)"1 k ^ m i x e d - g a s a b s o r p t i o n c o e f f i c i e n t , d i m e n s i o n l e s s koA ozone a b s o r p t i o n c o e f f i c i e n t , ( c m ) " 1 kv*x w a t e r v a p o r a b s o r p t i o n c o e f f i c i e n t , ( c m ) " 1 X ozone d e p t h , cm m a i r mass, d i m e n s i o n l e s s w w a t e r v a p o r d e p t h , cm o a e r o s o l w a v e l e n g t h e x p o n e n t , d i m e n s i o n l e s s fi t u r b i d i t y c o e f f i c i e n t , d i m e n s i o n l e s s 9 z e n i t h a n g l e , d e g r e e s a t m o s p h e r i c a l b e d o , d i m e n s i o n l e s s />g g r o u n d a l b e d o , d i m e n s i o n l e s s ta.^ a e r o s o l t r a n s m i t t a n c e , d i m e n s i o n l e s s m i x e d - g a s t r a n s m i t t a n c e , d i m e n s i o n l e s s v i i NOMENCLATURE T O J k ozone t r a n s m i t t a n c e , d i m e n s i o n l e s s iVx R a y l e i g h t r a n s m i t t a n c e , d i m e n s i o n l e s s r^x w a t e r v a p o r t r a n s m i t t a n c e , d i m e n s i o n l e s s o 0 s i n g l e s c a t t e r i n g a l b e d o , d i m e n s i o n l e s s X w a v e l e n g t h , i<ni SUBSCRIPT X. m o n o c h r o m a t i c v a l u e s GLOSSARY AIR MASS ,m, i s t h e p a t h l e n g t h o f t h e s o l a r r a d i a t i o n i n t h e a t m o s p h e r e r e l a t i v e t o t h e p a t h l e n g t h i f t h e sun were d i r e c t l y o v e r h e a d ( a t t h e z e n i t h ) . ZENITH ANGLE , 9 , i s t h e a n g l e between t h e l o c a l n o r m a l and t h e l i n e t o t h e sun p o s i t i o n . / r r \ - S e c e \ / / / / / / DIRECT IRRADIANCE i s t h e s o l a r r a d i a t i o n t h a t r e a c h e s t h e g r o u n d i n a d i r e c t l i n e w i t h t h e s u n . DIFFUSE IRRADIANCE under c l o u d l e s s s k i e s i s t h e r a d i a t i o n t h a t r e a c h e s t h e g r o u n d from t h e sky h e m i s p h e r e due t o 1) s c a t t e r i n g f r o m t h e d i r e c t r a d i a t i o n and 2) m u l t i p l e - r e f l e c t i o n s between t h e g r o u n d and t h e a t m o s p h e r e . RAYLEIGH-SCATTERED DIFFUSE IRRADIANCE i s t h e r a d i a t i o n s c a t t e r e d f r o m t h e d i r e c t r a d i a t i o n by t h e a i r m o l e c u l e s . GLOSSARY AEROSOL-SCATTERED DIFFUSE IRRADIANCE i s t h e r a d i a t i o n s c a t t e r e d f r o m t h e d i r e c t r a d i a t i o n by t h e a e r o s o l p a r t i c l e s s u s p e n d e d i n th e a t m o s p h e r e . MULTIPLY-REFLECTED DIFFUSE IRRADIANCE i s a component o f t h e d i f f u s e i r r a d i a n c e . Any r a d i a t i o n r e f l e c t e d by t h e g r o u n d and t h e n r e f l e c t e d by t h e a t m o s p h e r e down t o t h e g r o u n d a g a i n i s m u l t i p l y - r e f l e c t e d i r r a d i a n c e . T h i s r e f l e c t i o n p r o c e s s c o n t i n u e s i n d e f i n i t e l y . GROUND ALBEDO , , i s . t h e r a t i o o f t h e r a d i a t i o n r e f l e c t e d by t h e g r o u n d t o t h e r a d i a t i o n i n c i d e n t on t h e g r o u n d . ATMOSPHERIC ALBEDO , p^ , i s t h e r a t i o of t h e r a d i a t i o n r e f l e c t e d by t h e a t m o s p h e r e t o t h e r a d i a t i o n i n c i d e n t on t h e a t m o s p h e r e . X ACKNOWLEDGEMENTS T h i s r e s e a r c h was c a r r i e d o u t under t h e s u p e r v i s i o n o f D r . M. I q b a l , whose a d v i c e and g u i d a n c e was a p p r e c i a t e d . A l l c o m p u t i n g was done a t t h e U.B.C. Computing C e n t e r . F i n a n c i a l s u p p o r t o f t h e N a t i o n a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada i s g r a t e f u l l y a c k n o w l e d g e d . The d a t a t a p e p r o v i d e d by D r . J . V. Dave o f IBM S c i e n t i f i c C e n t e r , P a l o A l t o was a p p r e c i a t e d . •c 1 I INTRODUCTION An a c c u r a t e s o l a r s p e c t r u m , b o t h e x t r a t e r r e s t r i a l and t e r r e s t r i a l , i s n e c e s s a r y f o r t h e f u r t h e r d e v e l o p m e n t o f many s o l a r r e l a t e d e n e r g y p r o j e c t s . C h i e f among t h e s e a r e t h e p h o t o v o l t a i c s f o r power on s a t e l l i t e s , and t h e c o n t i n u e d d e v e l o p m e n t o f t e r r e s t r i a l l y b a s e d p h o t o v o l t a i c power c e l l s . The e x t r a t e r r e s t r i a l s p e c t r u m i s a t t e n u a t e d by t h e e a r t h ' s a t m o s p h e r e i n two d i s t i n c t ways: by s c a t t e r i n g and by a b s o r p t i o n . S c a t t e r i n g i s a c o n t i n u u m p r o c e s s , whereas m o l e c u l a r a b s o r p t i o n o c c u r s a t d i s c r e t e w a v e l e n g t h s and c a n v a r y g r e a t l y o v e r s m a l l w a v e l e n g t h i n t e r v a l s . S c a t t e r i n g by a c l e a n , d r y a t m o s p h e r e i s g e n e r a l l y r e f e r r e d t o a s R a y l e i g h s c a t t e r i n g ( r e f e r t o g l o s s a r y f o r a d e s c r i p t i o n of t e c h n i c a l t e r m s ) . A e r o s o l s a r e s u s p e n d e d m a t t e r , whether l i q u i d or s o l i d , s u c h as d u s t , v o l c a n i c a s h , p o l l e n , smoke, u r b a n p o l l u t i o n , o r smog. A b o s r p t i o n by a e r o s o l s a p p e a r s t o be a c o n t i n u u m p r o c e s s and i s u s u a l l y c h a r a c t e r i z e d as s u c h . T h e o r e t i c a l a n a l y s i s of t h e r a d i a t i v e t r a n s f e r e q u a t i o n by t h e s p h e r i c a l h a r m o n i c s a p p r o x i m a t i o n [ 1 ] has p r o d u c e d e x t e n s i v e d a t a s e t s of d i r e c t and d i f f u s e s o l a r i r r a d i a n c e . The n u m e r i c a l s o l u t i o n of t h e h i g h l y complex and r e a l i s t i c a t m o s p h e r e s i n c l u d e d a l l o r d e r s of s c a t t e r i n g and a b s o r p t i o n by t h e common a b o s r b i n g g a s e s and a e r o s o l s . The t h e r o r e t i c a l t r e a t m e n t assumes a p l a n e - p a r a l l e l , homogenous a t m o s p h e r e of i n f i n i t e e x t e n t i n t h e h o r i z o n t a l d i r e c t i o n . A l l n o n h o m o g e n e i t y due t o 2 s c a t t e r i n g a n d / o r a b o s r p t i o n i s c o n f i n e d t o t h e v e r t i c a l d i r e c t i o n and i s i n c o r p o r a t e d i n t o t h e d e f i n e d a t m o s p h e r e by d i v i d i n g t h e a t m o s p h e r e i n t o any number o f l a y e r s and u s i n g a p p r o p r i a t e amounts o f m o l e c u l e s o f a i r , a b s o r b i n g g a s e s , and a e r o s o l s f o r e a c h l a y e r . The d e f i n e d a t m o s p h e r e s , c a l l e d 'Model a t m o s p h e r e s ' , p r o g r e s s i n c o m p l e x i t y from p u r e l y R a y l e i g h s c a t t e r i n g , M odel A, t o R a y l e i g h s c a t t e r i n g p l u s m o l e c u l a r a b s o r p t i o n , M odel B. F u r t h e r c o m p l e x i t y i s added by i n c l u d i n g n o n - a b o s r b i n g a e r o s o l s , Model C. The number o f d a t a s e t s g e n e r a t e d by t h e n u m e r i c a l s o l u t i o n i s l i m i t e d due t o t h e computer t i m e i n v o l v e d and t h e c o m p l e x i t y o f t h e i n p u t d a t a , t h e number o f i t e r a t i o n s n e c e s s a r y f o r a c c u r a c y , and a l s o q u a n t i t y (1.9 x 1 0 6 ) of o u t p u t d a t a p o i n t s f o r e a c h d e f i n e d a t m o s p h e r e . A s i m p l e method t o o b t a i n t h e s p e c t r a l d i r e c t and d i f f u s e i r r a d i a n c e i s needed by e n g i n e e r s and s c i e n t i s t s o f a l l d i s c i p l i n e s . The d e s i r e d f o r u m u l a t i o n s h o u l d be b a s e d on a s i m p l e homogenous a t m o s p h e r e w i t h no l a y e r s but i n c o r p o r a t i n g t h e b e s t a v a i l a b l e knowledge of t h e m o l e c u l a r a b s o r p t i o n c o e f f i c i e n t s . F u r t h e r m o r e , t h e a e r o s o l a t t e n u a t i o n c o u l d be c h a r a c t e r i z e d t h r o u g h t h e A n g s t r o m t u r b i d i t y f o r m u l a . T h i s s i m p l e m o d el, once f o r m u l a t e d , s h o u l d t h e n be compared t o more a c c u r a t e d a t a s e t s [2] and a l s o any measured s p e c t r a l d a t a a v a i l a b l e f r o m t h e l i t e r a t u r e [ 3 , 4 , 5 , 6 ] , The c l o u d l e s s sky i s m o d e l l e d b e c a u s e t h i s c o n d i t i o n g e n e r a l l y p r o d u c e s t h e maximum e n e r g y a v a i l a b l e and t h e e f f e c t s o f c l o u d s c a n n o t e a s i l y be m o d e l l e d . T h i s maximum e s t a b l i s h e s t h e c r i t e r i a from w h i c h e n g i n e e r s c a n d e s i g n s o l a r c e l l s , s p a c e and p r o c e s s w ater h e a t i n g equipment and e s t i m a t e t h e t h e r m a l l o a d i n g on b u i l d i n g s , e t c . The maximum i s a l s o n e c e s s a r y i n t h e 3 d e s i g n o f m a t e r i a l s f o r s o l a r a p p l i c a t i o n s t o p r e v e n t e a r l y damage o r d e t e r i o r a t i o n o f p l a s t i c s . The d i r e c t o r beam i r r a d i a n c e p r o v i d e s most o f t h e e n e r g y a v a i l a b l e a t t h e g r o u n d . The d i r e c t s p e c t r a l i r r a d i a n c e has been s t u d i e d by s e v e r a l i n v e s t i g a t o r s . Moon [7] was p r o b a b l y t h e f i r s t t o combine a l l t h e i n d i v i d u a l s c a t t e r i n g and a b s o r p t i o n c h a r a c t e r t i e s t o g e t h e r i n a c o m p l e t e m o d e l . Thomas and T h e k a e k a r a [8] u s e d a f o r m u l a t i o n d e v e l o p e d by G a t e s [9] t o p r e d i c t t h e d i r e c t i r r a d i a n c e . H a t f i e l d e t a l [10] a l s o u s e d t h e G a t e s ' f o r m u l a t i o n but i n c l u d e d o t h e r e f f e c t s s u c h as l a t i t u d e , a l t i t u d e , s u r f a c e a l b e d o , s l o p e and s u r f a c e o r i e n t a t i o n . U n f o r t u n a t l e y t h e water v a p o r t r a n s m i t t a n c e c a l c u l a t e d by t h e G a t e s f o r m u l a t i o n i n t h e 0.8 t o 1.0jim band has been shown [11] t o be low and r e s u l t s i n a s h a r p d e c r e a s e i n i r r a d i a n c e i n t h i s band. T h i s method a l s o r e q u i r e s an i n t e r p o l a t i o n scheme t o match t h e w a v e l e n g t h o f t h e e x t r a t e r r e s t r i a l v a l u e s g e n e r a l l y r e c o r d e d i n t h e l i t e r a t u r e w i t h t h e a b s o r p t i o n c o e f f i c i e n t s . L e c k n e r ' s model [12] seems t o be t h e b e s t p r e s e n t l y a v a i l a b l e i n s i m p l e f o r m . He d e v e l o p e d m o n o c h r o m a t i c t r a n s m i t t a n c e f u n c t i o n s u s i n g r e c e n t d a t a [13] f o r c a l c u l a t i n g t h e d i r e c t s p e c t r u m . The r e s u l t s o f t h i s method show good agreement when compared t o measured s p e c t r a - and t h e t h e o r e t i c a l c a l c u l a t i o n s . W i t h L e c k n e r ' s m o d e l , a s e e m i n g l y a d e q u a t e , y e t s i m p l e method f o r c a l c u l a t i n g a d e t a i l e d d i r e c t s p e c t r u m i s a c h i e v e d . The d i f f u s e i r r a d i a n c e r e s u l t s from t h e i n t e r a c t i o n of s o l a r r a d i a t i o n w i t h t h e s c a t t e r i n g p a r t i c l e s o f t h e a t m o s p h e r e : a i r m o l e c u l e s and a e r o s o l s . The s c a t t e r i n g by t h e a i r m o l e c u l e s i s l i m i t e d t o v e r y s h o r t w a v e l e n g t h s of l e s s t h a n 1.0 nm. The 4 s c a t t e r i n g by a e r o s o l p a r t i c l e s , g e n e r a l l y c a l l e d M i e s c a t t e r i n g , i s i m p o r t a n t when p a r t i c l e s i z e i s o f t h e o r d e r o f t h e w a v e l e n g t h . The f i r s t impingement o f r a d i a t i o n on a p a r t i c l e i s c a l l e d p r i m a r y s c a t t e r i n g . The p r i m a r y s c a t t e r e d r a d i a t i o n i s f u r t h e r s c a t t e r e d by a i r m o l e c u l e s o r a e r o s o l p a r t i c l e s and t h i s p r o c e s s i s c a l l e d m u l t i p l e s c a t t e r i n g . The d e t a i l e d s t u d y o f m u l t i p l e s c a t t e r i n g i s complex, hence t h i s e f f e c t has n o t been t a k e n i n t o a c c o u n t i n t h i s work. T h e r e a r e s e v e r a l s i m p l e f o r m u l a t i o n s f o r p r e d i c t i n g t h e d i f f u s e s p e c t r a l i r r a d i a n c e i n t h e l i t e r a t u r e and t h e s e a r e r e v i e w e d i n t h e n e x t c h a p t e r . 5 II PAST FORMULATIONS A simple formula to p r e d i c t the s c a t t e r e d d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e was proposed by Berlage [14]. Th i s formula given below: D A =0.5 cos(e) (l 0 x• - ly ) (1) s t a t e s that the d i f f u s e i r r a d i a n c e i s the d i f f e r e n c e between the e x t r a t e r r e s t r i a l i r r a d i a n c e and the d i r e c t i r r a d i a n c e a"t the ground. For a R a y l e i g h atmosphere, t h i s does give c r e d i b l e r e s u l t s [15] when compared to more acc u r a t e a n a l y s i s [16] i f a subzone l a y e r r a d i a t i o n i n t e n s i t y i s s u b s t i t u t e d f o r I„A . T h i s f o r m u l a t i o n does not account f o r a l l molecular a b s o r p t i o n or any a e r o s o l e f f e c t s e x p l i c i t l y . Leckner [12] proposed a simple formula f o r the d i f f u s e i r r a d i a n c e = 0.5 I 6 \ C O S ( e ) T f t A T w A ( 1 - T V > T o ^ ) (2) Th i s expresses the d i f f u s e i r r a d i a n c e as the d i f f e r e n c e between the d i r e c t i r r a d i a n c e and a ' f i c t i t i o u s beam' subj e c t only to molecular a b s o r p t i o n . T h i s f o r m u l a t i o n takes i n t o account molecular a b s o r p t i o n but assumes that the a e r o s o l and the molecular s c a t t e r i n g f u n c t i o n s are i d e n t i c a l . H a t f i e l d et a l . [10] a l s o present a d i f f u s e r a d i a t i o n 6 model s i m i l a r t o t h a t o f L e c k n e r , i n c l u d i n g ozone a b s o r p t i o n and s c a t t e r i n g by a i r m o l e c u l e s and a e r o s o l . In t h i s model t h e d i f f u s e i r r a d i a n c e i s g i v e n by They have, however, o m i t t e d a l t o g e t h e r t h e a b s o r p t i o n due t o w a t e r v a p o r and t h e m i x e d - g a s e s . They i n c l u d e a f a c t o r k f o r t h e c i r c u m s o l a r component and d e f i n e t h e f o r w a r d s c a t t e r i n g f u n c t i o n f o r b o t h R a y l e i g h - s c a t t e r i n g and a e r o s o l s c a t t e r i n g as T h i s s c a t t e r i n g f u n c t i o n r e s u l t s i n v a l u e s o f F = 0.5 f o r 9 = 0.0° and and F = 0.398 f o r e = 6 0 . 