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UBC Theses and Dissertations

A new method for the analysis of ambient air data Zee, Christopher Chee Jang 1981

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A NEW METHOD FOR THE ANALYSIS OF AMBIENT AIR DATA by CHRISTOPHER CHEE JANG ZEE B.A.Sc. U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1977 A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f CHEMICAL ENGINEERING We a c c e p t t h i s t h e s i s as 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 A p r i l , 1981 © CHRISTOPHER CHEE JANG ZEE, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT The e m i s s i o n r a t e s o f p o l l u t a n t s o u r c e s and t h e r e s u l t i n g p o l l u t a n t ground l e v e l c o n c e n t r a t i o n (GLC) a t s u r r o u n d i n g l o c a t i o n s a r e t y p i c a l l y moni-. t o r e d on an h o u r l y b a s i s . These v a s t amount o f d a t a c o l l e c t e d a t s u b s t a n t i a l c o s t a r e n o r m a l l y used o n l y t o check f o r c o m p l i a n c e w i t h l o c a l g o v e r n m e n t a l s t a n d a r d s . The number o f t h e s e measurements p r e c l u d e s t h e i r a n a l y s i s by con-v e n t i o n a l methods. I n t h i s t h e s i s , a new CDFT ( C o n v o l u t i o n and D e c o n v o l u t i o n u s i n g F o u r i e r T r a n s f o r m s ) method i s p r e s e n t e d . T h i s method u t i l i z e s t h e emis-s i o n r a t e d a t a o f a s i n g l e s o u r c e and ground l e v e l c o n c e n t r a t i o n d a t a a t a s i n g l e r e c e p t o r e f f i c i e n t l y i n t h e form o f p r o b a b i l i t y d i s t r i b u t i o n s , t o d e t e r -mine a c h a r a c t e r i s t i c F u n c t i o n E and i t s a s s o c i a t e d p r o b a b i l i t y d i s t r i b u t i o n f ^ . T h i s f u n c t i o n c h a r a c t e r i z e s t h e l o c a l a i r shed between t h e s o u r c e and t h e E r e c e p t o r and can f u r t h e r be used t o p r e d i c t a new GLC p a t t e r n i n r e s p o n s e t o a new e m i s s i o n r a t e p a t t e r n . The method i s f i r s t d e v e l o p e d i n a n a l y t i c a l form i n t h e t h e o r y s e c t i o n . T h i s a n a l y t i c a l t e c h n i q u e i s a p p l i c a b l e t o t h o s e GLC and e m i s s i o n r a t e d a t a w h i c h can be f i t t e d by a n a l y t i c a l ( c o n t i n u o u s ) p r o b a b i l i t y d i s t r i b u t i o n s , s u c h as l o g n o r m a l d i s t r i b u t i o n s . T h i s method i s f u r t h e r a d a p t e d f o r n u m e r i c a l c o m p u t a t i o n , a p p l i c a b l e t o o t h e r GLC and e m i s s i o n r a t e d a t a w h i c h cannot be w e l l r e p r e s e n t e d by a n a l y t i c a l p r o b a b i l i t y d i s t r i b u t i o n s . A computer program t o c a r r y out t h e s e n u m e r i c a l a n a l y s e s i s p r e s e n t e d . I t i s f i r s t s u b j e c t e d t o v a r i o u s t e s t s t o e s t a b l i s h t h a t i t can be a c c u r a t e l y a p p l i e d t o d i s t r i b u t i o n s o f v a r y i n g c o m p l e x i t i e s . Then t h e s e n s i t i v i t y o f t h e program t o t h e e f f e c t o f i n t e r v a l s i z e , r ound o f f e r r o r s and i n s t r u m e n t e r r o r s were a s c e r t a i n e d . P r a c t i c a l a p p l i c a t i o n s o f b o t h t h e a n a l y t i c a l and n u m e r i c a l t e c h n i q u e s a r e d e m o n s t r a t e d i n examples. P o s s i b l e e x t e n s i o n s o f t h e method t o more complex s i t u a t i o n s a r e i a l s o d i s c u s s e d . ACKNOWLEDGEMENTS I s i n c e r e l y t h a n k my s u p e r v i s o r , P r o f e s s o r A x e l M e i s e n , who d e v o t e d many ho u r s t o g u i d e and encourage my e f f o r t s t h r o u g h o u t t h e c o u r s e o f t h i s work. Thanks a r e due a l s o t o t h e f a c u l t y and s t a f f o f t h e Department o f Ch e m i c a l E n g i n e e r i n g , whose e x c e l l e n c e i n t h e i r p r o f e s s i o n s had i n s p i r e d my g r a d u a t e work i n t h i s f i e l d . I would a l s o l i k e t o thank t h e B r i t i s h C o l u m b i a Government, t h e N a t i o n a l S c i e n c e 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 (NSERC) and The U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r t h e i r f i n a n c i a l a s s i s t a n c e . L a s t l y , I would l i k e t o e x p r e s s my t h a n k s t o my p a r e n t s and t e a c h e r s , and my a p p r e c i a t i o n t o my w i f e E l e n a f o r h e r s u p p o r t and t h e c o n f i d e n c e she has shown i n me t h r o u g h o u t t h i s s t u d y . T A B L E O F CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENT i i i LIST OF TABLES v i i LIST OF FIGURES v i i i 1. INTRODUCTION 1 2. LITERATURE REVIEW 5 2.1 Ground L e v e l C o n c e n t r a t i o n (GLC) Models 5 2.1.1 G a u s s i a n Plume D i s p e r s i o n Models 5 2.1.2 GLC Lognormal Model 10 2.1.3 D i s c u s s i o n 11 2.2 C o n v o l u t i o n , D e c o n v o l u t i o n and F o u r i e r T r a n s f o r m s 12 3. THEORY 13 3.1 Ground L e v e l C o n c e n t r a t i o n Models 13 3.2 A n a l y t i c a l T e c h n i q u e s ( f o r c o n t i n u o u s f r e q u e n c y d i s t r i b u t i o n s ) 14 3.2.1 C o n t i n u o u s F r e q u e n c y D i s t r i b u t i o n s 14 3.2.2 C o n v o l u t i o n 15 3.2.3 D e c o n v o l u t i o n lto 3.2.4 F o u r i e r T r a n s f o r m a t i o n s 17 3.2.5 L o g a r i t h m i c T r a n s f o r m a t i o n s 20 3.2.6 Summary o f A n a l y t i c a l T e c h n i q u e s 22 3.3 N u m e r i c a l T e c h n i q u e s ( f o r d i s c r e t e f r e q u e n c y d i s t r i b u t i o n s ) . 22 3.3.1 D i s c r e t e Frequency D i s t r i b u t i o n s 22 3.3.2 C o n v o l u t i o n and D e c o n v o l u t i o n 23 3.3.3 D i s c r e t e F o u r i e r T r a n s f o r m a t i o n s 25 3.3.4 F a s t F o u r i e r T r a n s f o r m s 30 v i v Page 3.3.5 L o g a r i t h m i c T r a n s f o r m a t i o n s 32 3.3.6 Summary o f t h e N u m e r i c a l T e c h n i q u e s 33 3.4 Summary 33 4. COMPUTER PROGRAMS 35 4.1 Program f o r A n a l y s i s o f D i s t r i b u t i o n s (P.A.D.) 35 4.2 Ground L e v e l C o n c e n t r a t i o n Data G e n e r a t i o n Program (D.G.P) 37 4.3 S t a t i s t i c a l T e s t i n g Program (S.T.P.) 39 5. RESULTS AND DISCUSSIONS 41 5.1 V a l i d a t i o n o f t h e CDFT method 41 5.1.1 S e r i e s 1: S i m p l e D i s t r i b u t i o n s 44 5.1.2 S e r i e s 2: E f f e c t o f I r r e g u l a r E m i s s i o n R ate 50 D i s t r i b u t i o n s 50 5.1.3 S e r i e s 3: E f f e c t o f Complex C h a r a c t e r i s t i c F u n c t i o n D i s t r i b u t i o n s ( P a r t 1) 56 5.1.4 S e r i e s 4 E f f e c t o f Complex C h a r a c t e r i s t i c F u n c t i o n D i s t r i b u t i o n ( P a r t 2) 56 5.2. S e n s i t i v i t y o f t h e CDFT Method 69 5.2.1 S e r i e s 5: E f f e c t o f T r u n c a t i o n o f a D i s t r i b u t i o n ( P a r t 1) 71 5.2.2 S e r i e s 6: E f f e c t o f T r u n c a t i o n o f a D i s t r i b u t i o n ( P a r t 2) 71 5.2.3 S e r i e s 7: E f f e c t o f I n t e r v a l S i z e o f D i s t r i b u t i o n s 82 5.2.4 S e r i e s 8: E f f e c t o f Round O f f E r r o r s on P r o b a b i l i t i e s 87 5.2.5 I n s t r u m e n t a l E r r o r s 92 5 . 3 Examples o f t h e CDFT Methods 9 5 5.3.1 P r a c t i c a l Example o f t h e A n a l y t i c a l CDFT Method 95 5.3.2 P r a c t i c a l Example o f t h e N u m e r i c a l CDFT Method 98 5.4 E x t e n s i o n o f t h e CDFT Method 99 5.4.1 The c a s e where C a Q n 100 5.4.2 The c a s e w i t h two s o u r c e s and one r e c e p t o r 100 v Page 6. CONCLUSIONS 103 7. RECOMMENDATIONS 105 NOMENCLATURE 1 0 6 REFERENCES 107 APPENDICES I . Examples o f C o n v o l u t i o n and D e c o n c o n v o l u t i o n 1-1 1.1 A n a l y t i c a l C o n v o l u t i o n o f two S i m p l e D i s t r i b u t i o n s ... 1-1 1.2 A n a l y t i c a l C o n v o l u t i o n u s i n g C o n t i n u o u s F o u r i e r T r a n s f o r m s 1-2 1.3 A n a l y t i c a l D e c o n v o l u t i o n u s i n g C o n t i n u o u s F o u r i e r .... 1-5 1.4 D i s c r e t e C o n v o l u t i o n o f two S i m p l e D i s t r i b u t i o n s 1-6 1.5 D i s c r e t e C o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m s 1-8 1.6 D i s c r e t e D e c o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m s I-11 I I . Computer Programs 11.1 Ground L e v e l C o n c e n t r a t i o n D a t a G e n e r a t i o n Program (D.G.P.) i i - i A. Program L i s t i n g . • . • 11 -1 B. E q u a t i o n s used i n t h e program D.G.P 11-16 11.2 Program f o r A n a l y s i s o f D i s t r i b u t i o n s (P.A.D.) 11-19 A. Program L i s t i n g 11-19 B. O u t l i n e o f Program L o g i c 11-30 11.3 S t a t i s t i c a l T e s t i n g Program (S.T.P.) 11-31 A. Program L i s t i n g 11-31 B. E q u a t i o n s used i n t h e program S.T.P 11-39 11.4 Program t o T r i m r o u n d o f f e r r o r s (TRM) . . . 11-41 I I I . T y p i c a l I n p u t and Output f i l e s DI t o D7, A l t o A7 r e f e r e n c e d i n F i g u r e 5.1 I I I - l v i LIST OF TABLES T a b l e No. T i t l e Page 4.1 Comparison Methods Used i n t h e Program S.T.P. .. 40 5.1 C o n d i t i o n s Used i n S e r i e s #1 t o #4 45 5.2 Comparison S t a t i s t i c s f o r P r o b a b i l i t y D i s t r i b u t i o n s i n S e r i e s #1 53 11.3.1 P r o b a b i l i t y T a b l e f o r CHI-SQUARE DISTRIBUTION 11-37 11.3.2 P r o b a b i l i t y T a b l e f o r K0LM0G0R0V-SMIRN0V One Sample S t a t i s t i c s 11-38 v i i L IST OF FIGURES F i g u r e No. T i t l e P a § e 1.1 C h a r a c t e r i z a t i o n and P r e d i c t i o n Schemes U s i n g t h e CDFT Method 3 2.1 P h y s i c a l C o n d i t i o n s d e s c r i b e d by t h e G a u s s i a n Plume D i s p e r s i o n E q u a t i o n 5 2.2 P h y s i c a l C o n d i t i o n s used i n t h e Model o f P a s q u i l l and Meade 9 3.1 S T r i m p l i f i c a t i o n o f D e c o n v o l u t i o n u s i n g F o u r i e r a n s f o r m s ( F , F~ ) 19 3.2 Schemes f o r CHARACTERIZATION and PREDICTION 21 3.3 Example o f D i s c r e t e F r equency D i s t r i b u t i o n 22 3.4 G r a p h i c a l Development o f D i s c r e t e F o u r i e r T r a n s f o r m 26 3.5 R e p r e s e n t a t i o n o f w i n t h e complex p l a n e 31 3.6 C o r r e s p o n d i n g c u m u l a t i v e f r e q u e n c y d i s t r i b u t i o n s F Q and F ^ L 33 4.1 M o d i f i c a t i o n s t o e l i m i n a t e " a l i a s i n g " e r r o r i n F0UR2 3 8 5.1 V a l i d a t i o n Scheme f o r program P.A.D 42 5.2 D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s , f q L 1 o f E m i s s i o n Rate #1, Q l ( G / s ) 46 5.3 D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s fQL2 f o r E m i s s i o n Rate #2, Q2(G/s) 47 5.4 D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f... o f Wind Speed, U (m/s) 48 U L 5.5 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f _ T , f „ T . o f C h a r a c t e r i s t i c F u n c t i o n , fcL hL E, E'( /m3) i n S e r i e s #1 49 5.6 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 1 » f C L l ' o f G r o u n d L e v e l C o n c e n t r a t i o n #1, C l , C l ' ( G / m 3 ) i n s e r i e s #1 51 v m Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 2 > f c L 2 ! ° f G r o u n d L e v e l C o n c e n t r a t i o n #2, C2,C2'(G/m 3) i n S e r i e s #1 ... D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f q ^ j o f E m i s s i o n Rate #1, Q l ( G / s ) i n s e r i e s #2 , Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f „ T , f„ T t o f C h a r a c t e r i s t i c F u n c t i o n EL EL E, E' (s/m ) i n S e r i e s #2 , Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 1 > ^CLl'°^ G r o u n c l L e v e l C o n c e n t r a -t i o n # 1, C I , Cl'CG/s) i n S e r i e s #2 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s ^-^2' ^CL2' °^ Ground L e v e l C o n c e n t r a t i o n #2, C2, C2' i n s e r i e s #2 P r o b a b i l i t y D i s t r i b u t i o n f g o f A t m o s p h e r i c S t a b i l i t y S i n S e r i e s #3 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s , f„,, f„ T» o f C h a r a c t e r i s t i c f u n c t i o n E,E (s/m 3) i n s e r i e s #3 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n £Q^> ^ C L l ' G r o u n d L e v e l C o n c e n t r a t i o n #1, C I , C l ' i n s e r i e s #3 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s , ^Q^' ^CL2' °^ Ground L e v e l C o n c e n t r a t i o n #2, C2, C2* i n s e r i e s #3 D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n f„ o f Wind p D i r e c t i o n , 8 (degree) i n s e r i e s #4 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n f,,, o f C h a r a c t e r i s t i c F u n c t i o n EL E (s/m 3) g e n e r a t e d by G a u s s i a n Model i n s e r i e s #4 Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n f ^ i ' ^ C L l ' °^ Ground L e v e l C o n c e n t r a t i o n #1, C l , C l (G/m ) i n s e r i e s #4 . . . . D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n f p o f C h a r a c t e r i s t i c f u n c t i o n E (s/m ) d e t e r m i n e d by CDFT method i n s e r i e s #4 i x X Comparison o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s ^^2' ^CL2* Ground L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #7, ( P a r t 1) Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f ^ ^ ' ^CL2' °^ G r o u n c ' L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #7 ( P a r t 2) Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s Q^2' CL2' °^ Ground L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #7, ( P a r t 3) . Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f ^ j ^ ' ^CL2' °^ Ground L e v e l C o n c e n t r a t i o n #1 C2, C2' (G/m 3) i n s e r i e s #8 ( P a r t 1) Comparison o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s F ^ ^ ' ^CL2' Ground L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #8, ( P a r t 1) Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f£L2» ^CL2' °^ Ground L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #8, ( P a r t 2) Comparison o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s F „ „ F ~ T O l o f Ground L e v e l C o n c e n t r a t i o n C2 3 C2* (G/m 3) i n s e r i e s #8 ( P a r t 2) A p p l i c a t i o n o f t h e A n a l y t i c a l CDFT method The c a s e i n v o l v i n g 2 s o u r c e s and one r e c e p t o r x i 1. 1MTR0VUCT10H A i r q u a l i t y r e g u l a t i o n s i n many a r e a s o f t h e w o r l d now s p e c i f y s t a n -d a r d s f o r b o t h t h e ground l e v e l c o n c e n t r a t i o n s CG-LC's] and s t a c k e m i s s i o n r a t e s o f v a r i o u s a i r p o l l u t a n t s . C o m p l i a n c e w i t h t h e s e s t a n d a r d s a r e en s u r e d by p e r i o d i c measurements. The GLC's a r e measured a t m o n i t o r i n g s t a t i o n s o r r e c e p t o r s l o c a t e d around t h e p o l l u t a n t s o u r c e s , and t h e e m i s s i o n r a t e s a r e measured a t t h e e x h a u s t s t a c k s . Readings a r e n o r m a l l y t a k e n e v e r y few m i n u t e s and t h e ave r a g e v a l u e s a r e computed and s t o r e d each hour. There a r e , t h e r e f o r e , 8760 d a t a p o i n t s c o l l e c t e d e v e r y y e a r f o r each measured v a r i a b l e , S u b s t a n t i a l c o s t s a r e i n c u r r e d i n t h e a c q u i s i t i o n , c o m p i l a t i o n and s t o r a g e o f such l a r g e amounts o f d a t a . I n most c a s e s , however, t h e s e d a t a a r e used o n l y t o check f o r c o m p l i a n c e w i t h t h e v a r i o u s g o v e r n m e n t a l s t a n d a r d s , i n d i c a t i n g how o f t e n each y e a r t h e s t a n d a r d s a r e approached o r v i o l a t e d . C l e a r l y , an a b i l i t y t o e x t r a c t more i n f o r m a t i o n f r o m t h e s e h i g h c o s t s d a t a w o u l d be v e r y d e s i r a b l e . We a s k e d t h e f o l l o w i n g two q u e s t i o n s ; 1. Can t h e s e measured d i s t r i b u t i o n s o f GLC's and e m i s s i o n r a t e s be used t o c h a r a c t e r i z e t h e l o c a l a i r shed between t h e s o u r c e and t h e r e c e p t o r ? 2, Can t h e s e measured d i s t r i b u t i o n s be used t o p r e d i c t new GLC's i f t h e r e i s a change i n e m i s s i o n r a t e p a t t e r n ? To d a t e , v a r i o u s t h e o r e t i c a l and e m p i r i c a l s i m u l a t i o n method have been p r o p o s e d t o r e l a t e GLC's t o e m i s s i o n r a t e s . However, s i n c e t h e t r a n s p o r t and d i s p e r s i o n o f a i r p o l l u t a n t s i n t h e atmosphere i s a f f e c t e d by numerous f a c t o r s , t h e s e models a r e u s u a l l y f a i r l y complex. They a r e o f t e n u n s u i t a b l e f o r a n a l y z i n g t h e r o u t i n e l y a c q u i r e d GLC and e m i s s i o n r a t e d a t a f o r two r e a s o n s : 1. t h e s e models t y p i c a l l y r e l a t e a s i n g l e v a l u e o f e m i s s i o n r a t e t o a 1 s i n g l e v a l u e o f GLC. A n a l y s i s o f h o u r l y a v e r a g e s o f one s e t o f e m i s s i o n r a t e and GLC o v e r a one y e a r p e r i o d would r e q u i r e 8760 s e p a r a t e a p p l i c a -t i o n s o f a model; 2. f o r each s e t o f GLC and e m i s s i o n r a t e measurement, t h e s e models r e q u i r e i n f o r m a t i o n on a d d i t i o n a l v a r i a b l e s , such as w i n d speed, w i n d d i r e c t i o n , c l o u d c o v e r , s o l a r r a d i a t i o n and a t m o s p h e r i c v e r t i c a l tempera-t u r e g r a d i e n t . Some o f t h e s e measurements a r e v e r y c o s t l y , such as t h e measurement o f t e m p e r a t u r e g r a d i e n t w h i c h r e q u i r e s t h e use o f e i t h e r a c o u s t i c r a d a r o r i n s t r u m e n t e d a i r c r a f t . I f such measurements must be made h o u r l y , t h e c o s t s o f 8760 measurements p e r v a r i a b l e p e r y e a r may become p r o h i b i t i v e . T h e r e f o r e , t h e use o f t h e s e c o s t l y models a r e o f t e n l i m i t e d t o t h e l a r g e r i n d u s t r i a l i n s t a l l a t i o n s . F o r t h e a n a l y s i s o f r o u t i n e l y a c q u i r e d GLC and e m i s s i o n r a t e d a t a , an a l t e r n a t e a p proach i s c l e a r l y n e c e s s a r y . The Method o& CQUVOLUTIOU and VECONVQLUTION uAtng FOURIER TRANSFORMS icvm I n t h i s t h e s i s , i t i s r e c o g n i z e d t h a t t h e l a r g e amounts o f GLC and e m i s s i o n r a t e d a t a can be c o n v e n i e n t l y h a n d l e d i n t h e fo r m o f f r e q u e n c y d i s t r i b u t i o n s . A new methodology i s p r e s e n t e d , t o q u a n t i t a t i v e l y r e l a t e t h e n o r m a l i z e d f r e q u e n c y d i s t r i b u t i o n s o f t h e e m i s s i o n r a t e s and t h e GLC's. A " C h a r a t e r i s t i c F u n c t i o n , E " , i s p r o p o s e d t o r e l a t e t h e e m i s s i o n r a t e (QJ o f a s o u r c e , t o t h e ground l e v e l c o n c e n t r a t i o n (C) a t a r e c e p t o r c a u s e d by t h a t s o u r c e , such t h a t C = EQ (1-1) The n o r m a l i z e d f r e q u e n c y d i s t r i b u t i o n s , o r p r o b a b i l i t y d i s t r i b u t i o n s f j - , , f g and f o f C, E and Q, r e s p e c t i v e l y a r e t h e n u n i q u e l y a s s o c i a t e d t o each o t h e r . By u s i n g t h e t e c h n i q u e s o f C o n v o l u t i o n , D e c o n v o l u t i o n , F o u r i e r T r a n s f o r m a t i o n and L o g a r i t h m i c T r a n s f o r m a t i o n , t h e n o r m a l i z e d f r e q u e n c y d i s t r i b u t i o n o f any one v a r i a b l e c a n b e u n i q u e l y d e t e r m i n e d f r o m t h e n o r m a l i z e d f r e q u e n c y d i s t r i b u t i o n s o f t h e o t h e r two v a r i a b l e s . S p e c i f i c a l l y , two a n o l o g o u s o p e r a t i o n s c a n be p e r f o r m e d : 1. C h a r a c t e r i z a t i o n : G i v e n s e t s o f C and Q d a t a , t h e n o r m a l i z e d f r e q u e n c y d i s t r i b u t i o n s f g o f t h e C h a r a c t e r i s t i c F u n c t i o n , E , c a n b e d e t e r m i n e d . T h i s scheme i s d e p i c t e d i n F i g . 1-1 ( a ) • 2. P r e d i c t i o n : Once f^ i s k n o w n , t h e e f f e c t o f new e m i s i o n r a t e p a t t e r n s o n t h e G L C ' s f o r t h e g i v e n s o u r c e - r e c e p t o r p a i r c a n be d e t e r -m i n e d ( s e e F i g . 1-1 ( b ) ) . C d a t a Q d a t a New Q d a t a f ' 1 c (a ) f : Q,new (b) CDFT M e t h o d CDFT M e t h o d f ' C ,new F i g u r e 1.1: C h a r a c t e r i z a t i o n and P r e d i c t i o n Schemes u s i n g t h e CDFT M e t h o d . Scope, ol ??izi>Qji£ WoxJk A f t e r a r e v i e w o f t h e r e l e v a n t l i t e r a t u r e , t h e me thod o f CONVOLUTION and DECONVOLUTION u s i n g FOURIER TRANSFORMS (CDFT-1 i s p r e s e n t e d i n two p a r t s . F i r s t , an a n a l y t i c a l t e c h n i q u e i s d e v e l o p e d f o r e m i s s i o n r a t e and GLC d a t a w h i c h c a n b e c o n v e n i e n t l y r e p r e s e n t e d b y c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n s , s u c h as n o r m a l o r l o g - n o r m a l d i s t r i b u t i o n s . E m p h a s i s i n 4 t h i s s e c t i o n i s p l a c e d on t h e t h e o r e t i c a l development o f t h e p r i n c i p l e s o f C o n v o l u t i o n , D e c o n v o l u t i o n , F o u r i e r T r a n s f o r m a t i o n s and L o g a r i t h m i c T r a n s f o r m a t i o n s . I n t h e s e c o n d p a r t , e m i s s i o n r a t e and GLC d a t a w h i c h ca n n o t be e a s i l y f i t t e d t o c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n s a r e con-s i d e r e d . N u m e r i c a l a p p r o x i m a t i o n s o f t h e a n a l y t i c a l t e c h n i q u e s a r e p r e s e n t e d and i n c o r p o r a t e d i n t o a FORTRAN computer program named "Program f o r t h e A n a l y s i s o f J j i s t r i b u t i o n s " o r P.A.D. The c h a r a c t e r i s t i c s , c a p a b i l i t i e s and l i m i t a t i o n s o f t h e program P.A.D. a r e t h e n examined. Two examples a r e t h e n g i v e n t o i l l u s t r a t e t h e a n a l y t i c a l and n u m e r i c a l methods. P o s s i b i l i t i e s o f e x t e n d i n g ? t h e t e c h n i q u e s t o more complex s i t u a t i o n s a r e a l s o c o n s i d e r e d . 2 . LITERATURE REVlEil 2 . 1 Ground LZVQJL Conczntnjjutton [&LCL MadeJU-V a r i o u s GLC mode l s h a v e b e e n d e v e l o p e d i n r e c e n t y e a r s t o s i m u l a t e t h e p r o c e s s o f t r a n s p o r t and d i s p e r s i o n o f a i r p o l l u t a n t s i n t h e a t m o s p h e r e . A l t h o u g h t h e y a r e n o t w e l l s u i t e d f o r a n a l y z i n g t h e r o u t i n e l y a c q u i r e d a m b i e n t a i r q u a l i t y d a t a , t h e y s h a r e a common f e a t u r e w h i c h f o r m s t h e b a s i s o f t h e CDFT m e t h o d . A b r i e f r e v i e w o f t h e s e m o d e l s i s , t h e r e f o r e , w a r r a n t e d . 2 , 1 . 1 GasjjatiAjxK PZimo. V*upvu>;£on. Model* I n t h e s t u d y o f a t m o s p h e r i c d i s p e r s i o n o f a i r p o l l u t a n t s f r o m p o i n t s o u r c e , t h e most w i d e l y e m p l o y e d s i m u l a t i o n m o d e l s a r e t h o s e b a s e d on t h e s e m i - e m p i r i c a l G a u s s i a n P lume D i s p e r s i o n E q u a t i o n , 1 8 C ! { x , y , z , H } = CQ/2TT a y u} exp { - 0 . 5 [ C y / c r y ) 2 + CH - Z ) 2 / a 2 2 J } C2-1D F i g u r e 2 . 1 : P h y s i c a l C o n d i t i o n s D e s c r i b e d b y t h e G a u s s i a n P lume D i s p e r s i o n E q u a t i o n . 5 6 The e q u a t i o n i s a p p l i c a b l e t o t h e s i t u a t i o n i l l u s t r a t e d i n F i g . 2.1. P a s q u i l l 1 6 was amongst t h e f i r s t t o use t h e e q u a t i o n t o c a l c u l a t e t h e c o n c e n t r a t i o n o f a n o n - r e a c t i n g a i r p o l l u t a n t a t l o c a t i o n ( x , y , z ) . The a i r p o l l u t a n t i s e m i t t e d by a p o i n t s o u r c e o f c o n s t a n t e m i s s i o n r a t e (QJ f r o m a s t a c k e x i t l o c a t i o n (O,0,HJ; t h e w i n d i s assumed t o blow a t c o n s t a n t v e l o c i t y (u) i n t o t h e x d i r e c t i o n . The p a r a m e t e r s O y{x} and o" z(x} a r e s o - c a l l e d d i s p e r s i o n c o e f f i c i e n t s i n t h e c r o s s - w i n d (y) d i r e c t i o n and t h e v e r t i c a l ( z ) d i r e c t i o n , r e s p e c t i v e l y . B o t h c o e f f i c i e n t s a r e f u n c t i o n s o f t h e downwind d i s t a n c e x. E x p e r i m e n t a l and t h e o r e t i c a l work have shown t h a t downwind p o l l u t a n t c o n c e n t r a t i o n p r o f i l e s a r e G a u s s i a n w i t h r e s p e c t t o t h e c e n t r e l i n e , a {x} and a {x} may, t h e r e f o r e , be y z r e g a r d e d as s t a n d a r d d e v i a t i o n s o f t h e c o n c e n t r a t i o n p r o f i l e s . E q u a t i o n 2-1 assumes u n r e s t r i c t e d d i s p e r s i o n i n t h e y and z d i r e c t i o n s . F o r a p o l l u t a n t w h i c h does n o t i n t e r a c t w i t h t h e ground, E q u a t i o n 2-1 can be m o d i f i e d by a r e f l e c t e d i m a g e 1 4 t o a c c o u n t f o r ground r e f l e c t i o n : C 2{x,y,z,H} = C!{x,y,z,H} + C j { x , y , z , - H } = CQ/2TTO a u\ exp {-0,5 ( y / a ) 2 } y z y {exp [-0.5 (H - Z l 2 / a 2 ] + exp [-,0.5 (H + Z ) 2 / a 2 ] } (2-2) I n s i t u a t i o n s where a t e m p e r a t u r e i n v e r s i o n e x i s t s , t h e p o l l u t a n t w i l l be t r a p p e d i n a l i m i t e d m i x i n g l a y e r o f h e i g h t L between t h e ground and t h e s t a b l e l a y e r a l o f t . The plume i s t h e n r e f l e c t e d by b o t h t h e ground and t h e l o w e r edge o f t h e s t a b l e l a y e r . E q u a t i o n 2-2 can t h e n be m o d i f i e d f u r t h e r 1 4 t o g i v e C 3{x,y,z,H} = ( Q/2Tra ya zul exp [-0.5 ( y / c y ) 2 ] 0 0 z-h + zth. + 7 x C 2 {exp [-0.5 C 2 L g. / L ) 2 ] + exp [-0.5 ( 2 L , ) ] } ) (2-3) » n=-» z' z ' To d e s c r i b e t h e m i x i n g c h a r a c t e r i s t i c s o f t h e atmosphere, P a s q u i l l p r o p o s e d s i x " s t a b i l i t y " c a t e g o r i e s . The d i s p e r s i o n c o e f f i c i e n t s were fo u n d t o be a f u n c t i o n o f s t a b i l i t y ( s ) and d i s t a n c e (x) downwind from t h e s o u r c e . G i l f o r d 5 p r e s e n t e d p l o t s o f a y { s , x } and a z { s , x } w h i c h were l a t e r m o d i f i e d by T u r n e r . 1 8 These p l o t s can be f i t t e d by e q u a t i o n s o f t h e t y p e a = ax*3 + c (2-4) where a,b,c, a r e c o n s t a n t s f o r each s t a b i l i t y and d i s p e r s i o n c o e f f i c i e n t , w i t h i n a g i v e n r a n g e o f x . 1 3 The s i m p l e G a u s s i a n Plume D i s p e r s i o n E q u a t i o n s s t a t e d above assume t h a t plume o r i g i n a t e s f r o m t h e s t a c k e x i t and p r o p a g a t e s o n l y i n t h e d i r e c t i o n o f t h e w i n d . I n r e a l i t y , t h e s t a c k gas n o r m a l l y l e a v e s t h e s t a c k a t a f i n i t e v e l o c i t y and a t an e l e v a t e d t e m p e r a t u r e . T h i s l e a d s t o a 'plume r i s e ' i n t h e z d i r e c t i o n due t o b o t h momentum and buoyancy e f f e c t s . The plume r i s e can be t a k e n i n t o a c c o u n t by t h e use o f t h e E f f e c t i v e S t a c k H e i g h t (He) i n p l a c e o f t h e p h y s i c a l s t a c k h e i g h t (H)• He = H + AH (2-5) where AH d e n o t e s t h e plume r i s e , . C o n s i d e r a b l e work has been done t o c h a r a c t e r i z e t h e plume r i s e b e c a u s e i t s t r o n g l y a f f e c t s t h e GLC v a l u e s c a l c u l a t e d by G a u s s i a n e q u a t i o n s . The plume r i s e has been f o u n d t o be a c o m p l i c a t e d f u n c t i o n o f t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e s t a c k , t e m p e r a t u r e and v e l o c i t y o f t h e s t a c k g a s , ambient t e m p e r a t u r e and p r e s s u r e and o t h e r m e t e o r o l o g i c a l c o n d i t i o n s . T u r n e r recommended u s i n g t h e d i m e n s i o n a l H o l l a n d ' s e q u a t i o n 1 8 ; 8 AH (V sd/uK1.5 + 0.00268 pd ( T s - T a ) / T j ) (2-6) Where AH - plume r i s e (m) V s -• s t a c k gas e x i t v e l o c i t y (m/s) d - i n s i d e s t a c k d i a m e t e r a t e x i t (m) u -- w i n d speed (m/s) P " ambient p r e s s u r e (mb) T s - s t a c k gas t e m p e r a t u r e (°K) T a - ambient t e m p e r a t u r e (°K) There i s no u n i v e r s a l plume r i s e e q u a t i o n . More t h a n 30 e m p i r i c a l and s e m i - e m p i r i c a l e q u a t i o n s have been p r o p o s e d . 7 A good r e v i e w o f t h e more common ones was p r e s e n t e d by Deniau. 