0 ° . R o b i n s o n [17] s t a t e s t h a t F s h o u l d be l a r g e r t h a n 0.5 f o r t u r b i d c o n d i t i o n s . He a l s o s t a t e s t h a t t h i s f o r m u l a f o r F was d e v e l o p e d f r o m d a t a f o r t h e Congo w i t h an a l b e d o o f 0.25 and t h a t F s h o u l d be a d j u s t e d f o r o t h e r v a l u e s of a l b e d o . He f a i l s t o m e n t i o n how i t c o u l d be a d j u s t e d . E q n s . (1) and (2) both,assume t h e f o r w a r d s c a t t e r i n g r a t i o t o be 0.5 and e q n . (4) r e s u l t s i n v a l u e s o f F l e s s t h a n 0.5. The v a l u e of 0.5 i s a p p r o x i m a t l e y t r u e f o r a R a y l e i g h - s c a t t e r i n g a t m o s p h e r e . However, t h e a e r o s o l s c a t t e r i n g i s v e r y s t r o n g l y b i a s e d i n t h e f o r w a r d d i r e c t i o n , and, b e c a u s e o f t h i s a s y m e t r y , t h e e f f e c t i v e s c a t t e r i n g r a t i o ( t o w a r d t h e g r o u n d ) i s a f u n c t i o n o f z e n i t h a n g l e . A m o d i f i c a t i o n o f L e c k n e r ' s f o r m u l a seems n e c e s s a r y t o a c c o u n t f o r t h e d i s s i m i l a r s c a t t e r i n g p r o p e r t i e s o f t h e a i r m o l e c u l e s and t h e a e r o s o l s . D A = I c > , cos (e ) T F T > K ( 1 - T V X T ^ X ) F k (3) F = 0.5 c o s (©) (4) 7 One a p p r o a c h , and t h e one f o l l o w e d h e r e , i s t o c o n s i d e r a R a y l e i g h - s c a t t e r i n g a t m o s p h e r e a n d an a e r o s o l s c a t t e r i n g a t m o s p h e r e a s a two s t r e a m a p p r o x i m a t i o n , a n d c a l u l a t e t h e d i f f u s e i r r a d i a n c e a s a sum o f t h e s e two c o m p o n e n t s . S u c h an a p p r o a c h h a s been u s e d i n b r o a d - b a n d d i f f u s e i r r a d i a n c e c a l c u l a t i o n s [ 1 8 ] . A t h i r d component o f d i f f u s e i r r a d i a n c e c a n be a d d e d t o t h e p r e v i o u s l y c a l c u l a t e d c o m p o n e n t s t o a c c o u n t f o r t h e m u l t i p l e r e f l e c t i o n e f f e c t s . The g r o u n d a l b e d o must be. s p e c i f i e d a n d a l s o a m o n o c h r o m a t i c a t m o s p h e r i c a l b e d o must be f o r m u l a t e d . 8 I I I MATHEMATICAL FORMULATION Mod e l o f D i r e c t I r r a d i a n c e The model f o r c a l c u l a t i n g t h e s p e c t r a l d i r e c t i r r a d i a n c e h as been a d a p t e d from L e c k n e r [ 1 2 ] . T h i s d i r e c t model u s e s e x p l i c i t t r a n s m i t t a n c e s f o r e a c h o f R a y l e i g h - s c a t t e r i n g , ozone a b s o r p t i o n , w a t e r v a p o r a b s o r p t i o n , m i x e d - g a s a b s o r p t i o n , a nd a e r o s o l e x t i n c t i o n . The 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 i s g i v e n by -r^ = ex p ( - 0 . 0 0 8 8 m x~« ) (5) The t r a n s m i t t a n c e due t o ozone a b s o r p t i o n i s T«X = exp(-k o X-^- m) (6) where t h e mo n o c h r o m a t i c a b s o r p t i o n c o e f f i c i e n t s , k,^ o b t a i n e d by V i g r o u x [ 19 ] and p r e s e n t e d i n Howard e t a l . [ 20 ] have been employed. The t r a n s m i t t a n c e due t o wa t e r v a p o r a b s o r p t i o n i s •^-0.2385 k w>w m/(1+20.07 k w V w m ) 0 - " 5 ^ = exp - m ) 0 - " 5 ^ (7) The t r a n s m i t t a n c e due t o mixed-gas a b s o r p t i o n i s T 3 > = exp £ - 1 . 4 1 k^m/( 1 + 1 18.93 k ^ m ) 0 - " 5 ^ (8) w h e r e a b s o r p t i o n c o e f f i c i e n t s , k g X a n d k „ j > a r e t a k e n f r o m 9 L e c k n e r [ 1 2 ] , The t r a n s m i t t a n c e due t o a e r o s o l e x t i n c t i o n is g i v e n by T ^ x = exp m X"*) (9) w h i c h i s t h e s i m p l e power law f i r s t p r o p o s e d by Angstrom [ 2 1 , 2 2 ] . The a i r mass, m, i s c a l c u l a t e d f r o m K a s t e n ' s f o r m u l a [23] m = 1/(cos(e) + 0.15(93.885- e ) " 1 - 2 5 3 ) (10) T h i s i s t h e r e l a t i v e a i r mass, s i n c e a l l c a l c u l a t i o n s a r e assumed t o be a t sea l e v e l . The d i r e c t s p e c t r a l i r r a d i a n c e on a h o r i z o n t a l s u r f a c e a t sea l e v e l a t t h e a v e r a g e e a r t h - s u n d i s t a n c e i s g i v e n by I> = I © * c o s ( e ) T K > raX r».A (11) The model o f d i f f u s e i r r a d i a n c e p r o p o s e d i n t h i s s t u d y i s now p r e s e n t e d . M o d e l o f D i f f u s e I r r a d i a n c e The d i f f u s e i r r a d i a n c e i s c o n s i d e r e d t o be composed of t h r e e d i s t i n c t components: (1) R a y l e i g h - s c a t t e r e d ; (2) a e r o s o l -s c a t t e r e d ; and (3) m u l t i p l y - r e f l e c t e d . The i n d i v i d u a l t r a n s m i t t a n c e s o f t h e d i r e c t model a r e u s e d t o g e n e r a t e e q u a t i o n s f o r e a c h o f t h e two s c a t t e r i n g components. The e q u a t i o n f o r t h e m u l t i p l y - r e f l e c t e d component was a d a p t e d from 10 [24] and t h e i n i t i a l i d e a f o r t h e f o r m u l a t i o n o f t h e a t m o s p h e r i c a l b e d o came from b r o a d - b a n d c a l c u l a t i o n s [ 1 8 ] . R a y l e i q h - s c a t t e r e d Component The R a y l e i g h - s c a t t e r e d d i f f u s e i r r a d i a n c e a r r i v i n g on a h o r i z o n t a l s u r f a c e i s D K\ = I e * cos (e ) rt>> r , A r^y, T ^ a ( 1 - T h X ) F r (12) T h i s i s s i m i l a r t o eqn. (2) e x c e p t t h e f a c t o r ( 1 - T V X T ^ X ) has been c h a n g e d t o (1-T , . X ) and i s t a k e n o u t s i d e t h e b r a c k e t e d q u a n t i t y . A n a l o g u s t o L e c k n e r ' s a p p r o a c h , t h e two beams o f r a d i a t i o n a r e t h e d i r e c t beam and a beam w h i c h i n c l u d e s m o l e c u l a r a b s o r p t i o n and a e r o s o l a t t e n u a t i o n . The d i f f e r e n c e between t h e s e i s t h e R a y l e i g h - s c a t t e r e d d i f f u s e i r r a d i a n c e . F r i s t h e R a y l e i g h f o r w a r d s c a t t e r i n g r a t i o , g e n e r a l l y t a k e n as 0.5. A e r o s o l - s c a t t e r e d Component S i m i l a r t o eqn.(12), t h e a e r o s o l - s c a t t e r e d d i f f u s e i r r a d i a n c e on a h o r i z o n t a l s u r f a c e i s .D».x = I«A cos (e ) T „ X T U > A T k X ( 1 - T ^ a ) u 0 Fa (13) H e r e , t h e d i f f e r e n c e between t h e d i r e c t beam and a beam w h i c h i n c l u d e s R a y l e i g h - s c a t t e r i n g and m o l e c u l a r a b s o r b e r s i s t h e a e r o s o l a t t e n u a t e d r a d i a t i o n . The s i n g l e s c a t t e r i n g a l b e d o o f t h e a e r o s o l , u e , p a r t i t i o n s t h e e n e r g y a t t e n u a t e d i n t o . s c a t t e r i n g and a b s o r p t i o n . A v a l u e o f o© = 1.0 i n d i c a t e s a c o m p l e t e l y s c a t t e r i n g a e r o s o l . The s i n g l e s c a t t e r i n g a l b e d o i s 11 assumed i n d e p e n d e n t of w a v e l e n g t h . F a , t h e e f f e c t i v e s c a t t e r i n g r a t i o ( t o w a r d t h e e a r t h , n o t n e c e s s a r i l y i n t h e f o r w a r d d i r e c t i o n ) i s a f u n c t i o n o f z e n i t h a n g l e , e. E x p e r i m e n t a l v a l u e s o f Fa were r e p o r t e d [25] and have been u s e d by o t h e r _;/-workers [ 1 8 ] . I t i s t o be e m p h a s i z e d t h a t t h e s e v a l u e s o f Fa g i v e o n l y a r o u g h e s t i m a t e o f t h e e f f e c t i v e f o r w a r d s c a t t e r i n g f u n c t i o n . The v a l u e s of Fa u s e d a r e : Fa = 0.923 f o r e = 0 . 0 ° , Fa = 0.78 f o r © = 6 0 . 0 ° , and Fa = 0.58 f o r e = 8 0 . 0 ° . L i k e t h e s i n g l e s c a t t e r i n g a l b e d o , t h e w a v e l e n g t h dependence o f Fa i s n e g l e c t e d . S e v e r a l o t h e r f o r m u l a t i o n s f o r t h e d i f f u s e i r r a d i a n c e have been i n v e s t i g a t e d . T h e s e f o r m u l a t i o n s a r e v e r y s i m i l a r t o t h e p r e c e e d i n g method i n t h a t t h e y a l l s e p a r a t e t h e e f f e c t s of t h e a e r o s o l s c a t t e r i n g and t h e R a y l e i g h - s c a t t e r i n g . One s u c h f o r m u l a t i o n i s T h i s f o r m u l a t i o n d o e s n o t t a k e i n t o a c c o u n t t h e i n t e r a c t i o n o f t h e a e r o s o l s c a t t e r i n g and t h e s c a t t e r i n g from t h e a i r m o l e c u l e s . T h i s was a c c o u n t e d f o r i n t h e p r e v i o u s f o r m u l a t i o n by m u l t i p l y i n g t h e R a y l e i g h s c a t t e r i n g component by t h e a e r o s o l t r a n s m i t t a n c e and m u l t i p l y i n g t h e a e r o s o l s c a t t e r i n g component by t h e R a y l e i g h t r a n s m i t t a n c e . The r e s u l t s o f t h e f o r m u l a t i o n of eqn (14) a r e g i v e n f o r c o m p a r i s o n t o an a c c u r a t e t h e o r e t i c a l model i n t h e n e x t c h a p t e r . M u l t i p l y - r e f l e c t e d Component The d i r e c t r a d i a t i o n r e a c h i n g t h e g r o u n d , t h e R a y l e i g h -(14) 12 s c a t t e r e d , and a e r o s o l - s c a t t e r e d d i f f u s e r a d i a t i o n r e a c h i n g t h e g r o u n d a f t e r t h e f i r s t p a s s t h r o u g h t h e a t m o s p h e r e a r e p a r t i a l l y r e f l e c t e d back by t h e e a r t h . T h i s u p w e l l i n g r a d i a t i o n i s p a r t i a l l y r e f l e c t e d by t h e a t m o s p h e r e downward a g a i n and t h i s p r o c e s s c o n t i n u e s . The r e s u l t i n g d i f f u s e i r r a d i a n c e a f t e r t h e s e m u l t i p l e - r e f l e c t i o n s i s g i v e n by D*X= d > + D ^ + D f c X ) ^ p ^ / ( 1 - ^ P^ ) (15) The g r o u n d a l b e d o , p^ , may be t a k e n i n d e p e n d e n t o f w a v e l e n g t h . The m o n o c h r o m a t i c a t m o s p h e r i c a l b e d o , p^^ , c a n be shown t o be ^ = T » A Tv* r ^ ^ ( l - T ' K A ) r ^ x F r + 0.22(1-rJLx>TKX o .j. (16) The p r i m e s on a l l t h e t r a n s m i t t a n c e s i n d i c a t e t h a t t h e y a r e a l l e v a l u a t e d a t m = 1.9. T h i s a i r mass, m = 1.9 i s t h e r e s u l t o f t u n i n g t o p r o d u c e t h e b e s t agreement w i t h t h e o r e t i c a l r e s u l t s [ 2 ] o f a t m o s p h e r i c a l b e d o . The f i r s t t e r m o f eqn. (16) r e p r e s e n t s t h e f r a c t i o n o f t h e u p w e l l i n g r a d i a t i o n r e f l e c t e d back t o t h e e a r t h due t o R a y l e i g h - s c a t t e r i n g . The s e c o n d t e r m r e p r e s e n t s t h e f r a c t i o n o f t h e u p w e l l i n g r a d i a t i o n r e f l e c t e d back t o t h e e a r t h by t h e a e r o s o l s . The f a c t o r 0.22 i s t h e e f f e c t i v e b a c k - s c a t t e r r a t i o of t h e a e r o s o l e v a l u a t e d a t m = 1.9. I n o t h e r words (1 - Fa) = (1 - .78) = 0.22. The t o t a l d i f f u s e i r r a d i a n c e f r o m a l l s o u r c e s i s g i v e n by D X = D_x + D ^ + D^x (17> S i m i l a r l y t h e g l o b a l i r r a d i a n c e i s t h e sum o f t h e d i r e c t 13 i r r a d i a n c e a n d t h e t o t a l d i f f u s e i r r a d i a n c e . T h i s i s g i v e n b y G X = IX + D A (18) T h e s e m o n o c h r o m a t i c v a l u e s o f d i r e c t i r r a d i a n c e , t o t a l d i f f u s e i r r a d i a n c e a n d g l o b a l i r r a d i a n c e c a n a l l b e s u m m e d s e p a r a t e l y t o g i v e t h e b r o a d - b a n d v a l u e s o f t h e s e q u a n t i t i e s . T h i s i s a c c o m p l i s h e d b y t a k i n g e a c h m o n o c h r o m a t i c i r r a d i a n c e v a l u e i n W m" 2(»im)" 1 a n d m u l t i p l y i n g b y t h e a p p r o p r i a t e w a v e l e n g t h i n t e r v a l i n pm. T h e s e v a l u e s a r e t h e n s u m m e d t o g i v e a b r o a d -b a n d v a l u e f o r a n y w a v e l e n g t h i n t e r v a l . 1 4 IV METHODOLOGY AND VALIDATION To v a l i d a t e t h e p r e s e n t f o r m u l a t i o n o f d i f f u s e s p e c t r a l i r r a d i a n c e , t h e r e s u l t s o b t a i n e d must be compared t o more a c c u r a t e t h e o r e t i c a l c a l c u l a t i o n s a nd t o measured s p e c t r a . The c o m p a r i s o n w i t h t h e o r e t i c a l s o l u t i o n s i s e a s y a s e x t e n s i v e n u m e r i c a l r e s u l t s a r e a v a i l a b l e [ 2 , 2 6 ] and t h e a t m o s p h e r i c i n p u t p a r a m e t e r s a r e e x p l i c i t l y known. A s i m i l a r c o m p a r i s o n w i t h m e a s u r e d s p e c t r a i s d i f f i c u l t . The measured d a t a a r e e i t h e r b a s e d on l o n g t e r m a v e r a g e s [ 3 , 4 , 5 ] o r a r e g i v e n i n g r a p h i c a l f o r m [ 6 ] w i t h i n s u f f i c i e n t i n f o r m a t i o n on t h e a t m o s p h e r i c p a r a m e t e r s . C o n s e q u e n t l y no c o m p a r i s o n w i t h measured s p e c t r a i s i l l u s t r a t e d . A l l t h e o r e t i c a l c o m p a r i s o n s between M o d e l A t m o s p h e r e s o f Dave [ 2 ] and t h e p r e s e n t work use t h e same e x t r a t e r r e s t r i a l s o l a r v a l u e s [ 2 0 ] , however t h e number and s p a c i n g o f w a v e l e n g t h i n t e r v a l s i s d i f f e r e n t between t h e two s t u d i e s . The s p e c t r u m i n v e s t i g a t e d e x t e n d e d from 0 . 2 9 t o 2 . 5 »»m. The e x t r a t e r r e s t r i a l s o l a r v a l u e s and t h e c o r r e s p o n d i n g a b s o r p t i o n c o e f f i c i e n t s u s e d i n t h i s s t u d y a r e i n A p p e n d i x I . The e x t r a t e r r e s t r i a l v a l u e s u s e d by Dave a r e i n A p p e n d i x I I . A c o m p a r i s o n between Model A ( p u r e R a y l e i g h a t m o s p h e r e ) and t h e p r e s e n t work f o r z e n i t h a n g l e s 0 . 0 , 6 0 . 0 and 8 0 . 0 ° i s p r e s e n t e d i n F i g . 1 . F o r a i r mass one, t h e agreement i s e x c e l l e n t . The major d i s c r e p a n c y ( . 3 5 t o . 4 0 ( im) i s due t o t h e d i f f e r e n t w i d t h s of t h e w a v e l e n g t h i n t e r v a l s u s e d i n t h e two s t u d i e s and hence t h e a v e r a g i n g o f t h e e x t r a t e r r e s t r i a l s p e c t r u m o v e r t h e s e w a v e l e n g t h i n t e r v a l s . A d e t a i l e d a n a l y s i s of t h i s a p p e a r s i n A p p e n d i x I I I . 1 5 — | — i —i— i — i — " — r~r~T— 1—r~1— r ~ > — r ~ i — r ~ T ~ ~ ] —r 450. _ 400. _ 350. _ HODEL fl 300. _ PRESENT riODEL 0:30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 WAVELENGTH (MICRONS) 1 C o m p a r i s o n of t h e d i f f u s e i r r a d i a n c e from Model A and t h e p r e s e n t work f o r z e n i t h a n g l e s 0.0, 60.0 and 8 0 . 0 ° . Ground a l b e d o i s 0.0. 16 F o r a i r mass two, t h e p r e s e n t work somewhat o v e r e s t i m a t e s t h e d i f f u s e i r r a d i a n c e a t s h o r t w a v e l e n g t h s 0.3 t o 0.35 jim. A t l o n g e r w a v e l e n g t h s t h e agreement i s q u i t e good. A t z e n i t h a n g l e 8 0 . 0 ° , o v e r e s t i m a t i o n o c c u r s up t o 0.45 nm , a f t e r w h i c h t h e agreement i s a g a i n good. The o v e r e s t i m a t i o n a t s h o r t w a v e l e n g t h s o f t h e p r e s e n t model c o u l d be a t t r i b u t e d t o t h e f a c t t h a t M o d e l A i n c l u d e d a l l o r d e r s o f s c a t t e r i n g whereas t h e p r e s e n t work does n o t . The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r Model A and t h e p r e s e n t f o r m u l a t i o n , and t h e p e r c e n t a g e d i f f e r e n c e between t h e two v a l u e s a p p e a r i n T a b l e 1. The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 1 a p p e a r i n A p p e n d i x IV. T a b l e 1: . I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 1 6 Model A T h i s Work % D i f f e r e n c e 0.0 64.25 62.72 2.4% 60.0 53.27 53.77 0.94% 80.0 33.92 35.76 5.4% A c o m p a r i s o n of Model B ( R a y l e i g h - s c a t t e r i n g p l u s a b s o r b i n g g a s e s ) w i t h t h e p r e s e n t work f o r z e n i t h a n g l e s 0 . 0 , 6 0 . 0 and 8 0 . 0 ° i s shown i n F i g . 2. The r e s u l t s a r e s i m i l a r t o t h o s e p r e s e n t e d i n F i g . 1. A t a i r mass one t h e agreement i s e x c e l l e n t , a t a i r mass two, t h e agreement i s v e r y good and a t 0 = 8 0 . 0 ° t h e o v e r e s t i m a t i o n o f t h e p r e s e n t work i s a g a i n n o t i c e d . 1 7 O C J o CO LU CJ> CE i — i a cr a: or LU CO ZD a 500. 450. 400. 350. 300. 250. 200. 150. 100 5 0 . 1 1 1 1 1 I " 1 ' I r 1 1 I | I I T nODEL B PRESENT nODEL 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 WAVELENGTH (MICRONS) F i g . 2 C o m p a r i s o n o f t h e d i f f u s e i r r a d i a n c e f r o m M o d e l B a n d t h e p r e s e n t w o r k f o r z e n i t h a n g l e s 0 . 0 , 6 0 . 0 a n d 8 0 . 0 ° . G r o u n d a l b e d o i s 0 . 0 , o z o n e c o n t e n t i s 0 . 3 1 8 c m a n d w a t e r v a p o r c o n t e n t i s 2 . 9 2 5 c m . 18 The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r Model B and t h e p r e s e n t f o r m u l a t i o n , and t h e p e r c e n t a g e d i f f e r e n c e between t h e two v a l u e s a p p e a r i n T a b l e 2. The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 2 a r e l i s t e d i n A p p e n d i x V I I . T a b l e 2: I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 2 e Model B T h i s Work % D i f f e r e n c e 0.0 57.41 56.91 0.87% 60.0 47.83 48.25 0.88% 80.0 30.16 31.30 3.8% T h e s e c o m p a r i s o n s i n d i c a t e t h a t a t z e n i t h a n g l e s f r o m 0.0° t o 60.0° t h e p r e s e n t work a c c u r a t e l y d e p i c t s s c a t t e r i n g and m o l e c u l a r a b s o r p t i o n by a c l e a n a t m o s p h e r e . T h i s i m p l i e s t h a t t h e f o r m u l a t i o n and t h e c o e f f i c i e n t s u s e d a r e c o r r e c t and t h a t t h e n e x t s t e p t o i n c l u d e t h e a e r o s o l s s t a r t s f r o m a s e c u r e p r o c e d u r e . A c o m p a r i s o n o f t h e r e s u l t s of M o d e l C a t m o s p h e r e ( R a y l e i g h - s c a t t e r i n g , a b s o r b i n g g a s e s p l u s a n o n - a b s o r b i n g a e r o s o l ) and t h e p r e s e n t p r o c e d u r e were c a r r i e d o u t as f o l l o w s . The d i r e c t i r r a d i a n c e was c a l c u l a t e d f o r s p e c i f i c ozone and w a ter v a p o r c o n t e n t c o r r e s p o n d i n g t o t h e v a l u e s u s e d i n Model C. The t u r b i d i t y p a r a m e t e r s o and fi were v a r i e d t o g i v e t h e b e s t f i t w i t h t h e d i r e c t s p e c t r u m o f M odel C. In t h i s manner, t h e v a l u e s o f t h e s e p a r a m e t e r s were d e r i v e d . An example o f s u c h an e x e r c i s e i s d e m o n s t r a t e d i n F i g . 3, where i t i s shown t h a t a*= 0.6 and fi • = 0.07 c o r r e s p o n d t o M odel C. 19 2000 1800 ~>—1—r ~i I i I i I i I i — I — i — I — r HODEL c DIRECT HODEL 0.3 0.5 0.7 0.9 1.1 WAVELENGTH (MICRONS) 1 .3 F i g . 3 C o m p a r i s o n o f t h e d i r e c t i r r a d i a n c e f r o m Model C and t h e p r e s e n t work f o r z e n i t h a n g l e s 0.0 and 6 0 . 0 ° . G r o u n d a l b e d o i s 0.0, ozone c o n t e n t i s 0.318 cm and water v a p o r c o n t e n t i s 2.925 cm, a = 0.6, p = 0.07 and o e = 1.0 20 The f i t o f t h e p r e s e n t d i r e c t model and Model C i s f o u n d t o be e x c e l l e n t f o r a i r mass one and two. The i n t e g r a t e d d i r e c t i r r a d i a n c e s f o r Model C and t h e p r e s e n t f o r m u l a t i o n , and t h e p e r c e n t a g e d i f f e r e n c e between t h e two v a l u e s a p p e a r i n T a b l e 3. The d i r e c t s p e c t r a l v a l u e s c o r r e s p o n d i n g t o t h e p a r a m e t e r s i n F i g . 3 a r e p r e s e n t e d i n A p p e n d i x V. T a b l e 3: I n t e g r a t e d D i r e c t I r r a d i a n c e f o r F i g . 3 9 Model C 0.0 969.27 60.0 393.64 H a v i n g d e t e r m i n e d v a l u e s f o r a and £ i n t h e t h i s manner, t h e d i f f u s e i r r a d i a n c e was c a l c u l a t e d u s i n g t h e p r e s e n t f o r m u l a t i o n , e q n s . ( 1 0 , 1 1 , 1 2 , 1 3 and 1 5 ) . A c o m p a r i s o n o f t h e s e r e s u l t s w i t h t h e d i f f u s e c a l c u l a t i o n s o f Model C a r e shown i n F i g . 4 f o r a i r mass one and F i g . 5 f o r a i r mass two. In o r d e r t o e l i m i n a t e t h e e f f e c t s o f m u l t i p l e r e f l e c t i o n s between g r o u n d and a t m o s p h e r e , g r o u n d a l b e d o i s assumed z e r o . P l o t s o f L e c k n e r ' s f o r m u l a t i o n , eqn. ( 2 ) , a r e a l s o shown i n t h e s e d i a g r a m s . In F i g . 4, up t o 0.6 >/m, t h e agreement between Model C a n d t h e p r e s e n t work i s good. The agr e e m e n t i s a l s o q u i t e good a t l o n g e r w a v e l e n g t h s , w i t h t h e p r e s e n t work g i v i n g g e n e r a l l y s l i g h t l y l o w e r v a l u e s t h a n t h o s e of Model C. O v e r a l l t h e agreement i s v e r y good a t a i r mass one. L e c k n e r ' s f o r m u l a t i o n a g r e e s up t o 0.40 ttrn and t h e n d r o p s o f f a b r u p t l y t o v a l u e s s u b s t a n t a i l l y below t h o s e of M o d e l C. T h i s Work % D i f f e r e n c e 973.76 0.46% 396.77 0.80% 2 1 a to U J cr i — i a CE cr: or: 500. 450. L i—1—i— i— i— i— i— i— i—1—i— l— i— I i I i j r HODEL C PRESENT I10DEL L E C K N E R L i i i I i I i I i I i I 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ].J0 1.20 3 . 30 W A V E L E N G T H ( M I C R O N S ) F i g . 4 C o m p a r i s o n of t h e d i f f u s e i r r a d i a n c e from Model C ,the p r e s e n t model and L e c k n e r ' s f o r m u l a t i o n f o r a i r mass one. G r o u n d a l b e d o i s 0.0, ozone c o n t e n t i s 0.318 cm and water v a p o r c o n t e n t i s 2.925 cm, o = 0.6, « = 0.07 and u0 = 1.0. 22 In F i g . 5, t h e p r e s e n t work i s c o n s i s t e n t l y s l i g h t l y below t h e v a l u e s of M o d e l C. L e c k n e r ' s f o r m u l a t i o n i s i n good agreement w i t h M o d e l C up t o .42 »<m, but t h e r e a f t e r d r o p s s i g n i f i c a n t l y below M o d e l C and below t h e p r e s e n t work. The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r M o d e l C, t h e p r e s e n t work and L e c k n e r ' s f o r m u l a t i o n a r e l i s t e d i n T a b l e 4. The d i f f u s e s p e c t r a l v a l u e s c o r r e s p o n d i n g t o t h e p a r a m e t e r s i n F i g . 4 and F i g . 5 a r e i n A p p e n d i x V I . T a b l e 4: I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 4 and 5 9 M o d e l C T h i s Work L e c k n e r 0.0 137.44 132.76 101.33 60.0 108.00 96.14 85.16 T h e s e c o m p a r i s o n s i n d i c a t e t h a t t h e p r e s e n t f o r m u l a t i o n c o n s i s t e n t l y p r o d u c e s b e t t e r r e s u l t s t h a n L e c k n e r ' s f o r m u l a t i o n up t o a i r mass two. Many o t h e r c o m p a r i s o n s have been made and t h e y a l l i n d i c a t e t h e same c o n c l u s i o n s . When a n o n - z e r o g r o u n d a l b e d o i s c o n s i d e r e d , t h e m u l t i p l e r e f l e c t i o n s between t h e g r o u n d and t h e a t m o s p h e r e add t o t h e t o t a l d i f f u s e i r r a d i a n c e . F i g . 6 shows a c o m p a r i s o n of M o d e l C and t h e p r e s e n t work f o r a i r mass one and g r o u n d a l b e d o 0.3. The p r e s e n t work u n d e r e s t i m a t e s somewhat i n t h e 0.33 - 0.40 »#m w a v e l e n g t h i n t e r v a l b u t i s o t h e r w i s e i n e x c e l l e n t agreement w i t h M o d e l C. A l s o i l l u s t r a t e d i s t h e s i m p l e f o r m u l a t i o n of L e c k n e r t h a t has no p r o v i s i o n f o r g r o u n d a l b e d o . I t i s much below t h e v a l u e s of Model C and t h e p r e s e n t work. 23 5 0 0 • r—i—|—i—\—i—i—i—f—i—r -!—i— 1—i— 1—r~i r - r WAVELENGTH (MICRONS) F i g . 5 C o m p a r i s o n o f t h e d i f f u s e i r r a d i a n c e from Model C , t h e p r e s e n t model and L e c k n e r ' s f o r m u l a t i o n f o r a i r mass two. Ground a l b e d o i s 0 . 0 , ozone c o n t e n t i s 0 . 3 1 8 cm and water v a p o r c o n t e n t i s 2.925 cm, a = 0 . 6 , fi = 0 . 0 7 and o 0 = 1 . 0 . T 1 1 1 1 1 1 1 1 1 1 1 1 I ' I 1 I r I i I i I i I I I 1 1 1 1 ' 1 1 1 1 1 1 1 0.30 0.40 0.50 0.60 0 .70 0 .80 0.90 3.00 3.10 3.20 3.30 WAVELENGTH (MICRONS) 6 C o m p a r i s o n o f t h e d i f f u s e i r r a d i a n c e f r o m M o d e l C , t h e p r e s e n t m o d e l a n d L e c k n e r ' s f o r m u l a t i o n f o r a i r m a s s o n e . G r o u n d a l b e d o i s 0 . 3 , o z o n e c o n t e n t i s 0 . 3 1 8 c m a n d w a t e r v a p o r c o n t e n t i s 2.925 c m , o = 0 . 6 , i = 0 . 0 7 a n d u 0 = 1 . 0 . 25 F i g . 7 p r e s e n t s t h e same c o m p a r i s o n a s F i g . 6 f o r a i r mass two. At t h i s a i r mass t h e p r e s e n t work i s g e n e r a l l y l o w e r t h a n t h e v a l u e s o f M o d e l C but s t i l l o v e r a l l much b e t t e r t h a n L e c k n e r ' s f o r m u l a t i o n . The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r Model C, t h e p r e s e n t work and L e c k n e r ' s f o r m u l a t i o n a r e l i s t e d i n T a b l e 5. The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 6 and F i g . 7 f o r t h e two g r o u n d a l b e d o s and t h e two z e n i t h a n g l e s 0.0° and 60.0° a r e l i s t e d i n A p p e n d i x V I I . T a b l e 5: I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 6 and 7 9 Model C T h i s Work L e c k n e r 0.0 171.40 161.58 101.33 60.0 122.88 108.54 85.16 S e v e r a l c o m p a r i s o n s w i t h t h e f o r m u l a t i o n o f H a t f i e l d e t a l [10] have been a t t e m p t e d . A l t h o u g h t h e s e c o m p a r i s o n s a r e n o t r i g o r o u s , t h a t i s , h i s a c t u a l s p e c t r u m and c o e f f i c i e n t s have n o t been u s e d , t h e r e s u l t s a r e p r o b a b l y i n d i c a t i v e o f t h e k i n d o f a c c u r a c y a v a i l a b l e w i t h t h e i r model . F i g . 8 i l l u s t r a t e s t h i s c o m p a r i s o n w i t h M o d e l C, t h e p r e s e n t work and H a t f i e l d f o r a i r mass one. The H a t f i e l d f o r m u l a t i o n p r o d u c e s r e s u l t s v e r y s i m i l a r t o t h o s e of L e c k n e r as shown e a r l i e r i n F i g . 4. The c o m p a r i s o n f o r a i r mass two i s shown i n F i g . 9. S i n c e H a t f i e l d e t a l o m i t t e d t h e a b s o r p t i o n due t o w a t e r v a p o r and t h e u n i f o r m l y - m i x e d g a s e s , t h e i r v a l u e s o f d i f f u s e i r r a d i a n c e show no a b s o r p t i o n d e p r e s s i o n s t h a t a r e so c h a r a c t e r i s t i c o f Model C and t h e p r e s e n t f o r m u l a t i o n . 26 500. 450. 400. 1 1 I — | — I — [ — 1 — i — i — l r i — I — | — I | I riODEL c PRESENT HODEL LECKNER 0 .30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ].)0 3.20 3.30 WAVELENGTH (MICRONS) 7 C o m p a r i s o n of t h e d i f f u s e i r r a d i a n c e f r o m Model C ,the p r e s e n t model and L e c k n e r ' s f o r m u l a t i o n f o r a i r mass two. Ground a l b e d o i s 0.3, ozone c o n t e n t i s 0.318 cm, wa t e r v a p o r c o n t e n t i s 2 .925 cm, a = 0.6, i = 0.07 and oo = 1.0. 27 F i g . 8 C o m p a r i s o n of t h e d i f f u s e i r r a d i a n c e f r o m Model C ,the p r e s e n t model and e q n . (3) o f H a t f i e l d f o r a i r mass one. Ground a l b e d o i s 0.0, ozone c o n t e n t i s 0.318 cm, w a t e r v a p o r c o n t e n t i s 2 . 9 2 5 cm, o = 0 . 6 , fi = 0.07 and Oo = 1 . 0 . 28 500. 450 . 400. 350. 