1* Even w i t h t h e m o d i f i c a t i o n s mentioned above, t h e G a u s s i a n models s t i l l do n o t p r e d i c t GLC's p r e c i s e l y . The main d e f e c t s a r i s e from t h e a s s u m p t i o n s o f c o n s t a n t e m i s s i o n r a t e (QJ , s t a b i l i t y (s)> wind speed ( U ) , and wi n d d i r e c t i o n ('S). The models a r e n o r m a l l y a p p l i e d t o ave r a g e v a l u e s o f t h e s e v a r i a b l e s t o p r e d i c t a l o n g t e r m a v e r a g e v a l u e o f GLC, S i n c e t h e dependence o f GLC o i l v a r i a b l e s such as s t a b i l i t y and wind d i r e c t i o n i s h i g h l y n o n - l i n e a r , t h e p r o c e d u r e i s n o t v e r y s a t i s f a c t o r y . Meade and P a s q u i l l 1 5 used s i x t e e n wind d i r e c t i o n s t o r e p r e s e n t 22 1/2° wi n d s e c t o r s and c a l c u l a t e d t h e GLC f o r each wind d i r e c t i o n . The f r e q u e n c y o f o c c u r r e n c e o f each w i n d s e c t o r (jfg) w a s t h e n used t o compute t h e average GLC u s i n g Eq. ( 2 - 6 ) . 16 16 C aver a g e = I C{x,y,Z,H} f 0 { e i } ] / [ E i = l i = l 16 = [ E i = l 16 [ S V B i } ] i = l N F i g u r e 2.2: P h y s i c a l C o n d i t i o n s Used i n t h e Model o f P a s q u i l l and Meade. The v a r i a b l e s R,a,&,B, a r e d e f i n e d i n F i g u r e 2.2, where g = a + 6 + IT x = R c o s 6 = R cos ( B - a - i r l , y = R s i n e = R s i n ( $ - a - T r ) . The model was u s e d t o examine t h e a v e r a g e s u l p h u r d i o x i d e GLC a t S t a y t h o r p e , U K . 1 5 F u r t h e r a l l o w a n c e f o r v a r i a b i l i t y o f w i n d speed and s t a b i l i t y was l a t e r p r o p o s e d by i n t r o d u c i n g t h e j o i n t - f r e q u e n c y d i s t r i b u t i o n f u n c t i o n f „ . A p p l i c a t i o n o f t h e s e models by Zimmerman i n A n k a r a , T u r k e y , 1 9 and s, p, u S t . L o u i s , U.S.A., 2 0 gave f a i r l y good agreement between e s t i m a t e s and measurements o f a v e r a g e SO2 g r o u n d l e v e l c o n c e n t r a t i o n s . A n o t h e r group o f m o d i f i c a t i o n s i n v o l v e s t h e i n c l u s i o n o f t i m e-dependence o f t h e d i s p e r s i n g plume t o p r e d i c t s h o r t t e r m GLC's e v e r y few m i n u t e s . I n 9 (2-6) t h e ' G a u s s i a n P u f f M o d e l ' , 1 8 s e q u e n t i a l p u f f s o f f l u e gas a r e c o n s i d e r e d t o be r e l e a s e d f r o m t h e s t a c k e x i t a t r e g u l a r t i m e i n t e r v a l s t o move downwind under p i e c e - w i s e c o n d i t i o n s o f w i n d s p e e d , w i n d d i r e c t i o n and s t a b i l i t y . The p o l l u t a n t d i s p e r s e s w i t h i n t h e p u f f volume a c c o r d i n g t o t h e t h r e e d i m e n s i o n a l G a u s s i a n d i s t r i b u t i o n . S i n c e t h e model keeps t r a c k o f t h e p o l l u t a n t c o n c e n t r a t i o n p r o f i l e i n each p u f f , i t r e q u i r e s e x t e n s i v e d a t a s t o r a g e . A p p l i c a t i o n o f t h i s model i s , t h e r e f o r e , l i m i t e d t o s i m p l e c a s e s . A m o d i f i c a t i o n o f t h e 'Plume Segment M o d e l ' 1 1 u s e s a compromise between t h e P u f f Model and t h e c o n v e n t i o n a l G a u s s i a n Plume Model t o r e d u c e s t o r a g e r e q u i r e m e n t s . T h i s model computes t h e GLC's by t h e Plume Model b u t t h e plume i s d i v i d e d i n t o segments w h i c h t r a v e l i n t h e p r e v a i l i n g w i n d d i r e c t i o n f o r s e q u e n t i a l t i m e s t e p . T h i s model has been s u c c e s s f u l l y a p p l i e d by t h e Tennesse V a l l e y A u t h o r i t y (TVA) t o p r e d i c t s h o r t t e r m SO2 c o n c e n t r a t i o n s o v e r 15 m i n u t e i n t e r v a l s . However, even s h o r t term models o f t h i s k i n d g e n e r a l l y r e q u i r e e x t e n s i v e c o m p u t i n g f a c i l i t i e s . 2.1.2 GLC LognoAmal Moduli A d i f f e r e n t i n t e r p r e t a t i o n o f GLC d a t a was f i r s t p r o p o s e d by L a r s e n e t a l 1 0 who f o u n d t h a t i n most o f t h e i r s t u d i e s , s h o r t t e r m GLC d a t a a r e w e l l a p p r o x i m a t e d by l o g n o r m a l p r o b a b i l i t y d i s t r i b u t i o n s . I n o t h e r words, a p l o t o f t h e c u m m u l a t i v e f r e q u e n c y d i s t r i b u t i o n o f GLC on l o g -p r o b a b i l i t y s c a l e s g i v e s n e a r l y a s t r a i g h t l i n e . The f r e q u e n c y o f o c c u r r e n c e o f any GLC v a l u e can t h e n be r e a d f r o m t h e p l o t . F u r t h e r s t a t i s t i c a l t e c h n i q u e s can a l s o be a p p l i e d t o p l a c e l e v e l s o f s i g n i f i c a n c e on f r e q u e n c y e s t i m a t e s . An i m p o r t a n t f e a t u r e o f l o g n o r m a l d i s t r i b u t i o n s i s t h a t t h e y p r e d i c t o n l y p o s i t i v e GLC's (as o b s e r v e d i n p r a c t i c e ) , whereas o t h e r d i s t r i b u t i o n s s u c h as n o r m a l d i s t r i b u t i o n s c a n l e a d t o n e g a t i v e GLC's. V a r i o u s s u b s e q u e n t s t u d i e s c o n f i r m e d t h e v a l i d i t y o f t h e 11 lognormal model.9 P o l l a c k 1 7 attempted to rationalize the findings of Larsen et a l by suggesting that GLC's can be regarded as tracers of the wind speeds, which are commonly observed to be lognormally distributed. He also pointed out that the lognormal distribution i s representative of a whole family of similar distributions such as the Weibull, Ganima and Pearson distributions, which can be f i t t e d to GLC data. Further refinements of the model were later presented by others and studies 6 , 7» 1 2 indicated that the lognormal approximation i s sometimes inappropriate. 2.1.3 ViAcuAAsLon The various Gaussian Plume Dispersion models just reviewed have one outstanding common feature, i.e. a linear dependence of GLC on emission rate. The refinements which are made in various models are merely to improve the dependence of GLC on physical and meteorological variables. The linear dependence of GLC on emission rate w i l l be u t i l i z e d later in the theory section. These various modifications of Gaussian models can be grouped into two types. The f i r s t type enables the prediction of the long term aver-age GLC value. Although this single estimate of long term GLC can be useful in certain design purposes, i t does not provide information on how the GLC values may vary with time. Models in the second type predict individual short term GLC values. Since GLC standards are often specified in terms of the permissible number of violations per year, the use of these models to predict violations would require the calculation of thousands of GLC data at substantial costs. Such data must later be grouped into 12 frequency d i s t r i b u t i o n s for compliance checks. The method to be presented i n t h i s thesis u t i l i z e s the frequency d i s t r i b u t i o n d i r e c t l y and, therefore, achieves a high degree of e f f i c i e n c y while minimizing the loss of d e t a i l . 2.2 Convdilution, Vtaonvolation, and Fowtlzn. 7'fian6fionmi. While the d e f i n i t i o n s and s p e c i f i c application of Convolution, Deconvolution, and Fourier Transforms w i l l be presented i n d e t a i l i n the theory section, t h e i r application i n other f i e l d s of study are b r i e f l y reviewed here. The concept of Convolution of two functions has found diverse ap p l i c a t i o n s 1 i n the studies of aperature d i s t r i b u t i o n of antennae,tele-v i s i o n image formation, energy and power spectra, two dimensional heat d i f f u s i o n from point souces, and p r o b a b i l i t y d i s t r i b u t i o n s i n s t a t i s t i c s . Deconvolution of functions has fewer applications, such as r e t r i e v a l of components of a power spectrum, removal of noise from a s i g n a l , and elimination of blur from holographic photographs. Although Fourier Transformation i s but one of many mathematical transformations av a i l a b l e , i t acquires importance f o r i t s applications to a wide variety of quite unrelated topics. I t i s useful for the stu d i e s 3 of l i n e a r systems, antennae, op t i c s , random processes, p r o b a b i l i t y , quantum physics, and boundary value problems. In cases where problems do not y i e l d a closed-form Fourier transform s o l u t i o n , the discrete Fourier Transforms may o f f e r solutions by discrete approximation. In 1965, Cooley and Tukey developed a Fast Fourier Transform (FFT) algorithm for e f f i c i e n t calculations by d i g i t a l computers. The speed of these calculations has since allowed the solution of problems i n many of the aforementioned areas. 3 . THEORY 3 .1 Gfiound Level Conc.e,nX/Uitlon Modtl& A l l t h e G a u s s i a n P lume D i s p e r s i o n E q u a t i o n s p r e s e n t e d i n t h e l i t e r a t u r e r e v i e w a r e o f t h e f o r m : C = EQ ( 3 . 1 ) where t h e g r o u n d l e v e l c o n c e n t r a t i o n C i s r e l a t e d t o t h e e m i s s i o n r a t e Q b y a f u n c t i o n E , w h i c h w i l l be d e f i n e d h e r e as t h e " C h a r a c t e r i s t i c F u n c t i o n " . T h i s f u n c t i o n r e p r e s e n t s t h e p h y s i c a l t o p o g r a p h i c a l and m e t e o r o l o g i c a l c o n d i t i o n s o f t h e s o u r c e - r e c e p t o r p a i r . Mos t r e f i n e m e n t s i n t h e G a u s s i a n P lume D i s p e r s i o n M o d e l s a r e m o d i f i c a t i o n s o f t h e c h a r a c t e r i s t i c f u n c t i o n . F o r any g i v e n s e t o f c o n d i t i o n s , E i s c o n s t a n t and C i s l i n e a r l y d e p e n d e n t on Q. The f u n c t i o n E i n t h e m o d e l s i s u s u a l l y a c o m p l e x e x p r e s s i o n ( i n v o l v -i n g v a r i o u s e s t i m a t e d o r m e a s u r e d p a r a m e t e r s ) w h i c h i s c o s t l y t o e v a l u a t e r e p e a t e d l y f o r p a r t i c u l a r c o n d i t i o n s . I n c a s e o f t h e s i m p l e G a u s s i a n p l ume d i s p e r s i o n e q u a t i o n ( 2 . 1 ) , E i s g i v e n b y : exp { - 0 . 5 [ ( - ^ ) + ] } " 27TU0- O — r - - - - I * . , , ; • v a y z y z S i n c e E q . 3 .1 r e l a t e s i n d i v i d u a l v a l u e s o f C , E a n d Q , i t w i l l be shown s u b s e q u e n t l y t h a t t h e f r e q u e n c y d i s t r i b u t i o n f ' , f ' , f ' o f l a r g e numbers o f L h \l C , E and Q v a l u e s a r e a l s o r e l a t e d . T e c h n i q u e s w i l l be d e v e l o p e d t o a l l o w : 1. C h a r a c t e r i z a t i o n : i . e . t h e d e t e r m i n a t i o n o f f^ g i v e n a s e t o f r e l a t e d C and Q d a t a ; t h e f r e q u e n c y d i s t r i b u t i o n f^ i s a n u m e r i c a l c h a r a c t e r i z a t i o n o f t h e a i r - s h e d b e t w e e n t h e s o u r c e and r e c e p t o r . 2 . P r e d i c t i o n : i . e . t h e e v a l u a t i o n o f t h e new GLC f r e q u e n c y d i s t r i b u t i o n f ' i n r e s p o n s e t o new e m i s s i o n r a t e f r e q u e n c y d i s t r i b u t i o n f ' a f t e r C new r M 3 Q new f g h a s been d e t e r m i n e d f r o m o r i g i n a l C and Q d a t a . 14 I t has a l r e a d y been s t a t e d t h a t some measured d a t a can be c l o s e l y f i t t e d by c e r t a i n c o n t i n u o u s d i s t r i b u t i o n s , most n o t a b l y t h e l o g n o r m a l d i s t r i b u t i o n s ; w h i l e o t h e r d a t a has t o be t r e a t e d as d i s c r e t e f r e q u e n c y d i s t r i b u t i o n s . The methods f o r t r e a t i n g c o n t i n u o u s and d i s c r e t e d a t a a r e analogous and t h e y w i l l be p r e s e n t e d i n sequence. 3 . 2 Analytical Techniques [|{tvi continuous ^fiequency dlitnlbutlons) 3 . 2 . 1 Continuous Vfiequency Vlittlbutions A d i f f e r e n t i a l f r e q u e n c y d i s t r i b u t i o n f u n c t i o n f {X} i s d e f i n e d f o r t h e i n t e r v a l (+<», -°°) s u c h t h a t t h e i n t e g r a l fXzf'{X}dX r e p r e s e n t s t h e f r e q u e n c y 1 t h a t X l i e s i n t h e range X j < X _ < X 2 . The c u m u l a t i v e f r e q u e n c y d i s t r i b u t i o n f u n c t i o n F'(X) i s d e f i n e d by: I n o t h e r words, F'{X} r e p r e s e n t s t h e f r e q u e n c y t h a t X has a v a l u e l e s s t h a n X 3. I t i s u s u a l t o n o r m a l i z e t h e f r e q u e n c y d i s t r i b u t i o n f u n c t i o n s by d i v i d -i n g by t h e t o t a l f r e q u e n c y , N, t h u s g i v e n t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s . The d i f f e r e n t i a l p r o b a b i l i t y d i s t r i b u t i o n f { X } i s d e f i n e d by: F'{X3} = / 3 f'{X}dX ( 3 . 2 ) — CO {x}dx = / 2 f'{X>dX/ / f ' {X}dX X / f'{X>dX/N ( 3 . 3 ) f { x } f'{X>/N ( 3 . 4 ) S i m i l a r l y t h e c u m u l a t i v e p r o b a b i l i t y f u n c t i o n F{X> i s d e f i n e d by: x 3 0 0 F { x J = / f ' { x > dX/ / f ' { x > d x 1° f ( X > d x / N -OO = F'{X)/N (3.5) I t i s o b v i o u s t h a t F{-°°} = 0 and F{°°} = 1. The v a l u e o f p r o b a b i l i t y d i s -t r i b u t i o n f u n c t i o n s a r e t h e r e f o r e c o n f i n e d t o a c o n v e n i e n t range o f 0 t o 1. P r o b a b i l i t y d i s t r i b u t i o n s w i l l be used i n t h e f o l l o w i n g s e c t i o n s : 3.2.2 ConvolivbLon The g e n e r a l e x p r e s s i o n , C = EQ, ( E q u a t i o n 3.1) can be r e - w r i t t e n a s : l o g C = l o g E + l o g Q (3.6) o r CL = EL + QL (3.7) where CL = l o g C, EL = l o g E, QL = l o g Q. The C o n v o l u t i o n Theorem s t a t e s t h a t l i n e a r a d d i t i o n o f two in d e p e n d e n t v a r i a b l e s i s a s s o c i a t e d w i t h t h e ' C o n v o l u t i o n ' o f t h e i r p r o b a b i l i t y d i s t r i b u t i o n s . Hence t h e p r o b a b i l i t y d i s t r i b u t i o n f ^ o f v a r i a b l e CL i s u n i q u e l y g i v e n by t h e c o n v o l u t i o n o f t h e p r o b a b i l i t y d i s -t r i b u t i o n s f „ T , f _ T o f t h e i n d e p e n d e n t v a r i a b l e s ( f a c t o r s ) EL and QL, i . e . : cL QL £ C L = £ E L ® f Q L (3.8) The symbol $ denote s t h e c o n v o l u t i o n o p e r a t i o n . E q u a t i o n 3.8 i s a s h o r t -hand r e p r e s e n t a t i o n o f t h e f o l l o w i n g e q u a t i o n oo f r T { C L . } = / f C T { E L } f n T { C L . - E L ) d E L (3.9) -oo S i n c e each v a l u e CL^ can be o b t a i n e d by summing v a r i o u s c o m b i n a t i o n s o f EL and QL, E q u a t i o n 3.9 s t a t e s t h a t t h e p r o b a b i l i t y o f CL^ I s g i v e n by t h e sum o f o a l l t h e j o i n t - p r o b a b i l i t i e s o f t h e s e EL, QL p a i r s . I t i s u n d e r s t o o d t h a t 16 t h e v a r i a b l e s EL and QL a r e i n d e p e n d e n t o f each o t h e r . V a r i o u s p r o p e r t i e s o f c o n v o l u t i o n and t h e i r p r o o f s can be f o u n d i n s t a t i s t i c a l t e x t b o o k s . 1 * 3 Some r e l e v a n t p r o p e r t i e s a r e : 1. C o n v o l u t i o n i s c o m m u n i c a t i v e : £ E L ® £QL = £QL ® £ E L ^ A ° 3 2. C o n v o l u t i o n i s d i s t r i b u t i v e : f E L ® [ £QL1 + £QL2^ = £ E L ® £QL1 + £ E L ® £QL2 3. Means o f f a c t o r s a r e a d d i t i v e : ^CL = y E L + y Q L C 3 ' 1 2 ) 4. V a r i a n c e s o f f a c t o r s a r e a d d i t i v e : *CL " °k + G Q L C3 .13) E v a l u a t i o n o f t h e c o n v o l u t i o n i n t e g r a l i s o f t e n d i f f i c u l t . An example o f c o n v o l u t i o n o f two s i m p l e f u n c t i o n s i s g i v e n i n A p p e n d i x I (Example 1 - 1 ) . 3.2.3 VQ.convolvJU.on When E q u a t i o n 3.8 a p p l i e s , i . e . £ C L = £ E L ® £QL C 3 . 8 D . and f„ T and f n i a r e known, f C I can be d e t e r m i n e d by ' D e c o n v o l u t i o n " . LiL I^ Li EL D e c o n v o l u t i o n a l w a y s i m p l i e s t h a t one o f t h e g i v e n p r o b a b i l i t y d i s t r i b u t i o n s i s dependent on t h e o t h e r . I n t h e c a s e o f plume d i s p e r s i o n , f ^ i s dependent on f n T . There i s no u n i v e r s a l l y a c c e p t e d symbol f o r D e c o n v o l u t i o n and t h e QL symbol © w i l l be u s e d h e r e : f E L " £ C L © £QL < 3- 1 4) o r , more e x p l i c i t l y , E q u a t i o n (3.9) can be r e w r i t t e n a s : £ E L ( E L } = d l L { f C L ( C L i ) } / £ Q L { C L i - E L } ( 3 ' 1 5 > 17 S i n c e t h e v a r i a b l e CL i s dependent on b o t h QL and EL, d i r e c t a n a l y t i c a l s o l u t i o n o f E q u a t i o n 3.15 f o r f . i s v e r y d i f f i c u l t and i n most c a s e s hL i m p o s s i b l e t o e v a l u a t e . T h i s p a r t l y a c c o u n t s f o r t h e l i m i t e d a p p l i c a t i o n o f D e c o n v o l u t i o n t e c h n i q u e s . However, t h e t e c h n i q u e o f F o u r i e r T r a n s f o r m a -t i o n o f f e r s an e a s i e r s o l u t i o n o f E q u a t i o n 3.15. 3.2.4 TovJvitn. Ttiani,lofimoutiom [F, F"1 ] The F o u r i e r T r a n s f o r m a t i o n s a r e amongst t h e many m a t h e m a t i c a l t r a n s f o r m a -t i o n t e c h n i q u e s a v a i l a b l e . They t r a n s f o r m f u n c t i o n s i n t h e r e a l ( x j , x 2 , ..) domain i n t o f u n c t i o n s i n t h e complex ( S j , s 2 , ..) domain and v i c e v e r s a . I t i s u s e f u l because many c o m p l i c a t e d p roblems i n t h e r e a l x-domain can be s o l v e d more s i m p l y i n t h e complex s-domain. I n t h e p r e s e n t a p p l i c a t i o n , t he one d i m e n s i o n a l F o u r i e r T r a n s f o r m s i n v o l v i n g x, s w i l l be employed. The F o u r i e r T r a n s f o r m F{s} o f a r e a l f u n c t i o n i s d e f i n e d a s : 0 0 F ( s ) = F{f(x)} = f { x ) e " i 2 7 T x s d x - 0 0 where F denotes t h e F o u r i e r T r a n s f o r m o p e r a t o r . The I n v e r s e F o u r i e r T r a n s f o r m f { x } o f a complex f u n c t i o n F{s} i s d e f i n e d as : f{x> = F _ 1 { F ( s ) } = / F { s } e i 2 7 T X S d s (3.17) - 0 0 where F * i s t h e I n v e r s e F o u r i e r T r a n s f o r m o p e r a t o r . f ( x ) , F{s) a r e known as t h e C o n t i n u o u s F o u r i e r T r a n s f o r m P a i r , and t h e symbol i s o f t e n used t o d enote t h e r e l a t i o n s h i p : f ( x ) F(s) The p r o c e s s o f F o u r i e r T r a n s f o r m a t i o n i s r e v e r s i b l e , t h a t i s : F-^FCfCx})} = F- 1(F(s)} = f ( x } (3.18) 18 F C F - ^ F t s . } } = F(£{x}} = F { s ) (3.19) F o u r i e r T r a n s f o r m s o f many common f u n c t i o n s and d i s t r i b u t i o n s a r e t a b u l a t e d and a c t u a l e v a l u a t i o n o f t h e i n t e g r a l s i n E q u a t i o n s 3.16 and 3.17 a r e o f t e n u n n e c e s s a r y . V a r i o u s p r o p e r t i e s and theorems r e l a t e d t o F o u r i e r T r a n s f o r m s can be f o u n d i n t e x t b o o k s . 1 ' 3 The S h i f t Theorem and t h e C o n v o l u t i o n Theorem a re most u s e f u l f o r t h i s t h e s i s . I. Shlfit Thtohem When f { x ) has t h e F o u r i e r T r a n s f o r m F { s } , t h e n f { x - a } has t h e F o u r i e r m r - i 2 i T a s _ r -1 T r a n s f o r m e F i s ) : P r o o f : - i a i T x s . f°° -f - i 2 i r ( x - a ) s - i 2 u a s ,, . j f i x i e dx =1 f l x - a j e ^ ' e d ( x - a ) -co / -co = e - i 2 7 r a s F { s } ( 3 > 2 0 ) I I . Convolution Tho.oh.zm When f ( x } has t h e F o u r i e r T r a n s f o r m F ( s } and g{x} has t h e F o u r i e r T r a n s f o r m G { s ) , t h e n f { x } % g{x} has t h e F o u r i e r T r a n s f o r m F ( s } G { S } ; i . e . , t h e c o n v o l u t i o n o f two f u n c t i o n s i s g i v e n by t h e m u l t i p l i c a t i o n o f t h e i r F o u r i e r T r a n s f o r m s . P r o o f : L e t h { x } = f { x } ® g ( x ) = / f{x'}g{x-x'}dx' <<x> Then: H{s} = F ( h { x l ) 00 f 00 = f [ f{x'}g{x-x'}dx] e - l 2 7 T x s d x /-co /-co = / £{x'} [ / g ( x - x } e - l 2 7 r X S d x ] d x /-co CO = r f(x'} [G(s> e - i 2 l T X S ] d x ' s-co = F { s } G ( s } (3.21) 19 Therefore, the use of Fourier Transforms reduces the complex process of convolution i n the x-domain to a simple m u l t i p l i c a t i o n i n the s-domain The s i m p l i f i c a t i o n i s i l l u s t r a t e d by Example 1-2, i n Appendix I. The major importance of the Fourier Transform l i e s with the recognition that since: H{s} = F{s}G{s} we can solve f o r F{s} by: F{s} = H{s}/G{s} (3.22) In other words, Deconvolution, an operation d i f f i c u l t to perform i n the x-domain, i s simply a d i v i s i o n i n the s-domain. Figure 3.1 depicts the Deconvolution of htx}, g W v i a the s-domain. An example of using the continuous Fourier Transforms to s i m p l i f y Deconvolution i s given i n Appendix I (Example 1-3). h ( x ) , g ( x } f ( x ) _ = h ( x } _ © _ g { x } d i f f i c u l t F s imp1e H(s}, G(s] F{s) = H{s)/G{ S} f ( x } F-' s imp1e F{s) Figure 3.1: S i m p l i f i c a t i o n of Deconvolution Using Fourier Transforms (F, F"1) .. -1 2 0 3.2.5 LoqaAlthmLC TKanifioAinatLorUlL, L ) When t h e p r o b a b i l i t y d i s t r i b u t i o n s f ^ o f v a r i a b l e X i s g i v e n ^ t h e p r o b a b i l i t y d i s t r i b u t i o n f ^ °f XL = l o g X can be e a s i l y o b t a i n e d by t h e f o l l o w i n g r e l a t i o n s h i p : f Y T { X L . } d X L = Prob [XL = XL.] A Li 1 X = Prob [XL = L o g { X i ) ] = Prob [X = X i ] •• f X L { X L i } = SL f X ( X i } S i n c e XL - l o g X , g ^ = i o r . g_ B x :. f v r { X L . } = X . f {X.} (3.23) XL l l x l C o n v e r s e l y : f x { X . } = ± - f Y T { X L . ) = e " X L i f Y T { X L . } (3.24) A l X- XL l XL l l The above t r a n s f o r m a t i o n s can be c o n c i s e l y r e p r e s e n t e d by L, t h e l o g a r i t h m i c T r a n s f e r o p e r a t o r and L \ t h e I n v e r s e L o g a r i t h m i c T r a n s f o r m o p e r a t o r : f x l - l t t x ) (3.25) The l o g a r i t h m i c t r a n s f o r m s a r e u s e f u l i n t r a n s f o r m i n g f ^ , f £ , f ^ i n t o f ^ * f g ^ , f q ^ a n<* v i c e v e r s a i n t h e a n a l y s i s o f C and Q d a t a . I . CHARACTERIZATION (C = EQ, CL = EL + QL, f E L = f C L © f Q L ] G i v e n t h e ground l e v e l c o n c e n t r a t i o n (C d a t a ) and The E m i s s i o n Rate F i t t i " g , f c F i t t i n g r L s_» t n  QL - HfQ L} F< fEL> " f E L I I . PREDICTION [ C = EQ', CL' = EL + QL', f C L < = f E L ® fQL'] EL ( f r o m above) A l t e r n a t e F i t t i n L F E m i s s i o n Rate s-» f 0, > f n I , » ^tfoi i ) Q' d a t a n IJL CQL' ® F" 1 L" 1 "* ^ f C L ' ^ ? f ™ ' * f ^ ' CL 1 F i g u r e 3-r2: Schemes f o r C h a r a c t e r i z a t i o n and P r e d i c t i o n 22 3.2.6 SummaAy oj thz Analytical Techniques The t e c h n i q u e s d e v e l o p e d so f a r e n a b l e t h e a n a l y s i s o f t h e f r e q u e n c y d i s t r i b u t i o n s o f ground l e v e l c o n c e n t r a t i o n s (C) and e m i s s i o n r a t e s (QJ . Two o p e r a t i o n s can be p e r f o r m e d , namely: 1. ChaAacteAlzatlon: i . e . t h e d e t e r m i n a t i o n o f t h e C h a r a c t e r i s t i c f u n c t i o n (E) l i n k i n g t h e s o u r c e - r e c e p t o r p a i r , i n t h e f o r m o f a p r o b a b i l i t y d i s t r i b u t i o n f ^ ; and 2. Vfizdlctlon: i . e . t h e e s t i m a t i o n o f t h e new g r o u n d l e v e l c o n c e n t r a -t i o n d i s t r i b u t i o n when t h e e m i s s i o n p a t t e r n i s a l t e r e d . F i g u r e 3.2 i s a summary o f t h e s e two o p e r a t i o n s . T h i s method o f a n a l y s i s i s h e r e b y c a l l e d t h e CDFT ( C o n v o l u t i o n and D e c o n v o l u t i o n u s i n g F o u r i e r T r a n s f o r m ) method. 3.3 Numznlcal TzchnlqueA {^on. VlscK.ztz T-nzquzncy VlstAlbutlons) 3.3.1 VlicJiztz T-nzquzncy VlitAlbutiom, A d i s c r e t e f r e q u e n c y d i s t r i b u t i o n f u n c t i o n £' i s d e f i n e d f o r x . <x<x s u c h t h a t f ' ( x . ) A x r e p r e s e n t s t h e f r e q u e n c y t h a t x l i e s i n t h e mm max x I R N J i n t e r v a l (x^-y^, X i + T ^ ' T h e i n t e r v a l s i z e i s d e n o t e d by Ax = 3 ^ - x ^ . I n o t h e r w ords, t h e x - a x i s between x . and x can be d i v i d e d i n t o N ' mm max i n t e r v a l s o f s i z e Ax^ and m i d p o i n t s x ^ ' s , as shown i n F i g u r e 3.3. ( X i ) ! i 1 1 i 1 1 l 1 • » 1 i ! — i — i i i l i I I X l Ax X i N x m i n ^ a x F i g u r e 3.3: Example o f A D i s c r e t e F r e q u e n c y D i s t r i b u t i o n 23 The f r e q u e n c y t h a t x has a v a l u e i n t h e i t h i n t e r v a l i s r e p r e s e n t e d by t h e a r e a o f t h e i t h r e c t a n g l e . I f t h e i n t e r v a l s Ax. a r e t h e same, an a l t e r n a t e 1 d i s c r e t e f r e q u e n c y d i s t r i b u t i o n f u n c t i o n f * can be d e f i n e d s u c h t h a t x f * ( x . ) = f'(x.)Ax.: The use o f f * a l l o w s more c o n c i s e r e p r e s e n t a t i o n o f t h e X X X 1 X f r e q u e n c i e s . A d i s c r e t e c u m u l a t i v e f r e q u e n c y d i s t r i b u t i o n f u n c t i o n F* can s i m i l a r l y be d e f i n e d by: F*{x.} = Z f * { x . } x i j = 1 x 3 As i n t h e c a s e o f c o n t i n u o u s f r e q u e n c y d i s t r i b u t i o n s , t h e d i s c r e t e f r e q u e n c y d i s t r i b u t i o n s a r e commonly n o r m a l i z e d by d i v i d i n g by t h e t o t a l f r e q u e n c y N. The d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n f = f*/N and X X d i s c r e t e c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n F = F*/N a r e t h u s X X o b t a i n e d . I t i s a g a i n o b v i o u s t h a t F ^ f" 0 0} = 0 and F ^ 0 0 } = 1. 3.3.2 Convolution and Vzconvolotion F o r d i s c r e t e f u n c t i o n s , t h e e x p l i c i t e x p r e s s i o n f o r t h e c o n v o l u t i o n h { x } = f ( x } @ g{x} i s g i v e n by N h { x . ) = Z f ( x . } g{x.-x.} (3.27) 1 j - 1 J 1 J D i s c r e t e c o n v o l u t i o n i s e a s i e r t o p e r f o r m t h a n c o n t i n u o u s c o n v o l u t i o n . B a s i c a l l y , i t r e q u i r e s o n l y t h e g r o u p i n g and summing o f t h e a p p r o p r i a t e j o i n t -p r o b a b i l i t i e s o f p a i r s o f x ' s . T h i s method, c a l l e d by B r a c e w e l l 1 as t h e method o f " S e r i a l P r o d u c t , ; , i n e f f e c t c o n s i d e r s t h e elements i n f { x ) , g ( x ) and h{x} t o be c o e f f i c i e n t s o f t h e p o l y n o m i a l s P j , P 2 and P 3 r e s p e c t i v e l y : P1 = f Q + f { y + f 2 y 2 + f 3 y 3 + f^ 1 P 2 = g 0 + g i y + g 2 y 2 + g 3 y 3 + s^y3 P = h + h.y + h 2 y 2 + h . y 3 + h. . y i + J 3 0 1+J ' where f Q = f{X Q} f ; = f{Xj} and so on. L e t t i n g P3 = P: x P 2, we have h = f g o o &o h 2 - f Q g 2 + f l S l + f 2 g Q h. . = f g. . + f,g. . , + f. . ,g. + f. .g^ Given the elements f\ and g^ from f{x} and g{x) respectively, the elements i n h{x) can thus be determined. An example of discrete convolution by this method of Serial Product i s given in Appendix I, example 1-3. The above relationship among coefficients of P :, P 2 and P3 can also be expressed in matrix form: h 0 r f 0 0 0 0 0 0 0 • 1 go *1 f 0 0 0 0 0 0 S i f 2 £ > f o 0 0 0 0 g 2 f. 1 f. in f i f 0 f. l 0 f 0 0 0 -- 0 f. I f. in - f. l f 0 0 -L. . 0 f. l f i - i - f °i or [H] = [F] [G] Convolution using matrices can be conveniently adapted to machine computation. However, i t requires large storage space of which a high fraction i s dedicated to the repeated storage of various f^'s . The (3.28) 25 c o m p u t i n g c o s t s t h e r e f o r e i n c r e a s e s h a r p l y w i t h t h e s i z e o f t h e m a t r i c e s A much more e f f i c i e n t method, ba s e d on D i s c r e t e F o u r i e r T r a n s f o r m , w i l l be c o n s i d e r e d s h o r t l y . From E q u a t i o n 3.28, i t i s c l e a r t h a t d i s c r e t e D e c o n v o l u t i o n c a n be p e r f o r m e d by u s i n g m a t r i x i n v e r s i o n : [F] = [H] [ G ] " 1 (3.29) Once a g a i n , t h i s m a t r i x o p e r a t i o n i s r a t h e r i n e f f i c i e n t compared t o u s i n g D i s c r e t e F o u r i e r T r a n s f o r m s . 3.3.3 ViAcAttz FouAleA Tfiani{>oAmotion* A n a l o g o u s t o t h e c o n t i n u o u s t r a n s f o r m , t h e D i s c r e t e F o u r i e r T r a n s f o r m (DFT) o f a sequence o f N numbers fy j = 0, 1, 2... N - l i s d e f i n e d by: N - l F.k = F { f } = Z f . w _ : , K k = 0, 1, 2, ... N - l (3.30) j=0 J where: w = e x p ( ^ ~ } and i = 7^1. S i m i l a r l y t h e I n v e r s e D i s c r e t e F o u r i e r T r a n s f o r m (IDFT) o f t h e complex sequence o f N numbers, Fk, k = 0, 1, 2, ... N - l i s d e f i n e d by f . = F-^F} - i V F k w + J k j = 0, 1, 2, N - l (3.31) J k=o The D i s c r e t e F o u r i e r T r a n s f o r m p a i r f_., F k a r e e s s e n t i a l l y m o d i f i c a t i o n s o f t h e C o n t i n u o u s F o u r i e r T r a n s f o r m p a i r f { . x ) , F ( s } . These m o d i f i c a t i o n s , i l l u s t r a t e d i n F i g . 