300. I 1 I 1 i i r -] r riODEL fl PRESENT HODEL HATFIELD 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3.0 3.) 3.2 3.3 W A V E L E N G T H ( M I C R O N S ) C o m p a r i s o n of t h e d i f f u s e i r r a d i a n c e from Model C, t h e p r e s e n t model and eqn. (3) of H a t f i e l d f o r a i r mass two. G r o u n d a l b e d o i s 0.0, ozone c o n t e n t i s 0.318 cm, wa t e r v a p o r c o n t e n t i s 2.925 cm, o = 0.6, B = 0.07 and 29 The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r Model C, t h e p r e s e n t work and H a t f i e l d ' s f o r m u l a t i o n a r e l i s t e d i n T a b l e 6. T a b l e 6: I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 8 and 9 9 Model C T h i s Work H a t f i e l d 0.0 137.44 132.76 105.76 60.0 108.00 96.14 83.64 A s i m i l a r c o m p a r i s o n f o r t h e f o r m u l a t i o n o f eqn. (14) i s shown i n F i g . 10. T h i s f o r m u l a t i o n o v e r e s t i m a t e s s t r o n g l y up t o 0.65 i i m , a f t e r w h i c h i t c o r r e s p o n d s a l m o s t e x a c t l y w i t h t h e p r e s e n t work. T h i s b e h a v i o u r , as s e e n i n F i g . 11, i s r e p e a t e d f o r a i r mass two. The i n t e g r a t e d d i f f u s e i r r a d i a n c e s f o r M o d e l C, t h e p r e s e n t ' w o r k and eqn. 14 a r e shown i n T a b l e 7. T a b l e 7; I n t e g r a t e d D i f f u s e I r r a d i a n c e f o r F i g . 1 0 and 11 9 0.0 60.0 Model C 137.44 108.00 T h i s Work 132.76 96. 1 4 Eqn. 14 1 50.25 1 20.93 30 5 D 0 - 1 — i — | — i — i — i — i — i — i — i — i — i i ' r ~ i r ^ i r WAVELENGTH (MICRONS) F i g . 10 C o m p a r i s o n o f t h e d i f f u s e i r r a d i a n c e f r o m Model C, t h e p r e s e n t model and eqn. (14) f o r a i r mass one. Ground a l b e d o i s 0.0, ozone c o n t e n t i s 0.318 cm, water v a p o r c o n t e n t i s 2 . 9 2 5 cm, a = 0 . 6 , * = 0.07 a n d u0 = 1.0. 31 O LJ O CO LU LJ CE i — i a cr CO ZD 500. 450. 400. 350. 300. 250. 200. 150 100. a 50. i — i — i — i — i i i r i — i — i — f — r nODEL c PRESENT. nODEL EQN. 14 i . 3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ].J !. 2 1.3 WAVELENGTH (MICRONS) F i g . 11 C o m p a r i s o n o f t h e d i f f u s e i r r a d i a n c e f r o m M o d e l C , t h e p r e s e n t m o d e l a n d e q n . ( 1 4 ) f o r a i r m a s s t w o . G r o u n d a l b e d o i s 0 . 0 , o z o n e c o n t e n t i s 0 . 3 1 8 c m , w a t e r v a p o r c o n t e n t i s 2 . 9 2 5 c m , o = 0 . 6 , B = 0 . 0 7 a n d u . = 1 . 0 . 32 V RESULTS In t h e p r e v i o u s c h a p t e r , t h e p r e s e n t work was compared t o an a c c u r a t e t h e o r e t i c a l model and i t was f o u n d t o p r o d u c e v e r y good r e s u l t s . The v e r s a t i l i t y o f t h e p r e s e n t method w i l l be i l l u s t r a t e d i n t h i s c h a p t e r . The s i x i n d e p e n d e n t i n p u t p a r a m e t e r s o f t h i s model a r e : a i r mass, ozone c o n t e n t , water v a p o r c o n t e n t , g r o u n d a l b e d o , t u r b i d i t y c o e f f i c i e n t , and w a v e l e n g t h e x p o n e n t . T h e s e p a r a m e t e r s a r e e a s i l y v a r i e d , so t h e e f f e c t s o f e a c h p a r a m e t e r c a n be i n v e s t i g a t e d i n d e p e n d e n t l y . The e x t r a t e r r e s t r i a l s o l a r s p e c t r u m p r e s e n t e d by T h e k a e k a r a [27] i s u s e d t o p r o d u c e a l l t h e r e s u l t s i n t h i s s e c t i o n . T h i s s p e c t r u m i s r e p r o d u c e d i n A p p e n d i x V I I I . I t i s u n d e r s t o o d t h a t u p d a t e d v a l u e s o f t h e s o l a r c o n s t a n t and i t s s p e c t r a l d i s t r i b u t i o n have been p r o p o s e d [ 2 8 ] . T h i s c u r r e n t s p e c t r u m i s p r e s e n t e d i n A p p e n d i x IX. However, t h e p r e s e n t method i s i n d e p e n d e n t of any p a r t i c u l a r e x t r a t e r r e s t r i a l s o l a r s p e c t r u m . A i r mass has t h e g r e a t e s t e f f e c t o f any o f t h e p a r a m e t e r s on t h e d i r e c t and g l o b a l i r r a d i a n c e , w hereas t h e d i f f u s e i r r a d i a n c e i s a f f e c t e d t o a much l e s s e r e x t e n t by a i r mass. T h i s e f f e c t i s d e m o n s t r a t e d i n F i g . 12 f o r a i r mass one and two. The b r o a d - b a n d (0.29 t o 4.0 »»m) d i r e c t i r r a d i a n c e d e c r e a s e s 60% from 925 W n r 2 t o 370 W n r 2 and t h e b r o a d - b a n d g l o b a l i r r a d i a n c e d e c r e a s e s 56% from 1073 W n r 2 t o 471 W n r 2 . The b r o a d - b a n d d i f f u s e i r r a d i a n c e d e c r e a s e s o n l y 32% f r o m 148 W n r 2 t o 100 W n r 2 and f o r w a v e l e n g t h s g r e a t e r t h a n a b o u t 1.0 vm t h e d i f f u s e i s e s s e n t i a l l y i n d e p e n d e n t of a i r mass. 3 3 1 I 1 I 1 I 1 I i I ' I i 1 i I > I r 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ] .0 J.I 1.2 1.3 UAVELENGTH (MICRONS) F i g . 12 The e f f e c t o f a i r mass on d i r e c t , d i f f u s e and g l o b a l i r r a d i a n c e on a h o r i z o n t a l s u r f a c e f o r o = 1.3, I = 0.07, f> = 0.0, o z o n e c o n t e n t i s 0.3 cm and water v a p o r c o n t e n t i s 2. 0 cm. C u r v e 1 g l o b a l m = 1 . 0 C u r v e 2 d i r e c t m = 1 . 0 C u r v e 3 g l o b a l m = 2. 0 C u r v e 4 d i r e c t m = 2. 0 C u r v e 5 d i f f u s e m = 1 . 0 C u r v e 6 d i f f u s e m = 2. 0 34 The s p e c t r a l c h a r a c t e r o f t h e c u r v e s f o r a i r mass one . and two a r e v e r y s i m i l a r and i t i s n o t i c e d t h a t t h e p e r c e n t a g e d e c r e a s e i n i r r a d i a n c e i s a l m o s t u n i f o r m a t a l l w a v e l e n g t h s ; even t h o u g h t h e a c t u a l amount a t t e n u a t e d a t s h o r t w a v e l e n g t h s (<1.0 nm) i s v e r y l a r g e . The t u r b i d i t y c o e f f i c i e n t ^ , has a s t r o n g i n f l u e n c e on b o t h d i r e c t and d i f f u s e i r r a d i a n c e . fi i s an i n d i c a t i o n o f t h e mass l o a d i n g of t h e a e r o s o l s and a l s o an i n d i c a t i o n of t h e v i s i b i l i t y f o r a g i v e n a e r o s o l p a r t i c l e s i z e d i s t r i b u t i o n . F i g . 13 shows t h e e f f e c t o f e on d i r e c t , d i f f u s e and g l o b a l i r r a d i a n c e f o r a i r mass one and a = 1.3. F o r fi = 0.2 ( V i s = 12 km) t h e b r o a d - b a n d d i r e c t i r r a d i a n c e i s a b o u t 19% l e s s t h a n f o r a = 0.07 ( V i s = 50 km). The b r o a d - b a n d d i f f u s e i r r a d i a n c e f o r fi = 0.07 i s a b o u t 49% l e s s t h a n f o r fi = 0.2. The b r o a d - b a n d g l o b a l i r r a d i a n c e f o r fi = 0.07 i s a b o u t 3% g r e a t e r t h a n f o r fi = 0.2. Thus even f o r a v e r y t u r b i d a t m o s p h e r e {fi = 0.2) t h e g l o b a l i r r a d i a n c e a v a i l a b l e i s o n l y m a r g i n a l l y a f f e c t e d a t a i r mass one. S p e c t r a l l y , fi has t h e g r e a t e s t e f f e c t on t h e d i f f u s e i r r a d i a n c e and p r e d o m i n a n t l y a t s h o r t w a v e l e n g t h s (<1.0 **m) . The s h a r p peak o f t h e d i r e c t c u r v e f o r fi = 0.07 has d i s a p p e a r e d and has become a wide p l a t e a u f o r fi = 0.2, i n d i c a t i n g t h e p r o n o u n c e d e f f e c t o f s c a t t e r i n g f o r an i n c r e a s e d mass l o a d i n g o f a e r o s o l s a t s h o r t (0.4 t o 0.7 itm) w a v e l e n g t h s . A l t h o u g h t h e g l o b a l i r r a d i a n c e i s d e c r e a s e d o n l y 3%, a l m o s t a l l of t h i s d e c r e a s e o c c u r s i n t h e v i s i b l e (0.4 t o 0.75 nm) band. 35 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 3.2 1.3 WAVELENGTH (HICRONS) F i g . 13 The e f f e c t of fi on d i r e c t , d i f f u s e and g l o b a l i r r a d i a n c e on a h o r i z o n t a l surface for a i r mass one,o = 1.3, 0.0, ozone content i s 0.3 cm and water vapor content i s 2.0 c m . Curve 1 g l o b a l fi = 0.07 Curve 2 g l o b a l fi = 0.2 Curve 3 d i r e c t fi - 0.07 Curve 4 d i r e c t fi = 0.2 Curve 5 d i f f u s e fi = 0.2 Curve 6 d i f f u s e fi = 0.07 3 6 F i g . 14 shows t h a t f o r a i r mass two, fi has a somewhat g r e a t e r e f f e c t on b r o a d - b a n d g l o b a l i r r a d i a n c e , a b o u t 9% d e c r e a s e , p a r t l y due t o t h e d e c r e a s e d v a l u e o f t h e e f f e c t i v e s c a t t e r i n g r a t i o o f t h e a e r o s o l . The b r o a d - b a n d d i r e c t i r r a d i a n c e d e c r e a s e d 3 1 % between fi = 0 . 0 7 and fl = 0 . 2 and t h e b r o a d - b a n d d i f f u s e i r r a d i a n c e d e c r e a s e d 43% between fi = 0 . 2 and fi = 0 . 0 7 . S p e c t r a l l y , t h e peak of t h e d i r e c t i r r a d i a n c e has moved from a p l a t e a u between 0 . 4 5 t o 0 . 7 >*m f o r fi = 0 . 0 7 t o a peak a t 0 . 7 »*m f o r fi = 0 . 2 . The d i f f u s e i r r a d i a n c e has c o r r e s p o n d i n g l y shown a l a r g e i n c r e a s e a t s h o r t ( 0 . 3 t o 0 . 6 >/m) w a v e l e n g t h s and a c t u a l l y e x c e e d s t h e d i r e c t i r r a d i a n c e f o r fi = 0 . 2 up t o 0 . 6 cm, i l l u s t r a t i n g v e r y w e l l t h a t what i s s c a t t e r e d f r o m t h e d i r e c t beam i s a c c o u n t e d f o r i n t h e d i f f u s e i r r a d i a n c e . A n o t h e r i n f l u e n c i n g f a c t o r o f t h e a e r o s o l s i s t h e w a v e l e n g t h e x p o n e n t , o , w h i c h i s r e l a t e d t o t h e s i z e d i s t r i b u t i o n o f t h e a e r o s o l p a r t i c l e s . A l a r g e v a l u e of a i n d i c a t e s a r e l a t i v e l y h i g h e r r a t i o o f s m a l l p a r t i c l e s t o l a r g e p a r t i c l e s . F o r most n a t u r a l a t m o s p h e r e s o i s a b o u t 1 . 3 . An a t m o s p h e r e c o n t a i n i n g a l a r g e amount of s m a l l p a r t i c l e s ( l a r g e a ) s c a t t e r s more t h a n an a t m o s p h e r e c o n t a i n i n g an e q u i v a l e n t mass of l a r g e p a r t i c l e s ( s m a l l o ) . In F i g . 1 5 , i t i s e v i d e n t t h a t o = 2 . 0 s c a t t e r s more e n e r g y from t h e d i r e c t beam, ab o u t 8%, t h a n d o e s o = 0 . 5 . The p l o t s o f t h e d i f f u s e i r r a d i a n c e show t h a t t h e e n e r g y s c a t t e r e d from t h e d i r e c t beam r e a p p e a r s as t h e d i f f u s e i r r a d i a n c e w i t h t h e d i f f e r e n c e between t h e two b r o a d - b a n d d i f f u s e v a l u e s b e i n g 3 2 % . The g l o b a l i r r a d i a n c e p l o t s show v e r y l i t t l e d i f f e r e n c e ( a b o u t 2%) between o = 2 . 0 and a = 0 . 5 , and t h e i n f l u e n c e o f o i s seen o n l y a t w a v e l e n g t h s l e s s t h a n 0 . 7 j»m. 3 7 1000. n r — j — i I i—I r 1 I 1 I 1 T I r BETA = 0.07 BETA = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3.0 1.1 1.2 1.3 WAVELENGTH (HICRONS) F i g . 14 The e f f e c t o f * on d i r e c t , d i f f u s e and g l o b a l i r r a d i a n c e on a h o r i z o n t a l s u r f a c e f o r a i r mass two, o = 1.3, />, = 0 .0, ozone w a t e r v a p o r c o n t e n t i s 2.0 cm C u r v e 1 g l o b a l B = 0.07 C u r v e 2 g l o b a l * = 0.2 C u r v e 3 d i r e c t * = 0.07 C u r v e 4 d i r e c t * = 0.2 C u r v e 5 d i f f u s e * = 0.2 C u r v e 6 d i f f u s e * = 0.07 38 F i g . 15 T h e e f f e c t o f o o n d i r e c t , d i f f u s e a n d g l o b a l i r r a d i a n c e o n a h o r i z o n t a l s u r f a c e f o r a i r m a s s o n e , * = 0 . 0 7 , />3 = 0 . 0 , o z o n e c o n t e n t w a t e r v a p o r c o n t e n t i s 2 . 0 c m . C u r v e 1 g l o b a l o ! = 0 . 5 C u r v e 2 g l o b a l 0 ! = 2 . 0 C u r v e 3 d i r e c t a : = 0 . 5 C u r v e 4 d i r e c t o = 2 . 0 C u r v e 5 d i f f u s e 0 = 2 . 0 C u r v e 6 d i f f u s e 0 = 0 . 5 39 The g r o u n d a l b e d o and m u l t i p l e r e f l e c t i o n s a r e c o n s i d e r e d n e x t . F i g . 16 shows t h e e f f e c t of t h e v a r i a t i o n o f g r o u n d a l b e d o on t h e d i f f u s e and g l o b a l i r r a d i a n c e . Two v a l u e s o f g r o u n d a l b e d o a r e i l l u s t r a t e d , p = 0.0 and p = 0.2. p = 0.2 i s r e p r e s e n t a t i v e o f most g r o u n d c o v e r s . Due t o m u l t i p l e r e f l e c t i o n s between t h e e a r t h and i t s a t m o s p h e r e , a l a r g e r g r o u n d r e f l e c t a n c e p r o d u c e s more d i f f u s e i r r a d i a n c e ( a b o u t 10%) and c o r r e s p o n d i n g i n c r e a s e s i n g l o b a l i r r a d i a n c e ( a b o u t 2 % ) . The e f f e c t s o f v a r i a t i o n o f w a t e r v a p o r c o n t e n t and ozone c o n t e n t a r e l i m i t e d t o s p e c i f i c w a v e l e n g t h bands and p r i m a r i l y t o t h e d i r e c t beam. T h e r e i s v e r y l i t t l e e f f e c t on t h e d i f f u s e i r r a d i a n c e due t o t h e s e p a r a m e t e r s . The g l o b a l i r r a d i a n c e i n d i c a t e s t h a t any d i f f e r e n c e i s due m a i n l y t o t h e v a r i a t i o n i n t h e d i r e c t i r r a d i a n c e . 40 i—l i | — i — | i | i | i—|—i—j—i—|—i—|—r 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1.0 ] . ! 1 .2 1 .3 U f l V E L E N G T H ( n i C R O N S ) F i g . 16 The e f f e c t o f g r o u n d a l b e d o on d i r e c t , d i f f u s e and g l o b a l i r r a d i a n c e on a h o r i z o n t a l s u r f a c e f o r a i r mass one, = 0.07, o = 1.3, ozone c o n t e n t i s 0.3 cm and water v a p o r c o n t e n t i s 2.0 cm. C u r v e 1 g l o b a l p^ = 0 . 2 C u r v e 2 g l o b a l p = 0 . 0 C u r v e 3 d i r e c t C u r v e 4 d i f f u s e p^ = 0.2 C u r v e 5 d i f f u s e p^ = 0.0 41 VI CONCLUSIONS The p r e s e n t method o f c a l c u l a t i n g t h e d i f f u s e and g l o b a l s p e c t r a l i r r a d i a n c e has been compared t o a c c u r a t e t h e o r e t i c a l a n a l y s i s and f o u n d t o g i v e good agreement f o r t h e model a t m o s p h e r e s i n v e s t i g a t e d . The major a d v a n t a g e o f t h i s f o r m u l a t i o n i s i t s s i m p l i c i t y . The computer t i m e i s m i n i m a l and t h e i n p u t p a p a m e t e r s a r e e a s i l y v a r i e d . The r e s u l t s p r e s e n t e d a r e i l l u s t r a t i v e of t h e f l e x i b i l i t y and t h e r a n g e o f e f f e c t s t h a t c a n be s t u d i e d w i t h t h i s f o r m u l a t i o n . C o m p a r i s o n s w i t h m e asured s p e c t r a s h o u l d be u n d e r t a k e n when s u f f i c i e n t i n p u t p a p a m e t e r s a r e s p e c i f i e d , t o c o m p l e t e t h e v a l i d a t i o n of t h i s f o r m u l a t i o n . 42 REFERENCES 1. J.V. Dave, ' E x t e n s i v e d a t a s e t s o f t h e d i f f u s e r a d i a t i o n i n r e a l i s t i c a t m o s p h e r i c m o d els w i t h a e r o s o l s and common a b s o r b i n g g a s e s ' , S o l a r E n e r g y 21, 361-369, ( 1 9 7 8 ) . 2. J.V. Dave, P r i v a t e c o m m u n i c a t i o n ( 1 9 8 1 ) . 3. C . J . Kok, ' S p e c t r a l i r r a d i a n c e of d a y l i g h t a t P r e t o r i a ' , J . P h y s . D: A p p l . Phy. 5, 1513-1520, ( 1 9 7 2 ) . 4. C . J . Kok, ' S p e c t r a l i r r a d i a n c e of d a y l i g h t f o r a i r mass 2', J . P h y s . D: A p p l . Phy. 5, L85-L88, ( 1 9 7 2 ) . 5. C . J . Kok and N.N. C h a l m e r s , ' S p e c t r a l i r r a d i a n c e o f d a y l i g h t a t Durban', N a t i o n a l P h y s i c a l L a b o r a t o r y SCIR R e s e a r c h R e p o r t 339, ( 1 9 7 8 ) . 6. R.E. B i r d and R.L. H u l s t r o m , ' S o l a r s p e c t r a l measurements and m o d e l l i n g ' , SERI/TR-642-1013, ( 1 9 8 1 ) . 7. P. Moon, ' P r o p o s e d s t a n d a r d s o l a r r a d i a t i o n c u r v e s f o r e n g i n e e r i n g u s e ' , J o u r n a l of t h e F r a n k l i n I n s t i t u t e 230, 583-617, ( 1 9 4 0 ) . 8. A.P. Thomas and M.P. T h e k a e k a r a , ' E x p e r i m e n t a l and t h e o r e t i c a l s t u d i e s on s o l a r e n e r g y f o r e n e r g y c o n v e r s i o n ' , S h a r i n g t h e Sun - A J o i n t C o n f e r e n c e o f t h e A m e r i c a n S e c t i o n o f ISES and t h e S o l a r E n e r g y S o c i e t y o f Canada I n c . 1, 338-354, ( 1 9 7 6 ) . 9. D.M. G a t e s , ' S p e c t r a l d i s t r i b u t i o n o f s o l a r r a d i a t i o n a t t h e e a r t h ' s s u r f a c e ' , S c i e n c e 51, 523-529, ( 1 9 6 6 ) . 10. J . L . H a t f i e l d , R.B. G i o r g i s J r . and R.G. F l o c c h i n i , 'A s i m p l e s o l a r r a d i a t i o n model f o r c o m p u t i n g d i r e c t and d i f f u s e s p e c t r a l f l u x e s ' , S o l a r E n e r g y 27, 323-329, ( 1 9 8 1 ) . 11. R.L. H u l s t r o m , ' I n s o l a t i o n m o d e l s , d a t a and a l g o r i t h m s ' , SERI/TR-36-110, ( 1 9 7 8 ) . 12. B. L e c k n e r , 'The s p e c t r a l d i s t r i b u t i o n o f s o l a r r a d i a t i o n a t t h e e a r t h ' s s u r f a c e - e l e m e n t s of a m o d e l ' , S o l a r E n e r g y 20, 143-150, ( 1 9 7 8 ) . 13. R.A. M c C l a t c h e y , R.W. F e n n , J . E . S e l b y , F . E . V o l z and J . S . G a r i n g , ' O p t i c a l p r o p e r t i e s o f t h e a t m o s p h e r e ' , 3 r d Edn. A i r F o r c e Cambridge R e s e a r c h L a b o r a t o r i e s , AFCRL-72-0497, ( 1 9 7 2 ) . 14. H.P. B e r l a g e , 'Zur t h e o r i e d e r B e l e u c h t u n g eine.r h o r i z o n t a l e n , F l a c h e d u r c h T a g e s l i c h t ' . Met. Zs. 4 5 ( 5 ) , 174-180, ( 1 9 2 8 ) . 43 15. S . I . S i v k o v , ' C o m p u t a t i o n s of s o l a r r a d i a t i o n c h a r a c t e r i s t i c s ' , I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , J e r u s a l e m , p63, ( 1 9 7 1 ) . 16. D. D e i r m e n d j i a n and Z. S e k e r a , ' G l o b a l r a d i a t i o n r e s u l t i n g f r o m m u l t i p l e s c a t t e r i n g i n a R a y l e i g h a t m o s p h e r e ' , T e l l u s V I , 382-398, ( 1 9 5 4 ) . 17. N. R o b i n s o n , ' S o l a r R a d i a t i o n ' , p 117-118, E l s e v i e r , Amsterdam, ( 1 9 6 6 ) . 18. J.A. D a v i e s and J . E . Hay, ' C a l c u l a t i o n o f t h e 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 ' , P r o c e e d i n g s of t h e F i r s t C a n a d i a n R a d i a t i o n D a t a Workshop, T o r o n t o , Ont., E d . By T. Won and J . E . Hay, ( 1 9 8 0 ) . 19. E . V i g r o u x , ' C o n t r i b u t i o n a l ' e t u d e e x p e r i m e n t a l e de l ' a b s o r p t i o n de 1'ozone', A n n a l s de P h y s . 8, 709-762, ( 1 9 5 3 ) . 20. J.N. Howard, J . I . F . K i n g and P.R. G a s t , 'Thermal r a d i a t i o n ' , .Chap. 16 o f Handbook of G e o p h y s i c s and Space E n v i r o n m e n t , Rev. E d . M a c M i l l a n , New Y o r k , N.Y., ( 1 9 6 5 ) . 21. A. A n g s t r o m , 'On t h e t r a n s m i s s i o n o f sun r a d i a t i o n and on d u s t i n t h e a t m o s p h e r e ' , G e o g r . Ann. 2, 156-166, ( 1 9 2 9 ) . 22. A. A n g s t r o m , 'On t h e a t m o s p h e r i c t r a n s m i s s i o n o f sun r a d i a t i o n I I ' , G e o g r . Ann. 2-3, 130-159, ( 1 9 3 0 ) . 23. F. K a s t e n , 'A new t a b l e and a p p r o x i m a t i o n f o r m u l a f o r t h e r e l a t i v e o p t i c a l a i r mass', A r c h . Met. G e o p h y s . B i o k l i m . B14, 206-223, ( 1 9 6 6 ) . 24. P.W. S u c k l i n g and J . E . Hay, ' M o d e l l i n g d i r e c t , d i f f u s e , and t o t a l s o l a r r a d i a t i o n f o r c l o u d l e s s d a y s ' , A t mosphere 14, 298-308, (1976) 25. G.D. R o b i n s o n , ' A b s o r p t i o n o f s o l a r r a d i a t i o n by a t m o s p h e r i c a e r o s o l , a s r e v e a l e d by measurments from t h e g r o u n d ' , A r c h . Met. Geophys. B i o k l i m . B12, 19-40, (1962) . 26. J.V. Dave, P. H a l p e r n and N. B r a s l a u , ' S p e c t r a l d i s t r i b u t i o n of t h e d i r e c t and d i f f u s e s o l a r e n e r g y r e c e i v e d a t sea l e v e l o f a model a t m o s p h e r e ' , I.B.M. P a l o A l t o S c i e n t i f i c C e n t e r T e c h n i c a l R e p o r t No. G320-3332, ( 1 9 7 5 ) . 27. M.P. T h e k a e k a r a , ' S o l a r e n e r g y o u t s i d e t h e e a r t h ' s a t m o s p h e r e ' , S o l a r E n e r g y 14, 109-127, ( 1 9 7 3 ) . 28. H. N e c k e l and D. L a b s , ' D i s t r i b u t i o n o f t h e s o l a r i r r a d i a n c e ' , P r o c e e d i n g s o f t h e 14th ESLAB Symposium on P h y s i c s of S o l a r v a r i a t i o n s . S c h e v e n i g e n , 16-19 September ( 1 9 8 0 ) . 44 APPENDIX I SOLAR SPECTRUM USED IN MODEL COMPARISONS The s o l a r s p e c t r u m l i s t e d below i s t a k e n f r o m Howard e t a l [ 2 0 ] . T h i s i s t h e s p e c t r u m u s e d i n a l l c o m p a r i s o n s t o Dave's M o d e l s [ 2 ] . The 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 ozone a r e t a k e n from V i g r o u x [19] as p r e s e n t e d i n Howard e t a l [20] and t h e water v a p o r and m i x e d - g a s a b s o r p t i o n c o e f f i c i e n t s a r e from L e c k n e r [ 1 2 ] . X. A X k&A k^A 0.2925 0.005 616.0 26.000000 0.0 0.0 0.2975 0.005 604.0 14.000000 0.0 0.0 0.3025 0.005 600.0 7. 120000 0.0 0.0 0.3075 0.005 700.0 3.660000 0.0 0.0 0.3125 0.005 780.0 1.800000 0.0 0.0 0.3175 0.005 810.0 1.150000 0.0 0.0 0.3225 0.005 930.0 0.675000 0.0 0.0 0.3275 0.005 1 100.0 0.244000 0.0 0.0 0.3325 0.005 1 120.0 0.090000 0.0 0.0 0.3375 0.005 1120.0 0.117000 0.0 0.0 0.3425 0.005 1160.0 0.017000 0.0 0.0 0.3475 0.005 1164.0 0.014000 0.0 0.0 0.3525 0.005 1180.0 0.007280 0.0 0.0 0.3575 0.005 1170.0 0.002620 0.0 0.0 0.3625 0.005 1230.0 0.000600 0.0 0.0 0.3675 0.005 1270.0 0.0 0.0 0.0 0.3725 0.005 1324.0 0.0 0.0 0.0 0.3775 0.005 1236.0 0.0 0.0 0.0 0.3825 0.005 1200.0 0.0 0.0 0.0 0.3875 0.005 1126.0 0.0 0.0 0.0 0.3925 0.005 1146.0 0.0 0.0 0.0 0.3975 0.005 1400.0 0.0 0.0 0.0 0.4025 0.005 1742.0 0.0 0.0 0.0 0.4075 0.005 1896.0 0.0 0.0 0.0 0.4125 0.005 1920.0 0.0 0.0 0.0' 0.4175 0.005 1934.0 0.0 0.0 0.0 0.4225 0.005 1894.0 0.0 0.0 0.0 0.4275 0.006 1760.0 0.0 0.0 0.0 0.4350 0.009 1910.0 0.001600 0.0 0.0 0.4450 0.010 2110.0 0.003200 0.0 0.0 0.4550 0.010 2178.0 0.003700 0.0 0.0 0.4650 0.010 2164.0 0.008500 0.0 0.0 0.4750 0.010 2150.0 0.010000 0.0 0.0 0.4850 0.010 2047.0 0.020000 0.0 0.0 0.4950 0.010 2050.0 0.021000 0.0 0.0 0.5050 0.010 1960.0 0.039000 0.0 0.0 0.5150 0.010 1925.0 0.040000 0.0 0.0 0.5250 0.010 1940.0 0.055000 0.0 0.0 0.5350 0.010 1975.0 0.071000 0.0 0.0 0.5450 0.010 1965.0 0.081000 0.0 0.0 0.5550 0.010 1930.0 0.090000 0.0 0.0 0.5650 0.010 1902.0 0.110000 0.0 0.0 4 5 APPENDIX I ( c o n t i n u e d ) A X. 0 . 5 7 5 0 0 . 0 1 0 1 9 1 4 . 0 0 . 1 2 0 0 0 0 0 . 0 0 . 0 0 . 5 8 5 0 0 . 0 1 0 1 9 0 4 . 0 0 . 1 1 1 0 0 0 0 . 0 0 . 0 0 . 5 9 5 0 0 . 0 1 0 1 8 7 8 . 0 0 . 1 1 1 0 0 0 0 . 0 0 . 0 0 . 6 0 5 0 0 . 0 1 0 1 8 0 2 . 0 0 . 1 2 7 0 0 0 0 . 0 0 . 0 0 . 6 1 5 0 0 . 0 1 0 1 7 6 4 . 0 0 . 1 1 2 0 0 0 0 . 0 0 . 0 0 . 6 2 5 0 0 . 0 1 0 1 7 3 2 . 0 0 . 0 9 7 0 0 0 0 . 0 0 . 0 0 . 6 3 5 0 0 . 0 1 0 1 6 9 0 . 0 0 . 0 8 3 0 0 0 0 . 0 0 . 0 0 . 6 4 5 0 0 . 0 1 0 1 6 6 1 . 0 0 . 0 6 9 0 0 0 0 . 0 0 . 0 0 . 6 5 5 0 0 . 0 1 0 1 6 3 8 . 0 0 . 0 5 7 0 0 0 0 . 0 0 . 0 0 . 6 6 5 0 0 . 0 1 0 1 6 2 5 . 0 0 . 0 4 8 0 0 0 0 . 0 0 . 0 0 . 6 7 5 0 0 . 0 1 0 1581 . 0 0 . 0 3 6 0 0 0 0 . 0 0 . 0 0 . 6 8 5 0 0 . 0 1 0 1 5 3 9 . 0 0 . 0 3 0 0 0 0 0 . 0 0 . 0 0 . 6 9 5 0 0 . 0 1 0 1 4 9 5 . 0 0 . 0 2 4 0 0 0 0 . 0 0 . 0 2 0 0 0 0 0 . 7 0 5 0 0 . 0 1 0 1 5 0 4 . 0 0 . 0 2 0 0 0 0 0 . 0 0 . 0 1 8 0 0 0 0 . 7 1 5 0 0 . 0 1 0 1441 . 0 0 . 0 1 7 0 0 0 0 . 0 1 . 0 0 0 0 0 0 0 . 7 2 5 0 0 . 0 1 0 1 4 2 0 . 0 0 . 0 1 3 0 0 0 0 . 0 0 . 9 0 0 0 0 0 0 . 7 3 5 0 0 . 0 1 0 1 3 9 0 . 0 0 . 0 1 1 0 0 0 0 . 0 0 . 4 6 5 0 0 0 0 . 7 4 5 0 0 . 0 1 0 1 3 4 7 . 0 0 . 0 1 1 0 0 0 0 . 0 0 . 0 3 0 5 0 0 0 . 7 5 5 0 0 . 0 0 7 1 3 5 9 . 0 0 . 0 0 8 7 0 0 0 . 0 0 . 0 0 0 5 0 0 0 . 7 6 0 0 0 . 0 0 7 1 3 2 5 . 0 0 . 0 0 7 0 0 0 3 . 0 0 0 0 0 0 0 . 0 0 0 0 1 0 0 . 7 7 0 0 0 . 0 1 0 1 2 7 5 . 0 0 . 0 0 4 0 0 0 0 . 2 1 0 0 0 0 0 . 0 0 0 0 1 0 0 . 7 8 0 0 0 . 0 1 0 1 2 4 5 . 0 0 . 0 0 . 0 0 . 0 0 0 6 0 0 0 . 7 9 0 0 0 . 0 1 0 1 2 2 3 . 0 0 . 0 0 . 0 0 . 0 1 7 5 0 0 0 . 8 0 0 0 0 . 0 1 0 1 1 9 9 . 0 . 0 . 0 0 . 0 0 , 0 3 6 0 0 0 0 . 8 1 0 0 0 . 0 1 0 1 1 7 4 . 0 0 . 0 0 . 0 0 . 3 3 0 0 0 0 0 . 8 2 0 0 0 . 0 1 0 1 1 5 1 . 0 0 . 0 0 . 0 1 . 5 3 0 0 0 0 0 . 8 3 0 0 0 . 0 1 0 1 1 2 3 . 0 0 . 0 0 . 0 0 . 6 6 0 0 0 0 0 . 8 4 0 0 0 . 0 1 0 1 0 8 7 . 0 0 . 0 0 . 0 0 . 1 5 5 0 0 0 0 . 8 5 0 0 0 . 0 1 0 1 0 6 1 . 0 0 . 0 . 0 . 0 0 . 0 0 3 0 0 0 0 . 8 6 0 0 0 . 0 1 0 1 0 3 6 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 0 . 8 7 0 0 0 . 0 1 0 1 0 1 4 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 0 . 8 8 0 0 0 . 0 1 0 1 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 2 6 0 0 0 . 8 9 0 0 0 . 0 1 0 9 7 8 . 0 0 . 0 0 . 0 0 . 0 6 3 0 0 0 0 . 9 0 0 0 0 . 0 1 0 9 4 5 . 0 0 . 0 0 . 0 2 . 1 0 0 0 0 0 0 . 9 1 0 0 0 . 0 1 0 921 . 0 0 . 0 0 . 0 1 . 6 0 0 0 0 0 0 . 9 2 0 0 0 . 0 1 0 9 0 7 . 0 0 . 0 0 . 0 1 . 2 5 0 0 0 0 0 . 9 3 0 0 0 . 0 1 0 8 8 8 . 0 0 . 0 0 . 0 2 7 . 0 0 0 0 0 0 0 . 9 4 0 0 0 . 0 1 0 8 6 6 . 0 0 . 0 0 . 0 3 8 . 0 0 0 0 0 0 0 . 9 5 0 0 0 . 0 1 0 8 4 0 . 0 0 . 0 0 . 0 4 1 . 0 0 0 0 0 0 0 . 9 6 0 0 0 . 0 1 0 8 2 0 . 0 0 . 0 0 . 0 2 6 . 0 0 0 0 0 0 0 . 9 7 0 0 0 . 0 1 0 8 1 2 . 0 0 . 0 0 . 0 3 . 1 0 0 0 0 0 0 . 9 8 0 0 0 . 0 1 0 7 8 8 . 0 0 . 0 0 . 0 1 . 4 8 0 0 0 0 0 . 9 9 0 0 0 . 0 1 0 761 . 0 0 . 0 0 . 0 0 . 1 2 5 0 0 0 1 . 0 0 0 0 0 . 0 3 0 7 4 2 . 0 0 . 0 0 . 0 0 . 0 0 2 5 0 0 1 . 0 5 0 0 0 . 0 5 0 6 6 7 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 1 . 1 0 0 0 0 . 0 5 0 6 0 2 . 0 0 . 0 0 . 0 3 . 2 0 0 0 0 0 1 . 1 5 0 0 0 . 0 5 0 5 2 9 . 0 0 . 0 0 . 0 2 3 . 0 0 0 0 0 0 1 . 2 0 0 0 0 . 0 5 0 4 7 6 . 0 0 . 0 0 . 0 0 . 0 1 6 0 0 0 1 . 2 5 0 0 , 0 . 0 5 0 4 2 3 . 0 0 . 0 0 . 0 0 7 3 0 0 0 . 0 0 0 1 8 0 1 . 3 0 0 0 0 . 0 5 0 381 . 0 0 . 0 0 . 0 0 0 4 0 0 2 . 9 0 0 0 0 0 1 . 3 5 0 0 0 . 0 5 0 3 4 0 . 0 0 . 0 0 . 0 0 0 1 1 0 2 0 0 . 0 0 0 0 0 0 1 . 4 0 0 0 0 . 0 5 0 3 0 8 . 0 0 . 0 0 . 0 0 0 0 1 0 1 1 0 0 . 0 0 0 0 0 0 4 6 APPENDIX I (continued) A X. k 9A 1 . 4 5 0 0 0 . 0 5 0 2 7 7 . 0 0 . 0 0 . 0 6 4 0 0 0 1 5 0 . 0 0 0 0 0 0 1 . 5 0 0 0 0 . 0 5 0 2 5 2 . 0 0 . 0 0 . 0 0 0 6 3 0 1 5 . 0 0 0 0 0 0 1 . 5 5 0 0 0 . 0 5 0 2 2 6 . 0 0 . 0 0 . 0 1 0 0 0 0 0 . 0 0 1 7 0 0 1 . 6 0 0 0 0 . 0 5 0 2 0 6 . 0 0 . 0 0 . 0 6 4 0 0 0 0 . 0 0 0 0 1 0 1 . 6 5 0 0 0 . 0 5 0 1 8 7 . 0 0 . 0 0 . 0 0 1 4 5 0 0 . 0 1 0 0 0 0 1 . 7 0 0 0 0 . 0 5 0 1 7 1 . 0 0 . 0 0 . 0 0 0 0 1 0 0 . 5 1 0 0 0 0 1 . 7 5 0 0 0 . 0 5 0 1 5 5 . 0 0 . 0 0 . 0 0 0 0 1 0 4 . 0 0 0 0 0 0 1 . 8 0 0 0 0 . 0 5 0 1 4 2 . 0 0 . 0 0 . 0 0 0 0 1 0 1 3 0 . 0 0 0 0 0 0 1 . 8 5 0 0 0 . 0 5 0 1 3 0 . 0 0 . 0 0 . 0 0 0 1 4 5 2 2 0 0 . 0 0 0 0 0 0 1 . 9 0 0 0 0 . 0 5 0 1 2 0 . 0 0 . 0 0 . 0 0 7 1 0 0 1 4 0 0 . 0 0 0 0 0 0 1 . 9 5 0 0 0 . 0 5 0 1 0 9 . 0 0 . 0 2 . 0 0 0 0 0 0 1 6 0 . 0 0 0 0 0 0 2 . 0 0 0 0 0 . 0 7 5 1 0 1 . 0 0 . 0 3 . 0 0 0 0 0 0 2 . 9 0 0 0 0 0 2 . 1 0 0 0 0 . 1 0 0 8 6 . 0 0 . 0 0 . 2 4 0 0 0 0 0 . 2 2 0 0 0 0 2 . 2 0 0 0 0 . 1 0 0 7 4 . 0 0 . 0 0 . 0 0 0 3 8 0 0 . 3 3 0 0 0 0 2 . 3 0 0 0 0 . 1 0 0 6 3 . 5 0 . 0 0 . 0 0 1 1 0 0 0 . 5 9 0 0 0 0 2 . 4 0 0 0 0 . 1 0 0 5 5 . 0 0 . 0 0 . 0 0 0 1 7 0 2 0 . 3 0 0 0 0 3 APPENDIX I I SOLAR SPECTRUM USED BY DAVE The s o l a r s p e c t r u m from Howard e t a l [20] u s e d by Dave h i s t h e o r e t i c a l M o d e l s i s p r e s e n t e d below. X. A X 0 .2875 0 .005 484. 0 0 .2925 0 .005 616. 0 0 .2975 0 .005 604. 0 0 .3025 0 .005 600. 0 0 .3075 0 .005 700. 0 0 .31 25 0 .005 780. 0 0 .31 75 0 .005 810. 0 0 .3225 0 .005 930. 0 0 .3300 0 .010 1110. 0 0 .3400 0 .010 1 140. 0 0 .3500 0 .010 1 172. 0 0 .3600 0 .010 1200. 0 0 .3750 0 .020 1257. 4 0 .3950 0 .020 1353. 4 0 .41 50 0 .020 1911. 0 0 .4350 0 .020 1865. 0 0 .4550 0 .020 2160. 0 0 .4750 0 .020 2142. 0 0 .4950 0 .020 201 4. 0 0 .5150 0 .020 1 935. 0 0 .5350 0 .020 1970. 0 0 .5550 0 .020 1929. 4 0 .5750 0 .020 1908. 0 0 .5950 0 .020 1868. 0 0 .61 50 0 .020 1765. 4 0 .6350 0 .020 1693. 4 0 .6550 0 .020 1640. 0 0 .6780 0 .026 1530. 2 0 .7035 0 .025 1 485. 6 0 .7250 0 .018 1417. 6 0 .7400 0 .012 1370. 0 0 .7525 0 .013 1 353. 8 0 .7625 0 .007 1308. 4 0 .7680 0 .004 1 295. 0 0 .7770 0 .014 1 249. 2 0 .7965 0 .025 1208. 4 0 .8255 0 .032 1 134. 2 0 .8455 0 .009 1071 . 0 0 .8675 0 .035 1022. 0 0 .8890 0 .008 982. 4 0 .9100 0 .034 924. 