3.4, a r e n e c e s s a r y t o p e r m i t c a l c u l a t i o n by d i g i t a l computer. P a r t (a) o f F i g u r e 3.4 shows t h e C o n t i n u o u s F o u r i e r T r a n s f o r m p a i r f { x } and F { s } . I n o r d e r t o a l l o w d i g i t a l c o m p u t a t i o n , t h e f u n c t i o n f { x } i s sampled by m u l t i p l y i n g i t by t h e s a m p l i n g f u n c t i o n A Q { x } shown i n p a r t ( b ) . F i g . 3.4 G r a p h i c a l Development o f t h e D i s c r e t e F o u r i e r T r a n s f o r m 27 The sampling interval i s T. The sampled function f{x) A Q{x} is shown in part (c). The transform of f{x} A Q{x} i s F{s} (g S"0^s^- When f ^ extends from +°° to an i n f i n i t e number of samples i s necessary which i s not suitable for machine use. The function f{x) must be truncated by multiplying i t by the truncation function y { x ) show in part (d) giving f { x } A Q { X ) y { x ) with associated transform F(s} @ ^{s} ® Y{s}. Lastly, the transform must also be sampled. The f i n a l Fourier Transform pair f., Fk is then an accept-able discrete approximation of the continuous transform pair f(x}, F(s). Various operational characteristics are notable in the discrete approximation. 1. PeAlodlclXy: Sampling of f{x} i n part (c) i s associated by the convolution of F{s} with a periodic function A Q { S } . This results in the period function F{s) t$). A Q { S } . . Similarly, in part (g) sampling in the s domain is associated by periodicity in the x-domain. 2. Aliasing: The sampling interval T of A Q { X } i n part (b) determines the interval size 1/T of the periodic function A Q { S } . Part C shows the possible "overlapping" or "aliasing" error that could occur when the sampling interval i s too large. This error can be reduced by using smaller sampling intervals. 3 . Rippling: The transform Y{s} in part (d) of the truncation function has "ripples" in i t . This introduces "Rippling" errors in parts(e) and (g). These errors can be decreased by increased truncation interval, T q in part (d). Mathematically speaking the periodic functions f y Fk obtained in part (e) do not exactly represent the functions f{x}, F(s); but in most applications, this property of periodicity does not affect the results. 28 S i m i l a r t o t h e C o n t i n u o u s F o u r i e r T r a n s f o r m t h e D i s c r e t e F o u r i e r T r a n s f o r m s can be a p p l i e d t o e v a l u a t e c o n v o l u t i o n and d e c o n v o l u t i o n o f d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n , u s i n g t h e f o l l o w i n g p r o p e r t i e s : (1) F o r D i s c r e t e C o n v o l u t i o n : G i v e n : h = f @ g t h e n : H k = F k G k (3.32) (2) F o r D i s c r e t e D e c o n v o l u t i o n : G i v e n : h = f © g H k = F k / G k (3.33) The f o l l o w i n g p r o c e d u r e i s u s e d t o e v a l u a t e t h e C o n v o l u t i o n h{x} = f { x } 0 g { x ) , when f { x } and g{x) c o n s i s t s o f N~ and N e n t r i e s , r e s p e c t i v e l y : (1) E x t e n d f { x ) and g { x ) t o N e n t r i e s each by a d d i n g t r a i l i n g z e r o s , when N = N f + N - 1 , ( e . g . i f g i v e n f { 2 0 } = 0.4, f { 3 0 } = 0.6, g { l 0 } = 0.3, g{20} = 0.7; t h e n N f = 2, N g = 2, N = 3. E x t e n s i o n o f f { X ) and g{X} w i l l g i v e f { 2 0 } = 0.4, f { 3 0 } = 0.6, f { 4 0 } = 0 and g { l 0 } = 0, g{20) = 0.7, g{30} = 0.0. (2) P l a c e t h e X v a l u e s o f f{X} i n t o t h e a r r a y X f ^ j j j = 1 , 2, ... N and f { x } v a l u e s i n t o t h e a r r a y f . , j = 1, 2, . . . N ( e . g . f ~- (10, 20, 3 0 ) , f = (0.4, 0.6, 0 1 ) . S i m i l a r l y p l a c e e n t r i e s o f g{x} i n t o a r r a y s x -and g., j = 1, 2, ... N. & >J j (3) E v a l u a t e t h e c o n v o l u t i o n h = f ® g u s i n g E q u a t i o n s 3.30, 3.31 and 3.32. (a) D i s c r e t e F o u r i e r T r a n s f o r m _ N ~ 1 _ - j k ( E q u a t i o n 3.3.0) k ~ ^ 0 j W k • 0, 1, 2, N (3.34) 29 ;k - z g . w j=0 J k = 0, 1, 2, N (3.35) (b) C o n v o l u t i o n i n t h e s domain ( E q u a t i o n 3.32) f o r : g H = FG i . e . H k = F k G k 0, 1, 2, N (3.36) (c) I n v e r s e D i s c r e t e F o u r i e r T r a n s f o r m ( E q u a t i o n 3.31) N-1 h. = Z H k w + J k  3 k=0 j = 0, 1, 2, N (3.37) (4) D e t e r m i n a t i o n o f t h e X.^  ^ v a l u e s f o r h{X} u s i n g t h e p r o p e r t i e s : a) X^ 1 = X_ 1 + X , (3.38) n , i t j J - gj b) AXy, = AX. = AX h f g = X g J " *g, j - 1 (3.39) C> X h , j = X h , l + J A X h ^ - 4 0 ^ I n t h e p r o c e d u r e f o r d i s c r e t e c o n v o l u t i o n t h e most n o t a b l e f e a t u r e i s t h a t t h e p r o b a b i l i t i e s h^'s and t h e v a r i a b l e X^ j i s o f h{X} a r e d e t e r m i n e d by two s e p a r a t e o p e r a t i o n s . T h i s i s d i f f e r e n t f r o m c o n t i n u o u s c o n v o l u t i o n , where h{X} i s d e t e r m i n e d by a s i n g l e o p e r a t i o n o f c o n v o l v i n g f { X } and g{X}. T h i s d i f f e r e n c e a r i s e s f r o m t h e f a c t t h a t f o r c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n s u c h as f { X } = k t e ^ 2 ^ , s p e c i f i c a t i o n o f X v a l u e s s i m u l t a n e o u s l y s p e c i f i e s t h e v a l u e o f f { X ) v i a t h e f u n c t i o n a l r e l a t i o n s h i p , w h e r e a s f o r d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s s u c h as [ f { l } = 0.1, f { 2 } = 0.7, f { 3 ) = 0.2, f { 4 } = 0.1], t h e r e i s seldom any d i s t i n c t r e l a t i o n s h i p between t h e v a l u e s o f v a r i a b l e [Xj = 1, X 2 = 2, X;3 = 3, X^ = 4] and t h e p r o b a b i l i t i e s [ f j = 0.1, f 2 = 0.7, f 3 = 0.2, f = 0.1]. D i s c r e t e c o n v o l u t i o n i n v o l v i n g 30 such d i s t r i b u t i o n s t h e r e f o r e r e q u i r e s t h e two p a r a l l e l o p e r a t i o n s , one f o r t h e p r o b a b i l i t i e s , f . ' s , and one f o r t h e v a r i a b l e s , X_.'s. P r o c e d u r e s f o r t h e D i s c r e t e D e c o n v o l u t i o n o f h{X} = f { X } @ g{x} i s a n a l o g o u s t o t h o s e f o r c o n v o l u t i o n e x c e p t f o r two a s p e c t s : 3) b) h = f © g H = F/G i . e . H k = F k / G k k = 0,1,2,....N' ( 3 . 4 1 ) 4) c) x h > 1 = X £ f l - X g j l (3.42) Examples o f d i s c r e t e c o n v o l u t i o n and d i s c r e t e d e c o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m a r e g i v e n i n Appendix I (Ex. 1.5 and Ex. 1.6). 3.3.4 Fa6t TouAJZA TA.an!>fiOAmi I n t h e a f o r e m e n t i o n e d p r o c e d u r e s f o r c o n v o l u t i o n , t h e m u l t i p l i c a t i o n N - l o f sequences F k = I f . w~^ and G k = E g. w~J^ a r e i n f a c t m u l t i p l i c a t i o n j=0 3 j=0 J o f p o l y n o m i a l s o f w. T h i s o p e r a t i o n c l o s e l y r e s e m b l e s t h e ' S e r i a l P r o d u c t ' method. The h i g h e r e f f i c i e n c y o f a p p l y i n g F o u r i e r T r a n s f o r m s i s a c h i e v e d by t h e s i m p l i f i c a t i o n o f t h e n u m e r i c a l r e p r e s e n t a t i o n o f powers o f w. The N - l -k t e r m w i s c o n v e n i e n t l y r e p r e s e n t e d i n p o l a r c o - o r d i n a t e s (R , 8 ) -k w t 2 T T k i - i _ r , ^-IIK-i e x p i - N — i = 11,- - j j p J where R = 1 and 6 = - 2 i T k / N 2 T T k - , (3.43) F o r example, when N = 6: -1 w = ( 1 , - - ) - 2 w = ( 1 , . 2 1 ) 3 J -3 w = ( 1 , - I T ) 31 w"6 = CI, -2TT) -7 7TT. -1 w = ( 1 , - 3 - ) = w In general, w ^N+J'^ = W ~ J . These values of w"^  describes a c i r c l e in the complex plane in the counter clockwise direction, as shown i n Figure 3.5. . Im Figure 3-5: Representation of w~ in the Complex Plane Numerical determination of w i s therefore greatly simplified by the geometric representation. The simplification forms the basis of the Fast Fourier Transform (FFT) algorithms for d i g i t a l computer calculation f i r s t developed by Cooley and Turkey2. The FFT algorithms reduces the computer time required by a factor of (log 2N)/N, a substantial savings for large values of N. Convolution and Deconvolution using Discrete Fourier Transform with FFT algorithms are therefore much more ef f i c i e n t than similar operations carried out using matrices. Various modifications of the FFT algorithms were later proposed by others 2. Some "Base 2" algorithms developed are 50% faster for sequences with number of entries N equal to 32 integer powers of 2. In the UBC computing f a c i l i t i e s Convolution and Deconvolution of sequences of 256 entries using a Base 2 FFT routine "F0UR2" i s approximately 100 times faster than using matrix solution by Gaussian Elimination (FSLE routine). These high speed FFT's form the basis of the CDFT method. 3.3.5 Logarithmic TKJXJU jowatloul, {L, L The definitions of f ^ , f £ L and f ^ require the frequencies to be specified for equal intervals of CL, EL and QL. The raw data of C and Q i s best handled by f i r s t taking logarithms to give log C and log Q and then grouping the data to give discrete frequency distributions f' , f' . Normalization then gives the discrete probability distributions fcL> fqi . In some situations, the input data available are already grouped into discrete frequency distributions f' and f' specified at equal intervals of L Q C and Q. In these cases, Logarithmic Transform ( L ) and Inverse Logarithmic Transform are necessary. If a discrete cumulative frequency distribution F Q i s given then: F^{QL.} = L ( F Q { Q . } ) = FQ{e Q L i} (3.44) since QL = log{Q}. Figure 3.6 provides an i l l u s t r a t i v e example: In Figure 3.6(a) are specified only at Q = 1, 2, 3, 4. Since QL = log Q, F* in Figure 3.6(b) i s only specified in QL = log 1, log 2, Q L log 3, log 4 which corresponds to unequal intervals. For our application F' T must be specified at equal intervals such as QL = 0.3, 0.6, 0.9, 1.2. QL Since, i n r e a l i t y , F^ i s continuous, F^ can be approximated by the dashed lines for small intervals. These dashed lines for F Q correspond to the dashed curves indicated for F m . For small intervals these curves can also Q L be approximated by straight lines. Therefore, by interpolating linearly 33 3 24 (a) Q 'QL 5 4 1  I 0.0 0.3 0.S 0.9 LOG i t 1 - h — L -•2 t 15 Q|_ LOG 2 LOG 3 LOG -4 (b) Figure 3-6: Corresponding Cumulative Frequency Distributions F' and F' between F^L given at QL = log 1, log 2, log 3, log 4. F^L at QL • 0.3, 0.6, 0.9, 1.2 can be obtained. The accuracy of the method increases with decreas-ing interval size. After F^L i s determined, normalization gives the discrete probability distribution f n i i s obtained by: QL VQV • v Q L i } • v ^ i - i * i 5 A 5 ) The Inverse Logarithmic Transform can be similarly performed by the interpolation method. 3.3.6 Summcuiy oj the. NumeAsLcal Technique* The process of Convolution and Deconvolution using numerical techniques follows the same principles as those outlined in the flow chart (Figure 3.2) for the analytical technique (The input data obviously need not be f i t t e d to analytical probability distribution i n this case). However, the numerical techniques requires e f f i c i e n t execution. This i s accomplished by the use of FFT routines on a d i g i t a l computer. 3.4 Summtvuj [TheaKy) For the probability distributions fg, and f^ of the variables 34 A, B, and C related by the linear equation: A = B + C (3.46) where B and C are independent. Convolution and Deconvolution allow any one of f., f D , f r to be determined, given the remaining two. The A B C procedure i s as follows: Convolution (3.47) Deconvolution (3.48) Deconvolution (3.49) The process of Convolution and Deconvolution are simplified by the use of Fourier Transforms. Analytical (continuous) Fourier Transforms can be used for analytical probability distributions and Discrete Fourier Transform for discrete probability distributions. Using logarithmic transformation for linearization, the techniques have been adapted to the non-linear equation: 1. A = B+C (B 2. B = A-C (A 3. C = A-B (A ,C independent) dependent on B,C) dependent on B,C) f B • ' f A ® f C f C - f H © f R C = EQ where E and Q are independent, for the analysis of ground level concentration and emission rate data. 4. COMPUTER PROGRAMS The n u m e r i c a l CDFT t e c h n i q u e s d e v e l o p e d f o r t h e a n a l y s i s o f d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s o f ground l e v e l c o n c e n t r a t i o n ( C ) and e m i s s i o n r a t e ( Q ) d a t a i s a d a p t e d t o d i g i t a l c o m p u t a t i o n u s i n g FORTRAN. FORTRAN i s chosen f o r s i m p l i c i t y and b e c a u s e e f f i c i e n t F a s t F o u r i e r T r a n s f o r m (FFT) s u b r o u t i n e s a r e a v a i l a b l e i n t h i s l a n g u a g e . Three programs were d e v e l o p e d , t h e main program "Program f o r A n a l y s i s o f D i s t r i b u t i o n s 1 (P.A.D.), p e r f o r m s t h e C o n v o l u t i o n and D e c o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m . The a c c u r a c y o f r e s u l t s c a l c u l a t e d w i t h t h i s program i s e v a l u a t e d by a p p l y i n g i t t o v a r i o u s d i s c r e t e d i s -t r i b u t i o n s o f C and Q. These d i s t r i b u t i o n s a r e g e n e r a t e d by a 'Ground L e v e l C o n c e n t r a t i o n D a t a G e n e r a t i o n Program' (p . G . P . ) , u s i n g a m o d i f i e d G a u s s i a n Plume D i s p e r s i o n Model D i s t r i b u t i o n s o b t a i n e d f r o m t h e s e two programs a r e compared u s i n g a ' S t a t i s t i c a l T e s t i n g p r o g r a m ' ( S . T . P . ) , The t h r e e programs, PAD, DGP, and STP, t o g e t h e r w i t h e q u a t i o n s , f l o w c h a r t s , and t y p i c a l d a t a i n p u t , a r e l i s t e d i n A p p e n d i x I I . These programs a r e e s s e n t i a l l y s e l f e x p l a n a t o r y . A l l i n p u t v a r i a b l e s a r e d e f i n e d and comment s t a t e m e n t s a r e i n s e r t e d ahead o f v a r i o u s c a l c u l a t i o n s t a g e s . S u b r o u t i n e s a r e u s e d e x t e n s i v e l y t o a c c e n t u a t e t h e main p a t h s ; o f ^ c o m p u t a t i o n s . The use o f s u b r o u t i n e s a l s o a l l o w s e a s y m o d i f i c a t i o n s i n f u t u r e o f t h e programs t o s u i t s p e c i f i c a p p l i c a t i o n s . F o l l o w i n g a r e b r i e f d i s c u s s i o n s o f t h e f u n c t i o n s and c h a r a c t e r i s t i c s o f t h e t h r e e p r o grams, 4 .1 PAoqwm fan AnaZy^AJb oj Vi&tAlbutLom CP.A.P.I. T h i s program p e r f o r m s t h e C o n v o l u t i o n and D e c o n v o l u t i o n o p e r a t i o n on g i v e n d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s . The i n d e p e n d e n t v a r i a b l e s o f t h e s e d i s t r i b u t i o n s must be s p e c i f i e d a t r e g u l a r i n t e r v a l s on a l i n e a r o r 35 l o g a r i t h m i c s c a l e . The d i s c r e t e F o u r i e r T r a n s f o r m s a r e e v a l u a t e d u s i n g t h e F a s t F o u r i e r T r a n s f o r m s u b r o u t i n e 'F0UR2' a v a i l a b l e f r o m t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a computing f a c i l i t i e s . I n p u t and o u t p u t d a t a can be l i s t e d and p l o t t e d i n t h e fo r m o f p r o b a b i l i t y d i s t r i b u t i o n s a nd/or c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s . A f l o w c h a r t l i s t i n g m ajor s t e p s p e r f o r m e d i n t h e program i s g i v e n i n A p p e n d i x I I . 2 . B . V a r i o u s i n d i c e s a r e u s e d t o c o n t r o l d a t a i n p u t , c a l c u l a t i o n p a t h , p r i n t ^ - o u t s and p l o t s . F o r example, t h e c a l c u l a t i o n p a t h i n each r u n can be t r a c e d by s e t t i n g ITRACE = 1, t h e r e b y c a u s i n g comment s t a t e m e n t s t o be p r i n t e d a t v a r i o u s j u n c t i o n s i n t h e program. T h i s i s u s e f u l f o r l o c a t i o n o f e r r o r when m o d i f i c a t i o n s t o t h e program i s b.eing made. One f e a t u r e t h a t r e q u i r e s e x p l a n a t i o n i s t h e 'F0UR2' s u b r o u t i n e . T h i s s u b r o u t i n e i s c a l l e d by t h e s t a t e m e n t : CALL F0UR2 (DATA, NDIM, NODIM, ISIGN, IFORM) The s u b r o u t i n e p e r f o r m s e i t h e r D i s c r e t e F o u r i e r T r a n s f o r m (ISIGN = -1) o r I n v e r s e D i s c r e t e F o u r i e r T r a n s f o r m (ISIGN = +1) on sequences s u p p l i e d i n t h e m u l t i d i m e n s i o n a l a r r a y 'DATA'. The d i m e n s i o n s o f 'DATA' a r e 'NDIM x NODIM' where NDIM must be a power o f 2. The s u p p l i e d d a t a can be complex (IFORM = 1 ) , r e a l (IFORM = 0) o r c o n j u g a t e s y m m e t r i c complex (IFORM = - 1 ) . I n t h e o u t p u t , t h e t r a n s f o r m e d d a t a a r e s t o r e d i n t h e a r r a y 'DATA', d i s p l a c i n g t h e i n p u t d a t a . F o r p r e s e n t a p p l i c a t i o n s , t h e d i s c r e t e F o u r i e r T r a n s f o r m i s p e r f o r m e d on a o n e - d i m e n s i o n a l a r r a y o f N r e a l d a t a p o i n t s . The c a l l i n g s t a t e m e n t i s , . t h e r e f o r e : CALL F0UR2 (DATA, N, 1, 0, -1) S i n c e t h e r e a l i n p u t d a t a a r e t r a n s f o r m e d i n t o c o n j u g a t e symmetric complex d a t a , t h e I n v e r s e d i s c r e t e F o u r i e r T r a n s f o r m can be c a l l e d by 37 CALL F0UR2 (DATA, N, 1, - 1 , 1) Modi, i c a t i o n s t o t h e a r r a y s o f i n p u t d a t a a r e n e c e s s a r y i n a p p l y i n g F0UR2 t o e v a l u a t e C o n v o l u t i o n and D e c o n v o l u t i o n . F i r s t , an a l i a s i n g e r r o r must be e l i m i n a t e d . T h i s i s i l l u s t r a t e d i n F i g . 4-1. F i g u r e s 4-1 ( a ) , ( b ) , and (c) i l l u s t r a t e t h e C o n v o l u t i o n f ^ = f ^ @ f g -The use o f D i s c r e t e F o u r i e r T r a n s f o r m s i n p l i e s t h e a p p r o x i m a t i o n o f d i s t r i b u t i o n s f ^ and f g by t h e p e r i o d i c d i s t r i b u t i o n s f ^ ' and f g ' . The c o n v o l v e d d i s t r i b u t i o n f^, d i f f e r s f r o m t h e o r i g i n a l f^, due t o o v e r l a p p i n g o r ' a l i a s i n g ' e r r o r . T h i s i s b e c a u s e , d u r i n g t h e t r a n s f o r m a t i o n , t h e f u n c t i o n s f ^ f g a r e assumed t o be r e p e t i t i v e . To a v o i d t h i s e r r o r , f ^ and f g can be a p p r o x i m a t e d by f ^ " and f g " w h i c h a r e f ^ f g w i t h N^, Ng t r a i l i n g z e r o ' s , r e s p e c t i v e l y . The c o n v o l v e d f ^ " t h e n a p p r o x i m a t e s f ^ w i t h o u t a l i a s i n g e r r o r . The minimum common l e n g t h , N, o f t h e t h r e e sequences i s , t h e r e f o r e N = = N^ + Ng - 1 S i n c e F0UR2 r e q u i r e s N t o be a power o f 2 ( i . e . N = 2 , k, 1, 2, 3,...) t h e sequences must be f u r t h e r m o d i f i e d by a d d i n g t a i l i n g z e r o ' s t o make up 2 e n t r i e s . F o r example, i f N^ = 5, N g = 15, = 1 9 . S i n c e N must be a power o f 2, t h e minimum common l e n g t h o f t h e sequences must be 32. 4.2 Ground Level Concent/utflon Vata GenoJuvtlon Psiogfuim (B.G.P.) T h i s program uses a m o d i f i e d s t e a d y s t a t e G a u s s i a n Plume D i s p e r s i o n Model t o g e n e r a t e ground l e v e l c o n c e n t r a t i o n (c) p r o b a b i l i t y d i s t r i b u t i o n s f o r a s i n g l e s o u r c e e m i t t i n g t o a s i n g l e r e c e p t o r . T h i s model i s s i m i l a r t o t h e one Zimmerman used i n A n k a r a , T u r k e y 1 9 - , and S t . L o u i s , U S A 2 0 . The model a l l o w s f o r v a r i a b i l i t y i n s t a b i l i t y ( s ) , w i n d d i r e c t i o n (J3), wind speed C U 3 and e m i s s i o n r a t e (Q), u s i n g d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s P s ' P g ' P u ' a n c * PQ' r e s P e c t i v e l y - These d i s t r i b u t i o n s can be s p e c i f i e d by r e a d i n g i n i n d i v i d u a l e n t r i e s o r t h e program can g e n e r a t e normal o f 0.4-0.2-0.4. 0.2 2 ' 4 A «-to N 0.3 0.2H o. i -J T f, Ni 6 4-B f 0.4. 0.2. (g) 8 ^ 4 • N <t») " B ~ * (C, N 'c 0.2H N fc 0.4 0.2i N 0 16 ( f ) N Aliasing Error ,fC = fAj'^B i i i 1 M 5 N C *-X t-tr f c " t NB HT* -0.4-0.2 ~ \ • (i) N fc« - fA» ®fB" N Nc FIGURE 4.1: Modifications to eliminate "Aliasing" error in F0UR2 l o g n o r m a l d i s t r i b u t i o n s b a s e d on s u p p l i e d p a r a m e t e r s s u c h as mean s t a n d a r d d e v i a t i o n and i n t e r v a l s i z e f o r each d i s t r i b u t i o n . I t i s assumed t h a t t h e f o u r v a r i a b l e s a r e i n d e p e n d e n t o f each o t h e r . However, t h e program c o u l d be e a s i l y m o d i f i e d t o a c c e p t d i s c r e t e j o i n t - p r o b a b i l i t i e s f o r v a r i a b l e s w h i c h a r e dependent on one a n o t h e r . S i n c e a l a r g e number o f ground l e v e l c o n c e n t r a t i o n (C) d a t a a r e c a l c u l a t e d f o r t h e v a r i o u s c o m b i n a t i o n s o f s t a b i l i t y , w i n d d i r e c t i o n , w i n d speed, and e m i s s i o n r a t e , i n d i v i d u a l C v a l u e s a r e n o t l i s t e d i n t h e o u t p u t . I n s t e a d , t h e C v a l u e s a r e s o r t e d i n t o d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s and d i s c r e t e c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s . A d i s t i n c t f e a t u r e o f t h i s program i s t h a t i t n o t o n l y c a l c u l a t e s t h e p r o b a b i l i t y d i s t r i b u t i o n s r e l a t e d t o C, t h e ground l e v e l c o n c e n t r a t i o n , b u t i t a l s o c a l c u l a t e s as i n t e r m e d i a t e r e s u l t s , t h e p r o b a b i l i t y d i s t r i b u t i o n r e l a t e d t o E, t h e c h a r a c t e r i s t i c f u n c t i o n . The program i s o r g a n i z e d i n t o v a r i o u s p a r t s , each headed by comment s t a t e m e n t s f o r c l a r i t y . S i n c e t h e a l g o r i t h m i s w r i t t e n t o a c h i e v e e f f i c i e n c y i n c o m p u t a t i o n , t h e d e t a i l e d f l o w c h a r t i s complex masking t h e main c a l c u l a t i o n p a t h . An o u t l i n e o f t h e v a r i o u s c a l c u l a t i o n s t e p s i s g i v e n i n s t e a d ( A p p e n d i x I I . l . B ) . I n a n t i c i p a t i o n o f p o s s i b l e e x t e n s i o n o f t h e CDFT method t o c a s e s i n v o l v i n g two e m i t t i n g s o u r c e s , t h e n e c e s s a r y a l g o r i t h m s has been i n c o r p o r a t e d i n t o v a r i o u s p a r t s o f t h i s program. 4.3 Statistical Touting Ptiogtum [ S . T , P . ) . G i v e n any two d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s , / t h i s program computes t h e a s s o c i a t e d d i s c r e t e c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s and p e r f o r m s s i x c o m p a r i s o n s o f t h e s e d i s t r i b u t i o n s . The methods o f c o m p a r i s o n s a r e g i v e n i n T a b l e 4-1. 40 T a b l e 4-1. Comparison Methods used i n t h e Program S.T.P. I . F o r t h e d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s : 1. G r a p h i c a l c o m p a r i s o n o f p l o t s 2. L e a s t s quare l i n e a r r e g r e s s i o n 3. C h i - s q u a r e goodness o f f i t t e s t . I I . F o r t h e d i s c r e t e c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s : 1. G r a p h i c a l c o m p a r i s o n o f p l o t s 2. L e a s t s quare l i n e a r r e g r e s s i o n 3. Kolmogorov - Smir n o v (K-S) goodness o f f i t t e s t . I n t h e g r a p h i c a l c o m p a r i s o n , t h e p l o t s o f two d i s t r i b u t i o n s a r e sup e r i m p o s e d t o r e v e a l any s i g n i f i c a n t d e v i a t i o n s . Any s i g n i f i c a n t d e v i a t i o n s can the n be q u a l i t a t i v e l y a s s e s s e d . F o r example, t h e d e v i a t i o n s may be predominant i n one t a i l o f d i s t r i b u t i o n s o r t h e y may be i n c r e a s i n g , d e c r e a s i n g o r f l u c t u a t i n g . I n t h e l e a s t s q u a r e l i n e a r r e g r e s s i o n , p a i r s o f d i s c r e t e p r o b a b i l i t i e s a r e grouped a c c o r d i n g t o t h e v a l u e s . They a r e t h e n f i t t e d by a s t r a i g h t l i n e . I f t h e two d i s t r i b u t i o n s a r e i d e n t i c a l , t h e s l o p e and i n t e r c e p t o f t h e r e g r e s s i o n l i n e s h o u l d be 1 and 0, r e s p e c t i v e l y ; and t h e c o r r e l a t i o n c o e f f i c i e n t s h o u l d a l s o be 1. The above two t e s t s a r e u s e f u l when a p a i r o f d i s t r i b u t i o n s a r e v e r y s i m i l a r . When l a r g e r d e v i a t i o n s o c c u r , t h e C h i - s q u a r e t e s t and Kolmogorov - Smirnov t e s t a r e used t o d e c i d e whether the d i s t r i b u t i o n s a r e s t a t i s t i c a l l y i d e n t i c a l a t a c e r t a i n l e v e l o f s i g n i f i c a n c e . The e q u a t i o n s u s e d i n t h e t e s t a r e summarized i n Ap p e n d i x I I . 3 . B . 5. RESULTS AMV DISCUSSIONS 41 As s t a t e d e a r l i e r , t h e C o n v o l u t i o n , D e c o n v o l u t i o n u s i n g F o u r i e r T r a n s f o r m (CDFT) method can be u s e d f o r b o t h t h e C h a r a c t e r i z a t i o n o f a s o u r c e - r e c e p t o r p a i r and P r e d i c t i o n o f new Ground L e v e l C o n c e n t r a t i o n p a t t e r n i n r e s p o n s e t o new e m i s s i o n r a t e s . The computer program "Program f o r A n a l y s i s o f D i s t r i b u t i o n " (P.A.D.) was s u b j e c t e d t o two k i n d s o f t e s t s : I t i s f i r s t v a l i d a t e d by a p p l i c a t i o n t o known c o r r e s p o n d i n g p r o b a b i l i t y d i s t r i b u t i o n s o f C and Q. The p r o c e d u r e w i l l be d e t a i l e d l a t e r . S i n c e t h e d i s t r i b u t i o n s c o m p i l e d f r o m f i e l d d a t a i n v a r i a b l y c o n t a i n i n s t r u m e n t and measurement e r r o r s w h i c h may c o m p l i c a t e t h e i n t e r p r e t a t i o n o f r e s u l t s , a m o d i f i e d G a u s s i a n D i s p e r s i o n Model i s u s e d t o g e n e r a t e t h e s e C and Q d i s t r i b u t i o n s . N e x t , t h e program i s c h e c k e d f o r a p p l i c a b i l i t y t o f i e l d d a t a by a p p l y i n g i t t o C and Q d i s t r i b u t i o n s g e n e r a t e d by t h e G a u s s i a n model and " c o n t a m i n a t e d " by a r t i f i c i a l i m p e r f e c t i o n w h i c h r e s e m b l e s measurement and i n s t r u m e n t l i m i t a t i o n s . A s e r i e s o f s e v e n s e p e r a t e computer r u n s were c a r r i e d o u t f o r each t e s t , u s i n g t h e t h r e e computer programs d e s c r i b e d e a r l i e r t o p e r f o r m i ) D a t a g e n e r a t i o n , i i ) A n a l y s i s o f D i s t r i b u t i o n s and i i i ) C o m p a r i s o n o f D i s t r i b u t i o n s . 