6 0 .9470 0 .040 850. 2 0 .9770 0 .020 794. 0 0 .9935 0 .013 755. 2 1 .0490 0 .098 678. 0 1 .1015 0 .007 678. 0 1 .1085 0 .007 528. 4 APPENDIX I I ( c o n t i n u e d ) x. 1 . 1 3 2 0 1 . 1 6 1 0 1 . 1 8 0 0 1 . 2 3 5 0 1 . 2 9 0 0 1 . 3 2 0 0 1 . 3 5 0 0 1 . 3 9 5 0 1 . 4 4 2 5 1 . 4 6 2 5 1 . 4 7 7 0 1 . 4 9 7 0 1 . 5 2 0 0 1 . 5 3 9 0 1 . 5 5 8 0 1 . 5 7 8 0 1 . 5 9 2 0 1 . 6 1 0 0 1 . 6 3 0 0 1 . 6 4 6 0 1 . 6 7 8 0 1 . 7 4 0 0 1 . 8 0 0 0 1 . 8 6 0 0 1 . 9 2 0 0 1 . 9 6 0 0 1 . 9 8 5 0 2 . 0 0 5 0 2 . 0 3 5 0 2 . 0 6 5 0 2 . 1 0 0 0 2 . 1 4 8 0 2 . 1 9 8 0 2 . 2 7 0 0 2 . 3 6 0 0 2 . 4 5 0 0 A X. 0 . 0 4 0 5 2 9 . 2 0 . 0 1 8 5 2 9 . 4 0 . 0 2 0 5 2 9 . 0 0 . 0 9 0 4 3 4 . 6 0 . 0 2 0 4 2 3 . 0 0 . 0 4 0 3 4 0 . 4 0 . 0 2 0 341 . 0 0 . 0 7 0 3 1 3 . 0 0 . 0 2 5 2 7 6 . 0 0 . 0 1 5 2 7 6 . 6 0 . 0 1 4 2 7 7 . 0 0 . 0 2 6 2 5 8 . 4 0 . 0 2 0 2 2 7 . 6 0 . 0 1 8 2 2 6 . 0 0 . 0 2 0 2 2 6 . 4 0 . 0 2 0 2 2 6 . 4 0 . 0 0 8 2 2 6 . 2 0 . 0 2 8 1 9 2 . 6 0 . 0 1 2 1 8 7 . 6 0 . 0 2 0 1 8 7 . 0 0 . 0 4 4 1 8 6 . 8 0 . 0 8 0 1 5 5 . 6 0 . 0 4 0 1 4 2 . 6 0 . 0 8 0 1 3 0 . 2 0 . 0 4 0 1 0 9 . 8 0 . 0 4 0 1 0 9 . 8 0 . 0 1 0 1 1 0 . 0 0 . 0 3 0 9 8 . 6 0 . 0 3 0 9 3 . 4 0 . 0 3 0 9 3 . 4 0 . 0 4 0 8 6 . 2 0 . 0 5 6 7 9 . 4 0 . 0 4 4 7 4 . 4 0 . 1 0 0 6 6 . 4 0 . 0 8 0 5 8 . 8 0 . 1 0 0 51 . 0 49 APPENDIX I I I EXPLANATION OF WAVELENGTH DISCREPENCY I t was n o t e d i n t h e d i s c u s s i o n o f F i g . 1 t h a t a d i f f e r e n c e between t h e i r r a d i a n c e v a l u e s of t h e p r e s e n t s t u d y and Model A was e v i d e n t . T h i s d i s c r e p e n c y i s due t o t h e d i f f e r e n t w i d t h s of t h e w a v e l e n g t h i n t e r v a l s u s e d i n t h e two s t u d i e s . Dave [2,25] u s e d w a v e l e n g t h i n t e r v a l s t h a t were much w i d e r a t t h i s p o i n t and hence a v e r a g e d o v e r a s u b s t a n t i a l d i p i n t h e e x t r a t e r r e s t r i a l s p e c t r u m between 0.3875 and 0.3925 nm. P r e s e n t s t u d y v a l u e s Dave's v a l u e s A. 0. 3575 1 170." 0. 3625 1 230.] 0. 3675 1270." 0. 3725 1 324. 0. 3775 1236. 0. 3825 1 200. 0. 3875 1 126.-0. 3925 1 146. 0. 3975 1400. 0. 4025 1742. 0. 4075 1896? 0. 41 25 1 920. 0. 4175 1934.1 0. 4225 1894. } 1200. 1257 1353. 191 1 0.36 0 . 3 7 5 I.A 1200. 1257. 0.395 1 353. 0.415 1911. In t h e f i r s t and s e c o n d columns a r e t h e w a v e l e n g t h and e x t r a t e r r e s t r i a l v a l u e s , r e s p e c t i v e l y , u s e d i n t h i s work. In t h e t h i r d column a r e t h e a v e r a g e d e x t r a t e r r e s t r i a l v a l u e s f o r t h e w a v e l e n g t h i n t e r v a l s u s e d by Dave. Columns f o u r and f i v e c o n t a i n t h e w a v e l e n g t h and e x t r a t e r r e s t r i a l v a l u e s , r e s p e c t i v e l y , u s e d by Dave. 50 APPENDIX IV SPECTRAL VALUES FOR F I G . 1 The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 1 a r e l i s t e d ' below f o r t h e t h r e e z e n i t h a n g l e s 0 . 0 ° , 6 0 . 0 ° and 8 0 . 0 ° . e = o . o ° 0 = 6 0 . 0 ° e = 8 0 . 0 ° X Dx Dx 0 . 2 9 2 5 2 1 5 . 3 8 0 5 1 4 0 . 0 0 6 0 5 3 . 4 9 7 8 0 . 2 9 7 5 2 0 3 . 7 4 2 2 1 3 4 . 9 3 7 4 5 2 . 4 2 0 0 0 . 3 0 2 5 1 9 5 . 0 6 2 4 1 3 1 . 5 5 8 2 5 2 . 0 2 3 2 0 . 3 0 7 5 2 1 9 . 1 3 2 2 1 5 0 . 4 1 8 5 6 0 . 6 1 5 3 0 . 3 1 2 5 2 3 4 . 9 3 1 5 1 6 4 . 0 3 0 6 6 7 . 4 2 7 7 0 . 3 1 7 5 2 3 4 . 5 6 6 5 1 6 6 . 4 8 0 7 6 9 . 8 6 7 9 0 . 3 2 2 5 2 5 8 . 7 8 1 3 1 8 6 . 5 7 9 3 7 9 . 9 9 8 3 0 . 3 2 7 5 2 9 3 . 9 5 6 5 2 1 5 . 1 5 9 7 9 4 . 3 0 2 3 0 . 3 3 2 5 2 8 7 . 3 1 2 3 2 1 3 . 3 5 0 6 95 .6261 0 . 3 3 7 5 2 7 5 . 6 9 9 2 2 0 7 . 5 6 5 4 9 5 . 1 6 4 7 0 . 3 4 2 5 2 7 3 . 9 1 8 5 2 0 8 . 9 5 0 6 9 8 . 0 0 8 3 0 . 3 4 7 5 2 6 3 . 6 0 4 5 2 0 3 . 6 1 3 5 9 7 . 7 0 8 7 0 . 3 5 2 5 2 5 6 . 2 3 1 2 2 0 0 . 2 8 8 2 9 8 . 3 2 1 9 0 . 3 5 7 5 2 4 3 . 5 6 9 2 1 9 2 . 5 5 7 7 9 6 . 6 8 1 4 0 . 3 6 2 5 2 4 5 . 4 6 1 4 1 9 6 . 1 5 2 5 1 0 0 . 7 0 3 4 0 . 3 6 7 5 2 4 2 . 9 3 9 0 1 9 6 . 1 3 0 4 1 0 2 . 9 2 3 2 0 . 3 7 2 5 2 4 2 . 7 6 4 9 1 9 7 . 9 0 0 4 1 0 6 . 1 1 0 0 0 . 3 7 7 5 2 1 7 . 2 3 2 5 1 7 8 . 7 2 4 7 9 7 . 8 6 6 8 0 . 3 8 2 5 2 0 2 . 1 6 8 5 1 6 7 . 7 9 0 9 9 3 . 7 8 6 9 0 . 3 8 7 5 1 8 1 . 8 5 5 2 1 5 2 . 1 8 8 6 8 6 . 7 8 5 2 0 . 3 9 2 5 177 .4464 1 4 9 . 6 7 1 9 8 7 . 0 2 6 3 0 . 3 9 7 5 2 0 7 . 8 5 2 4 1 7 6 . 6 3 2 3 1 0 4 . 6 5 8 6 0 . 4 0 2 5 2 4 8 . 0 1 5 0 2 1 2 . 2 6 0 5 1 2 8 . 0 8 9 0 0 . 4 0 7 5 2 5 8 . 9 0 2 6 2 2 3 . 0 7 2 9 1 3 7 . 0 1 5 0 0 . 4 1 2 5 2 5 1 . 5 0 1 7 2 1 8 . 0 8 2 8 1 3 6 . 2 5 7 6 0 . 4 1 7 5 2 4 3 . 0 6 1 8 2 1 2 . 0 4 4 3 1 3 4 . 6 8 6 5 0 . 4 2 2 5 2 2 8 . 4 2 5 7 2 0 0 . 4 2 4 7 129.3451 0 . 4 2 7 5 2 0 3 . 7 3 7 8 1 7 9 . 7 4 1 5 1 1 7 . 7 8 6 3 0 . 4 3 5 0 2 0 7 . 9 9 8 3 1 8 4 . 9 1 4 6 1 2 3 . 8 5 7 9 0 . 4 4 5 0 2 1 1 . 9 7 2 8 1 9 0 . 2 2 0 6 1 3 0 . 9 3 1 9 0 . 4 5 5 0 2 0 2 . 0 4 3 3 1 8 2 . 8 5 0 8 1 2 9 . 0 6 5 3 0 . 4 6 5 0 1 8 5 . 5 5 2 7 1 6 9 . 2 1 7 5 1 2 2 . 2 4 3 7 0 . 4 7 5 0 1 7 0 . 5 7 6 3 1 5 6 . 6 4 2 5 1 1 5 . 6 0 1 3 0 . 4 8 5 0 1 5 0 . 4 2 6 0 139.0101 1 0 4 . 6 2 4 4 0 . 4 9 5 0 1 3 9 . 6 8 2 5 129 .8221 9 9 . 4 9 1 4 0 . 5 0 5 0 1 2 3 . 9 6 1 8 •1 1 5 . 8 1 2 0 9 0 . 2 4 2 8 0 . 5 1 5 0 1 1 3 . 1272 1 0 6 . 1 9 1 5 84 .0221 0 . 5 2 5 0 106 .0472 9 9 . 9 7 6 7 8 0 . 2 2 6 8 0 . 5 3 5 0 1 0 0 . 5 2 6 4 9 5 . 1 4 6 7 7 7 . 3 4 7 5 0 . 5 4 5 0 9 3 . 2 2 6 8 8 8 . 5 5 6 9 7 2 . 8 5 5 7 0 . 5 5 5 0 8 5 . 4 3 6 6 8 1 . 4 2 5 6 6 7 . 7 3 0 6 0 . 5 6 5 0 7 8 . 6 3 9 8 7 5 . 1 7 5 5 6 3 . 1 7 0 3 0 . 5 7 5 0 7 3 . 9 8 5 9 7 0 . 9 2 3 7 60 .1591 0 . 5 8 5 0 6 8 . 8 7 6 5 6 6 . 1 9 5 0 5 6 . 6 3 6 7 0 . 5 9 5 0 6 3 . 6 3 7 2 61 .'3040 5 2 . 8 7 3 8 APPENDIX IV (continued) 9 = 0 . 0 ° © = 6 0 . 0 ° © = 8 0 . 0 ° X D x D A D A 0 . 6 0 5 0 5 7 . 2 5 1 7 5 5 . 2 7 2 5 4 8 . 0 2 6 4 0 . 6 1 5 0 5 2 . 5 9 5 5 5 0 . 8 7 9 0 4 4 . 5 1 3 6 0 . 6 2 5 0 4 8 . 5 0 6 9 4 7 . 0 1 0 8 41 . 3 9 1 9 0 . 6 3 5 0 4 4 . 4 9 7 0 4 3 . 1 9 8 5 3 8 . 2 6 0 5 0 . 6 4 5 0 • 4 1 . 1 5 0 5 4 0 . 0 1 3 0 3 5 . 6 3 4 0 0 . 6 5 5 0 3 8 . 2 1 6 1 3 7 . 2 1 4 4 3 3 . 3 1 0 9 0 . 6 6 5 0 3 5 . 7 3 3 1 3 4 . 8 4 3 8 31 . 3 3 7 2 0 . 6 7 5 0 3 2 . 7 9 3 0 3 2 . 0 1 7 5 2 8 . 9 2 2 6 0 . 6 8 5 0 3 0 . 1 3 4 4 2 9 . 4 5 6 4 2 6 . 7 1 8 7 0 . 6 9 5 0 2 7 . 6 5 5 0 2 7 . 0 6 2 5 2 4 . 6 4 1 4 0 . 7 0 5 0 2 6 . 3 0 3 6 2 5 . 7 6 6 3 2 3 . 5 4 5 3 0 . 7 1 5 0 2 3 . 8 4 4 3 2 3 . 3 7 9 5 2 1 . 4 3 5 7 0 . 7 2 5 0 2 2 . 2 4 7 2 2 1 . 8 3 2 8 2 0 . 0 8 0 3 0 . 7 3 5 0 2 0 . 6 3 3 4 2 0 . 2 6 5 9 1 8 . 6 9 3 6 0 . 7 4 5 0 1 8 . 9 5 7 9 1 8 . 6 3 4 7 1 7 . 2 3 6 1 0 . 7 5 5 0 1 8 . 1 4 6 8 1 7 . 8 5 0 4 1 6 . 5 5 3 0 0 . 7 6 0 0 1 7 . 2 3 7 8 1 6 . 9 6 2 1 1 5 . 7 4 8 5 0 . 7 7 0 0 1 5 . 7 5 2 8 1 5 . 5 1 1 1 1 4 . 4 3 4 9 0 . 7 8 0 0 1 4 . 6 1 7 6 1 4 . 4 0 2 1 1 3 . 4 3 2 1 0 . 7 9 0 0 1 3 . 6 5 3 9 1 3 . 4 6 0 5 1 2 . 5 7 9 7 0 . 8 0 0 0 1 2 . 7 3 6 1 1 2 . 5 6 2 5 1 1 . 7 6 3 2 0 . 8 1 0 0 1 1 . 8 7 2 2 1 1 . 7 1 6 4 1 0 . 9 9 0 8 0 . 8 2 0 0 1 1 . 0 8 7 5 1 0 . 9 4 7 3 1 0 . 2 8 6 9 0 . 8 3 0 0 1 0 . 3 1 0 6 1 0 . 1 8 4 7 9 . 5 8 5 8 0 . 8 4 0 0 9 . 5 1 7 3 9 . 4 0 5 2 8 . 8 6 5 5 0 . 8 5 0 0 8 . 8 6 3 8 8 . 7 6 2 9 8 . 2 7 1 9 0 . 8 6 0 0 8 . 2 6 2 5 8 . 1 7 1 5 7 . 7 2 4 1 0 . 8 7 0 0 7 . 7 2 4 4 7 . 6 4 2 1 7 . 2 3 2 9 0 . 8 8 0 0 7 . 2 7 9 8 7 . 2 0 4 7 6 . 8 2 7 1 0 . 8 9 0 0 6 . 8 0 7 2 6 . 7 3 9 1 6 . 3 9 3 3 0 . 9 0 0 0 6 . 2 9 2 0 6 . 2 3 0 9 5 . 9 1 7 5 0 . 9 1 0 0 5 . 8 6 8 7 5 . 8 1 3 5 5 . 5 2 6 7 0 . 9 2 0 0 5 . 5 3 3 8 5 . 4 8 3 2 5 . 2 1 7 7 0 . 9 3 0 0 5 . 1 8 9 9 5 . 1 4 3 8 4 . 8 9 9 2 0 . 9 4 0 0 4 . 8 5 0 6 4 . 8 0 8 6 4 . 5 8 4 0 0 . 9 5 0 0 4 . 5 1 1 0 4 . 4 7 3 0 4 . 2 6 7 6 0 . 9 6 0 0 4 . 2 2 3 9 4 . 1 8 9 2 4 . 0 0 0 0 0 . 9 7 0 0 4 . 0 1 3 7 3 . 9 8 1 6 3 . 8 0 4 5 0 . 9 8 0 0 3 . 7 3 9 3 3 . 7 1 0 1 3 . 5 4 7 6 0 . 9 9 0 0 3 . 4 6 8 1 3 . 4 4 1 6 3 . 2 9 3 1 1 . 0 0 0 0 3 . 2 4 8 8 3 . 2 2 4 6 3 . 0 8 7 5 1 . 0 5 0 0 2 . 4 0 4 5 2 . 3 8 8 5 2 . 2 9 3 2 1 . 1 0 0 0 1 . 8 0 2 8 1 . 7 9 1 9 1 . 7 2 4 2 1 . 1 5 0 0 1 . 3 2 6 8 1 . 3 1 9 4 . 1 . 2 7 1 7 1 . 2 0 0 0 1 . 0 0 7 4 1 . 0 0 2 1 0 . 9 6 7 3 1 . 2 5 0 0 0 . 7 6 0 6 0 . 7 5 6 9 0 . 7 3 1 4 1 . 3 0 0 0 0 . 5 8 5 8 0 . 5 8 3 0 0 . 5 6 3 9 1 . 3 5 0 0 0 . 4 4 9 6 0 . 4 4 7 6 0 . 4 3 3 3 1 . 4 0 0 0 0 . 3 5 2 2 0 . 3 5 0 7 0 . 3 3 9 7 1 . 4 5 0 0 0 . 2 7 5 3 0 . 2 7 4 2 0 . 2 6 5 7 1 . 5 0 0 0 0 . 2 1 8 7 0 . 2 1 7 9 0 . 2 1 1 2 1 . 5 5 0 0 0 . 1 7 2 1 0 . 1 7 1 4 0 . 1 6 6 2 APPENDIX IV ( c o n t i n u e d ) X. 1 . 6 0 0 0 1 . 6 5 0 0 1 . 7 0 0 0 1 . 7 5 0 0 1 . 8 0 0 0 1 . 8 5 0 0 1 . 9 0 0 0 1 . 9 5 0 0 2 . 0 0 0 0 2 . 1 0 0 0 2 . 2 0 0 0 2 . 3 0 0 0 2 . 4 0 0 0 9 = 0 . 0 ° 0 . 1 3 8 1 0 . 1 1 0 9 0 . 0 9 0 0 0 . 0 7 2 6 0 . 0 5 9 5 0 . 0 4 8 8 0 . 0 4 0 5 0 . 0 3 3 1 0 . 0 2 7 8 0 . 0 1 9 4 0 . 0 1 3 9 0 . 0 1 0 0 0 . 0 0 7 3 6 = 6 0 . 0 ° D> 0 . 1 3 7 6 0 . 1 1 0 5 0 . 0 8 9 7 0 . 0 7 2 4 0 . 0 5 9 3 0 . 0 4 8 6 0 . 0 4 0 3 0 . 0 3 3 0 0 . 0 2 7 7 0 . 0 1 9 4 0 . 0 1 3 8 0 . 0 0 9 9 0 . 0 0 7 3 0 = 8 0 . 0 ° 0 . 1 3 3 5 0 . 1 0 7 2 0 . 0 8 7 0 0 . 0 7 0 3 0 . 0 5 7 5 0 . 0 4 7 2 0 . 0 3 9 2 0 . 0 3 2 1 0 . 0 2 6 9 0 . 0 1 8 8 0 . 0 1 3 5 0 . 0 0 9 7 0 . 0 0 7 1 53 APPENDIX V SPECTRAL VALUES FOR F I G . 2 The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 2 a r e l i s t e d below f o r z e n i t h a n g l e s 0 . 0 ° , 60.0° and 8 0 . 0 ° . © = 0.0° © = 6 0 . 0 ° © = 8 0 . 0 ° X- Dx D> D A 0.2925 0.0553 0.0000 0.0000 0.2975 2.3746 0.0201 0.0000 0.3025 20.2702 1.4896 0.0004 0.3075 68.4286 15.0294 0.1416 0.3125 132.5409 52.8378 3.4250 0.3175 162.7211 80.7316 10.4099 0.3225 208.7908 122.0031 26.1677 0.3275 272.0103 184.5314 62.9637 0.3325 279.2058 201.6021 82.3887 0.3375 265.6299 192.8308 78.4069 0.3425 272.4417 206.7268 95.2884 0.3475 262.4333 201.8274 95.4701 0.3525 255.6387 199.3726 97. 1440 0.3575 243.3662 192.2405 96.2629 0.3625 245.4146 196.0784 100.6034 0.3675 242.9390 196.1304 102.9232 0.3725 242.7649 197.9004 106.1100 0.3775 217.2325 178.7247 97.8668 0.3825 202. 1685 167.7909 93.7869 0.3875 181.8552 152.1886 86.7852 0.3925 177.4464 149.6719 87.0263 0.3975 207.8524 176.6323 104.6586 0.4025 248.0150 212.2605 128.0890 0.4075 258.9026 223.0729 137.0150 0.4125 251.5017 218.0828 136.2576 0.4175 243.0618 212.0443 1 34.6865 0.4225 228.4257 200.4247 129.3451 0.4275 203.7378 179.7415 117.7863 0.4350 207.8926 184.7284 123.5302 0.4450 211.7571 189.8378 130.2401 0.4550 201.8058 182.4255 128.2771 0.4650 185.0519 168.3147 120.5356 0.4750 170.0347 155.6597 1 1 3.7033 0.4850 149.4723 137.2715 101.2169 0.4950 138.7528 128.1176 96.0919 0.5050 122.4339 113.0040 84.6003 0.5150 111.6973 103.5516 78.6383 0.5250 104.2085 96.5753 73.2445 0.5350 98.2821 90.9888 68.7699 0.5450 90.8561 84.1556 63.7126 0.5550 83.0260 76.9417 58.3548 0.5650 75.9366 70.1473 52.6531 0.5750 71 .2157 65.7647 49.3200 0.5850 66.4877 61.7286 47.1292 0.5950 61.4301 57.1676 43.9980 APPENDIX V ( c o n t i n u e d ) © = o . o ° e = 6 0 . 0 ° © = 8 0 . 0 ° X. DA 0 . 6 0 5 0 5 4 . 9 8 5 6 5 1 . 0 2 6 6 3 8 . 9 1 9 7 0 . 6 1 5 0 5 0 . 7 5 5 2 4 7 . 4 1 6 2 3 6 . 9 7 9 9 0 . 6 2 5 0 4 7 . 0 3 3 5 4 4 . 2 2 6 8 3 5 . 2 5 1 2 0 . 6 3 5 0 4 3 . 3 3 7 9 4 0 . 9 9 9 9 3 3 . 3 4 8 3 0 . 6 4 5 0 4 0 . 2 5 7 4 3 8 . 3 1 2 6 3 1 . 7 8 7 3 0 . 6 5 5 0 3 7 . 5 2 9 7 3 5 . 9 0 3 0 3 0 . 3 1 1 2 0 . 6 6 5 0 3 5 . 1 9 1 8 3 3 . 8 0 7 0 2 8 . 9 4 3 3 0 . 6 7 5 0 3 2 . 4 1 9 8 3 1 . 3 0 0 3 2 7 . 2 4 9 2 0 . 6 8 5 0 2 9 . 8 4 8 3 2 8 . 9 0 5 5 25 .4241 0 . 6 9 5 0 2 7 . 1762 2 6 . 2 2 9 6 2 2 . 9 5 1 5 0 . 7 0 5 0 2 5 . 9 0 0 8 2 5 . 0 6 4 6 2 2 . 1 1 9 4 0 . 7 1 5 0 2 1 . 2 3 0 0 19.6631 1 5 . 6 4 6 4 0 . 7 2 5 0 1 9 . 9 5 8 9 1 8 . 5 7 8 9 1 4 . 9 9 5 0 0 . 7 3 5 0 19.1321 1 8 . 0 9 9 6 1 5 . 2 1 2 6 0 . 7 4 5 0 1 8 . 6 4 0 2 1 8 . 1 1 4 4 1 6 . 2 5 6 0 0 . 7 5 5 0 1 8 . 0 9 0 4 1 7 . 7 4 0 9 1 6 . 2 8 6 7 0 . 7 6 0 0 1 2 . 7 4 1 9 1 0 . 8 9 0 9 7 . 1 8 8 5 0 . 7 7 0 0 1 4 . 6 9 3 2 1 3 . 9 8 8 9 1 1 . 9 9 5 7 0 . 7 8 0 0 1 4 . 6 1 1 5 1 4 . 3 9 0 5 1 3 . 4 0 3 3 0 . 7 9 0 0 1 3 . 5 3 3 2 1 3 . 2 6 3 6 1 2 . 2 2 1 9 0 . 8 0 0 0 1 2 . 5 4 5 8 1 2 .2671 1 1 . 2 5 1 6 0 . 8 1 0 0 1 1 . 1 8 8 9 10 .7338 9 . 4 0 9 2 0 . 8 2 0 0 9 . 6 3 6 4 8 . 9 1 3 8 7 . 1 5 9 8 0 . 8 3 0 0 9 . 4 4 4 2 8 . 9 5 2 3 7 . 6 3 1 9 0 . 8 4 0 0 9 . 1 6 0 8 8 . 8 8 3 4 8 . 0 0 5 3 0 . 8 5 0 0 8 . 8 4 6 6 8 . 7 3 1 0 8 . 2 0 1 3 0 . 8 6 0 0 8 . 2 6 2 5 8 . 1 7 1 4 7 . 7 2 3 8 0 . 8 7 0 0 7 . 7 2 4 3 7 . 6 4 2 0 7 . 2 3 2 6 0 . 8 8 0 0 7 . 2 6 7 5 7 . 1 8 1 6 6 . 7 7 5 0 0 . 8 9 0 0 6 . 6 5 9 8 6 .5161 6 . 0 1 2 3 0 . 9 0 0 0 5 . 3 2 3 3 4 . 8 7 8 2 3 . 8 4 3 8 0 . 9 1 0 0 5 . 0 8 2 7 4 . 7 0 9 3 3 . 8 1 2 0 0 . 9 2 0 0 4 . 8 8 1 8 4 . 5 6 2 7 3 . 7 7 3 0 0 . 9 3 0 0 2 . 6 2 0 4 1 .8946 0 . 8 4 3 9 0 . 9 4 0 0 2 .1261 1 .4405 0 . 5 4 8 8 0 . 9 5 0 0 1 .9089 1 .2728 0 . 4 6 6 7 0 . 9 6 0 0 2 . 1 6 2 8 1 .5750 0 . 7 1 4 4 0 . 9 7 0 0 3 . 2 6 2 0 2 . 9 3 9 6 2 . 2 2 9 0 0 . 9 8 0 0 3 . 2 5 8 2 3 . 0 3 2 2 2 . 4 8 5 4 0 . 9 9 0 0 3 . 3 5 3 6 3 . 2 7 2 4 3 . 0 0 8 2 1 .0000 3 . 2 4 3 5 3 . 2 1 4 7 3 . 0 6 4 6 1 .0500 2 . 4 0 4 5 2 . 3 8 8 4 2 .2931 1 .1000 1 .4598 1 .3159 1 .0007 1 .1500 0 . 7 0 9 7 0 . 5 2 8 7 0 .2541 ,1 . 2000 0 .9991 0 . 9 8 8 4 0 . 9 4 1 3 1 .2500 0 . 7 5 4 6 0 . 7 4 6 9 0 . 7 1 2 2 1 .3000 0 . 4 7 9 4 0 . 4 3 4 7 0 . 3 3 5 9 1 .3500 0 . 0 5 7 5 0 . 0 2 2 2 0 . 0 0 2 2 1 .4000 0 . 0 0 1 8 0 . 0 0 0 2 0 . 0 0 0 0 1 .4500 0 . 0 4 6 0 ' 0 .0200 0 . 0 0 2 6 1 .5000 0 . 1333 0 . 1056 0 . 0 5 8 9 1 .5500 0 . 1 7 0 2 0 . 1683 0 . 1 6 0 2 APPENDIX V (continued) X 1 . 6 0 0 0 1 . 6 5 0 0 1 . 7 0 0 0 1 . 7 5 0 0 1 . 8 0 0 0 1 . 8 5 0 0 1 . 9 0 0 0 1 . 9 5 0 0 2 . 0 0 0 0 2 . 1 0 0 0 2 . 2 0 0 0 2 . 3 0 0 0 2 . 4 0 0 0 e = 0 . 0 ° D X 0 . 1 3 3 5 0 . 1 1 0 1 0 . 0 8 3 4 0 . 0 5 7 2 0 . 0 1 1 7 0 . 0 0 0 0 0 . 0 0 0 1 0 . 0 0 4 2 0 . 0 1 6 8 0 . 0 1 7 2 0 . 0 1 3 1 0 . 0 0 9 2 0 . 0 0 4 1 0 = 6 0 . 0 ° D x 0 . 1 3 0 7 0 . 1 0 9 0 0 . 0 8 0 2 0 . 0 5 1 1 0 . 0 0 5 5 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 1 6 0 . 0 1 3 3 0 . 0 1 6 2 0 . 0 1 2 7 0 . 0 0 8 8 0 . 0 0 3 1 0 = 8 0 . 0 ° DjX 0 . 1 2 1 8 0 . 1 0 4 2 0 . 0 7 1 4 0 . 0 3 8 0 0 . 0 0 0 9 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 2 0 . 0 0 7 4 0 . 0 1 3 7 0 . 0 1 1 5 0 . 0 0 7 7 0 . 0 0 1 6 56 APPENDIX VI SPECTRAL VALUES FOR FIG.S 3,4 AND 5 The d i r e c t s p e c t r a l v a l u e s f o r F i g . 3 and t h e d i f f u s e s p e c t r a l v a l u e s f o r F i g . 4 and F i g . 5 g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r z e n i t h a n g l e s 0.0° and 60.0° a r e l i s t e d below. &•• = 0.0° 0 = 6 0 . 0 ° e= 0.0° e=60.0° X ix °x DX 0. 2925 0 .041 1 0. 0000 0. 0537 0 .0000 0. 2975 1 .9816 0. 0036 2. 3395 0 .0160 0. 3025 18 .8965 0. 31 47 20. 2514 1 .2005 0. 3075 70 .9155 3. 71 07 69. 3564 1 2 .2722 0. 3125 152 .0206 15. 1 085 1 36. 3387 43 .7366 0. 3175 205 .7225 26. 5178 169. 9405 67 .7816 0. 3225 289 .8806 45. 6963 221 . 4650 1 03 .9536 0. 3275 413 .3135 78. 2882 293. 1 367 159 .6472 0. 3325 462 .8469 96. 2948 305. 8059 177 .1814 0. 3375 479 .0107 103. 1259 295. 7832 172 .2384 0. 3425 532 .9998 1 23. 1636 308. 5168 187 .7445 0. 3475 555 .6123 1 33. 3389 302. 31 67 186 .4420 0. 3525 584 .3425 1 45. 4441 299. 661 1 187 .4077 0. 3575 599 .3027 1 54. 2550 290. 3643 183 .9421 0. 3625 649 .7549 1 72. 4370 298. 1096 191 .041 0 0. 3675 690 .2139 188. 4141 300. 5210 1 94 .6434 0. 3725 738 .8140 207. 0391 305. 8936 200 .1099 0. 3775 706 .9810 203. 0454 278. 8794 184 .1853 0. 3825 702 .4949 206. 4578 264. 4895 176 .2790 0. 3875 673 .6802 202. 31 58 242. 5025 1 63 .0356 0. 3925 699 .8103 214. 4745 241 . 2352 163 .5343 0. 3975 871 .5120 272. 2451 288. 1 348 196 .8795 0. 4025 1 1 04 .2070 351 . 1885 350. 6433 241 .4066 0. 4075 1 222 .4768 395. 4387 373. 3772 258 .9153 0. 41 25 1257 .9939 413. 4700 370. 0396 258 .3689 0. 41 75 1 286 .5110 429. 2529 364. 91 26 256 .4629 0. 4225 1 278 .0532 432. 5315 349. 9832 247 .5111 0. 4275 1 203 .7900 412. 9031 318. 6145 226 .6728 0. 4350 1 330 .6426 464. 8325 335. 3350 240 .4651 0. 4450 1 503 .2761 536. 9551 356. 1 1 47 257 .8904 0. 4550 1 583 .7427 577. 2900 353. 9851 258 .7295 0. 4650 1600 .5737 593. 3828 338. 6934 249 .3083 0. 4750 1616 .31 49 608. 9873 324. 8267 240 .8604 0. 4850 1557 .4497 593. 8660 298. 1 204 221 .9427 0. 4950 1580 .8647 610. 9075 288. 9917 216 .4790 0. 5050 1 521 .7932 592. 1 1 94 266. 3381 1 99 .5730 0. 51 50 151 1 .3262 594. 5776 253. 81 58 191 . 1 622 0. 5250 1 531 .7832 606. 0769 247. 3793 186 .3666 0. 5350 1566 .4331 622. 5996 243. 7480 183 .5484 0. 5450 1567 .2705 626. 441 2 235. 41 49 177 .4579 0. 5550 1547 .4172 621 . 7476 224. 7507 169 .5901 0. 5650 1526 .6345 614. 1 1 77 214. 7455 161 .5991 0. 5750 1 541 .9631 622. 5959 210. 3798 158 .3303 0. 5850 1548 .1270 630. 8142 205. 1 542 155 .2898 0. 5950 1 536 .0217 629. 5591 197. 9602 1 50 .2538 5 7 APPENDIX VI ( c o n t i n u e d ) e= 0 . 0 ° 9 = 6 0 . 0 ° e= 0 . 0 ° 9 = 6 0 . 0 ° X i A I A DA 0 . 6 0 5 0 1 4 7 4 . 4 3 3 6 6 0 4 . 5 9 2 8 1 8 5 . 0 2 9 1 1 4 0 , . 0 9 3 2 0 . 6 1 5 0 1 4 5 7 . 6 4 7 5 6 0 3 . 5 5 2 2 1 7 8 . 3 1 7 6 1 3 5 , . 9 5 9 3 0 . 6 2 5 0 1 4 4 4 . 8 8 5 7 6 0 3 . 9 0 6 7 1 7 2 . 4 9 0 2 1 3 2 , . 4 1 6 1 0 . 6 3 5 0 . 1 4 2 2 . 41 9 4 5 9 9 . 7 4 3 9 1 6 5 . 8 7 5 3 1 2 8 , . 1 4 8 8 0 . 6 4 5 0 1 4 1 0 . 0 6 7 6 5 9 9 . 5 9 0 1 1 6 0 . 7 7 6 9 1 2 4 , . 9 8 2 8 0 . 6 5 5 0 1 401 . 2 7 0 3 6 0 0 . 3 8 2 6 1 5 6 . 3 5 7 1 1 2 2 , . 2 1 0 7 0 . 6 6 5 0 1 3 9 9 . 1 9 8 0 6 0 3 . 3 4 5 7 1 5 2 . 9 1 3 6 1 2 0 , . 0 4 5 1 0 . 6 7 5 0 1371 . 1 7 5 5 5 9 5 . 4 8 4 6 1 4 6 . 8 8 2 0 1 1 5 , . 9 1 2 8 0 . 6 8 5 0 1 341 . 5 8 3 3 5 8 5 . 5 7 9 6 1 4 0 . 9 6 7 9 1 1 1 . 6 0 7 1 0 . 6 9 5 0 1 2 9 6 . 8 3 7 6 5 6 5 . 2 2 0 5 1 3 3 . 7 5 6 5 1 0 5 . 5 6 2 4 0 . 7 0 5 0 1 3 1 1 . 01 15 5 7 3 . 9 3 7 7 1 3 2 . 8 1 4 5 1 0 5 . 1 1 44 0 . 7 1 5 0 1 1 3 8 . 8 1 2 5 4 7 7 . 9 7 5 6 1 1 3 . 3 8 8 0 8 5 . 9 0 3 5 0 . 7 2 5 0 1 1 3 3 . 6 3 3 3 4 7 8 . 9 7 3 6 1 1 0 . 9 9 8 0 8 4 . 5 2 9 6 0 . 7 3 5 0 1 1 4 9 . 6 5 3 1 4 9 4 . 4 2 6 0 1 1 0 . 7 5 8 1 8 5 . 7 3 5 1 0 . 7 4 5 0 1 1 8 4 . 0 5 7 4 5 2 3 . 8 5 7 4 1 1 2 . 2 9 8 0 8 9 . 3 0 6 8 0 . 7 5 5 0 1 2 1 3 . 7 9 7 6 5 4 2 . 6 8 9 9 1 1 3 . 3 8 3 5 91 . 0 0 7 3 0 . 7 6 0 0 8 7 8 . 4 0 3 8 3 4 2 . 5 3 0 8 81 . 4 4 7 2 5 6 . 9 8 1 8 0 . 7 7 0 0 1 0 6 8 . 7 0 6 8 4 6 4 . 8 0 3 7 9 7 . 6 6 5 8 7 6 . 1 1 9 2 0 . 7 8 0 0 1 1 2 0 . 4 7 0 7 5 0 4 . 7 4 5 8 1 0 0 . 9 6 5 7 81 . 4 1 3 1 0 . 7 9 0 0 1 0 9 3 . 3 5 6 7 4 9 0 . 7 2 8 0 9 7 . 1 8 5 4 7 7 . 9 9 3 2 0 . 8 0 0 0 1 0 6 7 . 1 2 4 5 4 7 8 . 3 8 4 8 9 3 . 6 0 2 5 74 . 9 5 0 6 0 . 8 1 0 0 1 0 0 1 . 3 1 2 5 4 4 0 . 8 9 6 5 8 6 . 7 0 3 1 6 8 . 1 2 3 1 0 . 8 2 0 0 9 0 6 . 7 3 4 6 3 8 5 . 3 8 5 5 7 7 . 5 3 3 5 5 8 . 7 4 6 5 0 . 8 3 0 0 9 3 3 . 7 6 1 0 4 0 7 . 1 1 4 5 7 8 . 8 7 4 0 61 . 2 4 8 1 0 . 8 4 0 0 951 . 1 2 7 9 4 2 4 . 6 4 7 2 7 9 . 3 8 9 5 6 3 . 0 7 3 5 0 . 8 5 0 0 9 6 3 . 9 5 2 4 4 3 8 . 4 3 4 8 7 9 . 531 1 6 4 . 3 1 4 9 0 . 8 6 0 0 9 4 4 . 3 0 0 5 4 3 0 . 7 8 2 5 7 7 . 0 3 2 3 6 2 . 4 2 9 7 0 . 8 7 0 0 9 2 5 . 4 0 9 7 4 2 2 . 691 4 7 4 . 661 6 6 0 . 5 3 6 3 0 . 8 8 0 0 9 1 2 . 1 8 7 5 4 1 6 . 5 2 6 6 7 2 . 8 0 5 2 5 8 . 9 6 8 5 0 . 8 9 0 0 8 7 5 . 2 9 5 2 3 9 6 . 0 5 9 6 6 9 . 1 2 8 3 5 5 . 4 4 2 5 0 . 9 0 0 0 7 3 2 . 2 0 8 3 3 1 0 . 5 5 7 1 5 7 . 2 3 5 2 4 2 . 9 9 7 6 0 . 9 1 0 0 731 . 2 8 2 2 3 1 3 . 8 4 5 2 5 6 . 5 8 9 9 4 2 . 9 8 8 1 0 . 9 2 0 0 7 3 4 . 31 4 0 3 1 8 . 1 4 4 0 5 6 . 2 6 7 5 4 3 . 1 2 1 3 0 . 9 3 0 0 4 1 1 . 8 7 8 7 1 3 8 . 1 4 1 7 31 . 2 5 7 7 18 . 5 3 2 3 0 . 9 4 0 0 3 4 9 . 0 4 9 6 1 0 9 . 7 8 4 1 2 6 . 2 4 0 7 14 . 5 8 0 6 0 . 9 5 0 0 3 2 7 . 1 5 8 2 1 0 1 . 3 3 4 5 2 4 . 3 6 8 5 1 3 . 3 2 6 6 0 . 9 6 0 0 3 8 6 . 7 9 3 2 1 3 0 . 9 3 2 1 2 8 . 5 5 0 6 17 . 0 5 3 8 0 . 9 7 0 0 • 6 0 8 . 4 7 0 0 2 5 5 . 0 5 6 7 4 4 . 5 1 6 0 3 2 . 9 0 8 7 0 . 9 8 0 0 6 3 3 . 6 1 1 8 2 7 4 . 4 5 6 5 4 5 . 9 5 3 1 3 5 . 0 8 5 5 0 . 9 9 0 0 6 7 9 . 61 3 8 3 0 8 . 8 5 2 3 4 8 . 8 6 9 7 3 9 . 1 2 5 9 1 . 0 0 0 0 6 8 4 . 6 7 8 7 3 1 6 . 231 0 4 8 . 8 2 2 5 3 9 . 7 0 5 6 1 . 0 5 0 0 6 1 8 . 6 8 7 5 2 8 7 . 1 8 4 8 4 2 . 3 9 5 6 34 . 5 7 1 2 1 . 1 0 0 0 4 5 3 . 5 6 6 7 191 . 5 2 8 7 2 9 . 9 6 3 0 2 2 . 1 8 2 0 1 . 1 5 0 0 2 6 3 . 9 9 6 3 9 2 . 3 3 8 8 1 6 . 8 5 7 6 10 . 3 1 9 0 1 . 2 0 0 0 441 . 5 0 0 5 2 0 5 . 4 7 6 6 2 7 . 3 1 3 3 2 2 . 2 1 1 4 1 . 2 5 0 0 3 9 3 . 3 4 2 0 1 8 3 . , 4 8 2 4 2 3 . 6 2 1 7 19 . 2 2 6 4 1 . 3 0 0 0 2 9 2 . . 8 1 2 3 1 2 5 . , 3 6 2 0 1 7 . , 0 9 9 0 12 . 7 5 7 5 1 . 3 5 0 0 .40 . 9 0 8 4 7 . , 4 5 6 3 2 . 3 2 6 4 0 . 7 3 8 1 1 . 4 0 0 0 1 . , 5 2 0 0 0 . , 0 6 3 6 0 . , 0 8 4 3 0 . 0 0 6 1 1 . 4 5 0 0 4 3 . 6 5 8 3 9 . ,01 6 2 2 . , 3 6 3 7 0 . 8 4 8 1 1 . 5 0 0 0 . 1 4 5 . , 1 0 5 4 5 4 . 5 6 2 5 7 . , 6 7 8 4 5 . 0 1 1 6 1 . 5 5 0 0 2 1 1 . , 4 9 0 9 9 9 . , 3 9 0 2 1 0 . , 9 4 8 3 8 . 9 2 3 8 APPENDIX VI ( c o n t i n u e d ) © = 0 . 0 ° 0 = 6 0 . 0 ° 0 = 0 . 0 ° 0 = 6 0 . 0 ° A. 1 A IA D A D A 1 . 6 0 0 0 1 8 8 . 5 7 9 3 8 7 . 8 6 6 1 9 . 5 5 8 8 7 . 7 1 9 0 1 . 6 5 0 0 1 7 6 . 0 1 1 3 8 3 . 0 3 9 0 8 . 7 4 2 8 7 . 1 4 3 7 1 . 7 0 0 0 1 5 0 . 4 7 5 5 6 8 . 9 6 3 0 7 . 3 2 9 9 5 . 8 1 4 4 1 . 7 5 0 0 1 1 6 . 0 2 0 0 4 9 . 3 9 7 7 5 . 5 4 6 0 4 . 0 8 4 6 1 . 8 0 0 0 2 6 . 6 6 2 7 6 . 0 0 2 9 1 . 2 5 1 5 0 . 4 8 7 1 1 . 8 5 0 0 0 . 0 5 6 6 0 . 0 0 0 8 0 . 0 0 2 6 0 . 0 0 0 1 1 . 9 0 0 0 0 . 2 8 2 2 0 . 0 0 8 4 0 . 0 1 2 8 0 . 0 0 0 7 1 . 9 5 0 0 1 3 . 2 6 5 4 2 . 4 4 7 7 0 . 5 9 1 5 0 . 1 8 8 4 2 . 0 0 0 0 5 8 . 4 7 3 6 2 2 . 1 5 6 9 2 . 5 6 5 7 1 . 6 7 7 6 2 . 1 0 0 0 7 2 . 8 5 2 0 3 2 . 8 8 6 0 3 . 0 9 9 4 2 . 4 1 2 1 2 . 2 0 0 0 6 6 . 7 0 5 8 3 1 . 0 3 1 6 2 . 7 5 5 9 2 . 2 0 8 7 2 . 3 0 0 0 5 5 . 9 4 7 1 2 5 . 7 6 5 6 2 . 2 4 7 9 1 . 7 8 2 3 2 . 4 0 0 0 2 9 . 4 0 9 6 1 0 . 7 7 5 9 1 . 1 5 0 6 0 . 7 2 5 4 59 APPENDIX V I I SPECTRAL VALUES FOR FIG.S 6 AND 7 The d i f f u s e s p e c t r a l v a l u e s g e n e r a t e d by t h e p r e s e n t f o r m u l a t i o n f o r F i g . 6 and F i g . 7 f o r g r o u n d a l b e d o 0.0 and 0.2 a r e l i s t e d b e low. p = 0.0 P = 0.2 9 = 0.0° 0=60.0° 0 = 0.0° 0=60.0° X D> D Dx Dx 0. 2925 0. 0537 0. 0000 0. 0537 0, .0000 0. 2975 2. 3395 0. 0160 2. 3396 0, .0160 0. 3025 20. 2514 1 . 2005 20. 3050 1 , .2026 0. 3075 69. 3564 12. 2722 70. 8988 12, .4479 0. 3125 1 36. 3387 43. 7366 146. 1 482 45, .7385 0. 3175 169. 9405 67. 781 6 188. 8198 72, .5207 0. 3225 221 . 4650 103. 9536 255. 5820 1 13 .9383 0. 3275 293. 1367 159. 6472 354. 1758 180 .2055 0. 3325 305. 8059 177. 1814 377. 6360 202 .7376 0. 3375 295. 7832 172. 2384 365. 2244 196 .9180 0. 3425 308. 5168 187. 7445 387. 1 047 216 .7797 0. 3475 302. 31 67 186. 4420 380. 5667 215 .6085 0. 3525 299. 661 1 187. 4077 378. 5493 217 .1113 0. 3575 290. 3643 183. 9421 367. 8958 213 .4147 0. 3625 298. 1096 191 . 041 0 378. 5959 221 .9051 0. 3675 300. 521 0 194. 6434 382. 381 6 226 .2940 0. 3725 305. 8936 200. 1099 389. 8267 232 .8208 0. 3775 278. 8794 184. 1853 355. 8723 214 .4269 0. 3825 264. 4895 176. 2790 337. 8821 205 .3281 0. 3875 242. 5025 163. 0356 310. 0696 189 .9798 0. 3925 241 . 2352 1 63. 5343 308. 661 4 190 .6187 0. 3975 288. 1 348 196. 8795 368. 8530 229 .5333 0. 4025 350. 6433 241 . 4066 449. 01 54 281 .4758 0. 4075 373. 3772 258. 9153 478. 1990 301 .8958 0. 4125 370. 0396 258. 3689 473. 921 1 301 .2375 0. 4175 364. 91 26 256. 4629 467. 2832 298 .9697 0. 4225 ' 349. 9832 247. 5111 448. 0359 288 .4685 0. 4275 318. 61 45 226. 6728 407. 7097 264 .1023 0. 4350 335. 3350 240. 4651 428. 6509 279 .9707 0. 4450 356. 1 1 47 257. 8904 454. 4443 299 .9238 0. 4550 353. 9851 258. 7295 450. 8770 300 .5327 0. 4650 338. 6934 249. 3083 430. 2168 289 .0789 0. 4750 324. 8267 240; 8604 411. 5630 278 .8342 0. 4850 298. 1 204 221 . 9427 376. 2869 256 .3086 0. 4950 288. 991 7 216. 4790 363. 7583 249 .5622 0. 5050 266. 3381 199. 5730 333. 5640 229 .3371 0. 5150 253. 81 58 191 . 1622 316. ,9573 219 .2691 0. 5250 247. 3793 186. 3666 307. 4802 213 .1357 0. 5350 243. 7480 183. 5484 301 . ,5300 209 .2812 0. 5450 235. 41 49 1 77. 4579 290. ,0620 201 .8275 0. 5550 224. 7507 169. 5901 275. ,8643 1 92 .41 42 0. 5650 214. 7455 161 . 5991 262. ,2659 182 .7675 0. 5750 210. 3798 158. 3303 255. ,9612 178 .6435 0. 5850 205. , 1 542 1 55. ,2898 249. ,1928 175 .0350 0. ,5950 197. 9602 150. ,2538 239. ,8323 169 .0847 APPENDIX V I I ( c o n t i n u e d ) X 0 . 6 0 5 0 0 . 6 1 5 0 0 . 6 2 5 0 0 . 6 3 5 0 0 . 6 4 5 0 0 . 6 5 5 0 0 . 6 6 5 0 0 . 6 7 5 0 0 . 6 8 5 0 0 . 6 9 5 0 0 . 7 0 5 0 0 . 7 1 5 0 0 . 7 2 5 0 0 . 7 3 5 0 0 . 7 4 5 0 0 . 7 5 5 0 0 . 7 6 0 0 0 . 7 7 0 0 0 . 7 8 0 0 0 . 7 9 0 0 0 . 8 0 0 0 0 . 8 1 0 0 0 . 8 2 0 0 0 . 8 3 0 0 0 . 8 4 0 0 0 . 8 5 0 0 0 . 8 6 0 0 0 . 8 7 0 0 0 . 8 8 0 0 0 . 8 9 0 0 0 . 9 0 0 0 0 . 9 1 0 0 0 . 9 2 0 0 0 . 9 3 0 0 0 . 9 4 0 0 0 . 9 5 0 0 0 . 9 6 0 0 0 . 9 7 0 0 0 . 9 8 0 0 0 . 9 9 0 0 1 . 0 0 0 0 1 . 0 5 0 0 1 . 1 0 0 0 1 . 1 5 0 0 1 . 2 0 0 0 1 . 2 5 0 0 1 . 3 0 0 0 1 . 3 5 0 0 1 . 4 0 0 0 A = 0 . 0 p = 0 . 2 © = 0 . 0 ° D A 1 8 5 . 0 2 9 1 1 7 8 . 3 1 7 6 1 7 2 . 4 9 0 2 1 6 5 . 8 7 5 3 1 6 0 . 7 7 6 9 1 5 6 . 3 5 7 1 1 5 2 . 9 1 3 6 1 4 6 . 8 8 2 0 1 4 0 . 9 6 7 9 1 3 3 . 7 5 6 5 1 3 2 . 8 1 4 5 1 1 3 . 3 8 8 0 1 1 0 . 9 9 8 0 1 1 0 . 7 5 8 1 1 1 2 . 2 9 8 0 1 1 3 . 3 8 3 5 81 . 4 4 7 2 9 7 . 6 6 5 8 1 0 0 . 9 6 5 7 9 7 . 1 8 5 4 9 3 . 6 0 2 5 8 6 . 7 0 3 1 7 7 . 5 3 3 5 7 8 . 8 7 4 0 7 9 . 3 8 9 5 7 9 . 5 3 1 1 7 7 . 0 3 2 3 7 4 . 6 6 1 6 7 2 . 8 0 5 2 6 9 . 1 2 8 3 5 7 . 2 3 5 2 5 6 . 5 8 9 9 5 6 . 2 6 7 5 3 1 . 2 5 7 7 2 6 . 2 4 0 7 2 4 . 3 6 8 5 2 8 . 5 5 0 6 4 4 . 5 1 6 0 4 5 . 9 5 3 1 4 8 . 8 6 9 7 4 8 . 8 2 2 5 4 2 . 3 9 5 6 2 9 . 9 6 3 . 0 1 6 . 8 5 7 6 2 7 . 3 1 3 3 2 3 . 6 2 1 7 1 7 . 0 9 9 0 2 . 3 2 6 4 0 . 0 8 4 3 © = 6 0 . 0 ° DA 1 4 0 . 0 9 3 2 1 3 5 . 9 5 9 3 1 3 2 . 4 1 61 1 2 8 . 1 4 8 8 1 2 4 . 9 8 2 8 1 2 2 . 2 1 0 7 1 2 0 . 0 4 5 1 1 1 5 . 9 1 2 8 1 1 1 . 6 0 7 1 1 0 5 . 5 6 2 4 1 0 5 . 1 1 4 4 8 5 . 9 0 3 5 8 4 . 5 2 9 6 8 5 . 7 3 5 1 8 9 . 3 0 6 8 9 1 . 0 0 7 3 5 6 . 9 8 1 8 7 6 . 1 1 9 2 8 1 . 4 1 3 1 7 7 . 9 9 3 2 7 4 . 9 5 0 6 6 8 . 1 2 3 1 5 8 . 7 4 6 5 6 1 . 2 4 8 1 6 3 . 0 7 3 5 6 4 . 3 1 4 9 6 2 . 4 2 9 7 6 0 . 5 3 6 3 5 8 . 9 6 8 5 5 5 . 4 4 2 5 4 2 . 9 9 7 6 4 2 . 9 8 8 1 4 3 . 121 3 1 8 . 5 3 2 3 1 4 . 5 8 0 6 1 3 . 3 2 6 6 1 7 . 0 5 3 8 3 2 . 9 0 8 7 3 5 . 0 8 5 5 3 9 . 1 2 5 9 3 9 . 7 0 5 6 3 4 . 5 7 1 2 2 2 . 1 8 2 0 1 0 . 3 1 9 0 2 2 . 2 1 1 4 1 9 . 2 2 6 4 1 2 . 7 5 7 5 0 . 7 3 8 1 0 . 0 0 6 1 © = 0 . 0 ° 2 2 3 . 2 1 8 9 2 1 4 . 9 3 3 0 2 0 7 . 7 3 4 1 1 9 9 . 5 8 6 9 1 9 3 . 2 8 6 0 1 8 7 . 7 8 1 3 1 8 3 . 4 1 1 8 1 7 6 . 0 1 6 6 1 6 8 . 6 8 2 0 1 5 9 . 4 1 6 4 1 5 8 . 0 8 1 3 1 3 1 . 8 4 8 0 1 2 9 . 0 7 4 5 1 2 9 . 4 7 2 1 1 3 2 . 7 1 3 6 1 3 4 . 2 3 4 1 9 1 . 1 3 0 8 1 1 3 . 7 3 9 2 1 1 9 . 1 7 6 6 1 1 4 . 3 1 6 2 1 0 9 . 8 0 7 9 1 0 0 . 6 7 4 7 8 8 . 5 6 3 1 9 0 . 8 7 5 1 9 2 . 2 5 5 2 9 3 . 0 1 4 0 9 0 . 0 4 1 2 8 7 . 1 8 0 3 8 4 . 8 9 1 0 8 0 . 1 9 0 6 6 4 . 6 2 7 4 6 4 . 1 0 1 5 6 3 . 8 9 0 5 3 3 . 1 5 0 8 2 7 . 5 3 2 0 2 5 . 5 0 2 8 3 0 . 2 8 6 7 4 9 . 7 3 4 6 5 1 . 8 7 7 0 5 6 . 1 4 6 3 5 6 . 4 0 3 8 4 8 . 8 6 3 3 3 3 . 2 8 0 0 1 7 . 8 7 3 9 3 1 . 2 4 4 2 2 6 . 9 8 7 0 1 8 . 9 3 2 7 2 . 3 4 4 0 0 . 0 8 4 3 © = 6 0 . 0 ° DA 1 5 7 . 2 3 0 9 1 5 2 . 5 1 0 7 1 4 8 . 4 6 1 1 ' 1 4 3 . 5 9 8 3 1 3 9 . 9 7 8 0 1 3 6 . 7 8 8 6 1 3 4 . 2 5 9 4 1 2 9 . 5 6 6 0 1 2 4 . 6 4 0 0 1 1 7 . 5 9 3 9 1 1 6 . 9 9 7 7 9 4 . 2 1 6 3 9 2 . 7 1 3 7 9 4 . 3 4 9 1 9 8 . 9 6 3 2 1 0 0 . 9 6 2 9 6 1 . 0 1 2 4 8 3 . 5 7 3 5 9 0 . 1 5 2 4 8 6 . 1 7 6 6 8 2 . 6 7 5 9 7 4 . 6 5 9 6 6 3 . 7 2 3 4 6 6 . 7 9 8 9 6 9 . 1 6 2 5 7 0 . 8 1 0 9 6 8 . 7 1 18 6 6 . 5 8 5 2 6 4 . 8 0 2 8 6 0 . 7 3 1 1 4 6 . 3 0 8 3 4 6 . 3 9 0 2 4 6 . 6 0 4 7 1 9 . 2 0 1 6 1 5 . 0 0 8 5 1 3 . 6 9 6 6 1 7 . 6 7 2 4 3 5 . 2 1 0 1 3 7 . 7 8 3 9 4 2 . 6 0 1 7 4 3 . 3 8 4 4 3 7 . 7 1 9 1 2 3 . 6 4 8 1 1 0 . 6 9 0 4 2 4 . 1 2 0 6 2 0 . 8 6 2 5 1 3 . 5 7 4 8 0 . 7 4 1 4 0 . 0 0 6 1 APPENDIX V I I ( c o n t i n u e d ) X. 1.4500 1.5000 1.5500 1.6000 1 .6500 1 .7000 1.7500 1 .8000 1.8500 1 .9000 1.9500 2.0000 2. 1000 2.2000 2.3000 2.4000 0=0.0° Dx 2.3637 7.6784 10.9483 9.5588 8.7428 7.3299 5.5460 1 .251 5 0.0026 0.0128 0.5915 2.5657 3.0994 2.7559 2.2479 1 . 1 506 0 = 6 0 . 0 ° D* 0.8481 5.0116 8.9238 7.7190 7.1437 5.8144 4.0846 0.4871 0.0001 0.0007 0.1884 1.6776 2.4121 2.2087 1.7823 0.7254 0=0.0° D> 2.3895 8.2052 12.4415 10.8162 9.9327 8.2331 6.0863 1 .2684 0.0026 0.0128 0.5958 2.7364 3.4525 3.0981 2.5171 1.2180 0= 6 0 . 0 ° Dx 0.8536 5.2170 9.6509 8.3256 7.7246 6.2424 4.3223 0.4911 0.0001 0.0007 0.1892 1.7442 2.5763 2.3725 1.9097 0.7508 62 APPENDIX V I I I SOLAR SPECTRUM OF PARAMETER COMPARISONS The e x t r a t e r r e s t r i a l s o l a r s p e c t r u m u s e d t o g e n e r a t e t h e p l o t s i n C h a p t e r 5 i s from T h e k a e k a r a [ 2 7 ] . The a b s o r p t i o n c o e f i c i e n t s a r e f r o m L e c k n e r [ 1 2 ] . X A X ke»A k 3 * k IAJ 0.290 . 