5.1 VcUMicuUon thz CVFT MeXkod The v a l i d a t i o n o f t h e CDFT method c o n s i s t o f t h r e e segments ( F i g u r e 5-1). i ) D a t a G e n e r a t i o n : The program DGP i s u s e d i n Run #1 and #2 t o g e n e r a t e two s e t s o f p r o b a b i l i t y d i s t r i b u t i o n s f o r g r o u n d l e v e l c o n c e n t r a t i o n s c o r r e s p o n d i n g t o d i s t r i b u t i o n s o f e m i s s i o n r a t e and Q 2 Data G e n e r a t i o n A n a l y s i s o f D i s t r i b u t i o n s Run # (1) (2) (3) (4) Comparison o f D i s t r i b u t i o n s (5) (6) (7) GLC PAD Input Program £QL1 » C o n d i t i o n s j * \ G L C ( I n p u t f i l e DI) fQL2 > C o n d i t i o n s , , • (D2) f C L l . f Q L l > f Q L l - * (D3) f C L l > f Q L l > fQL2 • (D4) f E L > % L » (° 5) + f C L l » f C L l ' C°6) • f C L 2 » f C L 2 » C D 7 ) • PAD STP STP STP Output "* f Q L l > f E L > £ C L 1 (Output f i l e A l ) * £QL2 ' f E L > f C L 2 (A2) EL' > xChl (A3) "* fEL' . £CL2' C A 4) Compar i s o n (A5) -+ Compa r i s o n (A6) Comparison (A7) T y p i c a l I n p u t and Output d a t a f i l e s l i s t e d i n A p p e n d i x I I I . F i g u r e 5-1: V a l i d a t i o n Scheme f o r P,A.D. 43 and a s p e c i f i e d d i s t r i b u t i o n o f t h e C h a r a c t e r i s t i c F u n c t i o n E A l l t h e d i s t r i b u t i o n s a r e g r o u p e d i n t o l o g a r i t h m i c s c a l e t o f a c i l i t a t e s u b s e q u e n t a n a l y s i s . The C h a r a c t e r i s t i c F u n c t i o n E i s i n d i r e c t l y d e f i n e d by t h e f o l l o w i n g v a r i a b l e s : w i n d s p e e d , w i n d d i r e c t i o n , a t m o s p h e r i c s t a b i l i t y and e f f e c t i v e s t a c k h e i g h t . The n u m e r i c a l v a l u e s s e l e c t e d f o r t h e s e v a r i a b l e s were b a s e d on t h e example i n T u r n e r ' s "Workbook on A t m o s p h e r i c D i s p e r s i o n " : 1 8 . i i ) A n a l y s i s o f D i s t r i b u t i o n s : The p r o b a b i l i t y d i s t r i b u t i o n s g e n e r a t e d b y t h e program "DGP" were s u p p l i e d t o t h e program 'PAD* f o r C h a r a c t e r i z a t i o n and P r e d i c t i o n . I n C h a r a c t e r i z a t i o n , t h e d i s t r i b u t i o n s f ^ L l and f were u s e d i n Run #3 t o d e t e r m i n e f g L , • To c h e c k i f t h e r e i s any l o s s o f i n f o r m a t i o n i n d e t e r m i n i n g f g ^ , f r o m f ^ ^ a n d ^QH> t n e p r o c e s s was r e v e r s e d t o d e t e r m i n e a new d i s t r i b u t i o n f , f r o m f q ^ j a n d ^ELII* T Q ^E compared l a t e r w i t h f C L 1 • I n P r e d i c t i o n , a new d i s t r i b u t i o n f q L 2 was a p p l i e d t o f ^ t * n ^ u n t 0 d e t e r m i n e f^ L2» w i t h o u t any a d d i t i o n a l i n f o r m a t i o n . T h i s f ^ j ^ * W 1 H t h e n be compared w i t h f ^ . ^ g e n e r a t e d by DGP f r o m f Q L 2 , f £ L . • i i i ) C o m p a r i s o n o f D i s t r i b u t i o n s : T h r e e p a i r s o f d i s t r i b u t i o n s were compared u s i n g t h e program STP: a) t h e d i s t r i b u t i o n s f £ L and f £ L , a r e compared i n Run #5. Good agreement i n t h i s r u n v a l i d a t e s t h e a b i l i t y o f t h e program PAD t o d e t e r m i n e f ^ , t h e l o g r i t h m i c p r o b a b i l i t y d i s t r i b u t i o n o f t h e c h a r a c t e r i s t i c f u n c t i o n , f r o m t h e d i s t r i b u t i o n s f q L j a n d f ^ L l o n ^ y * b) t h e d i s t r i b u t i o n s f C L 1 and f C L 1 , a * e compared i n Run #6. Good agreement h e r e w o u l d f u r t h e r v a l i d a t e t h e a c c u r a c y o f f g L i . c) t h e d i s t r i b u t i o n s f C L 2 and f C L 2 » f r o m R u n # 7 - Good agreement h e r e v a l i d a t e s t h e a b i l i t y o f t h e CDFT method t o p r e d i c t f ^ L 2 u s i n g o n l y t h e d i s t r i b u t i o n s f q j ^ and t h e new f q L 2 44 S i x s e p a r a t e t e s t s were p e r f o r m e d f o r each c o m p a r i s o n : F i r s t t h e p a i r o f d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s were p l o t t e d on one s i n g l e g r a p h . T h i s t e s t r e v e a l s where t h e d e v i a t i o n s o c c u r i n r e l a t i o n t o t h e v a l u e s o f t h e v a r i a b l e s . Then t h e p a i r was f i t t e d by l i n e a r r e g r e s s i o n . T h i s t e s t shows i f t h e d e v i a t i o n s o c c u r when t h e d i s c r e t e p r o b a b i l i t y i s h i g h o r low. F o r example, i f t h e r e a r e r o u n d o f f e r r o r s t h e d e v i a t i o n s w o u l d be p r o m i n e n t f o r t h e l o w e r d i s c r e t e p r o b a b i l i t i e s . T h i r d l y , C h i -s q u a r e t e s t s was a p p l i e d t o t h e d i s c r e t e d i s t r i b u t i o n s . I n c a s e s where d e v i a t i o n s were o b s e r v e d i n t h e f i r s t two t e s t s , t h e C h i - s q u a r e t e s t w o u l d have a s c e r t a i n e d i f t h e s e d i s t r i b u t i o n s were s t a t i s t i c a l l y s i m i l a r w i t h i n a c e r t a i n c o n f i d e n c e l i m i t . The c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s were d e r i v e d f r o m t h e d i s c r e t e d i s t r i b u t i o n s and s i m i l a r c o m p a r i s o n s were c a r r i e d o u t , e x c e p t t h a t t h e C h i - s q u a r e t e s t a p p l i c a b l e o n l y t o d i s c r e t e d i s t r i b u t i o n s was r e p l a c e d by t h e K o l n o g o n o v - S m i r n o v t e s t a p p l i c a b l e t o t h e c u m u l a t i v e d i s t r i b u t i o n . A t o t a l o f 4 s e r i e s o f computer r u n s were c a r r i e d o u t , s t a r t i n g w i t h d i s t r i b u t i o n s t h a t a r e s i m p l e and smooth i n S e r i e s 1 and p r o g r e s s i n g t o t h e more c o m p l e x , i r r e g u l a r d i s t r i b u t i o n s i n S e r i e s 4. The c o n d i t i o n s u s e d i n t h e s e s e r i e s a r e summarized i n T a b l e 5.1. F o l l o w i n g a r e t h e r e s u l t s and d i s c u s s i o n s o f t h e f o u r s e r i e s o f t e s t s : 5.1.1 S&SLLZA 1. Simple. ViAtfU.buJU.ovUi S i m p l e l o g n o r m a l d i s t r i b u t i o n s f o r t h e e m i s s i o n r a t e s were u s e d . ( F i g u r e 5.2, 5 . 3 ) . A s i m i l a r l y s i m p l e d i s t r i b u t i o n f o r t h e c h a r a c t e r i s t i c f u n c t i o n i s a l s o u s e d by s p e c i f y i n g l o g n o r m a l l y d i s t r i b u t e d w i n d speed ( F i g u r e 5.4) and h o l d i n g t h e s t a b i l i t y and w i n d d i r e c t i o n c o n s t a n t . C o m p a r i s o n o f f C T g e n e r a t e d by t h e G a u s s i a n model and f „ T , d e t e r m i n e d by t h e CDFT method showed v e r y good agreement ( F i g u r e 5 . 5 ) . S i m i l a r c o m p a r i s o n s o f 45 T a b l e 5-1: C o n d i t i o n s Used i n S e r i e s #1 - #4 S e r i e s #1 S e r i e s #2 S e r i e s #3 S e r i e s #4 Wind Speed Log-normal Log-normal Log-normal Log-normal A t m o s p h e r i c S t a b i l i t y C o n s t a n t C o n s t a n t A r b i t r a r y F i g . 5-6 A r b i t r a r y Wind D i r e c t i o n C o n s t a n t C o n s t a n t C o n s t a n t Normal F i g . 5-7 E m i s s i o n Rate #1 Log-normal F i g . 5-3 I r r e g u l a r F i g . 5-5 Log-normal Log-normal E m i s s i o n Rate #2 Log-normal F i g . 5-4 Log-normal Log-normal Log-normal R e s u l t i n g C h a r a c t e r i s t i c F u n c t i o n S i m p l e Complex R e s u l t i n g GLC's S i m p l e Complex 0.40 H LOG Q2 F i g u r e 5.3: D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f q L 2 £ o r E m i s s i o n R a t e "2, Q2(G/s) 0.25 A 0.20 0.(5 0.10 0.05 0.00 1 x ™ot* GAUSSIAN METHOD 0 FROM CDFT METHOD -5.4 -5.1 -5 .0 LOG E* -4 .9 - 4 . 8 -5.3 -5.2 LOG E , Figure 5.5: Comparison of Discrete Logarithmic Probability Distributions f ^ , f ^ , of Characteristic Function, E, E'fs/m3) in Series #1 50 f C L l ' f C I l ' ^ f r o m R u n # 6 ) a n d f C L 2 ' a n d f C L 2 ' ^ f r o m R u n s h o w e d e q u a l l y good r e s u l t s ( F i g u r e 5-6, 5-7). The d e v i a t i o n s between t h e d i s t r i b u t i o n g e n e r a t e d by t h e G a u s s i a n model and CDFT method a r e n e g l i g i b l e . L i s t e d i n T a b l e 5-2 a r e t h e r e s u l t s o f t h e o t h e r t e s t s u s e d i n t h e c o m p a r i s o n s . I t i s seen t h a t i n e a c h c a s e , t h e e r r o r s i n t h e mean and s t a n d a r d d e v i a t i o n a r e n e g l i g i b l e . L i n e a r r e g r e s s i o n s f o r b o t h d i s c r e t e and c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s showed p e r f e c t c o r r e l a t i o n s . The C h i - s q u a r e t e s t and t h e Kolmogonov-Smirnov t e s t s f u r t h e r c o n f i r m e d t h a t each p a i r o f d i s t r i b u t i o n s i s i n d e n t i c a l . L i s t i n g s o f t h e v a r i o u s i n p u t f i l e s (D1-D7) and o u t p u t f i l e s (A1-A7) a r e i n c l u d e d i n A p p e n d i x I I I . S i n c e t h e s e f i l e s c o n t a i n n u m e r i c a l r e p r e s e n t a t i o n s o f p r o b a b i l i t y d i s t r i b u t i o n s , t h e y a r e r a t h e r l e n g t h y . L i s t i n g s f o r t h e i n p u t and o u t p u t f i l e s f o r l a t e r s e r i e s o f t e s t s t o t a l o v e r 400 page s . C o n s e q u e n t l y , t h e y a r e n o t i n c l u d e d i n t h i s t h e s i s . R e s u l t s o f t h e s e t e s t s w i l l be p r e s e n t e d o n l y i n t h e f o r m o f g r a p h s and s t a t i s t i c a l summaries. 5.1.2 SZAAJLA, 2. E ^ e c i IWIZQUZOJL EmLb&lon V.aZz VibthJJawtlon I n t h i s s e r i e s , t h e smooth l o g n o r m a l d i s t r i b u t i o n s f o r t h e e m i s s i o n r a t e u s e d i n S e r i e s #1 were r e p l a c e d by a r b i t r a r i l y s p e c i f i e d d i s t r i b u t i o n w h i c h were h i g h l y i r r e g u l a r , w i t h many " p e a k s " and " t r o u g h s " ( F i g u r e 5 - 8 ) . The p u r p o s e was t o examine i f t h e smoothness o f a d i s t r i b u t i o n i s a n e c e s s a r y c o n d i t i o n f o r t h e s u c c e s s f u l usage o f t h e CDFT method. R e s u l t s f r o m t h i s s e r i e s showed t h a t a l t h o u g h a change i n t h e e m i s s i o n r a t e d i s t r i b u t i o n d i d change t h e c h a r a c t e r i s t i c s o f t h e d i s t r i b u t i o n o f t h e g r o u n d l e v e l c o n c e n t r a t i o n ( F i g u r e 5-10), t h e CDFT method s t i l l a c c u r a t e l y d e t e r m i n e s t h e d i s t r i b u t i o n s f„.,, f 1 k } and f r T 9 , w h i c h 0.20 J 0.15 4 f c u . f c L i ' 0.10 0.05 H 0.00 X FROM GAUSSIAN METHOD o FROM CDFT METHOD -3.6 -3.4 -3.2 -3.0 -2.8 LOG Ci , LOG Cl' ^ * K * -2 .6 -2.4 Figure 5.6: Comparison of Discrete Logarithmic Probability Distributions f C L 1 . f C b i of Ground Level Concentration #1, C l , Cl'fG/m3) in series #1 X FROM GAUSSIAN METHOD O FROM CDFT METHOD Figure 5.7: Comparison of Discrete Logarithmic Probability Distributions f r i 9 , fr,->. of Ground Level Concentration #2, C2,C2'(G/m3) in series #1 T a b l e 5-2: Comparison S t a t i s t i c s f o r P r o b a b i l i t y D i s t r i b u t i o n s i n S e r i e s # 1. fEL' fEL» f C L l . f C L l ' f C L 2 > f C L 2 ' T o t a l Number o f I n t e r v a l s N 16 38 29 % E r r o r o f Mean AH 0.0150 % -0.0001 % 0.0002 % % E r r o r o f S t a in d a r d D e v i a t i o n Ao -0.1205 % -0.0006 % 0.0008 % S l o p e mi 0.9999 1.0000 1.0000 L i n e a r R e g r e s s i o n I n t e r c e p t b i 0.0000 0.0000 0.0000 D i s c r e t e C o r r e l a t i o n C o e f f i c i e n t r i 1.0000 1.0000 1.0000 P r o b a b i l i t y D i s t r i b u t i o n s X 2 T e s t # I n t e r v a l s X 2 n i ~ ' x r 13 0.0812 8 0.0000 16 0.0000 P r o b a b i l i t y t h a t D i s t r i b u t i o n s a r e D i f f e r e n t P i 0.0000 0.0000 0.0000 L i n e a r R e g r e s s i o n S l o p e ni2 1.0000 1.0000 1.0000 I n t e r c e p t b 2 0.0000 ~ 0.0000 0.0000 C u m u l a t i v e C o r r e l a t i o n C o e f f i c i e n t r 2 1.0000 1.0000 1.0000 P r o b a b i l i t y D i s t r i b u t i o n s Kolmogonov-Smir n o v K-S S t a t i s t i c s d 0.0002 0.0000 0.0000 O c c u r r e d a t I n t e r v a l # i 7 13 11 T e s t P r o b a b i l i t y t h a t D i s t r i b u t i o n s a r e D i f f . P 2 0.0000 0.0000 0.0000 CONCLUSION: THE PAIR OF DISTRIBUTIONS ARE: IDENTICAL IDENTICAL IDENTICAL 0.20 4 0.15 -0 .10-0 D 5 4 0.00. LOG Q l Figure 5.8: Discrete Logarithmic Probability Distributions f n i . of Emission Rate #1, Ql(G/s) in series «2 of Characteristic Function E, E' (s/m3) in Series #2. 56 a r e i d e n t i c a l t o t h e c o r r e s p o n d i n g d i s t r i b u t i o n s f ^ , £ C L , J a n < ^ ^CL2 ' ^ e g r a p h i c a l c o m p a r i s o n s a r e shown i n F i g u r e s 5-9 t o 5-11. 5.1.3 SQAIZA 3. Elject oj Complex CkaxacteAlAtlc Function Vl&&ilbutlon {Pant 7) I n t h i s s e r i e s , l o g n o r m a l e m i s s i o n r a t e d i s t r i b u t i o n s f r o m S e r i e s #1 were use d . However, a more complex d i s t r i b u t i o n f o r t h e C h a r a c t e r i s t i c F u n c t i o n was i n d i r e c t l y s p e c i f i e d by r e p l a c i n g t h e c o n s t a n t v a l u e f o r A t m o s p h e r i c S t a b i l i t y by a p r o b a b i l i t y d i s t r i b u t i o n ('Figure 5^12), i n a d d i t i o n t o t h e l o g n o r m a l d i s t r i b u t i o n f o r w i n d speed. The p u r p o s e was t o see how a c c u r a t e l y t h e CDFT method c o u l d t r e a t t h e s e more complex d i s t r i b u t i o n s . The r e s u l t i n g d i s t r i b u t i o n f„ T o f t h e C h a r a c t e r i s t i c bL F u n c t i o n i s shown i n F i g u r e 5-13. C o m p a r i s o n o f f £ L and f £ L , CFigure 5-13) showed t h a t v a l u e s o f f ^ , d e t e r m i n e d by t h e CDFT method " o s c i l l a t e " s l i g h t l y a r o u n d t h e f £ L f r o m t h e G a u s s i a n model. However, t h e s e o s c i l l a t i n g d e v i a t i o n s were s u f f i c i e n t l y s m a l l so t h a t t h e b a c k - c a l c u l a t e d f ^ n t^ i e p r e d i c t e d f ^ ^ ' s n o w e d good agreement w i t h f ^ L 1 and f C L 2 » r e s p e c t i v e l y , ( F i g u r e s 5^14, 5 - 1 5 ) . R e s u l t s o f t h e s t a t i s t i c a l c o m p a r i s o n s a l s o i n d i c a t e d t h a t t h e s e s m a l l d e v i a t i o n s were n e g l i g i b l e . 5.1.4 S o l a , 4, E ^ e c t 0^ Complex ChaAactefuAtic Function Vl&tnlhutlon [PaAt 2) I n t h i s s e r i e s , i n a d d i t i o n t o t h e p r o b a b i l i t y d i s t r i b u t i o n s o f w i n d speed and a t m o s p h e r i c s t a b i l i t y s p e c i f i e d i n S e r i e s #3, a n o r m a l d i s t r i b u t i o n o f w i n d d i r e c t i o n was f u r t h e r s p e c i f i e d fJFigure 5 - 1 6 ) , The GLC d i s t r i b u t i o n g e n e r a t e d by t h e G a u s s i a n model, t h e r e f o r e , a p p r o x i m a t e d t h e GLC v a l u e s t h a t w o u l d have been o b s e r v e d i n a c t u a l f i e l d measurement o v e r a p e r i o d o f t i m e G.IOO A LOG C l , LOG CJLB F i g u r e 5.10: C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f , f » o f Ground L e v e l C o n c e n t r a t i o n #1, C l , C l r (G/s) i n S e r i e s #2. o.ioo H 0 . 0 7 5 fcL2 , fcL2' 0 . 0 5 0 0 .025 X FROM GAUSSIAN METHOD <I> FROM CDFT METHOD 0 . 0 0 0 " m i -4.0 - 3 2 LOG C2 ' - 2 . 6 Figure 5.11: Comparison of Discrete Logarithmic Probability Distributions f C L 2» £CL2* o £ G r o u n t l Level Concentration #2, C2, C2' in series #2 Cn Oo 0.25 A 0.20 A 0.|5 fEL' 0.10 H 0.05 A 0.00 X FROM GAUSSIAN METHOD O FROM C D F T METHOD _Q_ -6.0 - 5 3 -5.6 -5.4 -d.2 LOG E , LOG E* Figure 5.13: Comparison of Discrete Logarithmic Probability Distributions, f p . , f p . , function E,E»(s/m3) in series #3 b - 4 . 8 of Characteristic 0.20 J 0.15 0.10 4 0.054 X FROM GAUSSIAN METHOD © FROM CDFT METHOD 0.00 i M M ^ f t i M y V -3.4 -2.6 -2.4 -3.2 -3.0 -2.-8 LOG C l , LOG C l ' Figure 5.14: Comparison of Discrete Logarithmic Probability Distribution S f c l , f C L , Ground Level Concentration #1, C l , C l ' in series #3 -4.0 -3.8 -3.6 -3.4 -32 -3,0 -2.8 -26 LOG C2, LOG C2' Figure 5.15: Comparison of Discrete Logarithmic Probability Distributions, f^.^* £CL2' °^ G r o u n t * Level Concentration #2, C2, C2' in series #3 0.100 i n w h i c h t h e w i n d speed, w i n d d i r e c t i o n , and a t m o s p h e r i c s t a b i l i t y a r e a l l s u b j e c t e d t o v a r i a t i o n s w i t h t i m e . The r e s u l t i n g C h a r a c t e r i s t i c s F u n c t i o n has a complex p r o b a b i l i t y d i s t r i b u t i o n shown i n F i g u r e 5-17. Under t h i s s i t u a t i o n , t h e GLC d i s t r i b u t i o n f ^ j . ^  g e n e r a t e d by t h e G u a s s i a n model spanned o v e r a l a r g e r a n g e o f 1 0 ~ 2 t o 10" 1 1* ( F i g u r e 5-18) S i n c e t h e h i g h e r v a l u e s o f GLC a r e more s i g n i f i c a n t i n terms o f c o m p l i a n c e w i t h c l e a n a i r r e g u l a t i o n s , t h e d i s t r i b u t i o n was " t r u n c a t e d " a t t h e l o w e r ends t o r e d u c e t h e number o f e n t r i e s t o be h a n d l e d . I n o t h e r words, GLC • s l e s s t h a n 1 ( T 1 1 + G/m3 and t h e i r p r o b a b i l i t i e s were n e g l e c t e d . T h i s t r a n s a c t i o n p r o d u c e d v e r y i n t e r e s t i n g r e s u l t s f o r t h e p r o b a b i l i t y d i s t r i b u t i o n s o f t h e c h a r a c t e r i s t i c f u n c t i o n f ^ , d e t e r m i n e d f r o m t h i s t r u n c a t e d f a n d t h e o r i g i n a l f Q L J • The d i s t r i b u t i o n f g e n e r a t e d by t h e G a u s s i a n model was shown i n F i g u r e 5-17, i n w h i c h E r a n g e d f r o m l O " 4 t o 1 0 " 1 8 and t h e h i g h e s t p r o b a b i l i t y o b s e r v e d was about 0.04. I n d i v i d u a l d a t a p o i n t s were n o t p l o t t e d , b e c a u s e t h e r e were 256 d a t a p o i n t s i n t o t a l . The d i s t r i b u t i o n f g ^ , d e t e r m i n e d by t h e CDFT method i s shown i n F i g u r e 5-19 and a p p e a r s t o be r a d i c a l l y d i f f e r e n t . However, c l o s e r e x a m i n a t i o n r e v e a l s t h a t f ^ i s an a p p r o x i m a t i o n o f f ^. F o r f g L t > l a r g e o s c i l l a t i o n s were o b s e r v e d f o r E < 1 0 ~ 1 6 w i t h v a l u e s o f p r o b a b i l i t y r a n g i n g f r o m -6 t o +7. S i n c e p r o b a b i l i t y by d e f i n i t i o n , a l w a y s f a l l between 0 and 1, t h e s e v a l u e s a r e o b v i o u s l y "wrong". However, c o m p a r i s o n o f p r o b a b i l i t i e s o f f ^ and f g L , f o r v a l u e s o f E > 1 0 " 1 6 i n expanded p r o b a b i l i t y s c a l e showed a p p r o x i m a t e agreement, w i t h b e t t e r agreement o b s e r v e d a t t h e h i g h e r end E * 1 0 - l f , A c o m p a r i s o n o f t h e c u m u l a t i v e p r o b a b i l i t i e s d e r i v e d f r o m f ^ and f ^ , i s shown i n F i g u r e 5-20, The d i s c r e p e n c y between f ^ a n d f p ^ , can be e x p l a i n e d as f o l l o w s . S i n c e t h e d i s t r i b u t i o n f i s g e n e r a t e d b y t h e G a u s s i a n model f r o m CL1 0.040 A 0.038 -J 0.024 0.016 H 0.008 0.000 LOG E F i g u r e 5.17: Discrete L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n EL o f C h a r a c t e r i s t i c s F u n c t i o n E (s/m ) g e n e r a t e d by G a u s s i a n Model i n s e r i e s #4 0 . 0 4 0 A LOG CI , LOG Cl' F i g u r e 5.18: Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f » £ C L l G r o u n d L e v e l Concentration-. #1, C I , C I ' G/m3 i n ^series.:#4 LOG E , LOG E' F i g u r e 5.20: C o m p a r i s o n o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y F ^ , F ^ , o f C h a r a c t e r i s t i c F u n c t i o n E, E' (s/m 3) i n s e r i e s #4 69 and f _ T, i n o r d e r t o f u l l y d e t e r m i n e f _ T by t h e CDFT method t h e QL1 EL bL f u l l d i s t r i b u t i o n s f q ^ j and £ ^ L 1 must be u s e d . T h i s was n o t done i n t h e p r e s e n t c a s e . I n s t e a d , a " t r u n c a t e d " d i s t r i b u t i o n , f ^ , ^ T was employed and p r o b a b i l i t y f o r s m a l l v a l u e s o f C l were n e g l e c t e d . The d i s t r i b u t i o n £ £ ^ t d e t e r m i n e d by t h e CDFT method, t h e r e f o r e , r e l a t e s f q L 1 t o f ^ , ^ T b u t n o t t o f _ . , . The o s c i l l a t i o n a t t h e l o w e r end o f f _ t , a r e ^ c o r r e c t i o n s " CL1 EL' by t h e F o u r i e r T r a n s f o r m s t o a c c o u n t f o r t h e m i s s i n g e n t r i e s i n f ^ ^ The s i g n i f i c a n c e o f t h i s " c o r r e c t i o n " i s t h a t f ^ , d e t e r m i n e d f r o m £QL1 a n c * £CL1 T ^S s V e c i £ i c t 0 t n e s e d i s t r i b u t i o n s . A l t h o u g h , when f ^ , i s combined w i t h f q L l b y u s i n g t h e CDFT method c o r r e c t l y b ack c a l c u l a t e s £CL1' T w n * c n a g r e e s f u l l y w i t h o r i g i n a l d i s t r i b u t i o n o f f Q l 1 • ( F i g u r e 5-18), a p p l y i n g f ^ , t o f q j ^ p r o d u c e d f ^ ^ ' w n i c n showed s i g n i f i c a n t d e v i a t i o n s a t t h e l o w e r end o f t h e d i s t r i b u t i o n ( F i g u r e 5-21). However, t h e d e v i a t i o n i s seen t o be l i m i t e d t o o n l y t h e l e a s t few e n t r i e s w i t h i n , t h e r a n g e o f s i g n i f i c a n t v a l u e s o f C2. Based on t h e r e s u l t s o f S e r i e s #1 t o #4, i t c a n be seen t h a t t h e CDFT method a c c u r a t e l y d e t e r m i n e s t h e d i s t r i b u t i o n f g L , g i v e n t h e d i s t r i b u t i o n s f q L 1 and f C L 1 ; f g L , can t h e n be u s e d t o p r e d i c t f ^ ^ t f r o m £QL2" ^ a^- s o o b s e r v e d t h a t t r u n c a t i o n o f t h e d i s t r i b u t i o n f ^ ^ l e a d s t o d i s t o r t i o n i n f ^ , and f ^ , d e t e r m i n e d by t h e CDFT method. T h i s and o t h e r o b s e r v a t i o n s were f u r t h e r i n v e s t i g a t e d s u b s e q u e n t l y . 5.2 SQ-vUi-LtlvAXy oj tkz CVFT MoAhod I n t h e n e x t 4 s e r i e s o f computer r u n s , t h e d i s t r i b u t i o n s u p p l i e d t o t h e CDFT method were m o d i f i e d t o i n c l u d e v a r i o u s i m p e r f e c t i o n s i n o r d e r t o e v a l u a t e t h e s e n s i t i v i t y o f t h e method. As shown l a t e r , t h e e f f e c t o f i n s t r u m e n t e r r o r does n o t r e q u i r e n u m e r i c a l c o m p u t a t i o n . -14.0 -12.0 -10.0 -8.0 -6.0 "4.0 "2.0 LOG C2 , LOG C2' F i g u r e 5.21: Comparison o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s , f C L 2 > f C L 2 * o f Ground L e v e l C o n c e n t r a t i o n #2, C2, C2' (G/m 3) i n s e r i e s #4 5.2.1 Sznloj> 5. E ^ z c t oj Tmnacution a. VibtAXbuJtion [VanX 7) 71 The effect of "truncation", or unavailable data at one end of the distribution f ^ i w a s f u r t h e r examined in this series.' The distributions ^QLl* ^CLl' a n d ^QL2 £ r o m Series 4 were again used. However, only 100 entries at the higher end of the 256-entry distribution f ^ j were considered. In other words, i t was assumed that data were available only for CI ranging from 10~2 to 10"7 G/m3. The truncated f C L I T (Figure 5-22) supplied to the CDFT method together with f q ^ * again produced f ^ , which showed pronounced os c i l l a t i o n with probability values outside the range of 0 to 1, especially at the lower end (Figure 5-23). Direct comparison with f ^ (Figure 5-17) again showed almost no apparent resemblance becuase of the difference in scale; but the cumulative probability distributions;in the same scale can be seen to match quite well for the significant values of E.(Figure 5-24). The deviations at the lower end appears to affect the entries more significantly than in Series 4. This i s logical becuase there was more information lacking at the lower end of f ^ i j used in this series. As expected, fQJJ , determined from f ^ , agreed very well with f ^ ^ j for a l l 100 entries CFigure 5-25). The prediction of f C L 2 i using f ^ , showed more pronounced deviations at the lower end. Small oscillations is seen for a l l other entries (Figure 5.26). 5.2.2 Sestiz* 6. EjjzcX oj Tmincation oj a. Vi&tAlbivbbon [Pcutt 2) The effect of truncation at the lower end of the f ^ ^ j distribution is further examined in this series. Only 60 entries corresponding to the highest CI values of 10" 5 to 10~2 G/m3 were used. Since most GLC measuring instruments have a range of about 10 3, the present distribution ^CLl w^** better represent f i e l d data than the distribution used in 0.040 H LOG Cl F i g u r e 5.22: T r u n c a t e d D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n , f ^ j ^ o f Ground L e v e l C o n c e n t r a t i o n #1 C1,T (G/m 3) i n s e r i e s #5 F i g u r e 5.24: C o m p a r i s o n o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s , F , F o f C h a r a c t e r i s t i c f u n c t i o n s E, E» (s/m 3) i n s e r i e s #5 -U 0.040 H X FROM GAUSSIAN METHOD <!> FROM CDFT METHOD C L l , LOG Cl , LOG C l ' F i g u r e 5.25: Co m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f , f C L ] , o f Ground L e v e l C o n c e n t r a t i o n #1, C l , C l ' (G/m 3) i n s e r i e s "5 0.040 A X FROM GAUSSIAN METHOD <!> FROM CDFT METHOD CL2 LOG C2 , LOG C2' F i g u r e 5.26: C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 2 » fcL2' o f G r o u n d L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #5 S e r i e s 4 and 5. O s c i l l a t i o n s a r e a g a i n o b s e r v e d i n f g ^ , ( F i g u r e 5-27). I t i s n o t a b l e t h a t t h e a m p l i t u d e o f t h e o s c i l l a t i o n s i n c r e a s e s as t h e number o f e n t r i e s i n t h e d i s t r i b u t i o n d e c r e a s e s . C o m p a r i s i o n o f f ^ a n c * f g ^ , i n F i g u r e 5-28 show l a r g e r d e v i a t i o n s t h a n i n t h e p r e v i o u s s e r i e s . However, f ^ n a n d £CL,1.I s t i l l show v e r y good agreement ( F i g u r e 5-29) . Two p r e l i m i n a r y c o n c l u s i o n s can be drawn a t t h i s s t a g e . F i r s t , t h e d i s t r i b u t i o n f _ T d e t e r m i n e d from f „ T , , and f_ T.. u s i n g t h e CDFT method EL QL1 C L l i s s p e c i f i c t o t h i s p a i r o f d i s t r i b u t i o n s . T h e r e f o r e , d e t e r m i n a t i o n o f f C L l * £ r o m £ £ L ' a n d f Q L l w i l 1 a l w a y s produce f C L 1 - , w h i c h i s i d e n t i c a l t o f c L l ' Second, t h e d i s t r i b u t i o n f g ^ , d e t e r m i n e d f r o m an i n c o m p l e t e o r t r u n c a t e d d i s t r i b u t i o n f O T , _ may d i f f e r f rom t h e G a u s s i a n g e n e r a t e d f „ T i n t h a t f C T , w i l l c o n t a i n o s c i l l a t i o n and d i s t o r t i o n s . The b e s t way t o a s s e s s t h e v a l i d i t y o f fg^> t h e r e f o r e , i s t o a p p l y i t t o p r e d i c t f ^ - ? from f Q ^ a n < * c o m p a r e t h e r e s u l t i n g f ^ ^ ' W l t n t n e G u a s s i a n g e n e r a t e d f ^ ^ 2 > I n a l l s u b sequent t e s t s emphasis w i l l be p l a c e d on co m p a r i s o n s o f f C L 2 a n d f C L 2 ' ' The c o m p a r i s o n o f f Q L 2 a n c ^ £ C L 2 ' 1 S s n o w n i n F i g u r e 5-30, w h i c h r e s e m b l e s t h e p r e v i o u s F i g u r e 5-29 f o r S e r i e s 5 i n t h a t t h e pronounced d e v i a t i o n s a t t h e l o w e r end o f t h e d i s t r i b u t i o n a f f e c t e d o n l y a f i x e d number o f e n t r i e s . T h i s number can be l i n k e d t o t h e number o f e n t r i e s i n t h e e m i s s i o n r a t e d i s t r i b u t i o n s : f q L 1 and f q L 2 ' * n o t n e r words, i f t h e d i s t r i b u t i o n f ^ j ^ n a s a r a n g e °f a n ^ f q j j a n c * £ Q L 2 ^ a s a r a n S e o £ 1 0 0 " 5 s t h e n t h e l o w e r I O 0 ' 5 r a n g e s o f v a l u e s o f f C L 2 t w i l 1 be d i s t o r t e d . However, i n r e g i o n s o f f ^ j ^ i n o t a f f e c t e d by t h i s d i s t o r t i o n , f ^ ^ 2 ' i s s t i l l o b s e r v e d t o o s c i l l a t e about f C L 2 - I t ; * s i n t e r e s t i n g t o n o t e t h a t t h e p e r i o d o f o s c i l l a t i o n i s 2 i n t e r v a l s , w h i c h suggests", t h a t i f each a d j a c e n t two v a l u e s a r e a v e r a g e d , t h e n f C L 2 , and f C L w i l l be i n good 140.0 100.0 1 60.0 20.0 20.0 - 6 0 . 0 LOG E' FIGURE 5.27: D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n , i ^ , o f C h a r a c t e r i s t i c f u n c t i o n E' (s/m 3) d e t e r m i n e d by CDFT method i n s e r i e s #6 i.o H 0.8 0.6 0.4 H 0.2 i 0.0 -8.0 * FROM GAUSSIAN METHOD <J> FROM CDFT METHOD 7.0 - 6 0 LOG E , LOG E' 1— -5.0 Figure 5.28: Comparison of Cumulative Logarithmic Probability Distributions, F^, F of Characteristic functions E, E' (s/m3) in series 116 0.4 X FROM GAUSSIAN METHOD • FROM CDFT METHOD -5 .2 - 4 . 6 -4 .2 - 3 . 8 -3 .4 -3.0 -2.6 LOG C l , LOG Cl 1 00 F i g u r e 5.29: Co m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f p » f p f , o f Ground ° Li IJ 1 Li IJ 1 L e v e l C o n c e n t r a t i o n #1 C l , C l ' (G/m 3) i n s e r i e s #6 0.4 I LOG C 2 , LOG C2* F i g u r e 5.30: C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 2 » £ C L 2 * o £ Ground L e v e l C o n c e n t r a t i o n W2 C2, C2' (G/m 3) i n s e r i e s #6 agreement. T h i s p r o c e d u r e w i l l be c a r r i e d o u t i n S e r i e s 7. 5.2.3 SeAi.es 7. E ^ z c t oj InteAval & l z z oj VM>tAA.bailon6 I n t h i s s e r i e s , t h e i n t e r v a l s i z e u sed i n t h e d i s t r i b u t i o n s f q ^ ' ^QL2' ^ C L l ' a n d ^CL2 w e r e r e d u c e d by h a l f , f r o m 0.10 t o 0.05 on t h e l o g j Q s c a l e . Each d i s t r i b u t i o n was, t h e r e f o r e , d e f i n e d b y t w i c e as many e n t r i e s as i n S e r i e s 6. The p u r p o s e was t o see i f t h e s m a l l e r i n t e r v a l s i z e h e l p s t o enhance t h e a c c u r a c y o f t h e CDFT method. Comparison o f t h e d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s ( F i g u r e 5-31) £ C L 2 a n d ^CL2' s n o w e d a p p a r e n t l y d i s t i n c t d e v i a t i o n s between them. T h i s was m a i n l y due t o t h e s m a l l s c a l e u s e d f o r t h e p r o b a b i l i t y a x i s . When t h e c o r r e s p o n d i n g c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s were compared ( f i g u r e 5-32). I t was seen t h a t t h e d e v i a t i o n s were alm o s t i n s i g n i f i c a n t . N o n e t h e l e s s , r e d u c t i o n o f t h e s e d e v i a t i o n s was d e s i r a b l e . When t h e a d j a c e n t v a l u e s o f p r o b a b i l i t y o f f ^ ^ ' w e r e added, so t h a t t h e i n t e r v a l s i z e d o u b l e f r o m 0.05 t o 0,10, a r e d u c t i o n o f t h e o s c i l l a t i o n o f f ^ j ^ ' a D 0 U t £QL2 w a s o b s e r v e d . ( F i g u r e 5-33). F u r t h e r d o u b l i n g o f t h e i n t e r v a l s i z e t o 0.2 p r o d u c e d a d i s t r i b u t i o n f ^ ^ ' w h i c h a g r e e d v e r y c l o s e l y w i t h f ^ ^ ( F i g u r e 5-34). However, d e v i a t i o n s a t t h e l o w e r end due t o t r u n c a t i o n s t i l l p e r s i s t . These r e s u l t s i n d i c a t e d t h a t i f t h e p r e d i c t i o n o f f ^ ^ t ^ s t 0 be a c c u r a t e f o r a c e r t a i n i n t e r v a l s i z e ( e g . 1 0 ° * 2 ) , t h e n t h e d i s t r i b u t i o n s ^ Q L l ' ^QL2 J a n d ^ C L l m u s t D e m e a s u r e d and s p e c i f i e d t o a t l e a s t h a l f t h a t i n t e r v a l s i z e Qi.e. 10°'- 1), and p r e f e r a b l y t o one q u a r t e r ( i . e . 10 ) . 0.2 H 0.16 0.12 H f f ' C L 2 , ' C L 2 ' 0.08 i 0.04 H 0.0 F i g u r e 5.31 -5 .0 - 4 . 5 -4.0 -3.5 -3.0 LOG C 2 , LOG C2' C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #7 ( P a r t 1) 2.5 2.0 CL2' fCL2« ° f G r O U n d 00 F F ' C L 2 , 1 CL2 ' -4.5 -4.0 - 3 . 5 - 3 . 0 LOG C 2 , LOG C2' F i g u r e 5.32: C o m p a r i s o n o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s 'Q^ ' ' c L 2 ' ° £ G r o u n t * L e v e l C o n c e n t r a t i o n C 2 , C2' (G/m 3) i n s e r i e s #7, ( P a r t - 1 ) 00 0.4 H 0.32 H 0.24 H f C L 2 f fcL2' 0.16 i 0.08 1 0.0 -5.2 <!> <2> <!> X FROM GAUSSIAN METHOD <> FROM CDFT METHOD <!> <I> -4.8 -4.4 -4.0 -3.6 LOG C 2 , LOG C2* -3.2 -2.8 F i g u r e 5.33: C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f ^ ^ ' £ C L 2 ' °^ Ground L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #7 ( P a r t 2) 00 X FROM GAUSSIAN METHOD <t> FROM CDFT METHOD 0.08 H 0.06 -CL2 ffcL2' 0.04 " 0.02 -0.0 F i g u r e 5.34: C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f r I 0 , f_._, o f Ground L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #7, ( P a r t 3) 5.2.4 SeAite S: EJJQJCZ oj Round OH EJViou In ?nabob WLtiiis 87 I n a l l p r e v i o u s s e r i e s , t h e p r o b a b i l i t y v a l u e s were g e n e r a t e d by t h e G a u s s i a n m o d e l , and t h e s e v a l u e s were a c c u r a t e t o a t l e a s t 5 d e c i m a l p l a c e s . When t r e a t i n g a c t u a l f i e l d measured d a t a , s u c h a c c u r a c y i s u s u a l l y i m p o s s i b l e . F o r example, i f h o u r l y d a t a o v e r a p e r i o d o f one y e a r i s u s e d , t h e l o w e s t p r o b a b i l i t y t h a t c a n be o b s e r v e d i s + 1/8760. When d a t a a r e c o l l e c t e d a t l e s s e r f r e q u e n c y , t h i s minimum o b s e r v a b l e p r o b a b i l i t y w i l l i n c r e a s e . In o r d e r t o examine t h e s e s i t u a t i o n s a s h o r t computer p r o g r a m "TRM" (A p p e n d i x I I - D ) was w r i t t e n t o t a k e t h e d i s t r i b u t i o n s F Q ^ J , FQT,2» F C L 1 F C L 2 f r o m s e r i e s 5> a n d r o u n d o f f t h e p r o b a b i l i t y v a l u e s t o t h e n e a r e s t 5/8760 and 10/8760, t h e r e s u l t i n g d i s t r i b u t i o n s were u s e d t o d e t e r m i n e f E L ' a n d f C L 2 ' a s i n P r e v i o u s s e r i e s . The c o m p a r i s o n o f f C L 2 » FQL2' A R E S N O W N i n F i g u r e 5.3 5 and F i g u r e 5.36 f o r t h e n e a r e s t m e a s u r a b l e p r o b a b i l i t y v a l u e o f 5/8760. F i g u r e s 5.37 and 5.38 a r e s i m i l a r g r a p h s f o r t h e 10/8760 c a s e . The good agreements i n t h e r e s u l t s showed t h a t even i n t h i s c a s e when t h e sample s i z e i s s m a l l , t h e CDFT method i s s t i l l r e l i a b l e . 0.0404 0.0324 0.0244 0.016 H 0.008H 0.000 X FROM GAUSSIAN METHOD ® FROM CDFT METHOD F i g u r e 5.35: LOG C2 , LOG C2' C o m p a r i s o n o f D i s c r e t e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s f C L 2 > £ C L 2 ' ° £ G r o u n d L e v e l C o n c e n t r a t i o n #2 C2, C2' (G/m 3) i n s e r i e s #8 ( P a r t 1) 00 00 LOG C2 , LOG C2 ' F i g u r e 5.36: Co m p a r i s o n o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s F C L 2 » F C L 2 ' ° f G r o u n d L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s 118, ( P a r t 1) X FROM GAUSSIAN METHOD <> FROM CDFT METHOD L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #8, ( P a r t 2) -4.8 r 3.6 -3.2 F i g u r e 5.38: -4.4 - 4 . 0 LOG C2 , LOG C2' C o m p a r i s o n o f C u m u l a t i v e L o g a r i t h m i c P r o b a b i l i t y D i s t r i b u t i o n s F L e v e l C o n c e n t r a t i o n C2, C2' (G/m 3) i n s e r i e s #8 ( P a r t 2) C L 2 1 -2.8 ^CL2' ° £ G r o u n d 92 5.2.5 Jnitmrnznt EAAOU One d i s t i n c t d i f f e r e n c e between t h e p r o b a b i l i t y v a l u e s g e n e r a t e d by computer models and t h o s e c o m p i l e d f r o m f i e l d measurements i s t h a t t h e l a t t e r a l w a y s c o n t a i n some i n s t r u m e n t e r r o r s . I t i s d e s i r a b l e t o know i f t h e CDFT method i s s e n s i t i v e t o t h e s e e r r o r s and i f i t compounds t h e s e e r r o r s . T h i s p r o b l e m can be examined w i t h o u t r e s o r t i n g t o n u m e r i c a l c o m p u t a t i o n s . O r d i n a r i l y , i n s t r u m e n t e r r o r s t e n d t o be n o r m a l l y d i s t r i b u t e d o v e r a range o f s a y + P%. T h i s e r r o r can a l s o be d e s c r i b e d by t h e use o f e r r o r f a c t o r R, where R ran g e s from ^ • = CI - y ^ ) t o R £ = (1 :+ -~) . F o r a v a r i a b l e o f t h e measured v a l u e , M, w i l l t h e n range from R X and R 2X w i t h c e r t a i n p r o b a b i l i t i e s . I f a c o l l e c t i o n o f measured v a l u e s a r e c o n s i d e r e d , t h e p r o b a b i l i t y d i s t r i b u t i o n o f M, R and X a r e r e l a t e d by t h e f o l l o w i n g r e l a t i o n -s h i p : M = RX (5 .1) l o g M = l o g R + l o g X C5.2) o r ML = RL + XL where ML = l o g M, RL = l o g R, XL = l o g X (5.3) S i n c e t h e e r r o r R i s random and t h e r e f o r e i n d e p e n d e n t o f x t h e c o n v o l u t i o n t h e o r y a p p l i e s , i . e . : f M L = £ R L ® £ X L ( 5 ' 4 ) From t h e t h e o r y s e c t i o n , we have f o r t h e p r e d i c t i o n f ^ 2 : £ C L 2 = F Q L 2 ® £ E L = £QL2 ® {£CL1 © £QL1} = F " 1 ( F { f Q L 2 } x F { f C L 1 > / F { f Q U > ) (5.5) 93 We can define f^.-, f r i _ to be the probability distributions in logarithmic QLt LLC scale for the instrument errors corresponding to the measurement of emission rate and GLC, then, £ QLl (with error) = = f* QLl ® f QLE (5.6) FQL2 (with error) = = f* QL2 £QLE (5.7) £ CLl (with error) = -. f* CLl £ CLE (5.8) £ RCL2 (with error) = : £ CL2 f CLE (5.9) If we replace fqL2> Q^LI» ^ CLI E cl u a t :^ o n 5.5 with corresponding ( f ^ L 2 with errors) ( f q L l with errors) and ( f ^ L 1 with errors). Equation (5.5) can be rewritten as: FCL2 " RL^£QL2 X £QLE} X HFCL2 X £CLE}/F{FQU X £QLE}) _ r l F ( R L^ { £QL2 } X F{£QLE}) X F F R^F { £CL2 } X F{£CL2} )^ _  rl F { £ QV X F{£QLE} X F{£^L2} X F{£CLE} F{£$L1} X F{£QLE} ' F"V{£QL2} x F[f* L 2} x F f f ^ } x F {£ C L E}/F {f* L 2» = £QL2 ® £CL2 -® £CLE © £QL1 = FCL2 ® £CLE C 5- 7 ) In Equation 5.7, the errors for the emission rate cancelled out so that the predicted f h a s the same error component as f ^ i * the measured distribution. It i s shown, therefore:, that the effect of instrument error i s not multiplied or accumulated by the use of the CDFT method. The distribution f >predicted by the CDFT method contains the same instrument errors as i f 94 i t was measured by t h e same i n s t r u m e n t u s e d f o r d e t e r m i n i n g f ^ L 1 . 5.3 Example* oj the CVFT Methods 5.3.1 Practical Example, 0& the Analytical CDFT Method 95 Situation: A power plant which emits SO2 from i t s boiler at the rate of Ql tons/day (hourly average) i s causing the GLC to be CI ppm (hourly average) at a nearby location. Ql and CI are measured over the period of 1 year. Both variables are lognormally distributed such that: Ql. Geometric Mean G^ = 100 tons/day Standard Geometric Deviation = 1.5 CI. Geometric Mean = 0.02 ppm Standard Geometric Deviation S^ = 1.75 The Governmental GLC standard for S0 2 i s C,. = 0.25 ppm (hourly average) not 8760J to be exceeded more than once in each year (i.e. Probability < RIL n)• Problem: 1. Is the plant currently meeting the GLC standard for SO2 at the given location? 2. If the boiler i s to be replaced by one that emits Q2 tons/day (hourly average), w i l l the plant s t i l l meet the standard? Q2 i s also lognormally distributed with a Geometric Mean of GQ2 = 200 tons/day and Standard Geometric Mean of = 1*3, respectively. Solution: Since the probability distributions can a l l be represented by lognormal distribution, Analytical solution can be used: Let QL1 = log Ql; CL2 = log CI, ELI = Log E l . Since Ql, CI are lognormally distributed, QL1, CLl are also normally distributed, such that: y C L l = m e a n ° f C L 1 = l 0 g GC1 ( 5" 8 ) u Q L 1 = mean of QL2 = log G Q 1 (5.9) o*CL1 = standard deviation of CLl = log (5.10) 96 CQL2 = standard deviation of QL2 = log (5.11) In the theory section, i t was shown that for: C = EQ (3.1) £CL " £EL ® £QL C3 . 8 ) m d yCL = UEL + yQL ( 3 - 1 0 ) a 2 = a 2 + a 2 (3.11) CL EL QL Since the normal distribution i s uniquely defined by i t s mean (u) and standard deviation (a), these equations can be used to solve the problem. Substituting (5.8), (5.9) into (3.10): log G c = log G E + log G Q or G C = G E ® G Q . (5.12) Substitute (5.10), (5.11) into (3.11) l o g 2 S c = l o g 2 S E + log 2 S Q (5.13) Assuming that the physical, meteorological and topographical conditions are unchanged from one period to another then the probability distribution of the Characteristic function would remain unchanged, i.e. G_ = constant E S_ = constant b Equations 5.12 and 5.13 then give GC1 / GC2 = VGQ2 ( 5 - 1 4 ) and l o g 2 S C 1 - l o g 2 S c 2 = l o g 2 S Q 1 - log 2 S Q 2 (5.15) Substituting numerical values: i ) The probability distribution of Cl i s plotted as a straight line on log-probability paper, (Figure 5 T 39.1, such that: % PROBABILITY C IS EXCEEDED F i g u r e 5 - 3 9 : A p p l i c a t i o n o f t h e A n a l y t i c a l CDFT Method C l ( 5 0 % ) = G C 1 = ° ' 0 2 p P m (^(15.87%) = G c l x 6 2 = 0.02 x 1.75 = 0.035 ppm I t i s seen t h a t f o r C = 0.25 ppm. Prob {C.>C } < 0.01% x — s — T h e r e f o r e , t h e number o f t i m e s C i s exceeded i n one y e a r i s l e s s s t h a n 8760 x 0.0001 = 0.876 h r . T h e r e f o r e , t h e s t a n d a r d i s c u r r e n t l y n o t v i o l a t e d , i i ) U s i n g E q u a t i o n ( 5 . 1 4 ) : GC2 = G C 1 X GQ2^ GQ1 = 0 , 0 2 X 2 0 ° / 1 0 0 = ° - 0 4 PPm U s i n g E q u a t i o n ( 5 . 1 5 ) : S c 2 = expClog2 S c l + l o g 2 S Q 2 - l o g 2 S Q 1 ) 1/2 1/2 = expClog2 1.75 + l o g 2 1.3 - l o g 2 1.5} J = 1.595 .'. C2(50%) = G c 2 = 0.04 ppm ' •. C2(15.87%) = G C 2 x S c 2 = 0.064 ppm The l i n e C 2 i s p l o t t e d on t h e same f i g u r e . I t i s seen t h a t : P r o b (C 2>C ) £ 0.01% T h e r e f o r e , t h e s t a n d a r d w i l l s t i l l be met when t h e new b o i l e r i s u s e d . 5.3.2 Practical example, oj the numerical CVFT method I n c a s e s where t h e d a t a c o l l e c t e d f o r e m i s s i o n r a t e s and GLC c a n n o t be f i t t e d by c o n t i n u o u s p r o p a b i l i t y d i s t r i b u t i o n s t h e n u m e r i c a l can be us e d . S i t u a t i o n : The e m i s s i o n r a t e s o f SO2 f r o m a s m e l t e r s t a c k was measured o v e r a p e r i o d o f 1 y e a r and 8760 h o u r l y a v e r a g e v a l u e s were c o l l e c t e d . I n a n e a r b y t o w n s h i p , t h e h o u r l y a v e r a g e s o f t h e ground l e v e l 99 c o n c e n t r a t i o n o f SO2 were s i m i l a r l y r e c o r d e d i n t h e same y e a r . The s m e l t e r i s now p l a n n i n g t o expand i t s P r o d u c t i o n , by r e p l a c i n g t h e o l d f u r n a c e s ' w i t h a new l a r g e r u n i t . The e m i s s i o n r a t e s o f SO2 f r o m an i d e n t i c a l new f u r n a c e i n a n o t h e r p l a n t had been m o n i t o r e d i n the l a s t y e a r i n terms o f h o u r l y a v e r a g e s . The l o c a l GLC s t a n d a r d f o r SO2 i s C g G/m3 ( h o u r l y a v e r a g e ) n o t t o be e x c e e d e d more t h a n once e v e r y y e a r . P r o b l e m : W i l l t h e new f u r n a c e f i r i n g a t t h e new r a t e meet t h e GLC s t a n d a r d ? A s s u m i n g t h e l o c a l w e a t h e r p a t t e r n does n o t change s i g n i f i c a n t l y f r o m t h i s y e a r t o n e x t ? S o l u t i o n : A p r o c e d u r e s i m i l a r t o t h e one o u t l i n e d i n t h e t h e o r y s e c t i o n ( F i g u r e 3.2 ) can be u s e d : 1. Group t h e d a t a f o r o l d e m i s s i o n r a t e s , o l d GLC and new e m i s s i o n r a t e s i n t o l o g a r i t h m i c p r o b a b i l i t y d i s t r i b u t i o n s f q L 1 > f r j L l ' £QL2 r e s p e c t i v e l y . 2. A p p l y t h e p r o gram P.A.D. t o d e t e r m i n e f £ L f o r t h e c h a r a c t e r i s t i c f u n c t i o n . 3. A p p l y t h e p r o g r a m P.A.D. t o d e t e r m i n e f ^ . f o r t h e new GLC's f r o m t h e d i s t r i b u t i o n s f q L 2 ^^h' 4. D e t e r m i n e f r o m t h e d i s t r i b u t i o n f ^ ^ 2 ' t n e p r o b a b i l i t y o f GLC>C s G/m3. I f t h e p r o b a b i l i t y i s l e s s t h a n 1/8760 o r 0.0001142, t h e n t h e new f u r n a c e w i l l meet t h e l o c a l GLC s t a n d a r d . 5.4 Extension the. CVFT method The CDFT method has so f a r been d e v e l o p e d f o r t h e s i t u a t i o n i n v o l v i n g one e m i t t e r and one r e c e p t o r i n w h i c h t h e GLC a r e d i r e c t l y p r o p o r t i o n a l t o t h e e m i s s i o n r a t e s . Two p o s s i b l e f u t u r e e x t e n s i o n s o f t h e method a r e o u t l i n e d b e low. 100 5.4.1 The. Cast whexe. C cc Q n I n some c a s e s , t h e GLC i s p r o p o r t i o n a l n o t t o t h e e m i s s i o n r a t e , Q, but t o Q n, where t h e v a l u e o f n f 1. T h i s i s u s e f u l i n t h e s t u d y o f some p h o t o c h e m i c a l l y a c t i v e p o l l u t a n t s . The p r o b a b i l i t y d i s t r i b u t i o n o f Q n i s d i f f i c u l t t o o b t a i n , s i n c e Q n i s n o t l i n e a r l y dependent on Q. However, t h e i n t r o d u c t i o n o f l o g a r i t h m i c s , l o g Q n = n l o g Q, overcomes t h i s p r o b l e m . G i v e n t h e p r o b a b i l i t y d i s t r i b u t i o n o f QL ( i . e . l o g Q) t h e p r o b a b i l i t y d i s t r i b u t i o n o f QL = n*QL i s e a s i l y o b t a i n e d by m u l t i p l i c a t i o n and i n t e r -p o l a t i o n s . The v a r i a b l e QL* and i t s p r o b a b i l i t y d i s t r i b u t i o n f * can t h a n QL be u s e d t o r e p l a c e QL and f n T i n a l l t h e e q u a t i o n s d e v e l o p e d i n e a r l i e r QL s e c t i o n s . 5.4.2 The Case. With Two SouAces and One R2.ce.pt0A. I n t h e t h e o r y s e c t i o n , i t was shown t h a t f o r A = B+C, f = f ® f^,, p r o v i d e d B and C a r e i n d e p e n d e n t . And f o r A = BxC, f ^ = f g ^ ® f ^ L where AL = l o g A, BL = l o g B, CL = l o g C. These p r o p e r t i e s can be u s e d t o e v a l u a t e t h e GLC and e m i s s i o n r a t e d a t a i n v o l v i n g two e m i t t e r s o f t h e same p o l l u t a n t and one r e c e p t o r , i n a s i t u a t i o n shown i n F i g u r e 5 . 4 0 . E m i t t e r #1 Plume #1 ^ — «^  ^ ^ ^ ^ Q R e c e p t o r ( M o n i t o r ) ^— ^ E m i t t e r #2 Plume #2 ^ F i g u r e 5 40 : The Case I n v o l v i n g Two S o u r c e s and One R e c e p t o r In t h i s c a s e , t h e p o l l u t a n t i s e m i t t e d by E m i t t e r # 1 a t e m i s s i o n r a t e Q and e m i t t e d by e m i t t e r # 2, a t e m i s s i o n r a t e Q 9, b o t h c o n t r i b u t i n g t o 1 z t h e GLC a t t h e r e c e p t o r . The i n d i v i d u a l c o n t r i b u t i o n s a r e C , and C 101 r e s p e c t i v e l y . T h e r e f o r e , t h e measured GLC a t r e c e p t o r , r = C + C b e f o r e t h e change (5.16) TI IA 2 C - = C O D + C 0 a f t e r t h e change (5.17) T2 2B ^ S i n c e : C J A = Q 1 A» E± (5-18) C 1 B = Q1B-E P.») C 2 . Q 2 - E 2 C5.20) t h e n ]T1 " C T 2 ~ C 1 A ' C 1 B = E 1 C Q 1 A " W •'• E l = ( C T 1 " C T 2 ^ 1 A " <W C 5' 2 1 ) = AC/AQ1 (5.22) H e r e , i n e q u a t i o n s 5.18 and 5.19, i t i s assumed t h a t t h e c h a r a c t e r i s t i c f u n c t i o n E, r e l a t i n g t h e f i r s t e m i s s i o n s o u r c e t o t h e r e c e p t o r r e m a i n s unchanged. T h i s can be a c h i e v e d by s e l e c t i n g p a i r s o f C ^ , and C 2 A , Q 2 A d a t a c o r r e s p o n d i n g o n l y t o s i m i l a r c o n d i t i o n s o f w i n d speed, w i n d d i r e c t i o n and a t m o s p h e r i c s t a b i l i t y . I n e q u a t i o n 5.22 t h e p r o b a b i l i t y d i s t r i b u t i o n o f C^, C ^ J ^ Q I A' GQ1B a r e a l l known, t h e r e f o r e , t h e p r o b a b i l i t y d i s t r i b u t i o n o f E, can be f o u n d by t h e f o l l o w i n g o p e r a t i o n : f A C = f C T l © fCT2 C5-23) £AQ1 = £Q1A © £Q1B C 5 - 2 4 )  £ E L 1 " £ A C L © £AQL1 ( 5 " 2 5 ) where EL = l o g E, AQL1 = l o g AQ1, ACL1 = l o g AC1, I n v e r s e l o g a r i t h m i c CE1 E L I t r a n s f o r m a t i o n d i s c u s s e d e a r l i e r a r e u s e d t o o b t a i n f . - L 1 { f A f t e r t h e d e t e r m i n a t i o n of f ,, f i s e a s i l y d e t e r m i n e d by t h e r e l a t i o n s h i p s : E l E2 102 E 2 = C2/Q2 = (CT1 - C ] A)/Q 2 = CC n - Q^D/Qj t h 6 n : £C2 = fCTl © £C1A C5'23> £CL1A = £QL1A ® £EL1 (5.24) £EL2 " £CL2 © £QL2 <5' 25> £E2 B L " 1 { £ E L 2 } C5-26^ again C L ^ = log C 1 A; QL 1 A = log Q 1 A > EL^ = log E2- Once both f and f 2 are both determined, the effect of any new emission rate Q, and Q. ' l,new 2,new can be determined by the following relationships: CT,new = Cl,new * C2,new = Ql,new E l + Q2,neti E2 ( 5' 2 7) ^ £CT,new " £C1,new ® £C2'new C5'28^ £Cl,new= r l [ L { £ Q l , n e w } ® L { f E 2 } ^ -+ ^ ^ e W * ® L { £E2 } 1 " C5- 2 9) This particular extension of the CDFT method should find useful application as i t can be used to distinguish the relative contributions of two pollutant sources simultaneously emitting to the same location. This a b i l i t y to assess relative contributions w i l l be a very useful tool in deciding at what rate should each emitter be allowed to release the pollutant, so that the combined GLC w i l l meet the standards. 6. CONCLUSIONS 103 The C o n v o l u t i o n , D e c o n v o l u t i o n u s i n g F o u r i e r T r a n s f o r m s (CDFT) method has been shown t o be a p p l i c a b l e t o e m i s s i o n r a t e and ground l e v e l c o n c e n t r a -t i o n d a t a f o r t h e p u r p o s e o f C h a r a c t e r i z a t i o n and P r e d i c t i o n , u s i n g t he r e l a t i o n s h i p C = EQ. The CDFT method can be a p p l i e d a n a l y t i c a l l y o r n u m e r i c a l l y d e p e n d i n g on the n a t u r e o f t h e e m i s s i o n r a t e and ground l e v e l c o n c e n t r a t i o n d a t a : 1. I f t h e s e d a t a can be f i t t e d by l o g - n o r m a l o r o t h e r s i m i l a r d i s t r i b u t i o n s , t h e n the f r e q u e n c y d i s t r i b u t i o n f £ ^ o f t h e C h a r a c t e r i s t i c F u n c t i o n E can be a n a l y t i c a l l y d e t e r m i n e d . I n a d d i t i o n , t h e new ground l e v e l c o n c e n t r a t i o n d i s t r i b u t i o n f ^ . ^ ^ n r e s P o n s e t 0 a new e m i s s i o n r a t e d i s t r i b u t i o n f q L 2 c a n a l s o be p r e d i c t e d . T h i s new p r o b a b i l i t y d i s t r i b u t i o n c a n t h e n be u s e d , f o r example, t o a n t i c i p a t e c o m p l i a n c e w i t h g o v e r n m e n t a l s t a n d a r d s f o r t h e new e m i s s i o n r a t e p a t t e r n i n f u t u r e . 2. E m i s s i o n r a t e s and GLC d a t a t h a t cannot be a p p r o x i m a t e d by l o g -n o rmal d i s t r i b u t i o n can be t r e a t e d n u m e r i c a l l y u s i n g t h e computer program "Program f o r A n a l y s i s o f D i s t r i b u t i o n s " (P.A.D) t o n u m e r i c a l l y d e t e r m i n e f and f ^ L 2 u s i n g t h e n u m e r i c a l CDFT method. T e s t i n g o f t h e n u m e r i c a l CDFT method i n d i c a t e d t h a t s e v e r a l p o s s i b l e e r r o r s can be i n t r o d u c e d by t h e n u m e r i c a l r e p r e s e n t a t i o n o f t h e v a r i o u s d i s t r i b u t i o n s : 1. The ma j o r s o u r c e o f e r r o r a r i s e s f r o m t h e l a c k o f d a t a a t t h e ends ( o r t a i l s ) o f t h e p r o b a b i l i t y d i s t r i b u t i o n s , e s p e c i a l l y t h e l o w e r end o f t h e GLC d i s t r i b u t i o n s due t o s e n s i t i v i t y r ange o f i n s t r u m e n t s . However, t h e s e e r r o r s a r e l i m i t e d t o a few e n t r i e s 104 a t t h e ends o f t h e d i s t r i b u t i o n s . I f t h e e r r o r can be c o n f i n e d t o t h e l o w e r end o f t h e d i s t r i b u t i o n s , i t s e f f e c t would be much r e d u c e d . T h i s can be a c h i e v e d by e n s u r i n g t h a t t h e i n s t r u m e n t s m e a s u r i n g GLC's a r e d e s i g n e d and c a l i b r a t e d so t h a t a l l o f t h e h i g h GLC v a l u e s t h a t might o c c u r would be w i t h i n the i n s t r u m e n t s ' measurement r a n g e . 2. The c h o i c e o f i n t e r v a l s i z e o f t h e d i s t r i b u t i o n has been found t o a f f e c t t h e a c c u r a c y o f t h e CDFT method a l t h o u g h t h e e f f e c t i s s m a l l . I n p r a c t i c e , s e v e r a l i n t e r v a l s i z e s s h o u l d be t r i e d t o d e t e r m i n e t h e optimum c h o i c e . 3. The e f f e c t o f sample s i z e has been found t o be n e g l i g i b l e , f o r sample s i z e s r a n g i n g from 876 t o 8760. However, h i g h e r c o n f i d e n c e l e v e l i s a l w a y s a s s o c i a t e d w i t h l a r g e r sample s i z e i n s t a t i s t i c a l method such as t h i s . 4. The e f f e c t o f i n s t r u m e n t e r r o r s a r e n o t compounded by t h e CDFT method. T h e r e f o r e , t h e e r r o r s a s s o c i a t e d w i t h t h e p r e d i c t e d d i s t r i b u t i o n f ^ ^ W 1 H be no l a r g e r t h a n t h o s e a s s o c i a t e d w i t h t h e measured d i s t r i b u t i o n f ^ j j • I t was a l s o shown t h a t t h e CDFT method can be r e a d i l y e x t e n d e d t o i n c l u d e s i t u a t i o n s when C a Q n. The p o s s i b l e e x t e n s i o n o f t h e CDFT method t o s i t u a t i o n s i n v o l v i n g two s o u r c e s and one r e c e p t o r appeared t o be e a s i l y a c h i e v a b l e . I t t h e r e f o r e w a r r a n t s f u r t h e r development. L a s t l y , i t must be emphasized t h a t t h e CDFT method i s a s t a t i s t i c a l method, w h i c h i s most u s e f u l i n d e s c r i b i n g a c o l l e c t i o n o f d a t a , and t h a t t h e r e l i a b i l i t y o f t h e method imp r o v e s as t h e sample s i z e i n c r e a s e s . 105 7. RECOMMENDATIONS I t i s recommended t h a t t h e CDFT method p r e s e n t e d be a p p l i e d t o f i e l d measured d a t a c o v e r i n g a v a r i e t y o f s i t u a t i o n s , i n o r d e r t o examine i f t h e method can be u s e f u l i n s i t u a t i o n s t h a t c o n t a i n u n c e r t a i n t i e s , e r r o r s and i m p e r f e c t i o n s t h a t may n o t have been i n c l u d e d i n the p r e s e n t d i s c u s s i o n . The a p p l i c a t i o n o f t h e CDFT method t o s i t u a t i o n s more complex t h a n s i n g l e s o u r c e - s i n g l e r e c e p t o r p a i r i s a l s o u r g e d . N O M E N C L A T U R E [in ohdvi o{ Equations) 106 a C Ground l e v e l c o n c e n t r a t i o n a t r e c e p t o r , ( G / m 3 ) E C h a r a c t e r i s t i c f u n c t i o n f o r t h e s o u r c e r e c e p t o r p a i r , ( s / m 3 ) Q S o u r c e e m i s s i o n r a t e o f p o l l u t a n t , ( G / s ) X Downwind s o u r c e t o r e c e p t o r d i s t a n c e , ( m ) y C r o s s w i n d s o u r c e t o r e c e p t o r d i s t a n c e , ( m ) R S o u r c e t o r e c e p t o r s t r a i g h t l i n e d i s t a n c e (m) Sou r c e t o r e c e p t o r direction,(° from N o r t h ) 3 Wind direction,(° fr o m N o r t h ) e e = (B-TT-OO H P h y s i c a l s t a c k h e i g h t , ( m ) A H £ Plume- r i s e f a c t o r , ( m ) . f S t a b i l i t y c o r r e c t i o n f a c t o r ( - ) S A t m o s p h e r i c s t a b i l i t y c a t e g o r i e s as d e f i n e d by P a s q u i l l , ( - ) V S t a c k gas e x i t v e l o c i t y , ( m / s ) D S t a c k e x i t d i a m e t e r T g S t a c k gas e x i t temperature,(°K) u Wind speed,(m/s) P A t m o s p h e r i c p r e s s u r e , ( m / s ) cl T & A t m o s p h e r i c temperature,(°K) a C r o s s w i n d d i s p e r s i o n c o e f f i c i e n t , ( m ) a V e r t i c a l d i s p e r s i o n c o e f f i c i e n t , ( m ) A , B , A , y y z ' r Curve f i t t i n g c o e f f i c i e n t s , ( - ) Z Z 107 REFERENCES 1. 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Zimmerman, J.R., "Some P r e l i m i n a r y r e s u l t s o f m o d e l l i n g from a i r p o l l u t i o n s t u d y o f A n k a r a , T u r k e y " P r o c . 2nd M e e t i n g o f t h e E x p e r t P a n e l on a i r m o n i t o r i n g " P a r i s , ( 1 9 7 1 ) . Zimmerman, J.R., "The NATO/CCMS a i r p o l l u t i o n s t u d y o f S t . L o u i s M i s s o u r i " , P r o c . 3 r d M e e t i n g o f t h e E x p e r t P a n e l on a i r p o l l u t i o n  m o n i t o r i n g " P a r i s , ( 1 9 7 2 ) . 109 1-1 A p p e n d i x I : Examples o f C o n v o l u t i o n and D e c o n v o l u t i o n Example I.1: A n a l y t i c a l ( C o n t i n u o u s ) C o n v o l u t i o n o f two s i m p l e d i s t r i b u t i o n s . G i v e n : P r o b a b i l i t y d i s t r i b u t i o n s , f ( X ) , and g ( X ) d e f i n e d as f o l l o w s : f ( X ) = 0<X<a ; z e r o o t h e r w i s e g(X) = K 2 0<X<a ; z e r o o t h e r w i s e F i n d : The c o n v o l v e d p r o b a b i l i t y d i s t r i b u t i o n h ( X ) s u c h t h a t : h ( X ) = f ( X ) $ g ( X ) S o l u t i o n : h ( X ) = f ( X ) $ g ( X ) oo = / f f X ^ g f X - X ^ d X ' CD (2) (3) (4) = f f(x^pc-x')dx' + f f(x ,)g(x-x ,)dx' + f nr^ia-x 0 ')dX' f ( X ' ) g ( X - X ' ) d X " (5) S i n c e f ( X ) = 0 f o r X<0 and f o r X>a. Now c o n s i d e r f r o m r a n g e s o f v a l u e s o f X: i ) X<0 (X-X')<0 f o r 0<X<a t h e n : g(X-X') = 0 h ( X ) = 0 f o r X<0 i i ) 0<X<a g(X-X') f 0 o n l y f o r 0<X-X'£a o r 0<X'<X .X _X h(X ) = ^  f ( X ' ) g ( X - X ' ) d X ' = f KxKzdX' = K i K 2 X 0 = 0<X<a (4) (5) 110 I .2 i i i ) a<x<2a g(X-X') f 0 o n l y f o r 0<X-X'<a o r X-a<x'<X X X h(x) = f f(x')gcx-x')dx' = f K aK 2dx' X-a X-a = K 1 K 2 [ a - ( X - a ) = K j K 2 ( 2 a - X ) f o r a<X<2a (6) i v ) X>2a a<(X-X') f o r 0 0<x'<a g(X-X') = 0 h(X) = 0 f o r X>_2a (7) Check; a t X=a e q u a t i o n (7) h(X) = K K X | V = K K a n J . i 2 1X=a • i 2 e q u a t i o n (8) h(X) = K j K 2 ( 2 a - X ) | x = & = K ^ a Example 1.2: C o n v o l u t i o n U s i n g C o n t i n u o u s F o u r i e r T r a n s f o r m s G i v e n : Normal D i s t r i b u t i o n s f ( X ) , g(X) I (X-u,) 2 f CX) = _ exp{- — } a £ J2n 2 a f 2 i gCX) = — l — e x p { - S—} a 7 2IT 2a- 2 g g F i n d : . C o n t i n u o u s P r o b a b i l i t y D i s t r i b u t i o n h ( X ) s u c h t h a t : h ( X) = f ( X ) ® g(X) I l l S o l u t i o n . 1. D i r e c t C o n v o l u t i o n : h(X) = f ( X ) ® g(X) = J f ( x ' ) g c x - x ' ) d x ' f ! ( x ' - y £ ) 2 ( x - x ' - v j I [ — e x p { - } e x p { - g — cr J2TT 2CT 2 a /*F 2a 2 r r 5 g (x'-y ) 2 ( X - x ' - y J 2 e x p { - i [ L _ + — £ — ] } d X 2TTCT.0 7 2- 2 2 f g d i f f i c u l t t o e v a l u a t e 2. C o n v o l u t i o n u s i n g C o n t i n u o u s F o u r i e r T r a n s f o r m : From F o u r i e r T r a n s f o r m t a b l e s , f o r : f ( X ) = A f e x p { - a 2 X 2} F ( s ) .tLHexp{. a f 4 a f 2 F ( s ) = F { f ( X - y . ) } = F ( s ) e x p { - i2TTy s} y £ t r Two theorems a r e u s e f u l h e r e : 1. S h i f t Theorem 2. C o n v o l u t i o n Theorem I f h(X) = f ( X ) ® g(X) t h e n H ( s ) = F ( s ) G ( s ) S i n c e : i ( X - y f ) 2 f (X) = — l — exp{- — ± — } a f J2n 2a£ 2 L e t f ' ( X ) = f f X + y J = — - exp{ } a £ /2TT 2 a f 2 f ( X ) = f ' ( X - u £ ) F* (S) = F { f ' (X)} = — — / T T • 2a e x p { - j - • 2 a £ 2 } = exp{ =—} F ( s ) = F { f ( X ) } = F T { f ' ( X - u £ ) > = e ' i 2 7 r u £ s F ' ( s ) s 2 a 2 F ( s ) = exp{ = — - i2Tru fs} S i m i l a r l y f o r : i (x-v2 g(X) i — e xp{- g — } a /27r 2a 2 g g 2 2 S a ( , G ( s ) = exp{ i 2 i r u s} 2 g U s i n g C o n v o l u t i o n Theorem: H ( s ) = F ( s ) G(s) S u b s t i t u t e ( 7 ) , ( 9 ) : s 2 ( a 2 + a 2 ) H( s ) = e x p { - 2 a - i 2 7 r s ( u £ + y g ) } H ( s ) c a n be r e w r i t t e n a s : 2 2 s a H(s) = exp{ Y~ - i27TUhs} Com p a r i s o n w i t h ( 8 ) , (9) g i v e s : i C x - V 2 h(X) = — L — : e x p { -a H /5F 2 a h 2 T h e r e f o r e h(X) i s a normal d i s t r i b u t i o n t o o , e q u a t i n g ( 1 0 ) , Mean: u, = u c + U K h p f K g 1-5 113 V a r i a n c e : o\ = a 2 f + a 2 g (13) The Normal d i s t r i b u t i o n h ( x ) i s , t h e r e f o r e , u n i q u e l y d e f i n e d w i t h t h e d e t e r m i n a t i o n o f t h e mean and v a r i a n c e Example 1.3: D e c o n v o l u t i o n U s i n g C o n t i n u o u s F o u r i e r T r a n s f o r m s  G i v e n : Normal D i s t r i b u t i o n f ( x ) , g ( x ) : _A Gx - P £ ) 2 f 00 = rr- exp • > •» 1 (x - •yf>-)2 g ( x ) = — — exp - -5= - S — a /2TT 2a 2 g g F i n d : C o n t i n u o u s P r o b a b i l i t y D i s t r i b u t i o n h ( x ) so t h a t : f ( x ) = g ( x ) $ h ( x ) h ( x ) = f ( x ) © g ( x ) S o l u t i o n ; 1. D i r e c t D e c o n v o l u t i o n : n e a r i m p o s s i b l e t o p e r f o r m 2. D e c o n v o l u t i o n u s i n g C o n t i n u o u s F o u r i e r T r a n s f o r m s : From example I - l , i f rCx) = — exp 1 ( x - y f )2 a "Z-f f s2 a 2 Then: F ( s ) = exp { 2 — " i 2 7 r u f s ^ s 2 a 2 S i m i l a r y : G( s ) = exp { — ^ - i2TT u s } h ( x ) = £(x) (?) g ( x ) F ( s ) H(s) G ( s ) E q u a t i o n ( 3 - 2 0 ) : I f : Then: H ( s ) = exp { T h i s c a n bw r e - w r i t t e n as: s 2 a h 2 H ( s ) = exp { 2" - i2ir y^s} s 2 ( a f 2 - a g 2 ) i 2 i r ( y f - V g ) s} .