0.005 482.0 38.000 0.0 0.0 0.295 0.005 584.0 20.000 0.0 0.0 0.300 0.005 514.0 10.000 0.0 0.0 0.305 0.005 603.0 4.800 0.0 0.0 0.310 0.005 689.0 2.700 0.0 0.0 0.315 0.005 764.0 1 .350 0.0 0.0 0.320 0.005 830.0 0.800 0.0 0.0 0.325 0.005 975.0 0.380 0.0 0.0 0.330 0.005 1059.0 0.160 0.0 0.0 0.335 0.005 1081 .0 0.075 0.0 0.0 0.340 0.005 1074.0 0.040 0.0 0.0 0.345 0.005 1069.0 0.019 0.0 0.0 0.350 0.005 1093.0 0.007 0.0 0.0 0.355 0.005 1083.0 0.0 0.0 0.0 0.360 0.005 1068.0 0.0 0.0 0.0 0.365 0.005 1132.0 0.0 0.0 0.0 0.370 0.005 1181.0 0.0 0.0 0.0 0.375 0.005 1157.0 0.0 0.0 0.0 0.380 0.005 1120.0 0.0 0.0 0.0 0.385 0.005 1098.0 0.0 0.0 0.0 0.390 0.005 1098.0 0.0 0.0 0.0 0.395 0.005 1189.0 0.0 0.0 0.0 0.400 0.005 1429.0 0.0 0.0 0.0 0.405 0.005 1644.0 0.0 0.0 0.0 0.410 0.005 1751 .0 0.0 0.0 0.0 0.415 0.005 1774.0 0.0 0.0 0.0 0.420 0.005 1747.0 0.0 0.0 0.0 0.425 0.005 1693.0 0.0 0.0 0.0 0.430 0.005 1639.0 0.0 0.0 0.0 0.435 0.005 1663.0 0.0 0.0 0.0 0.440 0.005 1810.0 0.0 0.0 0.0 0.445 0.005 1922.0 0.003 0.0 0.0 0.450 0.005 2006.0 0.003 0.0 0.0 0.455 0.005 2057.0 0.004 0.0 0.0 0.460 0.005 2066.0 0.006 0.0 0.0 0.465 0.005 2048.0 0.008 0.0 0.0 0.470 0.005 2033.0 0.009 0.0 0.0 0.475 0.005 2044.0 0.012 0.0 0.0 0.480 0.005 2074.0 0.014 0.0 0.0 0.485 0.005 1976.0 0.017 0.0 0.0 0.490 0.005 1950.0 0.021 0.0 0.0 0.495 0.005 1960.0 0.025 0.0 0.0 0.500 0.005 1942.0 0.030 0.0 0.0 0.505 0.005 1920.0 0.035 0.0 0.0 0.510 0.005 1882.0 0.040 0.0 0.0 0.515 0.005 1833.0 0.045 0.0 0.0 0.520 0.005 1833.0 0.048 0.0 0.0 0.525 0.005 1852.0 0.057 0.0 0.0 APPENDIX V I I I ( c o n t i n u e d ) A. I.* 0 . 5 3 0 0 . 0 0 5 1 8 4 2 . 0 0 . 0 6 3 0 . 0 0 . 0 0 . 5 3 5 0 . 0 0 5 1 8 1 8 . 0 0 . 0 7 0 0 . 0 0 . 0 0 . 5 4 0 0 . 0 0 5 1 7 8 3 . 0 0 . 0 7 5 0 . 0 0 . 0 0 . 5 4 5 0 . 0 0 5 1 7 5 4 . 0 0 . 0 8 0 0 . 0 0 . 0 0 . 5 5 0 0 . 0 0 5 1 7 2 5 . 0 0 . 0 8 5 v- 0 . 0 0 . 0 0 . 5 5 5 0 . 0 0 5 1 7 2 0 . 0 0 . 0 9 5 0 . 0 0 . 0 0 . 5 6 0 0 . 0 0 5 1 6 9 5 . 0 0 . 1 0 3 0 . 0 0 . 0 0 . 5 6 5 0 . 0 0 5 1 7 0 5 . 0 0 . 1 1 0 0 . 0 0 . 0 0 . 5 7 0 0 . 0 0 5 1 7 1 2 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 5 7 5 0 . 0 0 5 1 7 1 9 . 0 0 . 1 2 2 0 . 0 0 . 0 0 . 5 8 0 0 . 0 0 5 1 7 1 5 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 5 8 5 0 . 0 0 5 1 7 1 2 . 0 0 . 1 1 8 0 . 0 0 . 0 0 . 5 9 0 0 . 0 0 5 1 7 0 0 . 0 0 . 1 1 5 0 . 0 0 . 0 0 . 5 9 5 0 . 0 0 5 1 6 8 2 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 6 0 0 0 . 0 0 5 1 6 6 6 . 0 0 . 1 2 5 0 . 0 0 . 0 0 . 6 0 5 0 . 0 0 5 1 6 4 7 . 0 0 . 1 3 0 0 . 0 0 . 0 0 . 6 1 0 0 . 0 0 7 1 6 3 5 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 6 2 0 0 . 0 1 0 1 6 0 2 . 0 0 . 1 0 5 0 . 0 0 . 0 0 . 6 3 0 0 . 0 1 0 1 5 7 0 . 0 0 . 0 9 0 0 . 0 0 . 0 0 . 6 4 0 0 . 0 1 0 1 5 4 4 . 0 0 . 0 7 9 0 . 0 0 . 0 0 . 6 5 0 0 . 0 1 0 1 5 1 1 . 0 0 . 0 6 7 0 . 0 0 . 0 0 . 6 6 0 0 . 0 1 0 1 4 8 6 . 0 0 . 0 5 7 0 . 0 0 . 0 0 . 6 7 0 0 . 0 1 0 1 4 5 6 . 0 0 . 0 4 8 0 . 0 0 . 0 0 . 6 8 0 0 . 0 1 0 1 4 2 7 . 0 0 . 0 3 6 0 . 0 0 . 0 0 . 6 9 0 0 . 0 1 0 1 4 0 2 . 0 0 . 0 2 8 0 . 0 0 . 0 1 6 0 0 0 . 7 0 0 0 . 0 1 0 1 3 6 9 . 0 0 . 0 2 3 0 . 0 0 . 0 2 4 0 0 0 . 7 1 0 0 . 0 1 0 1 3 4 4 . 0 0 . 0 1 8 0 . 0 0 . 0 1 2 5 0 0 . 7 2 0 0 . 0 1 0 1 3 1 4 . 0 0 . 0 1 4 0 . 0 1 . 0 0 0 0 0 0 . 7 3 0 0 . 0 1 0 1 2 9 0 . 0 0 . 0 1 1 0 . 0 0 . 8 7 0 0 0 0 . 7 4 0 0 . 0 1 0 1 2 6 0 . 0 0 . 0 1 0 0 . 0 0 . 0 6 1 0 0 0 . 7 5 0 0 . 0 1 0 1 2 3 5 . 0 0 . 0 0 9 0 . 0 0 . 0 0 1 0 0 0 . 7 6 0 0 . 0 1 0 1 2 1 1 . 0 0 . 0 0 7 3 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 7 7 0 0 . 0 1 0 1 1 8 5 . 0 0 . 0 0 4 0 . 2 1 0 0 0 0 . 0 0 0 0 1 0 . 7 8 0 0 . 0 1 0 1 1 5 9 . 0 0 . 0 0 . 0 0 . 0 0 0 6 0 0 . 7 9 0 0 . 0 1 0 1 1 3 4 . 0 0 . 0 0 . 0 0 . 0 1 7 5 0 0 . 8 0 0 0 . 0 1 0 1 1 0 9 . 0 0 . 0 0 . 0 0 . 0 3 6 0 0 0 . 8 1 0 0 . 0 1 0 1 0 8 5 . 0 0 . 0 0 . 0 0 . 3 3 0 0 0 0 . 8 2 0 0 . 0 1 0 1 0 6 0 . 0 0 . 0 0 . 0 1 . 5 3 0 0 0 0 . 8 3 0 0 . 0 1 0 1 0 3 6 . 0 0 . 0 0 . 0 0 . 6 6 0 0 0 0 . 8 4 0 0 . 0 1 0 1 0 1 3 . 0 0 . 0 0 . 0 0 . 1 5 5 0 0 0 . 8 5 0 0 . 0 1 0 9 9 0 . 0 0 . 0 0 . 0 0 . 0 0 3 0 0 0 . 8 6 0 0 . 0 1 0 9 6 8 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 . 8 7 0 0 . 0 1 0 9 4 7 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 . 8 8 0 0 . 0 1 0 9 2 6 . 0 0 . 0 0 . 0 0 . 0 0 2 6 0 0 . 8 9 0 0 . 0 1 0 9 0 8 . 0 0 . 0 0 . 0 0 . 0 6 3 0 0 0 . 9 0 0 0 . 0 1 0 891 . 0 0 . 0 0 . 0 2 . 1 0 0 0 0 0 . 9 1 0 0 . 0 1 0 8 8 0 . 0 0 . 0 0 . 0 1 . 6 0 0 0 0 0 . 9 2 0 0 . 0 1 0 8 6 9 . 0 . 0 . 0 0 . 0 1 . 2 5 0 0 0 0 . 9 3 0 0 . 0 1 0 8 5 8 . 0 0 . 0 0 . 0 2 7 . 0 0 0 0 0 0 . 9 4 0 0 . 0 1 0 8 4 7 . 0 0 . 0 0 . 0 3 8 . 0 0 0 0 0 0 . 9 5 0 0 . 0 1 0 • 8 3 7 . 0 0 . 0 0 . 0 4 1 . 0 0 0 0 0 0 . 9 6 0 0 . 0 1 0 8 2 0 . 0 0 . 0 0 . 0 2 6 . 0 0 0 0 0 APPENDIX V I I I ( c o n t i n u e d ) X- A X i e x 0 . 9 7 0 0 . 0 1 0 8 0 3 . 0 0 . 9 8 0 0 . 0 1 0 7 8 5 . 0 0 . 9 9 0 0 . 0 1 0 7 6 7 . 0 1 . 0 0 0 0 . 0 3 0 7 4 8 . 0 1 . 0 5 0 0 . 0 5 0 6 6 8 . 0 1 . 1 0 0 0 . 0 5 0 5 9 3 . 0 1 . 1 5 0 0 . 0 5 0 5 3 5 . 0 1 . 2 0 0 0 . 0 5 0 4 8 5 . 0 1 . 2 5 0 0 . 0 5 0 4 3 8 . 0 1 . 3 0 0 0 . 0 5 0 3 9 7 . 0 1 . 3 5 0 0 . 0 5 0 3 5 8 . 0 1 . 4 0 0 0 . 0 5 0 3 3 7 . 0 1 . 4 5 0 0 . 0 5 0 3 1 2 . 0 1 . 5 0 0 0 . 0 5 0 2 8 8 . 0 1 . 5 5 0 0 . 0 5 0 2 6 7 . 0 1 . 6 0 0 0 . 0 5 0 2 4 5 . 0 1 . 6 5 0 0 . 0 5 0 2 2 3 . 0 1 . 7 0 0 0 . 0 5 0 2 0 2 . 0 1 . 7 5 0 0 . 0 5 0 1 8 0 . 0 1 . 8 0 0 0 . 0 5 0 1 5 9 . 0 1 . 8 5 0 0 . 0 5 0 1 4 2 . 0 1 . 9 0 0 0 . 0 5 0 1 2 6 . 0 1 . 9 5 0 0 . 0 5 0 1 1 4 . 0 2 . 0 0 0 0 . 0 7 5 1 0 3 . 0 2 . 1 0 0 0 . 1 0 0 9 0 . 0 2 . 2 0 0 0 . 1 0 0 7 9 . 0 2 . 3 0 0 0 . 1 0 0 6 9 . 0 2 . 4 0 0 0 . 1 0 0 6 2 . 0 2 . 5 0 0 0 . 1 0 0 5 5 . 0 2 . 6 0 0 0 . 1 0 0 4 8 . 0 2 . 7 0 0 0 . 1 0 0 4 3 . 0 2 . 8 0 0 0 . 100 3 9 . 0 2 . 9 0 0 0 . 100 3 5 . 0 3 . 0 0 0 0 . 1 0 0 31 . 0 3 . 1 0 0 0 . 1 0 0 2 6 . 0 3 . 2 0 0 0 . 1 0 0 2 2 . 6 3 . 3 0 0 0 . 1 0 0 1 9 . 2 3 . 4 0 0 0 . 1 0 0 1 6 . 6 3 . 5 0 0 0 . 1 0 0 1 4 . 6 3 . 6 0 0 0 . 1 0 0 1 3 . 5 3 . 7 0 0 0 . 1 0 0 1 2 . 3 3 . 8 0 0 0 . 1 0 0 1 1 . 1 3 . 9 0 0 0 . 1 0 0 1 0 . 3 4 . 0 0 0 0 . 1 0 0 9 . 5 k 3> 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 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 3 0 0 . 0 0 . 0 0 0 4 0 0 . 0 0 . 0 0 0 1 1 0 . 0 0 . 0 0 0 0 1 0 . 0 0 . 0 6 4 0 0 0 . 0 0 . 0 0 0 6 3 0 . 0 0 . 0 1 0 0 0 0 . 0 0 . 0 6 4 0 0 0 . 0 0 . 0 0 1 4 5 0 . 0 0 . 0 0 0 0 1 0 . 0 0 . 0 0 0 0 1 0 . 0 0 . 0 0 0 0 1 0 . 0 0 . 0 0 0 1 4 0 . 0 0 . 0 0 7 1 0 0 . 0 2 . 0 0 0 0 0 0 . 0 3 . 0 0 0 0 0 0 . 0 0 . 2 4 0 0 0 0 . 0 0 . 0 0 0 3 8 0 . 0 0 . 0 0 1 1 0 0 . 0 0 . 0 0 0 1 7 0 . 0 0 . 0 0 0 1 4 0 . 0 0 . 0 0 0 6 6 0 . 0 1 0 0 . 0 0 0 0 0 0 . 0 5 0 . 0 0 0 0 0 0 . 0 0 . 1 3 0 0 0 0 . 0 0 . 0 0 9 5 0 0 . 0 0 . 0 0 1 0 0 0 . 0 0 . 8 0 0 0 0 0 . 0 1 . 9 0 0 0 0 0 . 0 1 . 3 0 0 0 0 0 . 0 0 . 0 7 5 0 0 0 . 0 0 . 0 1 0 0 0 0 . 0 0 . 0 0 1 9 5 0 . 0 0 . 0 0 4 0 0 0 . 0 0 . 2 9 0 0 0 0 . 0 0 . 0 2 5 0 0 k w X 3 . 1 0 0 0 0 1 . 4 8 0 0 0 0 . 1 2 5 0 0 0 . 0 0 2 5 0 0 . 0 0 0 0 1 3 . 2 0 0 0 0 2 3 . 0 0 0 0 0 0 . 0 1 6 0 0 0 . 0 0 0 1 8 2 . 9 0 0 0 0 2 0 0 . 0 0 0 0 0 1 1 0 0 . 0 0 0 0 0 1 5 0 . 0 0 0 0 0 1 5 . 0 0 0 0 0 0 . 0 0 1 7 0 0 . 0 0 0 0 1 0 . 0 1 0 0 0 0 . 5 1 0 0 0 4 . 0 0 0 0 0 1 3 0 . 0 0 0 0 0 2 2 0 0 . 0 0 0 0 0 1 4 0 0 . 0 0 0 0 0 1 6 0 . 0 0 0 0 0 2 . 9 0 0 0 0 0 . 2 2 0 0 0 0 . 3 3 0 0 0 0 . 5 9 0 0 0 2 0 . 3 0 0 0 0 3 1 0 . 0 0 0 0 0 1 5 0 0 0 . 0 0 0 0 0 2 2 0 0 0 . 0 0 0 0 0 8 0 0 0 . 0 0 0 0 0 6 5 0 . 0 0 0 0 0 2 4 0 . 0 0 0 0 0 2 3 0 . 0 0 0 0 0 1 0 0 . 0 0 0 0 0 1 2 0 . 0 0 0 0 0 1 9 . 5 0 0 0 0 3 . 6 0 0 0 0 3 . 1 0 0 0 0 2 . 5 0 0 0 0 1 . 4 0 0 0 0 0 . 1 7 0 0 0 0 . 0 0 4 5 0 65 APPENDIX IX MOST RECENT SOLAR SPECTRUM The most r e c e n t s o l a r s p e c t r u m from N e c k e l and Labs [28] a l o n g w i t h t h e a b s o r p t i o n c o e f f i c i e n t s f r o m L e c k n e r [12] a r e l i s t e d b e low. X A X I tX kgA k uu 0.290 0.005 535.0 38.000 0.0 0.0 0.295 0.005 560.0 20.000 0.0 0.0 0.300 0.005 527.0 10.000 0.0 0.0 0.305 0.005 557.0 4.800 0.0 0.0 0.310 0.005 602.0 2.700 0.0 0.0 0.315 0.005 705.0 1 .350 0.0 0.0 0.320 0.005 747.0 0.800 0.0 0.0 0.325 0.005 782.0 0.380 0.0 0.0 0.330 0.005 997.0 0. 1 60 0.0 0.0 0.335 0.005 906.0 0.075 0.0 0.0 0.340 0.005 960.0 0.040 0.0 0.0 0.345 0.005 877.0 0.019 0.0 0.0 0.350 0.005 955.0 0.007 0.0 0.0 0.355 0.005 1044.0 0.0 0.0 0.0 0.360 0.005 940.0 0.0 0.0 0.0 0.365 0.005 1125.0 0.0 0.0 0.0 0.370 0.005 1165.0 0.0 0.0 0.0 0.375 0.005 1081.0 0.0 0.0 0.0 0.380 0.005 1210.0 0.0 0.0 0.0 0.385 0.005 931 .0 0.0 0.0 0.0 0.390 0.005 1200.0 0.0 0.0 0.0 0.395 0.005 1033.0 0.0 0.0 0.0 0.400 0.005 1702.0 0.0 0.0 0.0 0.405 0.005 1643.0 0.0 0.0 0.0 0.410 0.005 1710.0 0.0 0.0 0.0 0.415 0.005 1747.0 0.0 0.0 0.0 0.420 0.005 1747.0 0.0 0.0 0.0 0.425 0.005 1692.0 0.0 0.0 0.0 0.430 0.005 1492.0 0.0 0.0 0.0 0.435 0.005 1761.0 0.0 0.0 0.0 0.440 0.005 1755.0 0.0 0.0 0.0 0.445 0.005 1922.0 0.003 0.0 0.0 0.450 0.005 2100.0 0.003 0.0 0.0 0.455 0.005 2017.0 0.004 0.0 0.0 0.460 0.005 2032.0 0.006 0.0 0.0 0.465 0.005 2000.0 0.008 0.0 0.0 0.470 0.005 1980.0 0.009 0.0 0.0 0.475 0.005 2016.0 0.012 0.0 0.0 0.480 0.005 2055.0 0.014 0.0 0.0 0.485 0.005 1901.0 0.017 0.0 0.0 0.490 0.005 1920.0 0.021 0.0 0.0 0.495 0.005 1965.0 0.025 0.0 0.0 0.500 0.005 1862.0 0.030 0.0 0.0 0.505 0.005 1943.0 0.035 0.0 0.0 0.510 0.005 1952.0 0.040 0.0 0.0 0.515 0.005 1835.0 0.045 0.0 0.0 0.520 0.005 1802.0 0.048 0.0 0.0 APPENDIX IX ( c o n t i n u e d ) X A X. k<ix 0 . 5 2 5 0 . 0 0 5 1 8 9 5 . 0 0 . 0 5 7 0 . 0 0 . 0 0 . 5 3 0 0 . 0 0 5 1 9 4 7 . 0 0 . 0 6 3 0 . 0 0 . 0 0 . 5 3 5 0 . 0 0 5 1 9 2 6 . 0 0 . 0 7 0 0 . 0 0 . 0 0 . 5 4 0 0 . 0 0 5 1 8 5 7 . 0 0 . 0 7 5 0 . 0 0 . 0 0 . 5 4 5 0 . 0 0 5 1 8 9 5 . 0 0 . 0 8 0 0 . 0 0 . 0 0 . 5 5 0 0 . 0 0 5 1 9 0 2 . 0 0 . 0 8 5 0 . 0 0 . 0 0 . 5 5 5 0 . 0 0 5 1 8 8 5 . 0 0 . 0 9 5 0 . 0 0 . 0 0 . 5 6 0 0 . 0 0 5 1 8 4 0 . 0 0 . 1 0 3 0 . 0 0 . 0 0 . 5 6 5 0 . 0 0 5 1 8 5 0 . 0 0 . 1 1 0 0 . 0 0 . 0 0 . 5 7 0 0 . 0 0 5 1 8 1 7 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 5 7 5 0 . 0 0 5 1 8 4 8 . 0 0 . 1 2 2 0 . 0 0 . 0 0 . 5 8 0 0 . 0 0 5 1 8 4 0 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 5 8 5 0 . 0 0 5 1 8 1 7 . 0 0 . 1 1 8 0 . 0 0 . 0 0 . 5 9 0 0 . 0 0 5 1 7 4 2 . 0 0 . 1 1 5 0 . 0 0 . 0 0 . 5 9 5 0 . 0 0 5 1 7 8 5 . 0 0 . 1 2 0 . 0 . 0 0 . 0 0 . 6 0 0 0 . 0 0 5 1 7 2 0 . 0 0 . 1 2 5 0 . 0 0 . 0 0 . 6 0 5 0 . 0 0 5 1751 . 0 0 . 1 3 0 0 . 0 0 . 0 0 . 6 1 0 0 . 0 0 7 1 7 1 5 . 0 0 . 1 2 0 0 . 0 0 . 0 0 . 6 2 0 0 . 0 1 0 1 7 1 5 . 0 0 . 1 0 5 0 . 0 0 . 0 0 . 6 3 0 0 . 0 1 0 1 6 3 7 . 0 0 . 0 9 0 0 . 0 0 . 0 0 . 6 4 0 0 . 0 1 0 1 6 2 2 . 0 0 . 0 7 9 0 . 0 0 . 0 0 . 6 5 0 0 . 0 1 0 1 5 9 7 . 0 0 . 0 6 7 0 . 0 0 . 0 0 . 6 6 0 0 . 0 1 0 1 5 5 5 . 0 0 . 0 5 7 0 . 0 . 0 . 0 0 . 6 7 0 0 . 0 1 0 1 5 0 5 . 0 0 . 0 4 8 0 . 0 0 . 0 0 . 6 8 0 0 . 0 1 0 1 4 7 2 . 0 0 . 0 3 6 0 . 0 0 . 0 0 . 6 9 0 0 . 0 1 0 1 4 1 5 . 0 0 . 0 2 8 0 . 0 0 . 0 1 6 0 0 0 . 7 0 0 0 . 0 1 0 1 4 2 7 . 0 0 . 0 2 3 0 . 0 0 . 0 2 4 0 0 0 . 7 1 0 0 . 0 1 0 1 4 0 2 . 0 0 . 0 1 8 0 . 0 0 . 0 1 2 5 0 0 . 7 2 0 0 . 0 1 0 1 3 5 5 . 0 0 . 0 1 4 0 . 0 1 . 0 0 0 0 0 0 . 7 3 0 0 . 0 1 0 1 3 5 5 . 0 0 . 0 1 1 0 . 0 0 . 8 7 0 0 0 0 . 7 4 0 0 . 0 1 0 1 3 0 0 . 0 0 . 0 1 0 0 . 0 0 . 0 6 1 0 0 0 . 7 5 0 0 . 0 1 0 1 2 7 2 . 0 0 . 0 0 9 0 . 0 0 . 0 0 1 0 0 0 . 7 6 0 0 . 0 1 0 1 2 2 2 . 0 0 . 0 0 7 3 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 7 7 0 0 . 0 1 0 1 1 8 7 . 0 0 . 0 0 4 0 . 2 1 0 0 0 0 . 0 0 0 0 1 0 . 7 8 0 0 . 0 1 0 1 1 9 5 . 0 0 . 0 0 . 0 0 . 0 0 0 6 0 0 . 7 9 0 0 . 0 1 0 1 1 4 2 . 0 0 . 0 0 . 0 0 . 0 1 7 5 0 0 . 8 0 0 0 . 0 1 0 1 1 4 4 . 0 0 . 0 0 . 0 0 . 0 3 6 0 0 0 . 8 1 0 0 . 0 1 0 1 1 1 3 . 0 0 . 0 0 . 0 0 . 3 3 0 0 0 0 . 8 2 0 0 . 0 1 0 1 0 7 0 . 0 0 . 0 0 . 0 1 . 5 3 0 0 0 0 . 8 3 0 0 . 0 1 0 1 0 4 1 . 0 0 . 0 0 . 0 0 . 6 6 0 0 0 0 . 8 4 0 0 . 0 1 0 1 0 2 0 . 0 0 . 0 0 . 0 0 . 1 5 5 0 0 0 . 8 5 0 0 . 0 1 0 9 9 4 . 0 0 . 0 0 . 0 0 . 0 0 3 0 0 0 . 8 6 0 0 . 0 1 0 1 0 0 2 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 . 8 7 0 0 . 0 1 0 9 7 2 . 0 0 . 0 0 . 0 0 . 0 0 0 0 1 0 . 8 8 0 0 . 0 1 0 9 6 6 . 0 0 . 0 0 . 0 0 . 0 0 2 6 0 0 . 8 9 0 0 . 0 1 0 9 4 5 . 0 0 . 0 0 . 0 0 . 0 6 3 0 0 0 . 9 0 0 0 . 0 1 0 9 1 3 . 0 0 . 0 0 . 0 2 . 1 0 0 0 0 0 . 9 1 0 0 . 0 1 0 8 7 6 . 0 0 . 0 0 . 0 1 . 6 0 0 0 0 0 . 9 2 0 0 . 0 1 0 841 . 0 0 . 0 0 . 0 1 . 2 5 0 0 0 APPENDIX IX ( c o n t i n u e d ) A. I.* 0.930 0.010 830.0 0.0 0.0 27.00000 0.940 0.010 801 .0 0.0 0.0 38.00000 0.950 0.010 778.0 0.0 0.0 41 .00000 0.960 0.010 771 .0 0.0 0.0 26.00000 0.970 0.010 764.0 0.0 0.0 3.10000 0.980 0.010 769.0 0.0 0.0 1.48000 0.990 0.010 762.0 0.0 0.0 0.12500 1 .000 0.030 743.0 0.0 0.0 0.00250 1 .050 0.050 665.0 0.0 0.0 0.00001 1 .100 0.050 606.0 0.0 0.0 3.20000 1 . 1 50 0.050 551 .0 0.0 0.0 23.00000 1 .200 0.050 497.0 0.0 0.0 0.01600 1 .250 0.050 470.0 0.0 0.00730 0.00018 1 .300 0.050 437.0 0.0 0.00040 2.90000 1 .350 0.050 389.0 0.0 0.00011 200.00000 1 .400 0.050 354.0 0.0 0.00001 1100.00000 1 .450 0.050 319.0 0.0 0.06400 150.00000 1 .500 0.050 297.0 0.0 0.00063 15.00000 1 .550 0.050 274.0 0.0 0.01000 0.00170 1 .600 0.050 247.0 0.0 0.06400 0.00001 1 .650 0.050 234.0 0.0 0.00145 0.01000 1 .700 0.050 215.0 0.0 0.00001 0.51000 1 .750 0.050 187.0 0.0 0.00001 4.00000 1 .800 0.050 170.0 0.0 0.00001 130.00000 1 .850 0.050 149.0 0.0 0.00014 2200.00000 1 .900 0.050 1 36.0 0.0 0.00710 1400.00000 1 .950 0.050 1 26.0 0.0 2.00000 160.00000 2.000 0.075 118.0 0.0 3.00000 2.90000 2. 1 00 0. 1 00 93.0 0.0 0.24000 0.22000 2.200 0. 1 00 74.0 0.0 0.00038 0.33000 2.300 0. 100 63.0 0.0 0.00110 0.59000 2.400 0. 100 56.0 0.0 0.00017 20.30000 2.500 0. 100 48.0 0.0 0.00014 310.00000 2.600 0. 100 42.0 0.0 0.00066 15000.00000 2.700 0. 100 36.0 0.0 100.00000 22000.00000 2.800 0. 100 32.0 0.0 150.00000 8000.00000 2.900 0. 100 28.0 0.0 0.13000 650.00000 3.000 0. 100 24.0 0.0 0.00950 240.00000 3. 100 0. 100 21 .0 0.0 0.00100 230.00000 3.200 0. 100 19.7 0.0 0.80000 100.00000 3.300 0. 100 17.2 0.0 1.90000 120.00000 3.400 0. 100 15.7 0.0 1.30000 19.50000 3.500 0. 100 14.0 0.0 0.07500 3.60000 3.600 0.100 12.7 0.0 0.01000 3. 10000 3.700 0. 100 11.5 0.0 0.00195 2.50000 3.800 0.100 10.5 0.0 0.00400 1.40000 3.900 0. 100 9.5 0.0 0.29000 0.17000 4.000 0. 100 8.5 0.0 0.02500 0.00450 

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