*. h (x) = (x - y h ) 2  e x P { " 2a 2 } T h e r e f o r e h ( x ) i s a normal d i s t r i b u t i o n w i t h i t s v a r i a n c e and mean d e f i n e d by: a h 2 " a f 2 " a g 2 Example 1-4: D i s c r e t e C o n v o l u t i o n o f Two S i m p l e : D i s t r i b u t i o n s : G i v e n : D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n s f ( x ) and g ( x ) : x f f (x) X g g ( x ) 1 0.2 1 0.3 2 0.6 2 0.4 3 0.2 3 0.5 ZfCx) = = 1.0 2g(x) = 1.0 F i n d : D i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s h(.x) such t h a t \ " X f + X g i . e . h ( x ) = £(x) $ gCx) 115 1-7 S o l u t i o n : I t i s se e n t h a t s i n c e X, = X r + X , many c o m b i n a t i o n s o f X,. and X h f g' f g i s p o s s i b l e , and w i l l have v a l u e s r a n g i n g f r o m 2 t o 6. The d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n h ( x ) f o r x = 4, f o r example, has t o be d e t e r m i n e d f r o m t h e sum j o i n t - p r o b a b i l i t y o f f ( x ) and g ( x ) f o r a l l c o m b i n a t i o n s o f X r, X w i l l r e s u l t i n X, = 4. f g h X g X f X g f(x) gCx) f(x)gCx) h(x) 2 1 1 0.2 0.3 0.06 0.06 3 1 2 0.2 0.4 0.08 2 1 0.6 0.3 0.18 0.26 4 1 3 0.2 0.3 0.06 2 2 0.6 0.4 0.24 3 1 0.2 0.3 0.06 0.36 5 2 3 Q.6 0.3 0.18 3 2 0.2 0.4 0.08 0.26 6 3 3 0.2 0.3 0.06 0.06 Z = 1.00 Chec k s : i=l X. f C X J 1. Means: u s i n g y £ = n " n i=l f C V we g e t ] j f = 2, u = 2, U n = 4. t h e r e f o r e y n g M f ^g Means a r e a d d i t i v e 116 V a r i a n c e s . u s i n g <?£ 2 _ i = l CX i - u £ ) 2 f ( X . ) n i l l I , 1-8 we g e t a f 2 = 0.40; a g 2 = 0.60; a h 2 = 1.00 t h e r e f o r e a, 2 = aJ2- + a 2 .'.Variances a r e a d d i t i v e n f g Example 1-5: D i s c r e t e C o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m s G i v e n : D i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s f ( x ) , g ( x ) o f v a r i a b l e s X,., X s u c h t h a t : f g x f f x 1 0.2 2 0.6 3 0.2 I = 1.0 X g *x 1 0.3 2 0.4 , 3 0.3 Z = 1.0 F i n d : D i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s h ( x ) o f v a r i a b l e X^ s u c h t h a t = S £ + X g , X £ , X g i n d e p e n d e n t h ( x ) = f ( x ) ® g ( x ) S o l u t i o n : i ) P l a c e t h e g i v e n p r o b a b i l i t y d i s t r i b u t i o n s i n t o c o r r e s p o n d i n g sequences o f l e n g t h N. j f j X . g j g j 0 1 0.2 1 0.3 1 2 0.6 2 0.4 2 3 0.2 3 0.3 3 4 0.0 4 0.0 4 5 0.0 5 0.0 5 6 0.0 6 0.0 N - l N 0.0 N 0.0 117 1-9 i i ) F i n d t h e D i s c r e t e F o u r i e r T r a n s f o r m o f f j , g j u s i n g e q u a t i o n (3-30) N - l F K = j l 0 f j W' • j K = 0.2 + 0.6 W"K + 0.2 W"2K + 0. W"3K + o. w" ( n" 1 ) K (1) = A F K + W s a y 2 i r i where W = exp { - } J-l (2) S i m i l a r l y N - l G - j K = 0.3 + 0.4 W"K + 0.3 W"2K + 0. W"3K . + 0. W " C N - 1 ) K (3) = AGK + V (4) i i i ) F o r h. = f . ® g. e q u a t i o n 3-31 g i v e n Hv = Fv Gv 3 3 3 K K K. if nv xv AV H K = 0.06 + 0.26 W +0.36 W + 0.26 W + 0.06 W + 0. W"5K + 0. +0. W - ( N - l ) K (5) = ( A F K + B p K i ) ( A G K + B G K i ) AfflC + B H K i (6) i v ) C o m p a r i s o n o f e q u a t i o n s ( 1 ) . (.5) w i l l g i v e 118 1-10 h = 0.06 o hj, = 0.26 h 2 = 0.36 h 3 = 0.26 h 4 = 0.06 V l • ° I n d i g i t a l c o m p u t a t i o n , e q u a t i o n s C2), ( 4 ) , C6) a r e e v a l u a t e d n u m e r i c a l l y , and v a l u e s o f h j i d e n t i c a l t o t h o s e above ca n be f o u n d by a p p l y i n g e q u a t i o n C5-31) 1 N _ 1 IK h. = — E H WJ j N K=0 H K W v) The c o r r e s p o n d i n g X ^ i s e v a l u a t e d as f o l l o w s : = X £ + X i s a l i n e a r o p e r a t i o n and AX = X_ . - X_ . , = X . - X . , = c o n s t a n t f , J f , J - l g,J g , J - l X, = X. + X = 2 ho f o go " h i = ho + AX " 3 X h 2 = X h Q + 2AX = 4 X h . - + j A X = 2 + j v i ) The D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n h i s t h e r e f o r e d e t e r m i n e d : j h i h. J \ 1 h, " i x 0 2 0.06 2 I 0.06 1 3 0.26 3 •I 0.26 2 4 0.36 4 I 0.36 3 5 0.26 5 1 0.26 4 6 0.06 6 | 0.06 5 7 0.0 Z = 1.00 N - l N > 0.0 -119 1-11 Example 1-6: D i s c r e t e D e c o n v o l u t i o n u s i n g D i s c r e t e F o u r i e r T r a n s f o r m s G i v e n : D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n s hCx), f(.x) o f v a r i a b l e s X^ 3 X £ s u c h t h a t : \ h x 2 0.06 3 0.26 4 0.36 5 0.26 6 0.06 X f * • 1 X ! j i 0,2 \ 2 0,6 ! 0.2 : i and h i s dependent on f F i n d : D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n gC.x) o f v a r i a b l e X s u c h t h a t : g X = X, - X -g h f gCx) = hCx) © fCxl S o l u t i o n : 120 1-12 i ) S i m i l a r t o example 3,5 t h e f o l l o w i n g sequences can be f o u n d : 1 i h j f i i 11 f . J 0 2 0.06 1 i 0.2 1 3 0.26 2 j 0,6 2 4 0.36 3 0.2 3 5 0.26 4 0,0 4 [ 6 0.06 5 .1 Q.Q 5 s 7 0.0 6 •j 0,0 6 i 8 0.0 7 | 1 : N-1 N+l 0.0 N .] •ii } 0,0 1 i i ) D i s c r e t e F o u r i e r T r a n s f o r m o f h., f . c a n be f o u n d ; ' J J H K = 0.06 + 0.26 W"K + Q.36 W_2K + 0.26 w"4K + 0.06 W~5K + + 0. W •C.N-1)'K A H K + B,„,i JHK J CD C2) F K = 0.3 + 0,4 W"K + 0 . 3 W~2K + 0, W"3K + 0. W~4K + 0. W_5K . + 0. W •CN - in A F K + V C3) C.4) i l l ) F o r q- = h (?) f e q u a t i o n C3-32) g i v e s : X X X K l x x \K 0.06 * 0.26 W"K + 0.36 W~2K + 0.26 W"3K + K „ „ ,.,-3K . Q i 0 6 W - 4 K + Q 0.2 + 0.6 W"K + 0.2 W"2K + 0 I f •jv = 0.3 + 0.4 W + 0.3 W + 0 . . . C5) 121 1-13 A l t e r n a t e l y : Ajjjf + B„„i _ .IK HK A n - rs^ \ - ^ . B H K i • QK * V ( 6 ) i v ) C o m p a r i s o n o f e q u a t i o n ( 1 ) , (5) g i v e s q i = o.4 q 2 = 0.3 q 3 = 0.0 I n d i g i t a l c o m p u t a i t o n s , e q u a t i o n s C2), ( 4 ) , C6) a r e e v a l u a t e d n u m e r i c a l l y , and q. i d e n t i c a l t o t h o s e above can be f o u n d by I n v e r s e D i s c r e t e F o u r i e r T r a n s f o r m 1 N _ 1 i K * j = N KIO \ ^ v) The X . i s f o u n d by a p p l y i n g : 93 X = X, - X. = 2 - 1 = 1 qo ho f o X . = X + AX 2 q l qo X - = X + 2AX = 3 q2 qo X q : N - l " X o + C N " 1 ) A X " N v i ) The D i s c r e t e P r o b a b i l i t y D i s t r i b u t i o n q i s t h e r e f o r e d e t e r m i n e d : 122 nr - 1 q i q j 1 *, 1 1 • q i x 0 0.3 ! 1 j 0 . 3 ! 1 2 0.4 2 j 0.4 2 3 0.3 j 3 j 0.3 | 3 4 0.0 ] E = 1.0 \ 4 5 0.0 5 6 • N-1 . N 0.0. 1-14 123 I I -1 I I . l Ground L e v e l C o n c e n t r a t i o n D a t a G e n e r a t i o n P r o g r a m (D.G.P.) A. Progra m L i s t i n g S L G L C 1 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2 C ** ** 3 c * * GROUNO L E V E L C C N C E N T R A T I C N ( G L C ) *# 4 c * * D A T A G E N E R A T I O N P R O G R A M ** 5 c • * F G R ** 6 c * * S I N G L E R E C E P T O R ANO O N E OR TWO S O U R C E S ** 7 c ** ** 8 c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 9 . 1 0 c c T H I S PROGRAM G E N E R A T E S P R O B A B I L I T Y D I S T R I B U T I O N S OF 11 c C ANO E t G I V E N D I S T R I B U T I O N S O F S t U » B 12 c FOR E I T H E R O N E OR TWO S O U R C E S . 13 c 1 4 c I N P U T D I S T R I B U T I O N S C S , U , 8 , Q ) : 15 c I ) I F NORMAL D I S T R I B U T I O N , S U P P L Y MEAN , S T D . D E V . 1 6 c 2 1 I F L O G . N O R M A L D I S T R I B U T I O N . S U P P L Y ANT I L C G . OF 1 7 c M E A N , S T O . D E V . IN L O G . S C A L E . 18 c 3 ) D I S C R E T E V A L U E S OF P R O B A B I L I T Y D I S T R I B U T I O N MAY 19 c B E S U P P L I E D I N S T E A O . 2 0 c 21 c O U T P U T D I S T R I B U T I O N S ( C E ) : 2 2 c C A N BE G R O U P E D IN D I S T R I B U T I O N S OF E Q U A L L I N E A R 2 3 c I N T E R V A L OR E G U A L L C G . I N T E R V A L S . 2 4 c 2 5 c E A C H D I S T R I B U T I O N C A N B E P L O T T E D I N O N E OR A L L O F T H E 2 6 c F O L L O W I N G FOUR P L O T S : 2 7 c 1 ) V A R I A B L E V S . P R O B A B I L I T Y 2 8 c 21 V A R I A B L E V S . C U M U L A T I V E P R O B A B I L I T Y 2 9 c 31 L C G . OF V A R I A B L E V S . P R O B A B I L I T Y 3 0 c 4 1 L O G . C F V A R I A B L E V S . C U M U L A T I V E P R O B A B I L I T Y 3 1 c 3 2 c * * * * N O M E N C L A T U R E * * * * 33 c 3 4 c 1 1 i ! 3 5 c | GROUP 1 V A R I A B L E 1 S Y M B O L 1 U N I T I 3 6 c I I 1 1 3 7 c | | I 1 3 8 c I A T M O S P H E R I C 1 A T M O S P H E R I C T E M P E R A T U R E 1 T A 1 D E G . K I 3 9 c I C O N D I T I O N S ! A T M O S P H E R I C P R E S S U R E 1 PA 1 MB 1 4 0 c | | A T M O S P E R I C S T A B I L I T Y 1 S 1 — I 41 c I \ MINO S P E E D 1 U I M / S 1 4 2 c | | WIND D I R E C T I O N 1 B 1 D E G ( N ) ! 4 3 c 1 1 ' 1 4 4 c j « l S T A C K 4 5 c I S T A C K I S T A C K H E I G H T 1 H I I M 1 4 6 c 1 C O N D I T I O N S ! S T A C K O I A M E T S R 1 01 1 M 1 4 7 c | | S T A C K GAS E X I T V E L O C I T Y 1 V S l » M / S 1 4 8 c | | S T A C K G A S E X I T T E M P E R A T U R E 1 T S 1 1 D E G . K 1 4 9 c | | S T A C K G A S E M I S S I O N R A T E 1 0 1 1 G / S i 5 0 c ( R E C E P T O R 1 H O R I Z O N T A L D I S T A N C E 1 R l 1 M 1 51 c I L O C A T I O N 1 D I R E C T I O N FROM S O U R C E 1 A L P H A 11 O E G ( N ) 1 52 c 1 1 ' 1 53 c 54 c I S T A C K | S T A C K H E I G H T 1 H2 I M 1 55 c I C O N D I T I O N S ! S T A C K D I A M E T E R 1 0 2 ! M 1 124 I I-2 5 6 C 1 1 S T A C K G A S E X I T V E L O C I T Y 1 V S 2 1 M / S 5 7 C 1 1 S T A C K G A S E X I T T E M P E R A T U R E 1 T S 2 1 O E G . K 5 8 c I 1 S T A C K GAS E M I S S I G N R A T E 1 0 2 1 G / S 5 9 c I R E C E P T O R 1 H O R I Z O N T A L O I S T A N C E 1 R2 1 M 6 0 c 1 L Q C A T I C N 1 D I R E C T I O N PRCM S O U R C E 1 A L P H A 2 i O E G ( N ) 61 6 2 c 1 I 1 1 c | 1 1 63 c 1 GROUNO L E V E L C O N C E N T R A T I O N ( G L C l 1 C I G * M * * - 3 6 4 c | 1 1 6 5 c I C H A R A C T E R I S T I C F U N C T I O N S : 1 1 6 6 c I S T A C K #1 ~ R E C E P T O R 1 E l 1 S * M * * - 3 6 7 c I S T A C K « 2 — R E C E P T C * 1 E 2 1 S * M * * - 3 6 8 c I 1 1 6 9 c 7 C c O T H E R V A R I A B L E S U S E D IN T H I S P P C G R A f : 7 1 c ( X O E N O T E S ANY C F T H E S E V A R I A B L E S : S » U t B t O I t 0 2 t 6 1 . E 2 t C ) 7 2 c 7 3 c V A R I A B L E E X P L A N A T I O N S 7 4 c 7 5 c X L O G L O G . B A S E 10 C F X . 7 6 c P X D I F F E R E N T I A L P R O B A B I L I T Y O F X . 7 7 c C P X C U M U L A T I V E P R O B . OF X . 7 8 c I T R A C E I N D E X FOR T R A C I N G T H E P A T H O F C A L C U L A T I O N S 7 9 c l / T R A C E ; O / C C NCT T R A C E 8 0 c I P T X T Y P E O F I N P U T P R O B A B I L I T Y D I S T R I B U T I O N . 81 c ( E X C E P T 1 / N O R M A L 0 I S T R I B U T I 0 N : 2 / L C G N 0 R M A L D I S T R . 8 2 c I P T C E J FOR I P T X * l , 2 ? S U P P L Y X H , O X , I X , R M U X , S IGX 83 c 3 / I N 0 I V I 0 U A L E N T R I E S O F C I S T R . S U P P L I E D . 8 4 c I P T C E T Y P E OF O U T P U T D I S T R I B U T I O N S . 85 c l / E Q U A L L I N E A R I N T E R V A L S . 86 c 2 / E C U A L L C G A R I T H P I C I N T E R V A L S . 8 7 c I X NUMBER O F I N T E R V A L S IN A D I S T R I B U T I O N . 88 c XH H I G H E S T V A L U E O F X . F O R L C G N Q R M A L D I S T R . 8 9 c S U P P L Y H I G H E S T V A L U E OF X L C G . 9 0 c OX I N T E R V A L S I Z E O F X . 9 1 c FOR L O G N C M A L O I S T R . S U P P L Y O X L O G , 9 2 c T H E I N T E R V A L S I Z E IN L O G . S C A L E . 9 3 c RMUX MEAN O F X . A R I T H M A T I C MEAN FOR N O R M A L O I S T R . 9 4 c MEAN O F X L C G FOR L O G N O R M A L D I S T R . 9 5 c S I G X S T A N D A R D C E V I A T I C N O F X . 9 6 c F O R L C G N C R M A L O I S T R . S U P P L Y S I G L O G , 9 7 c T H E S T O . D E V . O F X L O G D I S T R . 9 8 c I F M T F O R M A T FOR R E A D S T A T E M E N T FOR I P T X = 3 » 9 9 c C H E C K S U B R O U T I N E R E A O l . 1 0 0 c I R E S C X I N D E X F C R R E S C A L I N G PRO B . O I S T R . 1 0 1 c 1 / R E S C A L E S O T H A T C U M . P R C B . = 1 . 0 ; 1 0 2 c C / O C NOT R E S C A L E . 1 0 3 c I P L O T X d -4) I N O E X FOR P L O T S . 1 / P L O T : 0/DO NOT P L O T . 1 0 4 c P L O T i l X V S . PX 1 0 5 c P L O T # 2 X V S . C P X 1 0 6 c PL0T#3 L O G . X - V S . PX 1 0 7 c PL0T#4 L O G . X V S . C P X 1 0 8 c X S I Z E X d -4) H O R I Z O N T A L S I Z E C F P L O T IN I N C H E S . 109 c Y S I Z E X I 1 -4) V E R T I C A L S I Z E C F P L O T I N I N C H E S . 1 1 0 c I S Y M B X ( 1 -4) I N O E X FOR P L C T S Y M B O L S . 1 1 1 c L B L X l L A B E L FOR X - A X I S F O R P L C T S # 1 , 2 ( X I 1 1 2 c L B L X 2 L A B E L FOR X - A X I S FOR P L C T S # 2 . 3 ( L Q G . X 1 1 1 3 c L B I P 1 L A B E L F O R Y - A X I S F O R P L 0 T S * 1 , 3 ( P X ) 1 1 4 c L 8 L P 2 L A B E L FOR Y - A X I S FOR PLCTS#2*»4 ( C P X ) 1 1 5 c f O F C H A R A C T E R S IN A L L A 8 C V E L A B E L S MUST B E =< 40. 125 II-3 1 1 6 1 1 7 1 1 8 1 1 9 1 2 0 121 1 2 2 1 2 3 1 2 4 1 2 5 1 2 6 1 2 7 1 2 8 1 2 9 1 3 0 1 3 1 1 3 2 1 3 3 1 3 4 1 3 S 1 3 6 1 3 7 1 3 8 1 3 9 1 4 0 141 1 4 2 1 4 3 1 4 4 1 4 5 1 4 6 1 4 7 1 4 8 1 4 9 1 5 0 151 1 5 2 1 5 3 1 5 4 1 5 5 1 5 6 1 5 7 1 5 8 1 5 9 1 6 0 1 6 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 7 1 6 8 1 6 9 1 7 0 1 7 1 1 7 2 1 7 3 1 7 4 1 7 5 C C c c c c c N X 1 . N X 2 , N P i , N P 2 » C P C H A R A C T E R S IN C O R R E S P O N D I N G L A B E L S . L H L X 3 H E A D I N G IN P R I N T . O U T F O R X . L 8 L X 4 H E A D I N G IN P R I N T OUT F O R L O G . X 0 OF C H A R A C T E R S IN L B L X 3 , L 8 L X 4 M U S T BE =< 1 6 . C C C D I M E N S I O N O I H E N S I O N 1 2 3 4 5 6 7 a 9 c l 2 3 4 5 6 7 8 9 * I L O G I C A L * ! 1 2 3 4 5 6 7 a S ( 6 ) » U l 5 1 I t 6 ( 1 4 4 ) , 0 1 ( 5 1 ) . 0 2 ( 5 1 ) * C E M 2 5 6 ) * C E 2 ( 2 5 6 ) , C C I 2 5 6 ) , 0 P C E K 2 5 6 J D P C C 1 2 5 6 ) , I P L 0 T S 1 4 ) I P L C T U I 4 ) , I P L O T B t 4 ) f I P L T Q K 4 ) , I P L T C 2 ( 4 ) . I P L T E l ( 4 ) , I P L T E 2 I 4 ) , I P L 0 T C I 4 ) , S F A C T R ( 6 ) , A Y ( 6 I t A Z 1 ( 6 ) * A Z 2 ( 6 ) , L B L S 1 ( 4 0 ) L B L U l ( 4 0 ) ? L B L 8 1 ( 4 0 ) , L B L C 1 ( 4 0 ) t L 8 L C 1 K 4 0 ) L B L Q 2 K 4 0 ) L B L E 1 1 ( 4 0 ) L 8 L E 2 K 4 0 ) L B L P 1 ( 4 0 ) * S L C G ( 6 I * U L C G 1 5 1 ) t 8 L C G 1 1 4 4 I Q 1 L C G ( 5 1 ) Q 2 L C G ( 5 l ) C E H K 2 5 6 ) C E H 2 ( 2 5 6 ) C C H ( 2 5 6 ) t , 0 P C E 2 ( 2 5 6 C P C C ( 2 5 6 , X S I Z E S U X S I Z E U ( 4 ) X S I Z E 3 ( 4 ) X S I Z Q H 4 ) X S I Z Q 2 ( 4 ) X S I Z E K 4 ) X S I Z E 2 ( 4 > X S I Z E C ( 4 ) 0 H U l ( 5 i ) t B Z K 6 I • 8 Z 2 ( 6 ) , , L B L S 2 ( 4 0 ) L B L U 2 I 4 0 ) t L B L B 2 ( 4 0 > » L B L C 2 ( 4 0 > , , L S L Q 1 2 ( 4 0 ) , L B L C 2 2 ( 4 0 ) , L 3 L E 1 2 ( 4 Q ) , L B L E 2 2 ( 4 0 ) L B L P 2 I 4 0 I P S ( 6 ) , P U ( 5 1 ) , P B l 1 4 4 ) , P Q l ( 5 1 ) , P Q 2 ( 5 1 ) , C E 1 L 0 G ( 2 5 6 ) C E 2 L 0 G ( 2 5 6 ) C C L C G I 2 5 6 I , . C P C E H 2 5 6 ) , , Y S I Z E S I 4 ) t Y S I Z E U ( 4 ) , Y S I Z E B U ) , Y S I Z Q K 4 ) , Y S I Z Q 2 ( 4 ) . Y S I Z E 1 ( 4 ) » Y S I Z E 2 ( 4 ) , Y S I Z E C I 4 ) , D H U 2 I 5 1 1 , C Z 1 ( 6)t C Z 2 ( 6 ) , L B L S 3 ( 1 6 ) , L B L U 3 I 1 6 ) . L B L B 3 ( 1 6 ) , L B L C 3 ( 1 6 ) , , L B L Q 1 3 ( 1 6 ) , , L 8 L C 2 3 ( 1 6 ) , , L B L E 1 3 ( 1 6 ) * , L 8 L £ 2 3 ( 1 6 ) , C P S ( 6 ) . C P U ( S l ) t C P B ( 1 4 4 > , C P Q K 5 1 1 , C P Q 2 ( 5 i ) , , C E H 1 L G ( 2 5 6 > , , C E H 2 L G ( 2 5 6 ) , C C H L C G ( 2 5 6 I« C P C E 2 ( 2 5 6 1 , I S Y M B S ( 4 ) , I S Y M 3 U ( 4 > , I S Y y B B ( 4 ) , I S Y M 0 U 4 ) , I S Y M 0 2 ( 4 ) , I S Y M E K 4 ) , I S Y M £ 2 ( 4 ) , I S Y M 8 C ( 4 ) , I P R I N T O ) , L B L S 4 ( 1 6 ) L B L U 4 ( 1 6 ) , L B L 3 4 ( 1 6 ) . L B L C 4 ( 1 6 ) , L 8 L Q 1 4 < 1 6 ) L B L Q 2 4 ( 1 6 ) L B L E 1 4 ( 1 6 ) L B L E 2 4 ( 1 6 ) D A T A I N P U T 1 3 3 3 3 : 3 3 3 3 3 3 3 3 3 3 3 3 3 3 R E A 0 ( 5 , 8 0 C ) R E A C ( 5 , 8 0 C ) R E A C ( 5 , 8 0 C ) R E A C ( 5 . 8 C C I R E A D ( 5 , a 0 C ) I T R A C E C A L L R E A C H 6 , I P T S , I S , t R E S C S . S H , O S , R H U S , S I G S , S , P S , I T R A C E ) C A L L R E A 0 1 ( 5 1 , I P T U , I U , I R E S C U , U H , O U , R M U U , S I G U , U , P U , I T R A C E ) C A L L R E A D 1 ( 1 4 4 , 1 P T B , I B , I R E S C 8 i B h , D B , R M U 8 , S I G B , B , P B , I T R A C E ) R E A C ( 5 , a O C ) T A , P A GAPHA N S O U R C K l , D l , V S l , T S l R 1 , A L P H A 1 C A L L R E A C l ( 5 1 , I P T Q 1 , 1 0 1 , I R S C Q 1 , 0 H 1 , O Q 1 , R M U Q 1 , S I G Q l , Q 1 , P Q I , 1 I T R A C E ) I F ( N S 0 U R C . N 6 . 2 ) GO T O 10 R E A 0 ( 5 , 8 C C ) H 2 , D 2 , V S 2 , T S 2 R E A C ( 5 , 8 0 0 ) R 2 , A L P H A 2 C A L L R E A C l ( 5 1 , I P T C 2 , I C 2 » I R S C Q 2 , 0 H 2 , 0 0 2 , R M U 0 2 , S I G 0 2 , 0 2 , P Q 2 , 1 I T R A C E ) 1 0 R E A C ( 5 , 8 C C ) I P T C E . O C E R E A C ( 5 , 6 C C ) I C E 1 I F ( N S Q U R C . N E . 2 ) GO TO 23 126 I I-4 176 REA0(5,aCC» ICE2 177 20 REAC ( 5 , 8 0 0 ) ICC 178 REAO(S .aCC) ICALC 179 REAC ( 5 , 8 0 C ) ( IPRINTI IJ , 1=1,3) 180 C — STANOARC OAT A INPUT FROM F I L E DCCT — 1 8 1 CALL REAC2IIPLOTS,X 5 IZES , Y S I Z E S ,ISYf3S, 182 I LBLS1.NSI,LBLS2 ,NS2,LBLS3.LBLS4,ITRACE) 183 CALL READ2lIPLaTU.XSIZEl,YSIZEL.ISY * B U , 184 L LBLU1 ,NUL,LBLU2 ,NU2,LBLU3,LBLU4 , ITRACE) 195 C A L L REA02IIPLCTB,XSIZEa,YSIZEe,ISY^BB, 186 L LBL3ItN BI,LBLB2,N 8 2 ,LBLB3,LBL34,ITRACE) 187 CALL R E A D 2 I I P L T C L . X S I Z Q I , Y S I Z C 1 , 1 S Y M Q l , 1 8 8 L LBL011 ,NQ11,LBLQ12 ,NC12,L8LC13,LBLC14,ITRACE) 189 CALL REA02(IPLTQ2,XSIZQ2,YSIZC2,ISYMa2, 190 L LBLG21,NC21*LBLQ22 ,NC22.LBLC23»L3LC24,ITRACE) 1 9 1 CALL REA02(IPLTEltXSIZElt YSIZ'Elt ISYMEl, 192 I LBLE11.NE11,LBLE12 ,NE12,LBLE13.LBLE14,ITRACE I 193 CALL REA02(IPLTE2,XSIZE2,YSIZE2,ISYME2, 194 L LBLE21,NE21,LBLE22 ,N£22,LBLE23,LBLE24,ITRACE) 195 CALL REA02UPL0TC.XSIZEC.YSIZECISYM9C, 196 L LBLCl,NCl,L8LC2.NC2,LBLC3.LBLC4,ITRACE) 197 R E A 0 ( 5 , 3 C C ) ( A Y ( I ) , I = 1 , 6 ) t"98 R E A C ( 5 , 3 0 C ) BY 199 REACl5,flCC> (AZK I ) , I s 1 , 6 1 2 0 0 REAC ( 5 , 8 0 0 ) ( 5 2 1 ( 1 1 , 1 = 1 , 6 ) 201 REA0 (5,80C) (CZ1( I 1,1 = 1 , 6 1 2 0 2 REAC!5,80C) ( A Z 2 ( I ) , 1 = 1 , 6 ) 2C3 REA0 (5,80C) (BZ2( 1 1 » 1= I » 6 ) 204 REAC(5,8CC) ( C Z 2 ( I ) , I = 1 , 6 ) 2C5 R E A C ( 5 , 8 i C ) NPl.LBLPl 206 REAC15,81C) NP2.LBLP2 207 C 3 3 3 3 3 3 3 = 3 3 = 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 208 C M * CALCULATE PROBABILITY DISTRIBUTIONS 209 C 3 3 » 3 3 3 3 3 3 3 3 = = 3 3 = 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 210 30 CALL ALCGZ(IS»S »SLGG»I TRACE) 2 1 1 CALL CUMUL(IS.PS,CPS.CUMS,ITRACE) 2 1 2 C 213 GO T O ( 4 0 , 6 C , 5 0 l , I P T U 214 4 0 CALL NCfiMAL( I U , U H , C U,RMUU,SIGU , U , P U , 0 , 0 . 0 , 1 TRACE ) 215 50 CALL ALCGZ I IU,U,ULQG,ITRACE) 216 GO TO 70 217 60 CALL LGNORMI IU,UH,0U.RMUU , S IGU .U,ULOG ,PU, ITRACE) 218 70 CALL CUMUL(IU.PU,CPU,CUMU.IT RACE) 219 C 2 2 0 GO 70(80,8 C ,120),IPTB 221 80 0ELB=180.-RMU8 2 2 2 eH»6H*0ELB 223 I F I 8 H . L 6 . 0 . ) 3H=BH+360. 2 2 4 IFt I P T B . E 0 . 2 ) GO T O 90 225 C A L L NORMAL ( I B , B H , D 8 ,180.,SIGB,8 , P B , 1,360 . 0 ,ITRACE) 2 2 6 GO T O ICC 227 90 CALL LGNORMI IB ,BH,08 ,180.,SIGB,B,BLCG.PB,ITRACE) 228 100 CQ 110 1 = 1* IB 229 8(I)=8(I)-0ELB 230 I F ( B ( I ).LE . O . ) B ( I)=8 ( I)+360. 231 110 IF ( B ( D.GT.360.) B ( I )=B ( IJ-360. 232 120 CALL ALCGZ(18,3,BLOG.ITRACE) 233 CALL CUMUL(IE,PB,CPfi,CUMB»ITRACE) 234 C 235 GO TO(130,15C,140),IPTQ1 127 II-5 2 3 6 1 3 0 C A L L N O P f A L I I G I , C H I , 0 C 1 , R X U C I . S I G C 1 , C I , P Q 1 , 0 , 0 . 0 , I T R A C E ) 2 3 7 1 4 0 C A L L A L C G 2 1 1 0 1 , 0 1 , Q I L C G , I T R A C E ) 2 3 8 G C T C 1 6 C 2 3 9 1 5 0 C A L L L G N C R H ( I Q 1 , Q H 1 , 0 0 1 , R r » U C l » S I G C l r C l , C I L Q G » P ( i l , I T R A C E ) 2 4 0 1 6 0 C A L L C U M U L ( I C I , P Q 1 , C P Q I , C U M C 1 , I T R A C E ) 2 4 1 I F I N S 0 U R C . N E . 2 ) G C TO 2 1 0 2 4 2 G O T O 1 1 7 0 , 1 9 0 , 1 8 0 ) , I P T C 2 2 4 3 1 7 0 C A L L N O P f A L ( I C 2 , C H 2 , 0 C 2 , R M U C 2 , S I G C 2 , Q 2 t P Q 2 , 0 , 0 . 0 , I T R A C E ) 2 4 4 1 8 0 C A L L A L C G Z I I C 2 , 0 2 , Q 2 L C G , I T R A C E ) 2 4 5 G C TO 2 0 C 2 4 6 1 9 0 C A L L L G N C R M ( 1 0 2 , C H 2 , 0 C 2 , R M U 0 2 , S I G 0 2 » 0 2 , 0 2 L 0 G , P Q 2 , 1 T R A C E ) 2 4 7 2 0 0 C A L L C U M U L ( 1 0 2 , P Q 2 , C P 0 2 , C U M C 2 , I T R A C E ) 2 4 8 C . =3M!==3=3S3==3=333=33=3=333=33=33=3=33333333=33 2 4 9 C M R E S C A L E P R O B A B I L I T Y D I S T R I B U T I O N S II 2 5 0 c « » a 3 i 3 a =33333 3=3 3333=3=333=3333=3 »3 3 383 333 3 33 3 3= 2 5 1 2 1 0 I F l I R E S C S . E Q . l ) C A L L R E S C A L ! I S , P S « C P S , C U P S f I T R A C E ) 2 5 2 I F U R E S C U . E C - l ) C A L L R E S C A L < I U . P U , C P U , CUMU , I T 3 A C E ) 2 5 3 I F U R E S C B . E Q . l ) C A L L R E S C A L I I B » P B , C P 8 , C U M B , I T R A C E ) 2 5 4 I F ( I R S C Q l . E O . i ) C A L L R E S C A L ( 1 0 1 , P Q I . C P O l , C U M O I, I T R A C E ) 2 5 5 I F I N S 0 U R C . N E . 2 ) G C TO 2 2 0 2 5 6 I F I l R S C 0 2 . E Q . i l C A L L R E S C A L ( 1 0 2 , P Q 2 . C P Q 2 . C U M 0 2 , I T R A C E ) 2 5 7 c •33*3333333333=3333======333=333=333=3=3=3= 2 5 8 c II C A L C U L A T E E 1 M A X , E 2 M A X £ C M A X II 2 5 9 c SI3333 8333S3'SSS33332SSS3323S3333: 333333S333 2 6 0 2 2 0 I F ( I C A L C . E C . O ) GO TO 7 2 0 2 6 1 0 0 2 3 C 1 = 1 , I S 2 6 2 2 3 0 $ F A C T R ( l ) = > l - C . 2 * < S U > - 4 > / 3 . 2 6 3 0 H F 1 = V S 1 * C 1 * U . 5 + 0 . C G 2 6 8 * P A * 0 1 * . ( T S l - T A ) / T S l ) 2 6 4 0 0 2 4 C 1 = 1 , I U 2 6 5 2 4 0 O H U i - ( I J - O K F l / U ( I ) 2 6 6 I F I N S 0 U R C . N E . 2 ) G C TO 2 6 0 2 6 7 0 H F 2 = V S 2 * C 2 * < 1 . 5 + 0 • C 0 2 6 8 * P A * D 2 * < T S 2 — T A ) / T S 2 ) 2 6 8 0 0 2 5 0 1 = 1 , I U 2 6 9 2 5 0 0 H U 2 ( I ) = 0 H F 2 / L ( I ) 2 7 C c 2 7 1 2 6 C I F d T R A C E . E C . i l P R I N T 2 7 0 2 7 2 2 7 0 F O R M A T ( • • , ' M A I N — 2 0 8 ' ) 2 7 3 C M A X » C . 2 7 4 E 1 M A X = 0 . 2 7 5 X 1 * R 1 2 7 6 X 1 K » X I / 1 C C C . 2 7 7 c — FOR 1 S T S O U R C E — 2 7 8 2 8 0 0 0 3 1 C K < a l , I S 2 7 9 N S M F I X l S ( K S ) ) 2 8 C S I G Y I = A Y ( N S ) * < X 1 K * * B Y 1 2 8 1 I f I X L G T . 1 0 C C . ) GO T C 2 9 C 2 8 2 S I G 2 1 = A Z 1 ( N S ) * ( X 1 K * * 8 Z 1 ( N S ) ) + C Z l I N S I 2 8 3 G O T C 3 C C 2 8 4 2 9 0 S I G Z I = A Z 2 < N S ) * ( X 1 K * * B Z 2 ( N S ) J + C Z 2 ( N S ) 2 6 5 3 0 C S I G Y Z l » S I G Y l * S I G Z l * 3 . 1 4 1 5 5 3 2 8 6 0 0 3 1 0 K U » 1 , I U 2 8 7 H E F F l = H l + C H U l ( K U ) * S F A C T R C K S ) 2 8 8 E X I * I H E F F 1 / S I G Z 1 ) * » 2 2 8 9 E 1 = I E X P ( - C . 5 * E X 1 ) ) / < S I G Y Z l * U l K U J ) 2 9 0 3 1 0 I F I E I M A X . L T . E l ) E 1 M A X = E 1 2 9 1 I F ( N S G U R C . E Q . 2 ) GO TO 3 2 0 2 9 2 C M A X » E 1 M A X * G 1 d C l ) 2 9 3 G O T O 3 7 0 2 9 4 c — FOR 2ND S O U R C E — 2 9 5 3 2 C I F U T R A C E . E Q . l ) P R 1 N T 3 3 0 128 I I-6 2 9 6 3 3 0 F C P M A T I 1 ' t , M A l N - 2 2 2 « 1 2 9 7 E2MAX=C. 2 9 8 X2=R2 29 9 X 2 K = X 2 / I C C C . 3 0 0 00 3 6 0 K S = l , I S 301 N S « I F I X ( S ( K S ) 1 3 0 2 S I G Y 2 = A Y ( N S ) * ( X 2 K * * 8 Y ) 3 0 3 I F ( X 2 . G T . 1 0 0 0 . ) GO TO 340 3 0 4 S I G Z 2 = » A Z 1 ( N < ) * < X 2 K « * 8 Z 1 ( N S ) ) * C Z 1 ( N S ) 3 0 5 GC TC 3 5 C 3 0 6 3 4 0 S I G Z 2 = A Z 2 ( N S ) * ( X 2 K * * 8 Z 2 ( N S )) + C Z 2 ( N S ) 3 0 7 3 5 0 S i G Y Z 2 = S I G Y 2 * S I G Z 2 * 3 . l 4 1 5 9 3 3 0 8 OC 3 6 C K L = l » I L 30 9 H E F F 2 = t - 2 * C H U 2 ( K U ) * S F A C T R ( K S ) 3 1 0 E X 2 » ( H E F F 2 / S I G Z 2 >*«2 3 1 1 E 2 = ( E X P ( - C . 5 - E X 2 ) > / ( S I G Y Z 2 * U ( K U ) ) 312 3 6 C I F ( E 2 f A X . L T . E 2 ) E2PAX=E2 3 1 3 C M A X = E 1 M A X * G 1 U Q 1 ) + E 2 M A X * C 2 ( I C 2 ) 3 1 4 C » » 3 £ » 3 s s : s : : 3 : : : s 3 3 s : 3 a s 3 3 = 3 3 3 s s : s : : s 3 : = = = = = = = = = = = = 3 1 5 C 1 1 C L A S S 1 F I C A T I C N OF E l , c 2 £ C D I S T R I B U T I O N II 3 1 6 . C S 3 3 S 3 3 3 3 3 S 3 3 3 3 3 X 3 3 3 S 3 3 3 3 3 3 3 3 3 3 3 3 S 3 3 3 3 3 3 3 3 3 5 3 3 =33 35 33 3 3 1 7 37C I F d T R A C E . E C . i l P R I N T 3 8 0 3 1 8 3 8 0 FORMAT( • • i ' M A I N - 2 3 5 1 ) 31 9 C A L L C L A S S K I P T C E , I C E l , C C E , E 1 M A X , C E I , C E h l , 3 2 0 1 C E 1 L 0 G . C E H I L G • O P C E 1 , C P C E 1 , I T R A C E ) 3 2 1 I F ( N S C U R C . N E . 2 ) GC TC 390 322 C A L L C L A S S K I P T C E , I C E 2 . C C E . E 2 M A X . C E 2 . C E H 2 , 3 2 3 1 C E 2 L 0 G , C E H 2 L G , C P C E 2 , C P C E 2 , I T R A C E ) 3 2 4 3 9 C C A L L C L A S S I ( I P T C E , I C C O C S . C P A X . C C C C H , 3 2 5 1 C C L O G , C C H L C G , D P C C , C P C C , I T R A C E ) 3 2 6 C 3= 3 3 3 3 3 3 3 3 3 3 3 3 3 3 = 3 3 3 = 3 3 = = 3 3 = 3 3 3 3 = 3 3 3 3 3 3 2 7 C II C A L C U L A T E AND SORT E L C II 3 2 8 C 3 3 3 3 3 3 3 3 3 3 3 3 3 = = = 3 3 3 3 3 = = = = 3 3 3 3 = = 3 3 3 = = = = = 3 2 9 C 3 3 0 C TO IMPROVE E F F I C I E N C Y OF THE C A L C U L A T I C N S » A C R T I C A L ANGLE 3 3 1 C GAMMA I S USED. GAMMA IS C E F I N E D AS THE ANGLE F R C * PLUME 332 C C E N T R E L I N E BEYONO WHICH THE G . L . C . IS N E G L I G I B L E . 3 3 3 C FOR MOST C A S E S , I T IS REASONABLE TO ASSUME GAMMA=60 DEG. 3 3 4 C 3 3 5 c 3 3 6 4 0 0 I F ( I T R A C E . E C . l ) P R I N T 4 1 0 3 3 7 4 1 0 FORMAT ( • • , ' M A I N - 2 5 C ) 3 3 8 CPNZC=G. 33 9 C P N Z E 1 = C . 3 4 0 I F ( N S C U R C . E 0 . 2 ) GO TO 470 341 C 3 3 3 3 3 3 3 3 3 S I N G L E SOURCE 3 3 = 3 = 3 3 3 = 342 00 4 6 0 KB = 1 , I B 3 4 3 P P B = P 8 ( K E ) 3 4 4 8 1 8 C = 8 ( K 8 J - 1 8 0 3 4 5 THETAl=sABS( ALPHA 1-B18C) 3 4 6 I F ( T H E T A l . G E . 3 6 0 . ) T H E T A l = T H E T A 1 - 3 6 0 . 3 4 7 I F ( T H E T A 1 . G T . G A M M A ) GO T C 460 3 4 8 T R A 0 l = C . C 1 7 4 5 3 2 9 * T H E T A l 3 4 9 X 1 » R 1 * C C S ( T R A C 1 ) 3 5 0 Y l * A B S ( R l * S I N ( T R A 0 l ) ) 3 5 1 X 1 K - X 1 / 1 C C 0 . 3 5 2 4 2 0 OC 45C K S = 1 , I S 353 N S = I F I X S ( K S ) ) 3 5 4 S I G Y 1 * A Y ( N S ) * ( X I K * « 8 Y ) 3 5 5 I F I X 1 . G T . 1 0 C 0 . 1 GO TO 4 3 0 i 129 I I - 7 3 5 6 S I G Z l = A Z l ( N S ) * ( X l K * * 8 Z l ( N S n * C Z K N S > 3 5 7 G C TO 4 4 C 3 5 8 4 3 0 S I G Z l = A Z 2 < N S ) * ( X l K * * e Z 2 ( N S I ) * C Z 2 ( N S ) 3 5 9 4 4 0 Y U = ( Y 1 / S I G Y I M * 2 3 6 0 S I G Y Z 1 » S I G Y 1 » S I G Z 1 * 3 . 1 4 1 5 9 3 3 6 1 P P 8 S * P F e * P S ( K S » 362 0 0 4 5 C K U = l , I U 3 6 3 H E F F 1 = H H - 0 H U 1 ( K U J * S F A C T R ( K S I 3 6 4 E X I = Y Y 1 - M H E F F I / S I G Z 1 ) * * 2 3 6 5 E l » ( E X P ( - 0 * 5 * E X l i ) / ( S I G Y Z l * U ( K U ) ) 3 6 6 P E « P P B S * P U K L > 3 6 7 C P N Z E 1 = C P N Z E 1 * P E 3 6 8 C A L L S C R T I E l , P E . I C E 1 » C E h W C P C E i I 3 6 9 0 0 4 5 0 K Q l = l , 1 0 1 37C C » E l * G K K C n 3 7 1 P C * P E * P G 1 ( K C I ) 3 7 2 C P N Z C = C P N Z C - » P C 3 7 3 C A L L S O f t T ( C . P C . I C C , C C > - , O P C C ) 3 7 4 4 5 0 C O N T I N U E 3 7 5 4 6 C C O N T I N U E 3 7 6 GO TO 6 9 0 377 C s » a = » s s s = TWO S O U R C E S = « « « « 3 7 8 4 7 C I F ( I T R A C E . E O . 1 ) F P I N T 4 8 0 3 7 9 4 8 0 F C R M A T ( * • » 'MA I N - 3 5 0 • I 3 8 0 C P N Z E 2 * C . 3 8 1 0 0 6 8 0 K E = 1 , I E 3 8 2 P P e » P 8 ( K E I 3 8 3 B 1 8 0 - = E ( K e ) - 1 8 C 3 8 4 ' 7 H E T A 1 » A B S ( A L P H A l - 8 1 8 0 1 3 8 5 I F I T H E T A l . G E . 3 6 0 . ) T H E 7A L = T H E T A 1 - 3 6 0 . 3 8 6 K E 1 » 1 3e7 I F ! T H E T A I . L E . G A M P A I GO T C 4 9 0 3 8 8 K E 1 » 0 3 8 5 E l » 0 . 3 9 0 GC TO 5 0 C 3 9 1 4 9 0 T R A D I N G . C 1 7 4 5 3 2 9 * T h E T A l 3 9 2 X 1 = R 1 * C C S ( T R A 0 1 > 3 9 3 " Y l * A 8 S ( R l * S I N ( T R A C U ) 3 9 4 X l K = X i / l C C C . 3 9 5 C 3 9 6 5 0 0 T h 6 T A 2 = A 6 S ( A L F H A 2 - 8 1 8 C ) 3 9 7 I F U H E T A 2 . G E . 3 6 0 . ) T H E T A 2 = T H E T A 2 - 3 6 0 . 3 9 8 K E 2 = l 3 9 9 I F ( T H E T A 2 . L E . G A M M A ) G C TC 5 1 C 4 C C E 2 * 0 . 4 C 1 K E 2 = 0 4 0 2 GO T O 5 2 Q 4 0 3 5 1 C T P A C 2 = C . 0 1 7 4 5 3 2 9 * T H E T A 2 404 X 2 = R 2 * C C S ( T R A 0 2 ) 4 C 5 Y 2 = A 8 S ( R 2 * S I N ( T R A 0 2 ) ) 4 0 6 X 2 K = X 2 / 1 C C C . 4 0 7 C 4 C 8 5 2 0 I F I K E 1 . E C . 1 . A N 0 . K E 2 . E C . I ) GC T O 5 3 0 4 0 9 I F < K E 1 . E C . 1 . A N C . K E 2 . E C . G ) GC TO 5 9 0 4 1 C I F l K E l . E C 0 . A N 0 . K E 2 . E C . i l GC T O 6 3 0 4 1 1 I F ( K E 1 . E G . Q ~ A N 0 . K E 2 . E Q . 0 ) GO TO 6 7 0 4 1 2 C C A S E l U — T H E T A 1 . T H E T A 2 > GAMMA — 4 1 3 C — K E 1 , K E 2 = 1 4 1 4 5 3 0 0 0 5 8 0 K S = l , I S 4 1 5 P P 8 S = P P 8 * P S I K S ) 130 II-8 4 1 6 N S = I F I X ( S ( K S 1 ) 4 1 7 S I G Y 1 = A ^ ( N S ) * ( X 1 K * * 8 Y ) 4 1 8 I F U L G T . I O O O . ) GC TG 5 4 0 4 1 9 S I G Z 1 = A Z I I N S ) * ( X I K * * 8 2 1 ( N S » l - C Z l ( N S » 4 2 0 G C T C 55C 4 2 1 5 4 C S I G Z 1 = A Z 2 ( N S > * C X 1 K * * B Z 2 I N S > I + C Z 2 ( N S » 4 2 2 5 5 0 Y Y I = ( Y 1 / S IGY1 ) * * 2 4 2 3 S I G Y Z 1 = * S I G Y I * S I G Z 1 * 3 . 1 4 1 5 9 3 4 2 4 C 4 2 5 S I G Y 2 » A Y ( N S ) * ( X 2 K * * e Y ) 4 2 6 I F ( X 2 . G T . 1 0 0 C . ) G C TO 5 6 0 4 2 7 S I G Z 2 = A Z 1 ( N S > * ( X 2 K * * 8 Z U N S ) l + C Z M N S l 4 2 8 GO T G 5 7 C 4 2 9 5 6 0 S I G Z 2 = A Z 2 ( N S J * ( X 2 K * * e Z 2 ( N S ) ) + C Z 2 ( N S I 4 3 0 5 7 C Y Y 2 * ( Y 2 / S I G Y 2 > » * 2 4 3 1 S I G Y 2 2 = S I G Y 2 * S I G Z 2 * 3 . 1 4 1 5 9 3 4 3 2 C 4 3 3 0 0 5 S C KU = i , I t 4 3 4 H E F F l * H l + O h U l ( K U J * S F A C T R ( K S ) 4 3 5 E X 1 = Y Y W ( H E F F I / S I G Z 1 >**2 4 3 6 E 1 » ( E X P ( - 0 . 5 * E X 1 I ) / ( S I G Y Z I * U ( K U ) I 4 3 7 P E = P P B S * P U ( K U l 4 3 8 C P N Z E 1 = C P N Z E 1 * P E 4 3 9 C A L L S C R T I E 1 , P E , I C E 1 , C E H , 0 F C E U 4 4 0 H E F F 2 = H 2 * 0 h U 2 ( K U ) * S F A C T R ( K S ) 4 4 1 E X 2 = Y Y 2 - ( H E F F 2 / S I G Z 2 > * * 2 4 4 2 E 2 = ( E X P ( - 0 . 5 * E X 2 ) ) / ( S I G Y Z 2 * U ( K U ) > 4 4 3 C P N Z E 2 = C P N Z E 2 * P E 4 4 4 C A L L S C R T ( E 2 , P E , 1 C E 2 , C E H 2 , 0 F C E 2 > 4 4 5 ' CO 5 8 0 K 0 1 = 1 , I C 1 4 4 6 C l = E l « O l ( K C i » 4 4 7 0 0 5 8 0 K Q 2 = 1 , I C 2 4 4 8 C 2 = E 2 * C 2 ( K C 2 I 4 4 9 P C = P E * P C 1 ( K C 1 ) * P C 2 ( K C 2 > 4 5 0 C P N Z C = C F N Z C * P C 4 5 1 C = C 1 * C 2 4 5 2 5 8 0 C A L L S C R T ( C , P C , I C C , C C H , C P C C J 4 5 3 GO T C 6 7 0 4 5 4 C C A S E # 2 — T H E T A 1 > G A M M A : T H E T A 2 < G A M M A 4 5 5 C — K E 1 » 1 ; K E 2 = 0 4 5 6 5 9 C 0 0 6 2 0 K S = l , I S 4 5 7 N S = I F I X ( S ( K S ) ) 4 5 8 S I G Y I = A Y ( N S » * ( X I K * * 8 Y > 4 5 9 I F ( X 1 . G T . 1 0 0 0 . ) G C TO 6 0 0 4 6 0 S I G Z l ' A Z l ( N S » * I X l K * * e Z l ( N S ) ) * C Z 1 ( N S I 4 6 1 GO T C 6 1 0 4 6 2 6 0 C S I G Z 1 * A Z 2 ( N S ) * ( X 1 K * * B Z 2 ( N S J ) * C Z 2 ( N S I 4 6 3 6 1 C Y Y 1 = ( Y 1 / S I G Y 1 ) * * 2 4 6 4 S I G Y Z 1 = S I G Y I * S I G Z 1 * 3 . 1 4 1 5 9 3 4 6 5 P P B S = P P 6 * P S ( K S ) 4 6 6 0 0 6 2 C KU=1,IU 4 6 7 H E F F l = h l ~ O H L l ( K U » * S F A C T R I K S > 4 6 8 E X I » Y Y 1 * ( H E F F I / S I G Z 1 ) * * 2 4 6 9 E i » ( E X P l - C . 5 * E X l l I / ( S I G Y Z 1 * U ( K U I I 4 7 0 P E « P P B S * P U ( K U ) 4 7 1 C P N Z E 1 = C P N Z E I + P £ 4 7 2 C A L L S O R T ( E 1 , P E , I C E 1 , C E H 1 , O P C E 1 J 4 7 3 0 0 6 2 C K Q l M . I C i 4 7 4 C » E 1 * 0 1 ( K C 1 ) 4 7 5 P C » P E * P Q I ( K O I I 131 i i - y 476 CPNZC-CPNZC+PC 477 62C C A L L SCRT(C,PC, ICC,CCH,CPCC) 478 60 TO 670 479 C CASE#3 — THETAKGAMMA : THETA2>GAMMA — 480 c — KE1 = 0 S KE2 * 1 481 630 00 660 KS=l, IS 482 N S » I F I X ( S ( K S ) ) 483 SIGY2=AY<NS)*<X2K**SY) 484 I F ( X 2 . G T . 1 0 0 C . » GC TC 64C 485 SIGZ2»AZ1(NS !*<X2K**8Z1(NS J )+CZl(NS) 486 GC TC 65C 487 640 SIGZ2=AZ2(NS)*(X2K**8Z2(NS)> +CZ2INS) 488 650 YY2=(Y2/SIGY2)**2 489 SIGYZ2=SIGY2*SIGZ2*3.141553 490 * PP8S=PPe*PS(KS) 491 DO 66C KL=l , IU 492 HEFF2=H2*DHt)2( KU ) *SF ACTR < KS) 493 EX2»YY2+(HEFF2/SIGZ2>»*2 4 9 4 E 2 » I E X P ( - 0 . 5 * E X 2 ) ) / ( S I G Y Z 2 * L ( K U ) ) 495 PE=PPBS*PU(KU) 496 CPNZE2=CPNZE2*PE 497 CALL SCRT(E2,PE, ICE2,CEH2,DPCE2) 498 OC 66C KQ2=l.IQ2 499 C«E2*Q2(KC2) 500 PC»FE*PC2(KC2> 501 CPNZC=CPNZC*PC 502 66C CALL SCRTICPCICCtCCHtCPCCI S03 67C CONTINUE 504 680 CONTINUE 5C5 c 506 69C CPZ£l=CLNe*CUMS*CUf<U-CPNZEl 5C7 CALL GROUP(ICE1,CPCEl .CFCEl .CPZE1• ITRACE) 508 IFINS0URC.NE.2I GO TO 700 509 CPZE2=CUr*e*CUMS*CUHU-CPNZE2 510 CALL GROUP(ICE2»OPCE2»CPCE2«CFZE2,ITRACE) 511 CPZC»CUMe*CUMS*CUMU*CUMOl*CUfQ2-CPNZC 512 GC TO 71C 513 700 CPZC=CUWe*CUMS*CUMU*CUMQl-CPNZC 514 710 CALL GROUP 1 ICC,CPCC.CPCCtCPZCtITRACE) 515 516 c S 3 3 S 3 2 3 S 3 5 S 5 3 3 3 S 3 S S 2 5 3 S c I | PRINT OUT 1 1 517 c a a M 3 a a = = s = 3 s = = = s = = s = s = 518 720 IF I IPRINTID.EQ.OI GC TC 730 519 PR INTE2C 520 730 IF( IPPINT(2) .6a.O) GO TC 740 521 CALL P R I N T Z U S , S , S L C G , P S , C P S . L E L S 3 , L 8 L S 4 ) 522 C A L L PRINTZ( IU,U,ULGG,PU,CPU,LBLU3,L8LU4» 523 CALL PRINTZ< Ie ,e ,8LCG.P8 ,CPe,LeLe3 , L8LB4) 524 CALL PR INT Z(101 ,Q1.0ILCG.PC 1 ,CPCI .LBLC13.L8L014) 525 IFINSCURC.NE.2) GC TO 740 526 CALL PRINTZIIC2,C2,Q2L0G,PC2,CPC2,LBLC23,LBLQ24I 527 740 I F ( I C A L C . N E . l ) GO TO 760 528 IF I IPRINT(2) .EC.0l GO TC 760 529 CALL PRINTZJICE1,CE1,CE1LCG,CPCE1,CPCE1,LBLE13,L3LE14) 530 IF(NSCURC.NE.2) GO TO 750 531 CALL PRINTZ( ICE2,CE2.CE2LCG,0PCE2,CPCE2,L8LE23,LBLE24) •32 750 CALL FRINTZ( ICC.CC,CCLGG,0PCC,CPCC,LBLC3,LBLC4) 533 c 3C3S3S3C 3E33333335 3333 3331 33333! 33 3333 534 c II PLOTS 1 1 535 c » 3 3 3 3 3 3 3 3 3 3 = = 5 = = : = = -132 I I - I C 536 537 538 535 540 541 542 543 544 545 S46 547 548 54* 550 551 552 553 554 555 556 557 558 559 560 561 562 S63 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 760 CALL P L C T Z U I S , S , S L O G , P S , C P S , 1 4 , IPLCTStXSIZES.YSIZES, ISYM8S. 2 L8LS1,NS1,L3LS2,NS2,LBLP1,NP1,LBLP2.NP21 CALL P L C T Z K I U r U . U L C G . P U . C P U , 1 4 , IPLCTU,XSIZEL.YSIZEU. ISYMBU, 2 LBLUl »NU1.LBLU2.NU2.LeL PI ,NPl .L3LP2.NP2) CALL P L C T Z l ( i e , 8 , 8 L C G . P 8 , C P e , 1 4 , IPLCTB,XSIZEB,YSIZEe, ISYM8B. 2 L 8 L B l , N B l , L B L B 2 . N B 2 » L e L P l . N P l .LBLP2.NP2) CALL PLCTZ1(IC1,Q1,<J1L0G,PQI,CPC1, 1 4 , IPLTGI ,XSIZC1,YSIZCI , ISYMC1. 2 L 8 L 0 U f N C l l , L a L 0 1 2 , N C 1 2 t L S L P l . N P l , L 8 L P 2 , N P 2 ) IFCNSCURC.NE.2) GC TO 770 CALL PL0TZ1( I02 .02,C2LCG.PC2,CPC2. 1 4, IPLTQ2.XSIZC2.YSIZC2, ISYMQ2. 2 LBLC21,NC21,LBLC22,NC22.LeLPl ,NPl ,LBLP2,NP2 ) 770 I F C I C A L C . N E . i l GC TC 790 CALL P L O T Z U I C E l , C E i . C E l L C G , C P C E l , C P C E l » 1 4 , IPLTE1 ,XS IZE1 ,YS IZE1 , ISYME1, 2 L B L E l l . N E l l . L 8 L E 1 2 , N E l 2 , L e L P l . N P l . L B L P 2 , N P 2 l IF(NS0URC.NE.2) GO TO 780 CALL PLCTZUICE2,CE2,CE2LOGtOPCE2.CPCE2. 1 4 . IPLTE2.XSIZE2.YSIZE2. ISYME2. 2 LBLE21 ,NE21 ,LBLE22 .NE22 ,LBLPl .NPl .L8LP2 .NP2 l 78C CALL P L G T Z H I C C C C C C L C C C F C C C P C C , 1 4 , I P L C T O X S I Z E C Y S I Z E C , ISYMBC, 2 L B L C l , N C l , L B L C 2 . N C 2 , L B L F l , N P l , L B L P 2 , N P 2 l 790 CALL PLCTNO C C c I 3 Z S 3 3 3 3 S 5 S S S 3 S S S 3 : II FORMATS 800 FORMA712CG20.7I 810 FOflMAT(G10.4t40Al) 820 F O R M A T ( • I • » * • / / • • a x , l 2 3 4 5 6 7 8 9 * G . L . C . DATA GENERATION PROGRAM VARYING STA8ILITY (SI WINO SPEEC (U) HI NO OIRECTION (BI EMISSION RATE (0) , a x . • , / • • ,3X, • , /« • ,8X. ' , / • • ,3X, «,/• • ,3X. • , / • • ,3X, . , / . t . a x . . , / . . . a x , . , / • . ,3X, • / / / / ) STCP ENC C C C C C C C c c c c c c c S U B R O U T I N E S II SUBROUTINE REAO1 SS SS 11 SS 133 11-11 596 C«* TO REAO IN PARAMETERS FOR A PROBABILITY DISTRIBUTION. 597 C 598 SUBROUTINE REAOIIN, IPTX, IX, IRESCX.XI-.OX.RMUX, S IGX, X ,PX, 599 1 ITRACE) 600 DIMENSION X ! M , P X ( N ) 601 IF{I TRACE.EO.1) PRINT10 602 10 FORMAT! * « ,«—REA01 —•• I 603 REA015,8C> IFTX 604 REAC(5,80) Ix 6C5 REA0(5,8C) IRESCX 606 GO T0(2C,2C,3C) , IPTX 6C7 20 REAC(5,8C) Xh.OX,RMUX.SIGX 6C8 RETURN 609 30 RE4C(5,80) IFMT 610 GO T C ! 4 0 , 6 C ) , IPMT 611 4C DO SC 1=1. IX 612 SO REAC(S.8C) X d J . P X d ) 613 RETURN 614 60 00 70 1 = 1, IX 615 70 REAC(5,9C) I I , X d ) ,XX, PX d ) 616 80 FQPMAT(2CG2C.7J 617 90 F0RMAT(I4,3E16.7) 618 RETURN 619 ENO 62C C SS -SS 621 C II . SUBROUTINE REAC2 II 622 C SS • SS 623 C * * TO READ IN PARAMETERS FOR PLCTS AND PRINT CUTS. 624 C 625 SUBROUTINE R E A 0 2 ( I P L C T , X S I Z £ , Y S I Z E , I S Y M B • 626. 1 LBL1.N1,L8L2,N2,L8L3,LBL4, ITRACE) 627 . DIMENSION IPLCT14), X S I Z E ( 4 » , YSIZE14) , ISYMB.4» 628 LOGICALM LBL1 (40) , LSL2 (4C ), LBL3 (16 ) , LBL4(16) 629 I F U T R A C E . E C . l ) PRINT10 630 10 FORMAT!• • , •==REA02==« ) 631 PEAC<5,20) IPLCT 632 REA0(5,2C) XSIZE 633 REJC(5,2C) YSIZE 634 REA0(5,2C) ISYMB 635 REAC15.3C) N1.L3L1 636 REACI5.3C) N2,LBL2 637 REACI5,4C) LBL3 638 REAC(5,4C) LBL4 639 20 FGRMATI4G2C.7) 640 30 F0RMAT(G1C.4,40A1) 641 4C FORMAT!16A1) 642 RETLRN 643 ENO 644 C SS SS 645 C II SUBROUTINE NORMAL II 646 C SS SS 647 C * * TO GENERATE A NORMAL PROBABILITY DISTRIBUTION. 648 C 649 SUBROUTINE NORMAL!IX,XH,OX,RMUX,SIGX, X ,PX, INDEX,CYCLE, 65C 1 ITRACE) 651 DIMENSION X ( I X ) . P X d X ) 652 IF( ITRACE.EC.1) PRINT10 653 10 FORMAT! • • , • NORMAL-*-*- • » 654 SS IG*S IGX* l . 4 l42 655 00 30 1=1,IX 134 11-12 656 J » I X - I * l 657 X ( J ) = X H - C X » ( I - 1 > 658 IF( INDEX.ECO 1 GC TO 20 659 I F ( X ( J ) . L E . C . ) X (J ) = X ( J ) 4 C Y C L E 660 I F I X U ) .GT.CYCLE 1 X ( J l = X ( J ) - C Y C L E 661 20 P l=ERF( (X(J ) -C .5*0X-RMUX) /SSIG) 662 P2=ERF(I X I J I « C . 5 * C X - R M L X ) / S S I G ) 663 30 P X ( J ) a ( P 2 - P l » / 2 . 664 RETURN 665 ENC 666 C SS- •. • • — — S S 667 C II SUBRCUTINE LGNGRK 1 1 668 c — SS 665 c * * TO GENERATE A ICGNCRMAL PRCGAEILTY DISTRBUTION, USING THE 670 c •1 NCRMAL* SUERCUTINE ABOVE 671 c 672 SUBROUTINE LGNORMIIX,XH ,0X,RPUX,SIGX,X ,XLOG,PX, ITRACE) 673 DIMENSION X( IX) ,XLCG<IX)tPX( IX) 674 IF( I TRACE . E O . l t PRINTIO 675 10 FCRMATI • '» ' • •LCGNGSM**• ) 676 c XHLCG=AL0G10IXH) 677 c 0XLCG*ALCG10(CX) 678 c RMULQG 3ALCGlC(Rr*U) 679 c SIGLOG-ALCGIO(SIG) 680 CALL NCRMAL(IX,XH*CX,RMUX.SIGX,XLOG,PX,0,0.0) 681 CO 20 1=1,IX 682 20 X( I) = 1C.* *XLCGI I ) 683 RETURN 684 ENC 685 c SS — — _ _ _ _ _ - S S 686 c II SUBRCUTINE ALCGZ 11 687 c ss - S S 688 c * * TC COMPUTE ALOGIO.IXI FOR AN ARRAY. NOTING ERRORS IF IT OCCURS. 689 c 650 SUBROUTINE ALCGZUX. X , XLCG. ITRACE 1 691 0IMENSIQN X ( I X ) , X L C G U X ) 692 I F I I T R A C E . E Q . i l PRINTIO 693 10 FORMAT!' : : A L C G Z : s ' l 694 00 40 1=1,IX 695 IF IX I I I .GT .C . I GC TC 30 696 IF(X( I I . L T . O . ) GO TO 20 697 PRINT50,I 698 x m - i . o 699 GC TC 3C 700 20 PRINT6C, I ,X( I ) 7C1 XII ) = A8S(X( I) ) 7C2 3C XLOGII )=ALCG1G(X(I)l 7C3 40 CONTINUE 704 50 FCPMAT (' • , 'ERRCRs X ( I ) . E C O . ; I = ' , I 3 , « OEFAULT X( I )=0. 7C5 60 FORMAT ( • ' , ' E R R C R : X U I . L T . C : I=>« , 13, » XII ) = • , E 1 5 . 7 / • • . 7C6 1 • DEFAULT X I I ) = A 8 S X { I I « ) 707 RETURN 708 END 7C9 c SS - S S 710 c II SUBROUTINE CUMUL 11 711 c SS -SS 712 c * * TO CCMPUTE CUMULATIVE PROBABILITIES FROM DIFFERENTIAL PRC8. 713 c 714 SUBROUTINE CUMULI IX ,PX ,CPX,CW, ITRACE) 715 CI PENSION PX( IX) ,CPX( IXJ 135 11-13 716 I F d T R A C E . S 0 . i l PRINTIO 717 10 FCRMAT(* ' , ' — C U M U L — • ) 718 CUM=0. 719 00 20 1=1,IX 72C CUM=CUM*PX(I) 721 2C CPXd)=CLM 722 RETURN 723 ENC 724 C SS ^SS 725 c 1 1 SUBROUTINE KESCAL 11 726 c SS SS 727 c*« ' TO RESCALE A CISTRIBUT1CN SO THAT CUMULATIVE PROBABILITY = 1.0 728 c 729 SL'EROLTINE RESCAU IX,PX,CPX,CUM, ITRACE) 730 DIMENSION PX(IX) , C P X d X ) 731 I F d T R A C E . E Q . i l PRINTIO 732 10 FORMAT(• • , » > - R E S C A L - < ' 1 733 CUM=0. 734 SUM=CPX(IXI 735 I F ( S U M . L E . C ) GO TC 30 736 00 20 1=1,IX 737 PX( I l = PX(11/SUM 738 CIM=CUM*PXCI) 739 2C CPX<II=CUM 740 RETLRN 741 30 PRINT40,IX,CPX( IXI 742 40 FORMAT(• «,'ERRCR IN R E S C A L , C P X ( I X ) . L E . 0 ' / • 743 I «IX= • , 15, • CPX(IX)= » , E 1 6 . 7 / I 744 RETURN 745 ENC 746 c SS —SS 747 c 11 SUBROUTINE CLASS1 J 1 748 c SS SS 749 c** TO CREATE CLASSIFICATIONSFOR ARRAY X,GIVEN MAXIMUM X,INTERVAL 750 c SIZE S NUMBER CF INTERVALS: 751 c FOR IPTX=1 X WILL HAVE EQUAL LINEAR INTERVALS; 752 c IPTX=2 X WILL HAVE EQUAL LOGARITHMIC INTERVALS. 753 c 754 SUBROUTINE CLASSI( IPTX, I> ,DX.XMAX,X,XH,XLCG,XHLCG. 755 1 OPCX,CPCX,ITRACEI 756 DIMENSION XIIX I, X»- (IX) ,XLCG(IX I ,XHLCG(IX1, 757 1 CPCX(IX),CPCX( IX ) 758 IF( ITRACE.EC.1) PRINTIO 759 10 FORMAT! • • , ' — C L A S S 1—• ) 760 GO TC(20,4C)• IPTX 761 20 CALL CLASS21IX,XMAX,OX,XH,X,ITRACE) 762 CO 30 1=1,IX 763 XLOG (I )=ALCG10!Xm 1 764 30 XHLCG(I)*AL0G10(X(I)) 765 RETURN 766 4C XMXLOG=ALOGIO(XMAX) 767 CXLCG=CX 768 CALL CL ASS2II X,XMXLOG,OXLCCXHLOG,XLCG,ITRACEI 769 00 50 1=1,IX 770 X H d l » 1 0 . * * X H L G G ( I 1 771 50 X d I=IC.**XLCG(I ) 772 OC 60 1=1,IX 773 OPCX(I )=C. 774 60 C P C X ( I ) » C . 775 RETLRN 136 11-14 776 777 C 778 C 779 c 780 c** 781 c 782 c 783 c 784 c 785 c 786 c 787 c 788 789 790 751 792 793 794 795 796 797 798 799 c 8CC c 801 c 802 c** 8C3 c 804 c 8C5 806 8C7 8C8 809 810 811 812 813 814 c 815 c 816 c 817 c** 818 c 819 c 820 821 822 823 1 824 825 826 827 828 829 830 831 c 832 c 833 c 834 c** 835 c ENC SS — SS II" SU8RCUTINE CLASS2 I I TO CREATE CLASSIFICATIONS FCR ARRAY OF X, BASED CN SUPPLIED MAXIMUM X(XMAX) ,INTERVAL SIZE(DX) £ NUMBER OF INTERVALS REQUIRED! IX ) . XII » CORRESPONDS TC MIOPCINT OF THE CLASS, X H U ) THE UPPER LIMIT. X U X J IS MAOE TC BE ONE CLASS GREATER THAN XMAX SO THAT LATER THE DIFFERENTIAL PROBABILITY DISTRIBUTION WILL END WH> A ZERO ENTRY, AS DESIRED FCR OICRETE FCURIER TRANSFORMATICN. SUBROUTINE CLASS2!IX,XMAX,OX,XH,X,I TRACE) OIMENSICN X(IXI,Xh(lX) I F ( l T R A C E . E O . l ) PRINTIO 10 FORMAT(• • ,«==CLASS2==« » 0X2=0X/2. 00 20 1*1, IX J=IX-I+l XCJI -XMAX-CX*!1-2) 2C XH!J )=X1J»-CX2 RETURN ENC I t SUBROUTINE SORT I I SS SS TC SCRT AND PLACE X,PX IN APPROPRIATE CLASS IN OISTRIBLTICN CF CXH.CFCX SUBROUTINE SORT IX,PX, ICX.CXH,CPCX) DIMENSION CXH(ICX),0PCX(ICX) 00 10 1 = 1, ICX J » I C X - I - H I F « X . G T . C X h ! J ) l GO TO 20 10 K=J 20 OPCXIK)=CPCX!K)+PX RETURN ENC SS SS 11 SUBROUTINE GROUP I I SS—• SS TC GROUP DIFFERENTIAL PROBABILITIES TC FCRM CUMULATIVE PROBABILITIES. PROBABILITY CF X . L E . X ( l ) IS GROUPED IN CPCXI1) SUBROUTINE GRCUF!ICX,OPCX,CFCX.CPZEPO,ITRACE) DIMENSION CPCX!ICX),CPCX(ICX) IF! ITRACE.EC.1 ) FRINTIO 0 FORMAT!• • , • . . G R C U P . . « ) SUM=»CPZERC 00 2C 1=1,ICX SUM=SUM4CPCXII) CPCXII)=SUM CPCXU)=DPCX< l)*CPZERO RETURN ENC I I SOUROUTINE PRINTZ I I SS SS ! PRINT A DISTRIBUTION. 137 11-15 836 SUBROUTINE PR INTZI N , X , X L C G , FX ,CFX, L E U ,L8L2 > 837 OIHENSICN X(N),XLCG<N>,PX(N),CPX(N) 838 LOGICAL*! LBLL(16) ,L8L2(16 I 839 P R I M 4 C , L e L l , L 3 L 2 840 J * l €41 00 10 1*1 fN 842 IF (CPXI I ) .NE.O.O) GC TO 20 843 10 J=I 844 2C CO 30 l * J . N 845 K = I - J * l 846 30 PRINT5C»K»X(11 ,XLOG! I I,PX(I I ,CPX(I) 847 40 FORMAT(• • , / / / 2 2 X , • * * - * - * - * - * - * - * - * - * - * - * - * * ' , / / • • , 848 1 2 X , ' J *«32A1 , 'PROBABIL ITY ' ,5X»'CUMULATIVE * / • ' , 849 2 54X, 'PROBABILITY' / ) 85C 50 FORMAT!• ' , I 3 , 4 £ l 6 . 7 l 851 RETLRN 852 ENC 853 C SS > SS 854 C It SUBROUTINE PL0TZ1 II 855 C SS SS 856 C * * TC PLCT 1 TC 4 DISTRIBUTIONS USING PLCTZ2 SUBROUTINE 157 C 858 SUBROUTINE P L C T Z K I X , X , A L O G X , P X , C P X , 859 I N . IPLOT,XSIZE,YSIZE, ISYMB, 860 2 LBLXI ,NXI ,L3LX2,NX2.LBLY1,NY1,LBLY2,NY2) 861 01 MENS ION X(IX),ALCGX(IX),PX<IX ) ,CPX! IX ), 862 1 IPLOTIN),XSIZE(N) ,YSIZE(N).ISYMB!N) 863 LOGICAL*! LBLXK NX1 ), LBLX2 !NX2) .LBLY1! NY1) ,L8LY2! NY2 ) 864 IFCIPLQTtI ) .EC .C ) GO TO 10 865 CALL ALAXIS(LELX1,NX1,LBLY1,NY1) €66 CALL P L C T Z 2 U X , X , P X , X S I Z E ! l l , Y S l Z E ( l l , l S Y M B i l l l . 867 10 IF! IPLOT12J.EC.C) GC TC 20 868 CALL ALAXIS(LELXl ,NXL,LeLY2.NY2) €69 CALL PL0TZ21I>,X,CPX,XSIZE(2) ,YSIZEI2) , ISYMB!2)) 87C 20 IF! IPLOT!3 ) •EC.01 GC TC 30 871 CALL ALAXISILBLX2,NX2,LELY1,NYU 872 CALL PLCTZ2! IX ,AL0GX,PX,XSIZE(3) ,YSIZE!3) , ISYM3I3) ) 873 3C I F U P L 0 T ( 4 ) . E C . C > GC TC 40 874 CALL ALAXIS!LBLX2,NX2,LBLY2,NY2) 875 CALL PLCTZ2( IX,ALCGX,CPX,XSIZE!4) ,YSIZE(4) , ISYMB(4) ) 876 4C RETLRN 877 ENC 878 C SS SS 879 C II SUBROUTINE PLQTZ2 II 880 C SS — S S 881 C * * TO PLCT A DISTRIBUTION USING UBC ALGRAF ROUTINES 882 C 883 SUBROUTINE PL0TZ21N,X,Y,XSIZE,YSIZE ,1SYMB) 884 OIMENS ION X!N) ,Y!N) €85 CALL ALSIZElXSIZE,YSIZEl 886 CALL A L S C A L I O . C O . 0 , 0 . 0 , 0 . 0 ) 887 CALL ALGR4F(X«Y,N, ISYMB) 888 RETLRN 889 ENC ENO CF FILE APPENDIX Ii-T-B E q u a t i o n s u s e d i n t h e Program (D.G.P.) N Wind D i r e c t i o n F i g u r e 2.2: P h y s i c a l C o n d i t i o n s f o r t h e G a u s s i a n Plume D i s p e r s i o n E q u a t i o n The ground l e v e l c o n c e n t r a t i o n i s g i v e n by: d i a g r a m : C = EQ The C h a r a c t e r i s t i c f u n c t i o n i s d e f i n e d b y: y z y z x = R cos 6 y = R s i n 8 6 = JB-TV-aj H = H + f -AH. e s f 0.2 , f s = ! + _ _ ( s _4) CD (2) (3) (4) (5) (6) (7) 139 II-V D T -T A H £ = - j j - (1.5 + 0.00268 pD S T a j a = A xEy y y B a = A X z + C z z z C o e f f i c i e n t s A , B , A , B , C y y z z z (8) (9) (10) s 1 2 3 4 6 1 i A y 213 156 104 . 68 50.5 34 1 i B y 0.894 | j F o r X<1000 M i j i A z j 440. 8 106.6 61.0 s 33.2 22.8 14. 35 | i i B z 1. 941 1.149 0.911 i : 0.725 0.678.; 0. 740^ C z 9. 27 3.3 ' o. i i -1.7 -1.3 -0. 35 ; \ \ F o r X>1000 M i A z 459. 7 i f 108.2 61.0 44.5 55.4 I 62. 6 i B z i 2. 0961 i 1.098 0.911 , 0.516 ; i 0.305 0. 180; c z -9. 6 j 2.0 o I 1 -13.0 | i -34.0 ; 1 l - 4 8 . 0 • j ** Cu r v e f i t t i n g by D.O. M a r t i n 1 3 o f p l o t s by D. T u r n e r 1 8 N o t e s : * N o m e n c l a t u r e as p e r page (106) Appendix I I - T - C C a l c u l a t i o n S t e p s i n GLC Program 1. G i v e n o r g e n e r a t e : P g C S j ) , " P y C U . ) , P B ( B K ) i = l , N s j = l , Nu K=1,N B 1=1, N„ 140 n-18 2. C a l c u l a t e : E . = f ( S . , U., B„, o t h e r d a t a ) f o r i = 1, Ng, J = 1, Ny, K = 1, N B a s s o c i a t e d P r o b . = P_ ^ = P g ( S D *Py(U\)'PgCB^) 3. C a l c u l a t e : C, ... = E. ..-Q, f o r i = l , N g, J = l , Ny, K=1,N B, £-1, N g o a s s o c i a t e d P r o b = P C = P E Lijk£ E i j k Q 1 4 . F i n d E = max[E. .. 1 max L i j k J o C = max[C. .. ] max L i j k J 5. Form N c l a s s e s o f E, C, w i t h c l a s s - i n t e r v a l ACE o o N max N max E.. = E - (N-U)ACE C X f = C - (N-U)ACE U max N max v •* o o 6. S o r t E i j k , Cijk£ i n t 0 P r o b a o i l i t y " D i s t r i b u t i o n P'„, P c p _ , ,_ ACE ACZ, P ^ = ProbCEy - -j- <E<_EU * —) N N N = Prob { E S E U E B(E - - £ _ < £ . . , < E + ^P) } m 2 i j k — m 2 J i = l j = l k = l S i m i l a r l y : •p . P r o b ( E - ^ E < E < E m + ^ ) E. ., m 2 — m 2 13k N N N A r p A r p - Prob ffi S ZU S B ( E _ - ^ <E ^ E . + ^ ) > i = l j = l k = l N N N N . A C E S i m i l a r l y : P f = Pro b { E S Z U Z B E Q (C - — < C i i k £ l C m C i j k l i - l j = l k = l £=1 m 2 ^ m 1 141 11-19 II.2 Program F o r A n a l y s i s f o r D i s t r i b u t i o n s ( P - A . D J A. Program L i s t i n g $1 PAD 1 C 2 C 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 c 11 c 12 c 13 c 14 c 15 c 16 c 17 c 18 c 19 c 20 c 21 c 22 c 23 c 24 c 25 c 26 c 27 c 28 c 29 c 3C c 31 c 32 c 33 c 34 c 35 c 36 c 37 c 38 c 39 c 40 c 41 c 42 c 43 c 44 c 45 c 46 c 47 c 48 c 49 c 50 c 51 c 52 c 53 c 54 c 55 c ( P . A . D . ) PRCGP.AM FOB ANALYSIS CF DISTRIBUTIONS. THIS PROGRAM PERFORMS CCNVUCLUTICN OR C ECCNVOLUTION ON TWO GIVEN PROBABILITY DISTRIBUTIONS ( A , B ) , TC YIELD A THIRO PROBABILITY DISTRIBUTION ( C ) . SPECIF ICALLY,TUS PRCGRAM 1) CAN TREAT PRCEABILITY CISTRI BUT ICNS IN EITHER LINEAR OR LCG. SCALE : 21 CAN CONVOLVE DISTRI BUT ICNS, FCR C=A-B CR C=AB: 3) CAN DECONVOLVE 0 I STR I BUT ICNS, FCR C=A-8 OR C=A/B; 4) CAN DECONVOLVE DISTRIBUTIONS. FCR C=8-A OR C=e /A AFTER C HAS BEEN OETERMINED,THERE ARE CFTICNS TC: 1) CONVOLVE DISTRIBUTIONS, FOR £=C*0 OR E=CD 21 DECCNVCLVE 01STR IELT ICNS, FCR E=C-D CR E=C/C THE PATH OF CALCULATICNS IS CONTROLLED BY THREE INCICES: ICPT1, ICPT2.IOPT3. ( I I -I ICPT1 I I (2 )-1(1) I C = A + B LOGIC)= LCG(A)+LOG(B) (C=ABl I0PT2 I L 1(2) I C»A-B LCG(C)= LCGJA)-LOG(B) (C=A/B> 1(3) I C=8-A LOG(C)= LQG(B)-LOGIA) (C=B/A( ICPT3 I (1) 1(1) I E*C*0 LCG(E)= LQG(C)+LOG(0) (E*CD) 1(2) I E=C-0 LQG(E ) = LCG(C)-LCG(D) (E*C/C» 1(C) I STOP STOP 142 11-20 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c C c c c c c c c c c c c C c c c c DATA INPUT ITRACE I0PTI . ICPT2, IOFT3 IREADA*IR EACE IREACC NA,N8,NG ITRIMC.ITRIME O E L C C E L E IRESCCtIRESCE IPLOTXU-4) XSIZEXC1-4) Y S I Z E X U - 4 ) ISYMBX!1-4) LBLX1 L8LX2 LBLPl LBLP2 LBL>3 LBLX4 ISTATX INDEX FOR TRACING PATH CF CALCULATIONS; 1/TRACE CALCULATIONS : C / 00 NOT TRACE. INDEXES FCR PATh OF CALCULATIONS AS DEFINED ABCVE. INDEX FOR REAC FORMATS FCR DISTRIBUTIONS CF A,B,D ; SEE SUBROUTINE READZl . NUMBER CF ENTRIES OF DISTRIBUTIONS OF A , 3 . 1/TPIM OFF SMALL FLUCTUATICNSl < CElC.OELE» OF PROB.DISTRIBUTION OF C,E ; 0/ DC NOT TRIM. MAX. FRACTIONAL FLUCTUATION ALLOWED IF ITRIMC,ITRIME =1 l/RESCALE DISTRIBUTION SO THAT CUM.PR0B=1.0 C/DC NOT RESCALE. INOEX FOR PLCTS ; l / P L O T : C / NO PLOT. IFLCTXCl ) : X VS. PROBABILITY IPL0TXI2): LOGIX) VS. PROBABILITY IPLCTX(3I: X VS . CUMULATIVE PROB. IPL0TX14) LCGIX) VS. CUMULATIVE PROB. HORIZONTAL PLGT SIZES { INCHES) VERTICAL FLCT SIZE(INCHES) INDEX FCR PLCT SYMBOLS <0 ! POINTS CNLY; >0 : PCINTS C L INE. »0 J LINE CNLY. LINEAR X-AXIS LABEL!FLCT01.3 I ( M CF CHARACTERS=NX1,MAX=40) LOG.X-AXIS LAeEL{PL0T*2#4 I 14 OF CHAR.^NXll PROBABILITY Y-AXIS LABEL!PLCT#1,2 I (# OF CHAR.=NP1) CUMULATIVE PRCB.Y-AXIS LABEL!PLOT#3,4) (* OF CHAR.=NP2) PRINT OUT HEAOING,LINEAR SCALE,MAX=16 CHAR. AS LBLX3,LCG.SCALE. INDEX FCR CALCULATIONS ANC PRINT CUT CF PARAMETERS OF A DISTRIBUTION :l /YES;0/NO. * * * * * * * * * 4 4 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CIMENSICN A !512) ,ALCG(512),PA! 1024),CPA(5 1 2 ) ,FPA( 1C24) , 1 B !512) .BLCG(512».P6 (1024),CPB!512 I , F P e ( 1 0 2 4 ) , 2 C ! 5 1 2 ) , C L 0 G ! 5 1 2 I,PC(1 0 2 4 ) ,CPC(512)TFPC(1 C 2 4 ) , 3 DI512),DLCG15 12>,PCI1C24I ,CPC I 512 ) , FPC U 0 2 4 > , 4 E!512) ,ELCG(512),PE!1024),CPEI 5 1 2 ) , F P E ( 1 C 2 4 ) OIMENSICN I PLCTA(4) ,XSIZEA!4),YSIZEA(4) , ISYM8A!4 I, 1 I P L C T B(4) , X S I Z E a(4).YS I Z EE!4),IS Y M B B(4), 2 IPLCTC(4) ,XS IZEC(4J ,YS IZEC14 ) , ISYMBC<4), 3 IPLCTCI4) , X S I Z E D!4 ) , Y S I Z E 0(4) , I S Y M b O(4), 4 I P L C T E!4 ) , X S I Z E E(4I , Y S I Z E E(4 ) , I S Y M B E(4) COMPLEX TRPAI513) ,TRP8(513 ),TRPC1513 ) , 1 TRPCI513),TRPE(513) EQUIVALENCE IFPA.TRPAI , (FPB.TRPB) , 1FPCTRPC) , 1*3 H - 2 1 116 1 ( F P C T R P O ) , ( F P E . T R P E I 117 LGGICAL*1 L B L A l ( 4 0 ) , L e L A 2 ( 4 0 ) , L B l A 3 ( l 6 ) , L B L A 4 ( 1 6 > , 118 1 L B L 8 1 ( 4 0 ) F L B L B 2 ! 4 C ) * L B L B 3 ! 1 6 ) , L B L S 4 ( 1 6 ) , 119 2 L f l L C l ( A O ) « L B L C 2(40 ) « L E L C 3 ( 1 6 ) « L B L C 4 ( 1 6 ) * 120 3 L B L 0 l ( 4 0 ) . L B L D 2 ( 4 C ) . L 8 L D 3 ( l 6 ) , L e L 0 4 U 6 ) , 121 4 L B L E l!40 ) , L B L E 2!40 ) , L e L E 3 I 1 6 ) , L B L E 4 ( 1 6 1 , 122 5 LBLP1I40) , LBLF2(4C • .T ITLE(10.40) 123 C • + + 124 C DATA INPUT 125 C — 126 REAC(5,14C> NTITLE 127 CO 10 1 = 1, NTITLE 128 10 REAC(5,16C) (T ITLE! I.J ) ,J=1«4C) 129 REAO( f . lAC) ITPACE 130 REAC15.14C) IGPT1,I0PT2,I0PT3 131 READ(5,14C) IREAOA, IREACB 132 REAC(5,14C) NA,NB 133 NN=NA+NE 134 CO 2C 1=1, S 135 NP=2**I 136 IF(NP.GE.NN) GO TC 3C 137 20 CONTINUE 138 30 N=NP 139 P=2*N 140 M = N*1 141 CALL REAC2KN ,NA, A .ALOG,PA ,CPA, IPLOTA, XSI2EA, YSIZEA, ISYMBA, 142 1 1STATA,L6LA1,NA1,LBLA2,NA2,L8LA3,LBLA4,IREAOA,ITRACE) 143 CALL PEACZl(N,Ne,B ,BLCG,Fe ,CPe , IPLOTB,XS I ZE8 ,YS IZEB, ISY^B3 , 144 1 ISTATB,LELB1,N81,LBLB2.N82.LBLB3,LBLB4, IREACB,ITRACE) 145 IREACC=3 146 C A L L REACZ1(N .NC.C.CL0G,FC,CPC. IFL0TC,XSIZEC,YSIZEC. ISYMBC, 147 1 I S T A T C L E L C l , NCI, LBLC2•NC2,LBLC3.LBLC4,1 REAOC , ITRACE) 148 REA0I5.14C) ITRIMC,DELC , IRESCC 149 IF( I 0 P T 3 . E C 0 ) GO TC 4C 150 READ15.14C) IREAOC 151 REACI5.14C) NO 152 CALL FEACZ1(N ,N0 .D.CLCG,FC ,CFC. IPL0T0,XSIZE0,YSIZED, ISYMBD, 153 1 ISTATC,LBLC1,NC1,LBL02,N02.LBL03,LBLC4, IREACC,ITRACE) 154 IREADE=3 155 C A L L R E A C Z H N , N E , E . E L C G , P E , C F E , I P L O T E , X S I Z E E , Y S I Z E E , ISY«3 = , 156 1 ISTATE,LBLE1,NE1,LBLE2,NE2,LBLE3,LBLE4, IREADE, ITRACE) 157. REA0I5,14C) ITRIME,DELE, IRESCE 158 40 REAC(5,15C) NPl .LBLPl 159 REAC(5,15C) NP2.LBLP2 160 C — 161 C CONVOLUTION S DECCNVCLLTICN A,E —> C 162 C • + 163 IFTA=1 164 C A L L CNVLTMM ,N 1 ,N , P A, PB , PC , FFA , FPB, FPC, TRP A, TRPB , TRPC, 165 1 . 10PT2,IFTA,ITRIMC,CELC,IRESCC,ITRACE) 166 GO T0(5C,6CJ, ICPT1 167 50 CALL >AXI£(N»A ,NA,8»N8,C ,NC»N*C ,10PT2»ICPT1,ITRACE) 168 CALL LCGZ(N,NNC,C,CLCGi 169 GO TC 7C 170 60 CALL XAXISIN,ALOG,NA ,8L0G,NB.CLCG,NC,NNC,I0PT2,I0PT1,ITRACE) 171 CALL E X P Z U ,NNC .CLCG.C) 172 70 CALL CUMUL(N,NC,PC,CPC,0.0, 1, ITRACE) 173 C •+ 174 C ** CONVOLUTION £ DECONVC LUT ICN C C —> E 144 1 1 - 2 2 176 IF I ICPT3.EC.C) 60 TC 110 177 IFYC*2 178 CALL CNVLTNtM,N l ,N .PC.PD,PE,FFC,FPD,FPE,TRPC.TRPO.TRPE, 179 1 IGPT3, IFTC,ITRIME,OELE.IRESCE,ITRACE) 180 GO T C t B C S C ) , ICFT l 181 80 CALL XAXISIN,C,NC,C,NO,E,NE,NNE, I0PT3, I0PT1, ITRACE) 182 CALL LCGZ(N»NNE»E»ELCG) 183 GO TO ICC 184 90 CALL XAXIS(N,CLCG,NC,OLCG,NO,ELCG.NE.NNE,I0PT3, ICPT 1,I TRACE) 185 CALL EXPZlN.NNE.ELCG.El 186 100 CALL CUMUL1N, NE.PE, CPE, 0 . 0 , 1, ITRACE) 187 C •»» ~ •*-<• 188 C •»•» PRINTOUT £ PLOTS 189 C * * 190 11C 00 12C I=1,NTITLE 191 120 P R I N T 1 7 C , ( T I T L E ( I . J ) . J - 1 , 4 0 ) 192 CALL PRTPLT(N,NA,A,ALGG,FA.CFA. 193 1 LBLAI ,NA1 ,L8LA2,NA2,LELA3,LBLA4,LBLPl ,NP1 .LBLF2 .NP2 , 194 2 IPLCTA,XSIZEA,YSI2EA, ISYfEA, ISTATA, ITRACE) 195 CALL F R T P L T ( N , N e , B , B L O G , F B , C P E , 196 1 LBLB1,NB1,LBLS2,NB2,LBLB3,LBLB4,LBLPl ,NP1,L8LP2,NP2, 197 2 IPLCTB,XSIZEB,YSIZEB,ISYKBB,ISTATe, ITRACE) 198 CALL FRTPLTIN,NC,C,CLCG,FC,CPC, 199 1 L B L C l , N C l , L B L C 2 , N C 2 , L B L C 3 . L B L C 4 , L e L P l , N P l , L E L P 2 , N P 2 , 200 2 IPLCTCXSIZEC, YSIZEC, ISYKeCISTATC, ITRACE) 2CI IF ! IQPT3.EC.C) GO TC 130 202 CALL FRTPLT(N,NC,0,CLOG,PC,CPO, 203 1 LBLOl ,NCI ,LBL02 ,N02 ,LELC3 ,LBL04 ,LBLPl ,NP1 ,LELP2 ,NP2 , 204 2 IPLCTCXSIZED.YSIZEO, I SYKBD , I ST AT C , ITRACE) 205 CALL F R 7 P L T | N , N E , E , E L C G , F E . C P E . 206 1 LBLE 1»NE1,LBLE2,NE2,LBLE3»LBLE4,LBLPl ,NP1,LBLF2«NP2» 207 2 IPLCTE.XSIZEE,YSIZEE,ISYMBE,ISTATE,ITRACE) .208 130 CALL FLCTNO 209 140 FORMATI20G20.7) 210 15C F0F>AT(G2C.7,40A1I 211 16C FORMAT(4CA1) 212 170 FCFMATC « ,40A1) 213 STCP 214 ENC C >XSSS*a83CSS383tSSSS33SSSS>SiaSS333333aSSS3SSSSS33S=33«31IS 216 C 217 C S U B R O U T I N E S 218 C 219 C 22C C 221 C SS SS 222 C II SUBROUTINE REA0Z1 II 223 C SS SS 224 C**TO REAC IN CATA FOR EACH DISTRIBUTION. 225 C 226 SUBROUTINE READZl (N ,NX ,X ,XLCG,P X,CP X , 227 1 IPLOT,XSIZE,YSIZE, ISYMB,ISTATS, 228 2 LBLX1,NX1,LBLX2.NX2.LBLX3,LELX4, IREAC,ITRACE) 229 DIMENSION X IN),XLCG<N),PXIN I .CPXIN ), 230 1 I P L 0 T ( 4 » , X S ! Z E ( 4 I . Y S I Z E ( 4 ) , I S Y M B < 4 J 231 LOGICAL*1 L E L X l ( 4 C ) , L 8 L X 2 ( 4 0 ) « L B L X 3 ( 1 6 ) , L 8 L X 4 ( 1 6 ) 232 C 233 IFI ITRACE.EO.l) PRINT 10 234 10 FCF-M AT (• — R E A C Z 1 — • ) 235 REAC(5,8C) IPLOT 2 3 6 R E A C ( 5 , 8 C 1 X S I Z E 2 3 7 R E * C t « , E C I Y S I Z E 2 3 8 R E A C ( 5 , e C l I S Y M B 2 3 9 R E 4 C 1 5 . 8 C , I S T A T S 2 4 0 R E A C ( 5 , 9 0 ) N X 1 . L B L X 1 2 4 1 R E A C I 5 , 9 C I N X 2 . L B L X 2 2 4 2 C E A C ( 5 , 1 0 Q ) L B L X 3 2 4 3 R E A 0 ( 5 , 1 C C J L B L X 4 2 4 4 I F U R E A 0 . E C . 3 I R E T U R N 2 4 5 NNX=N-NX-H 2 4 6 MNXA=NNX-1 2 4 7 CO 20 I = 1 , N N X A 24e P X ( I I = C . 2 4 9 2 0 C P X I I 1 = 0 . 2 5 0 GO T 0 U C 5 C I , I R E A D 2 5 1 3 C CO 4C I=NN>,N 2 5 2 4 0 RE/C{5*IIO i x , x m , x L C G ( i ) 9 P x m , C P x m 2 5 3 R E T U R N 2 5 4 5 0 C C 6 0 I=NNX,N 2 5 5 6 0 R E A D ( 5 , 8 C J X ( I ) , P X I I l 2 5 6 S U H = 0 . 2 5 7 OC 7C I=NNX,N 2 5 8 XLCGm = A L C G 1 0 ( X C I J» 2 5 9 SUM=SUM-»PX(II 2 6 C 7 C C P X I I I = S U M 2 6 1 8 0 F C R M A T I 2 0 G 2 0.7) 2 6 2 9 0 F C P M T ( G 1 6 . 7 , 4 0 A U 2 6 3 I C O F O R M A T ( 1 6 A 1 ) 2 6 4 1 1 0 F 0 R W A T 1 I 4 , 4 E 1 6 . 7 , I 5 ) 2 6 5 R E T U R N 2 6 6 E N C 2 6 7 C SS — S S 2 6 8 C 11 S U B R O U T I N E C N V L T N 1 1 2 6 5 c S S — — • - • SS 2 7 0 C * * T C P E P F O R M F C U R I E R TRANSFORM , C C N V C L U T I C N , 0 E C 0 N V O L U T I C N , 2 7 1 C F O P D I S T R I B U T I O N S SC TH AT : { C=A-»B ) , {C= A-B I OR (C=8-A) 2 7 2 C I TH C P T I C N S TO T R I M O F F S M A L L F L U C T U A T I O N S AND T C R E A C A L E • 2 7 3 C 2 7 4 S U E R O L T I N E C N V L T M M , M , N , P A , P E , P C , F F A , F P B , F P C , T R P A , T R P 8 . 2 7 5 1 T R P C , I C O N V , I F TA, I T RIM,D E L T A ,IRE S C , I T R A C E ) 2 7 6 D I M E N S I O N P A I M 1 . P e i M I , P C ( M | , F P A I M 1 , F P B C M 1 , F P C ( M I 2 7 7 C O M P L E X T R P A I N l J , T R F B I M ) . T F P C ( M ) 2 7 8 C 2 7 9 I F C I T P A C E . E C . l t F R I N T 1 0 2 8 0 10 F O R M A T ( • * , ' • • C N V L T N * * ' ) 281 I F ( I F T A . N E • 1 ) GO T C 4C 2 8 2 OC 20 1=1,N 2 8 3 20 F P A ( I 1 = P A ( I I 2 84 0 0 2C I=N1,H 2 8 5 30 F P A C I ) = C . 2 8 6 GO TO 6 C 2 8 7 4 0 DC 50 1=1.M 2 8 8 50 F P A I I > = P A ( I ) 2 8 9 6C I F I I T R A C E . E O . i l PR I N T 7 C 2 9 C 70 F C R M A T ( * • , ' — F T . P A — • ! 2 9 1 C A L L F C U R 2 I F P A , M , 1 , - 1 , 0 I 2 9 C CQ 80 1 = 1 , N 2 5 3 80 F P 8 ( I ) = P B ( I I 2 9 4 CO 50 I » M , M 2 9 5 9 0 F P 8 i I I = C . 296 IF( ITRACE.EQ.1) PRINT100 297 100 FORMAT!• — F T.Pe—«) 296 CALL F 0 U R 2 ! F P E » M , 1 , - 1 , 0 ) 299 I F U C C N V . N E . l ) GO TO 130 300 I F I I T R A C E . E Q . i l PRINTllO 301 110 FORMAT(' « , ' — C C N V — • ) 302 00 12C 1 * 1 , M 303 120 TPPC!I)=TRPA!I)*TRPB(I) 3C4 GC TC 23C 305 130 IF IICCNV.NE.2) GO TC 180 3C6 IF! ITRACE.EQ. i ) PRINT140 307 14C FORMAT t ' ' , •—OECCNV A/B — • S 308 CO 17C 1*1,Nl 3C9 AX*CABS!TRPA!1)1 310 EX*CABS!TRPE(I)) 311 I F I A X . E C . C . ) GO TC 150 312 I F I 8 X . E C C . ) GO TO 16C 313 TRPCII)=TRP/( I ) /Tf iPB(I) 314 GC TC 17C 315 150 TRPCII )=(0.0,0.0) 316 GC TO 17C 317 160 PRINT28C.I ,TRPA!I) ,TRFB I1 ) 318 T R P C I I ) * ( 1 . 1 1 1 1 , 1 . l i l l ) 319 17C CONTINUE 32C GC TC 23C 321 18C IF( ICCNV.NE.31 GO TC 270 322 IF I ITRACE.EO. l ) PPINT150 323 15C FORMAT(* • , • — C E C C N V E/A — 324 00 22<J I=1,N1 325 AX=CABS1TRPACI)) 326 8X=CABSITRPEII)) 327 IF<BX.EG.C) GC TC 200 328 IF IAX.EC.O) GO TC 21C 329 TRPCII)=TRP6!I) /TRPA(I) 330 GC TC 22C 331 2CC TRPC!I )=IC.C,C.O) 332 GC TC 22C 333 210 PRINT2 8C >1 , TR PAI I) ,TR P8 11) 334 TRPC! I )=C1.1111,I.1111) 335 22C CONTINUE 336 230 CC 240 1 = 1,M 337 240 PC(I)=FPC!I) 338 IF! ITRACE.EO.1) PRINT250 335 250 FORMAT (* INV.FT. PC — •) 340 CALL FCUR2IPCM.1, 1,-11 341 CQ 260 1=1,M 342 260 P c m = p c i n / M 343 CALL RARRG<M,N,PC,ICGNV,ITRACE) 344 I F I I T R I M . E C l ) CALL TR IM (M ,PC .CELT A, I TRACE) 345 IF! I R E S C . E C l ) CALL RE SCAL(M, PC,ITRACE) 346 RETURN 347 27C PRINT29CICCNV 348 280 FORMAT(• • , 'ERROR IN COMPLEX CIVISION . I .TRPA, TRPB ARE: • / 349 1 • •tI5 , 4 E 1 6 . 7 ) 350 290 FORMAT(• 'ERROR, ICCNV.NE.1,2 OR 3: ICONV=', 15) 351 RETURN 352 END 353 C 354 c II SUBROUTINE RARRG 1 1 355 c SS — SS 1 4 7 11-25 356 C * * 10 1 SHIFT CATA IK ARRAY PX<N): 357 C FOR PX FRCM CONVOLUTION, ROTATE DATA FORWARD BY 358 C (K+n- POSITIONS; 359 c FCR PX FRCM CECONVOLUTICN, POTATE CATA FORWARD BY 36C c ( N - lJ POSITIONS. 361 c 362 SUBROUTINE RARRGIM.N.PX,ICONV,ITRACE1 363 OIMENSION PXIM 364 IF( ITRACE.EQ.1) PRINTIO 365 10 FCRMATI* ' , ' — RARRG—•» 366 MA=M-l 367 GO TG<20,4C,4Q),ICCNV 368 20 XX=PX(M) 369 CO 30 1=1,fA 370 J = M-I +1 371 30 PX(JI = PXU-U 372 PXll)=XX 373 GC TO 6C 374 40 XX=PX(1) 375 CO 50 1 = 1,MA 376 50 p x n )=PXI i * i ) 377 PX(MI=XX 378 6C CO 70 1=1,N 379 J=N*I 380 XX=PX(Ii 381 PX(I)=PX(J1 382 7C PX(J»=XX 383 RETURN 384 ENO 385 C SS 386 c II SUBROUTINE TRIM 11 387 c S S  388 C**T0 TRIM OFF FLUCTUATIONS IN PROBABILITY WHICH ARE LESS THAN 389 c SPECIFIED FRACTION CF THE HIGHEST PEAK. 390 c 391 SLBRQLT INE TRIMIM,PX,0ELTA,ITRACEI 392 DIMENSION PX(M) 393 IF ( I TRACE•EQ.1J PRINTIO 394 10 FCPMAT I • — T R I M — • » 395 PXMAX=0. 396 00 20 1=1,M 397 2C IFtPXMAX.LT.PX( I )) PXMAX=PXUJ 398 OC 30 1=1,M 399 XX=ABS(P>(IJ/PXMAX1 400 30 IFIXX.LT.CELTA) PXU» = 0 . 4C1 CALL RESCALIM.PX,ITRACE) 402 RETURN 403 ENC 404 c 405 c I | SUBROUTINE RESCAL 1 1 406 c SS • SS 407 C**T0 RESCALE CISTRIBUTION SO THAT CUMULATIVE PROBABILITY = 1.0 4C8 C 409 SU8R0LTINE RESCAL(M ,PX. I TRACE) 41C OIMENSICN PX(M) 411 IF ( ITRACE.EQ. l ) PRINTIO 412 10 FCRMATt• — R E S C A L — • ) 413 SUM=0. 414 00 20 1=1,M 415 20 SLM=SUM*PXII1 416 00 30 1*1,M 417 30 PXUI=PXUI/SUM 418 RETLRN 415 ENC 420 C SS — S S 421 c II SUBRCUTINE XAXIS II 422 c SS SS 423 C**TC CETERMINE THE SCALES ON THE X-AXIS 424 C -425 SUBROUTINE XAXI S( N ,A ,N'A ,B ,N8 ,C ,NC ,NNC . I CPT1, I 0PT2. ITRACE) 426 DIMENSION A(N) ,8(N),C(N) 427 IFIITRACE.EC.il PRINTIO 428 10 FOPMATI• » , ' — X A X I S — • ) 429 NNA»N-NA4l 430 NNe=N-N8*l 431 GC TCI2C3C ,4C) ,IOPTl 432 20 CVA>=A(M-eiNI 433 CMIN=A(NNA )•E1NNB) 434 GC TC 6C 435 30 CMAX=AIN)-E(N) 436 GC TC 5C 437 4C CMA>=EIN)-AINI 438 50 IF( I0PT2.EC.1) CMIN=l.E-75 439 IF( ICPT2.EC.2I CMIN=-75. 44C 60 GRC=A(N)-A(N-1) 441 CCNSTC=CMAX-GRC*N 442 DO 70 I-l.N 443 NC=I 444 j= \-1 w> i 445 C!J)=GRC*J*CCNSTC 446 IFICIJ).LT.CMIN» GC TC 8C 447 70 CONTINUE 448 8C NC*NC-1 449 NNC=N-NC-H 450 RETURN 451 END 452 C SS- : SS 453 C 11 SUBROUTINE CUMUL 11 4 54 c s s SS 455 C**CALCULATIGN CF CUMULATIVE PROBABILITIES 456 C 457 SUBROUTINE CUMUL<N,NX,PX,CPX,CPX2,ICUM,ITRACE) 458 CIMENSICN PXIN),CFXINI 459 IF(ITRACE.EO.1) PRINTIO 460 1C FCRMATI* — CUMUL—') 461 NNX=N-NX*1 462 NNXB = NNX-»l 463 IF! ICLM.NE.ll GO TC 30 464 CPXINNX )=PX(NNX)*CPXZ 465 DO 2C I=NNXB,N 466 20 CPXI i>=CPX(I-II*PX(i» 467 RETURN 468 3C PX«NN>l=nCP>INNX»-CPXZ 469 CO 40 I=NNX8,N 470 4C p x n i * c p x ( i ) - c p x i i - i ) 471 RETLRN 472 END 473 C 474 C 11 SUBRCUTINE PRTPLT 11 475 C SS ss 149 11-27 476 C**TC PRINT OUT AND PLOT DATA FOR A GIVEN DISTRIBUTION. 477 C 478 SUBROUTINE PRTPLTIN,NX,X,XLCG,PX,CPX, 479 I L B L X l , N X l , L B L X 2 , N X 2 , L E L X 3 . L B L X 4 , L B L P l . N P l . L B L P 2 . N P 2 480 2 IPLOT,XSIZE,YSIZE, ISYMB,ISTATS. ITRACE I 481 DIMENSION X(N) ,XLGG(N),PXCN ) ,CFXIN 1, 482 1 IPLCTI4) .XSIZEI4) ,YSIZE(4) , ISYMfi(4) 483 LCGICAL*1 LELXl INX1) ,LBLX2(NX2>,LBLXj{16) .LBLX4I16) , 464 1 L B L P K N P l >.LBLF2(NP2) 485 CALL FRINTZIN.NX.X,XLOG,PX,CPX.LBLX3»LBLX4) 486 I F U S T A T S . N E . l ) GC TC 10 487 CALL STATS1(N,NX,X,PX,LELX3) 488 CALL STATSHN .NX.XLCG, PX , LBLX4 I 489 IC 00 2C 1=1,4 45C I F I I P L C T d U E C . l ) GC TO 30 491 2C CONTINUE 492 30 CALL S U FT IN , NX , X , XLCG, PX ,C FX, I TRACE) 493 CALL FLCTZ1(NX,X,XLCG,PX,CPX,4 , IPLOT.XSIZE,YSIZE. ISYMB, 494 1 LBLX1,NX1,LBLX2,NX2,LBLPl ,NP1,LBLP2,NP2) 495 4C RETURN 496 ENC 498 C H SUBROUTINE STATS1 II 499 C SS SS 500 C**TO CCMFUTE STATISTICAL PARAMETERS: 501 C MEAN,STANOARO DEVIAT ICN,ASSYMETRY,CO EFFICIENT OF KURTCSIS 5C2 C 503 SUBROUTINE STATSl lN .NX,X .PX,LABEL) 504 CIMENSICN X(N),PX(NI 505 LOGICAL* I LABELU6I 5C6 RMU=>0. 507 VAP=0. 5C8 S 3 * C . 5C9 S4=C. 51C NNX=N-NX+l 511 CO 10 I=NNX,N 512 IC RML=RMU*X(I)*PX<I) 513 00 20 I=NNX,N 514 SS=XII)-PMU 515 VAR=VAR+{SS**2>*PXII» 516 S3=S3*(SS**3)*PX(I ) 517 2C S4=S4+lSS**4>*PXm 518 IFIVAR.GT . C . ) GO TC 30 519 S I G - C . 520 G1 = 0 . 521 G2=C. 522 GO TO 4C 523 30 SIG=SGRT(AES(VAR) ) 524 G1=S3/(SIG**3I 525 G2=S4/CS IG**4i 526 4C PRINT50,LABEL 527 PRINT60,RMU,SIG,G1,G2 528 50 FORMAT(' • , « • / • « , • *=* STATISTICAL PARAMETERS FCR • . 529 1 •DISTRIBUTION O F ' , 2 X . 1 6 A 1 , • * = * • / / • * , 8 X . ' M E A N ' . 1 2 X . 530 2 'STO.DEV. ' . 8X , 'SKEWNESS' ,5X , •COEFF.KURTOS IS • ) 531 60 FORMAT!' • .E2C.7.3E16.7> 532 RETLRN 533 END 534 C SS SS 535 C II SOURCUTINE PRINTZ II 150 H - 2 8 536 C SS -SS 537 C**TO PRINT A DISTRIBUTION. 538 C 539 SUBROUTINE PRINTZI N,NX,X,XLCG,PX,CPX,LBL1.LBL21 540 DIMENSION X(NI.XLCG(NI,PX(NI,CFX(N I 541 LCG ICAL*1 LBL1116),LBL21 16J 542 PRINT20.LeLl .LBL2 543 NNX=N-NX*1 544 OC 10 I=NNX,N 545 J=I-NNX*1 546 10 PRINT30 ,J ,X ( I I ,XL0G( I ) ,PX( I I ,CPX( I1 , I 547 20 FORMAT! • * »* • / • , . 1 5 X , , * = * = * = * = *=* = *=* =*=*=* =*=* = *=*=*% 548 I • • / / / • • , 2 X , » J ' , 32A1 , 'PROBABIL ITY* ,5X , 549 2LMLLATIVE« , 7 X , « I * , / • • , 54X , •FRCEABIL ITY• / I 550 30 FORMAT I * • , 13 ,4E16.7 , I 5 ) 551 RETLRN 552 ENC 553 C SS —SS 554 C Ii SUBROUTINE SHIFT II 555 C SS — S S 556 C**TO SHIFT DATA IN ARRAYS GF SIZE N FROM (NNX,NI TO I I,NX) 557 C (TC REDUCE SIZE OF ARRAY TO BE PLOTED) 558 C 559 SUeROUTINE SHIFTIN,NX,X,XLCG,FX,CPX,ITRACE) .560 DIMENSION X ( N ) , X L C G ( N ) , P X ( N ) , C F X ( N ) 561 I F d T R A C E . E C . i l PRINTIO 562 10 FORMAT(• ' , ' — S H I F T — ' ) 563 *NX*N-NX* l 564 J * l 565 00 20 I=NNX,N 566 X(J)=X(II 567 XLCG(J ) = >LCG( I) 568 PX(JI=PX(I> 565 CPX(JJ=CPX(I) 57C 20 J=J*1 571 RETLRN 572 ENC 573 C SS SS 574 C 11 SUBRCUTINE PL0TZ1 II 575 C SS — SS 576 C * * T C PLOT CISTRIBLTICNS USING UEC ALGRAF ROUTINE 577 C 578 SUBROUTINE PLCTZ1(I X,X,XLCG.FX ,CPX , 579 1 N, IPLCT,XSIZE,YSIZE, ISYMB, 580 2 LBLXI,NX1»LBLX2»NX2.LBLY1.NY1,LBLY2,NY2 I 581 DIMENSION X( IX ) ,XLCG(I X ),PX( I X ) ,CPX( I X) , 582 I IPLCT(N).XSIZE(N).YSIZE(N), ISYM8IN) 583 L0GICAL*1 LBLX1(NX 1),LBLX2(NX2),LBLY1(NYi) ,LBLY2(NY21 584 IF( I F L C T ( l ) . E C . C ) GC TC 10 585 CALL ALAXIS ILELXl .NXULBLYl .NY l ) 5e6 CALL PLCTZ2(I X,X,PX,XSIZE(1 I,YSIZEII), ISYMBI11 ) 587 10 IF ( IFL0T(2 ) .EC.CI GC TC 20 588 CALL ALAXIS(LELX1,NX1,LBLY2,NY2) 585 CALL FLCTZ2IIX,X ,CPX,XSIZEI2I ,YSIZE!2I . ISYMBI2II 590 20 IF ! IPLQT(31.EC.O) GC TC 30 551 CALL ALAXI<(L£LX2,NX2,LELY1,NY1) 592 CALL FLCTZ2!IX,XL0G,PX ,XSIZE(3) ,YSIZE!3) , ISYMB(3)) 553 30 IF ! IPLCT(4 I .EC.C) GC TC 40 554 CALL ALAXIS(LELX2.NX2.L£LY2,NY2 ) 595 CALL PLCTZ2(1X,XL0G,CPX,XSIZE(4I ,YSIZE(4) , ISYMB(4)) 151 11-29 596 4C RETLRN 597 ENC 558 C SS- :  -SS 599 C II SUBRCUTINE PLCTZ2 II 6CC C SS 601 SUBROUTINE PLCTZ2 IN .X ,Y ,XS IZE ,YS IZE» ISYMB) 602 DIMENSION X ( M , Y ( N ) 6C3 CALL ALSIZElXSIZE,YSIZEi 604 CALL A L S C A L ( 0 . 0 , 0 . 0 , 0 . 0 , 0 . 0 » 6C5 CALL ALGRAf(X,Y,N, ISYMB) 6C6 RETURN 6C7 ENC 6C8 C SS — —ss 6C5 C II SUBROUTINE EXPZ 1 1 610 C SS :  -SS 611 C * * T C COMPUTES 1G.**X ,PRINTING ERROR MESSAGE WHEN X.GT.75 612 C 613 SUBROUTINE EXPZIN.NNX,XLCG,X» 614 DIMENSION X(N),XLCG(N) 615 DO 3C I=NNX,N 616 IPIXLOGII > .LT . -75 .J GC TC 10 617 IF IXLCGI I ) .GT.75 .1 GC TO 20 616 XII )=1C.**XLCG(I) 619 GC TC 3C 62C 1C XII M C . C 621 GC TO 30 622 20 PRINT4C ,XLCG( 11 623 XLOG(I)=75. 624 X ( I » = 1 C . * * 7 5 625 3C CONTINUE 626 40 FORMAT! • , • ERROR IN EXPZ XLCG .GT . 75 . ; XLOGl I ) = ' , E 16 .7 / • 627 I •DEFAULT XL0G(II=75 ; XI11=10.**75 •» 628 RETURN 625 ENO -SS 630 C SS—*"—— — — — — - - — 631 C | | SUBROUTINE LOGZ 11 632 C S S — — — - — — — — — — -SS 633 C**T0 CALCULATE AL0G1QIXI : PRINT ERROR MESSAGE IF X . L T . C . 634 c 635 SU8RCUT INE LOGZIN,NNX,X,XLOG1 636 DIMENSION X(N),XLCG(N) 637 00 2C I=NN>,N 638 IF(X( I) . G T . C . ) GC TO 20 635 IFIXI I ) . E C . 0 . 1 GC TC 10 640 - PP INT30 , X(I ) 641 x c i i =Aes ix i i ) ) 642 GO TO 20 643 10 PRINT4C 644 XII 1=1 .CCC 645 20 XLCG(I)=ALCG1C(X(I)1 646 30 FORMAT( ' «,«EPRCR IN LOGZ:X(I> = • , E l 6 . 7 , • O E F A U L T X=ABSIX. • J 647 40 FORMAT( • •» * ERRGR IN LOGZ ; X(II=0-C0'1 648 RETURN 645 END ENO CF FILE APPENDIX TI-2-B; Outline of Program Logic for P.A.D. 152 11-30 Z READ DATA < I0PT1 I0PT1=? OF U =1 j READ (A, PA ) ( B , PB ) / i <( IOPT3 • 3? y-^ NO READ (D,PD) I0PT2 -.'.? .-1 2 < ' = 3 c = A + B C - A - B C = B - A fc- fB@fA < IOPT3 = ? = 1 -= 2 E = = G + D E = • C - D f E= ' fG^D • *C@f D > - 3 > /READ (LOG A,P A)(LOGB,P B)/ <IOPT3 » 3? / -NO / READ (LOG D, P D) / > < I0PT2 = ? > 1 3 C - AB • C - A/B C - B/A -LOG.C LOG C LOG C =LOG A+LOG B =LOG A-LOG B =LOG B-LOG A fC- *A&B fC= fA@fB fC fB@fA 1 I *• 1 IOPT3 = ? ___ • - 2 lr S3 E = C D E = C / D L O G E =LOGC+LOG D L O G E =LOG C - L C G D f E= f c®f D fE= W D / PRINT OUT • p / PLOTS / c STOP 153 11-51 I I . 3 S t a t i s t i c a l T e s t i n g Program (S.T.P.) A. Program L i s t i n g SI STP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 25 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 S3 54 55 C C c c c c C C c C c c C c c c c C c C c C C c C c c c C c C c c C c c c C c C c C C c c c c c C ( S . T . P . ) STATITICAL TESTING PROGRAM _ _ _ _ _ _ _ FOR COMPARING ThC PROBABILITY DISTRIBUTIONS USING 1) LEAST SCUARE LINEAR REGRESSION AND CHI—SQUARED GOCDNESS OF FIT TEST FCR DIFFERENTIAL PRQBAEILIT IES: 21 LEAST SQUARE LINEAR REGRESSION AND KQLMLGQROV—SMIRNQV 60CCNESS OF FIT TEST FOR CUMULATIVE PROBABILITIES. TWO MODIFICATIONS ARE MACE IN THE TESTS: I I DIFFERENTIAL PRCEAEILITIES ARE RESCALE SO THAT SLM=1.0;, I I I CUMULATIVE PROBABILITIES ARE JUSTIFIED TO THE UPPER END. NTITLE TITLE NFREC IREAD I P L C T U - 6 I INPUT CATA « OF LINES IN THE TITLE TITLE CF OUTPUT,NTITLE LINES OF 40 CHARACTERS EACH TOTAL FREQENCY FOR THE DISTRIBUTIONS. I USED IN CHI-SCUAREO TEST 1 INDEX FCR REAC FORMATS X S I Z E d - 6 ) Y S I Z E U - 6 ) ISYMBA ISYMBB ISY*BC LBL > N PA( I ) ,P6( I ) INDEX FOR PLOTS: (1 / PLCT; 2 / NC FLCT; - 1 / OVERLAP LAST PLOT,CNLY PLCT#i PA VS. I 12 PB VS. I 03 PA VS . PB #4 CPA VS. I §5 CPB VS. I #6 CPA VS. CPB HORIZONTAL SIZES CF PLOTS IN VERTICAL SIZES CF PLOTS IN INCHES. INDEX FCR PLCT SYMBOL FCR PLOTS # 1 , ¥ 4 . INOEX FCR PLCT SYMBCL FOR PLOTS #2,#5 INOEX FCR PLCT SYMCOL FCR PLCTS # J , * 6 . X-AXIS LABEL FCR PLCTS M CF ENTRIES . DIFFERENTIAL PRCBAEILITIES OF A , 8 . OIMENSICN PM512) ,PB(512 ),CPA (512 I ,CPE (5 12 ! , XI 512 I, 1 IPLCT(6) LCGICAL*1 T ITLE( IC,40) ,LBLX(20) REAC(5,25C) NTITLE 00 10 1 = 1,NTITLE 154 11-32 56 10 REIC(5,30C) ( T I T L E ( I « J ) « J * 1 , 4 0 ) 57 REAC(5,25C) NFREQ 58 M,= NFPEC 55 PCRIT=5./M 60 REAC(5,29C) IREAO 61 REA0(5,29C) ILIST 62 REAC15,25C> (IPLOT.C I >. 1=1.6 ) 63 REAC(5,29C1 XS IZE,YSIZE, ISYMBA , ISYMB8,ISYMBC 64 R£AC(5,3GC) LB LX 65 REAC(5,2SC) N 66 GO TO(20,5C),IREAO 67 20 CO 3C 1=1,N 68 30 RE AC(5 *31C) I0LM1,0UM2,0UM3,PAC11 69 00 40 1=1,N 70 40 REAC(5,31CI ICLM1,0UM2,0UM3,PB(11 71 GO TC 8C 72 50 CO 60 1 = 1,N 73 6C REAC(5»25C) FA(I) 74 DO 7C 1 = 1,N 75 70 REAC(5,29C) PB(I) 76 C 77 C * * CALCULATE MEANS,STD.DEV. • S, ANO 2 ERROR * * 78 C (IN TERMS CF I, THE CLASS NUMBER) 79 C 8C 80 SUMA=0. 81 SUMB=C. 82 RMUA=C. 83 RMLE=0. 84 SIGA=0. 85 SIGB=C. 86 00 SC 1=1,N 87 SUMA=SUMA*PA( I) 88 SUM8=SUME-PBl I) 89 CPA(I)=SLMA 90 CPB(II=SLMB 91 RMLA=RMLA+PA( 11*1 92 90 RMUE=RMUE + FB( I )*I 93 0A=1.C-SLMA 94 CB*1.C-SUME 95 RMUA=PMUA/SUMA 96 RMUE=RMUe/SUMB 97 DC IOC 1=1,N 98 VARA=VARA+((I-RMLA)**2)*PA(I)/SUMA 99 VAR8=VARE*((I-RMUB)**2 )*PBII)/SUM8 100 CPAII)=CPA(I)fOA 101 100 CPE (I )=CPE( I MCE 102 SIGA=SORT(ABS(VARA) I 103 SICB=SORT(ABS(VARe) ) 104 PERRM.U=lCC.*(RMOA-fiMUB)/RMUA 105 PERSIG=1CC.*(SIGA-SIG8)/SIGA 107 C * * DIFFERENTIAL PROBABILITIES * * 108 C 109 C 110 C 1) LEAST SOUARE LINEAR REGRESSION 111 C 112 SUMA=0. 113 SUMB=C. 114 SUMA2=0. 115 SUMB2»C. 155 11-33 116 SUMAB*0. 117 CO 110 1 = 1,N 118 SUMA«SLMA*PA(I» 119 suMe=suye-rPe( n 120 SLMA2=SUfA2+PA(I )**2 121 SLMB2=SUyB2*Pe(I)**2 122 110 SUMAB = SUME + PA( 1 1 * P fi I I ) 123 7ERM1=SUMAE-SLMA*SL>»8/N 124 TERM2=SUMA2-(SUMA**2)/N 125 TERM3=SUMB2-(SUMB**2I/N 126 COEFFl=TERr'l/TERM2 127 COEFF2 = SUPS/N-CCEFFl*SUf'A/N 128 R2»(TERMl**2 ) /CT£RM2*TERM3l 125 R=SQRT(A8S(R2I) 130 C 131 C 21 CHI-SQUARED GCCDNESS CF FIT TEST. 132 C (DISTRIBUTION IS RECLASSIFIED SO THAT FREQUENCY IN EACH 133 C CLASS IS >= 5 ,AS RECUIR ED FOR THE TEST.I 134 C 135 XSCR-C. 136 IFIAG=0 137 NCLASS=0 138 FALAST=C. 139 F8LAST=C. 140 CO 150 1=1,N 141 FA=PA(I>*FALAST 142 FB=PB( IJoFBLAST 143 I F ( I F L A G . E C . l ) GC TO 120 144 IF (FA .LT .PCRIT .OR.FB.LT .PCPIT ) GC TC 130 145 GC TO 14C 146 120 IF IFA.GE.PCPIT .ANO.FE.GE.PCRIT l CO TO 140 147 130 IFLAG=l 146 FALAST=FA 149 F8LAST=Fe 15C GC TO 15C 151 14C >SCR=SQR*((FA-FBI**2)/FA 152 IFLAG=C 153 FALAST = C . 154 FELAST=0. 155 NCLASS=NCLASS*l 156 150 CONTINUE 157 CHISCR=NFFEC*XSCR 159 C * * CUMULATIVE PROEABIL IT IES * * 160 C 161 C 162 C 1) LEAST SQUARE LINEAR REGRESSION 163 C 164 SUPA=C. 165 SUM8=C. 166 SUMA2=C. 167 SUME2-0. 168 SUMAB=0. 169 00 16C 1 = 1, N 170 SLMA=SUMA+CPA(II 171 SUM8 = SUfE-»CF8(I» 172 SLMA2=SUfA2+CPA(I)**2 173 SLME2 = SUf B2-»CPB( I 1**2 v 174 160 SUMAB=SUMAe+CPA(I»*CP8(II 175 TERMl=SUMAE-SUMA*SUM8/N 176 TERK2=SUMA2-( SOMA**2) /N 177 TER*3=SUH82-{ SUKB * *2 I/N 178 CCEFF3MERM1/TERM2 179 COEFF4=SUMe/N-CCEFF3*SUMA/N 180 R2C=(TERf* 1**2 )/( TERf2*TERM3 I 181 RC = SQRT(AES(R2C) I 162 C 183 C 21 KCLVOGCPOV-SfIRNOV GOODNESS OF FIT TESTo 184 C 185 0PMA>=0« 166 IHAX=G. 187 00 17C IM.N 168 CP*A8S(CFA( I I -CPBUI 1 169 IFICP.LE.DPMAX) GC TO 170 190 CPHAX=0P 191 I fAXM 192 17C CONTINUE 153 SCRTN = SCRT(N *1 . ) 194 CVKS2CM.C7/SCRTN 195 CVKS1GM.22/SCRTN 156 CVKS5M.36/SCRTN 197 CVKS1M.63/SCRTN 198 C 199 c ** PR INT-CLT * * 2C0 c 201 CC 18C 1=1,NTITLE 202 180 PRINT320, (T ITLE( I ,J),J=1,40I 2C3 I F d L I S T . N E . i l GO TC 2C0 204 PRINT33C 205 OC ISC I M , N 206 190 PRINT34C, I ,PA(I),PB(II ,CPA( I ) ,CP8( I I 2C7 200 PRINT25C 208 PRINT36C,RI<UA,SIGA 209 PRINT370,RfUB,SIGB 210 PRINT38C,PERRI,U,PERSIG 211 PRINT390 212 PRINT400,CCEFFl,CCEFF2 213 PR INT4 I C R 214 PRINT420,f 215 PR INT43CNCLASS 216 PRINT44CCHSCR 217 PRINT450 218 PRINT46CCCEFF3.C0EFF4 219 PRINT47CRC 220 PRINT48C.N 221 PR INT49C,CFPA>,INAX 222 PRINT50CC\(KS20,CVKS1CCVKS5,CVKSI 223 c 224 *5 o tz C * * PLCTS * * 226 ^ ——- 00 21C IM,N 227 210 J l l I I= I * l . 228 C 229 220 IF( I F L C T l l I . E C.CI GC TC 23C 230 CALL ALS IZEIXSIZE,YSIZE1 231 CALL ALAXISILELX,2 0 , 'PROBABILITY' , I11 232 CALL A L S C A H C O , 0 . 0 , 0 . 0 , 0 . 0 1 233 CALL ALGRAF(X,PA,N,ISYNBA) 234 230 IF( IPLCTI2 I .ECCI GC TO 240 235 NNIMFL0T(2I*N 157 11-55 236 CALL AL0ASMC.5C0 .C .125 .0 .SCO.0 .125 ) 237 CALL ALGRAF(X»PE»NM«ISYM8BI 236 240 I F ( I F L C T ( 3 ) . E C . C . ) GC TC 25C 239 CALL ALAX IS I•EXPECTED PRCBAeILITY• ,20, 24C I 'CALCLLATED PRCBAB ILITY«,221 241 CALL A L S C A L ( 0 . 0 , 0 . 0 , 0 . 0 , O . O I 242 CALL ALOASMC.O , 0 . C , C . 0 . 0.0 ) 243 CALL ALGRAF(PA.PB.N.ISYMBC) 244 250 IF< IPL0T<4J.EC.O) GC TC 260 245 CALL ALAXISILeiX , 20 , 'CUMULATIVE PR08AEILITY•,22» 246 CALL A L S C A L t O . 0 , 0 . 0 , 0 . 0 , 0 . 0 ) 247 CALL ALGRAF(X,CPA,N,ISYM8A) 248 260 I F ( I P L O T ( 5 ) . E C 0 ) GC TO 27C 249 NN2-IFLCT(2) *N 250 CALL A L 0 A S M C 5 C 0 , C.125,C.500.0 .125) 251 CALL ALC-RAFIX .CPB.NN2. ISYM8E J 252 27C IF I IPLCT16 ) .EC .0 ) GO TO 280 253 CALL ALAXISI'EXPECTEC CUMULATIVE PROBABILITY' .31, 254 1 'CALCULATED CUMULATIVE PROBABILITY',331 255 CALL A L S C A H O . C O . 0 , 0 . 0 , 0 - 0 ) 256 CALL A L C A S M C G . O . C C C O . O I 257 CALL ALGRAFICPA.CPB.N,ISYM8C) 258 280 CALL FLCTNC 259 C 260 C * * FORMATS * * 261 C • 262 25C FORMAT(1CG2C.7) 263 300 F0FMATI4CA1) 264 31C FORMAT! I4.4E16.7) 265 320 FORMAT! • ' ,4CA1) 266 330 FORMAT( • ' , • I • , 3X , •PIEXPECTEC I • ,4X»•P(CALCULATED ) ' , 2X , 267 1 'CPiEXPECTED)« ,3X , 'CP1CALCULATE0) • / ) 26E 340 FORMAT!' « , I 3 , 4 E 1 6 . 7 ) 269 35C.FORMAT!' ' , « • / / / ' « , ' • • / ' « , 27C 1 * * * STATISTICAL PARAMETERS * * • / • ' , 271 2 • ' / • 272 3 ' CIN TERMS CP I, THE CLASS * ) « / / • ' , 23X, 'MEAN° 273 4 U X . ' S T O . D E V . ' / ) 274 360 FORMAT!' ' . ' EXPECTED S 2 F 1 6 . 7 ) 275 37C FORMAT 1 * • , ' CALCULATED »,2F16.7) 276 380 FORMAT!* ' , ' X ERRCR ' , F 1 4 . 5 , ' S ' , F 1 4 . 5 , ' * ' / / / ) 277 390 FORMAT! * ' , ' • / • ' , 278 1 ' * * DIFFERENTIAL PRCBA8ILIT IES * * • / • ' , 280 3 'S3 LEAST SCL'ARE LINEAR REGRESSION %i> /) 281 400 FORMAT(* • , 'P(CALCULATED )=SLCPE * PIEXPECTED) + INTERCEPT', 282 1 / ' ' , ' SLCPE = ' .F13 .4 , 283 2 / • ' , ' INTERCEPT » ' , F 1 3 . 4 ) 284 41C FORMAT!' • , »CCRRELATION COEFFICIENT = ' , F 1 6 - 7 / / ) 285 42C FORMATI• ' , ' ? ? CHI-SQARE GCCDNESS CF FIT TEST S 3 ' / / ' ' , 286 1 ' TCTAL FREQUENCY - * « , I 1 6 ) 287 430 FORMAT!* ' , ' « CF REGRCUPED CLASSES =', I16) 268 440 FORM AT(• ' , ' CHI-SOARED =',F16.7//) 289 450 FORMAT! * ' , ' ' / • ' , 290 1 • * * CUMULATIVE PROEA8ILITIES * * • / • ' , 291 2 • • / / • ' . 292 3 * J? LEAST SQUARE LINEAR REGRESSION « • / ) 293 460 FORMAT!' • , •CPICALCULAT ED> = SLCPE *CP(EXPECTED) • INTERCEPT', 294 1 / • ' , ' SLCPE - ' . F 1 3 . 4 , 295 2 / ' ' , ' INTERCEPT « ' , F 1 3 . 4 I 158 I I - 3 6 296 297 256 299 3CC 301 302 303 3C4 305 306 3C7 3 C 8 ENO CF FILE 470 FORMAT(• 48C FORM AT C' 1 • •90 FORMAT(• 1 • 2 • 500 FORMAT(• 1 • 2 • 3 8 4 • STCP ENC CORRELATION CCEFFICIENT = « , F l 6 . 7 / / > %% KOLNCGORCV-SMIPNOV GOODNESS OF FIT TEST • / / • » , ' # OF CLASSES IN» « ' » I 1 6 > MAX.DEVIATION IN CUMULATIVE PROBABILITY' / « • , F 1 6 . 7 / OCCURED AT CLASS * « , I 4 / > — CRITICAL K-S VALUES FOR N>40 A R E : ' / 20% . . . . « . F 1 6 . 7 / 10? . . . . . ' . F 1 6 . 7 / 1% « , F 1 6 . 7 / 1% « , F 1 6 . 7 I 159 I I -CMI-SQUARE D1STRIUUTION the ° n a " d f ' i n C S ^ « h e V a k ' C S ° f a t h i — « r a n d o m -iub . c for which the fight-tui. probabi.uy is as Biven on dl I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .99 .(XX)l(i .02 .12 .30 .55 .87 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 .98 .95 .90 Night-Tail Probability .80 .70 .50 .30 .20 .10 .05 MW .01 ft .04 .10 .21 .18 • .35 .58 .43 •71 . 1.06 .75 1.14 1.61 1.13 1.56 2.03 2.53 3.06 3.61 4.18 4.76 5.37 5.98 1.64 2.17 2.73 3.32 3.94 4.58 5.23 5.89 6.57 7.26 2.20 2.83 3.49 4.17 4.86 5.58 6.30 7.04 7.79 8.55 .064 .15 .4ft 1.07 1.64 2.71 3.84 .45 .71 . 1.39 2.41 3.22 4.60 5.99 1.00 1.42 2.37 3.66 4.64 6.25 7.82 1.65 2.20 3.36 4.88 5.99 7.78 9.49 2.34 3.00 4.35 6.06 7.29 9.24 11.07 3.07 3.83 5.35 7.23 8.56 10.64 12.59 3.82 4.67 6.35 8.38 9; 80 12.02 14.07 4.59 5.53 7.34 9.52 11.03 13.36 15.51 5.38 6.39 8.34 10.66 12.24 14.68 16.92 6.18 7.27 9.34 11.78 13.44 15.99 18.31 6.99 8.15 10.34 12.90 14.63 17.28 19.68 7.81 9.03 11.34 14.01 15.81 18.55 21.03 8.63 9.93 12.34 15.12 16.98 19.81 22.36 9.47 10.82 13.34 16.22 18.15 21.06 23.68 10.31 11.72 14.34 17.32 19.31 22.31 .25.00. .02 5.41 7.82 9.84 11.67 13.39 15.03 16.62 18.17 19.68 21.16 22.62 24.05 25.47 26.87 28.26 .01 6.64 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72. 26.22 27.69 29.14 30.58 (Continued) .001 10.83 13.82 16.27 18.46 20.52 22.46 24.32 26.12 27.88 29.59 31.26 32.91 34.53 36.12 37.70 Right-Tail Probability df .99 .98 .95 .90 .80 .70 .50 .30 .20 .10 .05 .02 .01 .001 17 18 19 20 21 22 23 24 25 26 27 28 29 30 6.41 7.02 7.63 8.26 8.90 9.54 10.20 10.86 11.52 12.20 12.88 13.56 14.26 14.95 6.61 7.26 7.91 8.57 9.24 9.92 10.60 11.29 11.99 12.70 13.41 14.12 14.85 15.57 16.31 7.96 8.67 .9.39 10.12 10.85 11.59 12.34 13.09 13.85 14.61 15.38 16.15 16.93 17.71 18.49 9:31 10.08 10.86 11.65 12.44 13.24 14.04 14.85 15.66 16.47 17.29 18.11 18.94 19.77 20.60 11.15 12.00 12.86. 13.72 14.58 15.44 16.31 17.19 18.06 18.94 19.82 20.70 21.59 22.48 23.36 12.62 13.53 14.44 15.35 16.27 17.18 18.10 19.02 19.94 20.87 21.79 22.72 23.65 24.58 25.51 15.34 16.34 17.34 18.34 19.34 20.34 21.34 22.34 23.34 24.34 25.34 26.34 27.34 28.34 29.34 18.42 19.51 20.60 21.69 22J8 23.86 24.94 26.02 27.10 28.17 29.25 30.32 31.39 32.46 33.53 20.46 21.62 22.76 23.90 25.04 26.17 27.30 28.43 29.55 30.68 31.80 32.91 34.03 35.14 36.25 23.54 24.77 25.99. 27.20 28.41 29.62 30.81 32.01 33.20 34.38 35.56 36.74 37.92 39.09 40.26 26.30 27.59 28.87-30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.88 40.11 41.34 42.56 43.77 29.63 31.00 32.35 93.69 35.02 36.34 37.66 38.97 40.27 41.57 42.86 44.14 45.42 46.69 47.96 32.00 33.41 34.80 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89 39.29 40.75 42.31 43.82 45.32 46.80 48.27 49.73 51.18 52.62 54.05 55.48 56.89 58.30 59.70 For df > 30, the probabilities based on the asymptotic distribution are approximated as follows: Let Q be a chi-square random variable with degrees of freedom df. A right- or left-tail probability for Q is approximated by a right- or left-tail probability, respectively, from Table A for z, where r = - ^2(df) - 1 T a b l e I I . 3 . 1 . P r o b a b i l i t y T a b l e f o r CHI-SQUARE DISTRIBUTION 160 11-38 KOLMOGOROV-SMIRNOV ONE-SAMPLE STATISTIC Table entries for any sample size M are the values of a Kolmogorov-Smirnov one-sample random variable for which the right-tail probability for a two-sided test is as given on the top row, and the right-tail probability for a one-sided test is as given on the bottom row. Right-Tail Probability for Two-Sided Test .V .200 .100 .050 .020 .010 ,Y .200 .100 .050 .020 .010 I .900 .950 .975 .990 .995 21 .226 .259 .287 .321 .344 2 .684 .776 .842 .900 .929 2? .221 .253 .281 .314 .337 J .565 .636 .708 • .785 .829 23 .216 .247 .275 .307 .330 4 .493 .565 .624 .689 .734 24 .212 .242 .269 .301 .323 > .447 .509 .563 .627 .669 25 .208 .238 .264- .295 .317 .410 .468 .519 .577 .617 26 .204 .233 .259 .290 .311 7 .381 .436 .483 .538 .576 27 .200 .229 .254 .284 .305 i .358 .410 .454 .507 .542 28 .197 .225 .250 .279 .300 ) .339 .387 .430 .480 .513 29 .193 .221 .246 .275 .295 10 .323 .369 .409 .457 .489 30 .190 .218 .242 .270 .290 11 .308 .352 .391 .437 .468- 31 .187 .214 .238 .266 .285 •s2 .296 .338 .375 .419 .449 32 .184 .211 .234 .262 .281 13 .285 .325 .361 .404 .432 33 .182 .208 .231 .258 .277 14 .275 .314 .349 .390 .418 34 .179 .205 .227 .254 .273 :5 .266 .304 .338 .377 .404 35 .177 .202 .224 .251 .269 :6 .258 .295 .327 .366 .392 36 .174 .199 .221 .247 .265 •7 .250 .286 .318 .355 .381 37 .172 .196 .218 .244 .262 18 .244 .279 .309 .346 .371 38 .170 .194 .215 .241 .258 19 .237 .271 .301 .337 .361 39. .168 .191 .213 .238 .255 ZO .232 .265 .294 .329 .352 40 .165 .189 .210 .235 K-> .100 .050 .025 .010 .005 .100 .050 .025 .010 .005 Right-Tail Probability for One-Sided Test For /V > 40, the table entries based on the asymptotic distribution are ap-proximated by calculating the following for the appropriate value of N: Right-Tail Probability for Two-Sided Test •200 .100 .050 .020 .010 1.07. V',V. 1.22 v'-V 1.36.\,V 1.52. 1.63 v77 .100 .050 .025 .010 .005 Right-Tail Probability for One-Sided Test T a b l e I I I . 3 . 2 . P r o b a b i l i t y T a b l e f o r KOLMOGOROV-SMIRNOV One Sample S t a t i s t i c s 161 I I - 3 9 Appendix I I - 3 - 5  Appendix I I - 2 - C E q u a t i o n s 1. L i n e a r L e a s t Square R e g r e s s i o n To a p p r o x i m a t e N s e t s o f p o i n t s (X^, Y^) by a s t r a i g h t l i n e : y = mX + b where EX.Y. - (EX.EY.)/N i i i i m = E X . 2 - ( E X . ) 2 / N l l EY. EX. , i i b = - r ; m'-rr-[EX.Y. - ( E X . E Y . ) / N ] 2 i i l l J t h e r 2 _ • [ E X . 2 - C 2 X . ) 2 / N ] [ E Y . 2 - ( E Y . ) 2 / N ] r 2 , t h e c o e f f i c i e n t o f d e t e r m i n a t i o n i n d i c a t e s goodness o f f i t : r 2 = 1 = good f i t r 2 = 0 = no f i t 2. C h i - S q u a r e d Goodness o f F i t T e s t G i v e n : E x p e c t e d D i s c r e t e P r o b a b i l i t i e s PA^' i = 1,2, ....N' C a l c u l a t e d D i s c r e t e P r o b a b i l i t i e s P B ^ i = 1,2, ....N' T o t a l number o f c l a s s N' T o t a l F r e q u e n c y M The X 2 - t e s t r e q u i r e t h e g i v e n d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s t o be r e g r o u p e d so t h a t f r e q u e n c y i n each c l a s s i s >_ 5, i . e . Regroup i n t o d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n s , PA.., PB^, j = 1, 2, ....N, 162 11-40 so t h a t m 'PA j >_ 5 , j = 1 , 2 , N , t h e n N (m-PA. - m « P B . ) 2 N (PA . - P B . ) 2 X 2 = .1, —JTT 1- = m I 3 n A — 2 _ 1=1 m ' P A . . . P A . 1 3=1 j D e g r e e o f F reedom ( d . f . ) = N - l T h i s X 2 v a l u e i s t h e n c h e c k e d a g a i n s t X 2 , f r o m t h e t a b l e I I - 3 - 1 a t t a c h e d f o r d . f . = N - l . 3 . K o l m o g o r o v - S m i r n o w Goodness o f F i t T e s t G i v e n : E x p e c t e d D i s c r e t e C u m u l a t i v e P r o b a b i l i t i e s FA^ i = 1 , 2 , , .N C a l c u l a t e d D i s c r e t e C u m u l a t i v e P r o b a b i l i t i e s F B ^ i = 1 , 2 , N T o t a l N u m b e r - o f C l a s s e s N K-S t e s t v a r i a b l e : D = m a x ( ( F A i - F B ^ ) } T h i s D v a l u e i s t h e c h e c k a g a i n s t K - S v a l u e s f r o m T a b l e I T - 3 - 2 , 163 11-41 I I . 4 Program t o T r i m Round O f f E r r o r s (TRM) 1 DIMENSION X1512) .XLOGI512I , PX I 512 >.CPXI512 », 2 i IFREQ!512) , ICFRQ!512. 3 REA0I5,1J N * ' REAO!5,l> IFTTL,IFMIN 5 REA0I5.1) CPX2 6 Y » ( 1 . 0 * I F T T L I / I F M I N 7 ICFX2=IFIX(Y*CPXZ) 8 I A » I F I X ! 1 0 . * Y * C P X Z J - l O * I F I X t Y * C P X Z \ 9 I F U A . G E . 5 ) ICFXZ-ICFXZ+1 10 ISUM-ICFXZ 11 Y A = L / Y 12 PRINT20 13 PRINT21, N 14 PRINTZ2,IFTTL 15 PRINT23,IFMIN 16 PRINT24-YA 17 00 10 1=1,N i s R E A O ( 5 , 2 » K , x m , x i . a G r i . , P x n i . c p x m 19 I F R E Q ! I ) » I F I X ! Y * P X ! I I I 20 I B » I F I X U O . * Y * P X l I J ) - l O * ! F l X ( Y * 0 X l I I ) 21 I F U B . G E . 5 ) IFREQ! I ) * IFREOt I ) *1 22 ISUH*IS>JM + IFREQ! I ) 23 10 ICFROUJ--TSUM 24 10F« IF IXII I . * IFTTLJ/ IFMIN- ISUMI 25 00 15 1*1,N 26 ICFROID - r C F R O C I I + IDF 27 P X U )»1 . * I F R E Q ! I ) / Y 28 C P X ( I )=1 . * ICFR0( I1 /Y 29 I F » I F R E Q m * ! F M I N 30 I C F * I C F R Q ( t » * T F M I N 31 15 PRINT 3, T ,X ( I ) ,XLQG( I1 ,PX( I ) ,CPX( I ) , I F , I C F 32 1 FORMATI20G20.7) 33 2 FORMAT(14,4E16-7) 34 3 FORMAT!' • , 1 3 , 4 E 1 6 . 7 , 2 15 J 35 20 FORMAT!• ' , ' •) 36 21 FORMAT (• • , ' * * TOTAL NUMBER OF ENTRIES -<*tI12. 37 22 FORMAT!« ' , » * * TOTAL FREQUENCY » « , I L 2 ) 38 23 FORMAT ( • • ' , ' * * MINIMUM ACCURATE FREQUENCY * « , I 1 2 ) 39 24 FORMAT!• • , » * * MINIMUM ACCURATE PROBABILITY * » , E 1 6 . 7 / ' »> 40 STOP 41 ENO ENO OF FILE APPENDIX I I I : 164 m 1 T y p i c a l I n p u t and Ou t p u t f i l e s DI t o D7, A l t o A7 R e f e r e n c e s i n F i g u r e 5.1 JL 01 1 C, . . ITRACE 2 3 , .0 IP7S 3 1» IS 4 0, . . IRESCS 5 1. . . IFMT 6 4 , 1 . O C , . . S U J . P S U J 7 2, . . IPTU 8 21 , . . IU 9 1, IRESCL 10 1. 2 0 , C . C 5 , 0 . 7 0 , 0 . 1 2 5 , . .UH,0U,RMLU,SIGl 11 3 , . . IPT8 12 I t . . I B 13 1, . . I R E S C e 14 1, IFMT 15 180 .0 ,1 .00 , . .BH,D8,RMUB,SIGB 16 293,970, • • TA,PA 17 6C. ..GAMMA 18 1, . .KSCUPC 19 3 0 . 0 , 1 . 5 , 1 3 . 0 , 3 9 4 . 0 , . .H1 ,01 ,VS1 ,TS1 20 1 5 C C . C 0 . O , . .R1.AIFHA1 21 2, . . IPTQ1 22 21, 101 23 1, . . IRSCC1 24 2 . 5 0 , 0 . 0 5 , 2 . 0 0 , 0 . 0 8 , ..Q1H,CC1,RMUC1,SIGC1 25 2 , C.050, IPTCE.CCE 26 41 , . . ICE1 27 4 1 , . . ICC 28 1, . . I C A L C 29 1 , 1 , 1 , . . IPPIN7(1-3) (TC e E FOLLOWED BY OGLCTJ ENO OF F I L E $L 02 — — - • 1 0 , . . ITRACE 2 3 , I P T S 3 I t . . I S 4 0, . . IRESCS 5 1, • . IFMT 6 4 . 1 . 0 C , . . s m ,PS( 1 ) 7 2, . . IPTU 8 21, . . IU 9 I t . . IRESCU 10 1 . 2 C C . C 5 . 0 . 7 C , 0 . 1 2 5 , . .UH,OU,RfL'U,SICL 11 3 , . . I P T B 12 1 , . . IB 13 1, IRESCE 14 1, . . IFMT 15 180 .0 ,1 .00 , . .en.OB.^MUB.SIGB 16 293,970, . . T A , P A 17 6C. ..GAMMA 18 1 , ..NSCURC 19 3 0 . 0 , 1 . 5 , 1 3 . 0 , 3 9 4 . C , . .H1 ,01 ,VS1 ,TS1 20 1500 .C ,0 .0 , . .R1,ALPHA1 21 2 , . . I FTC1 21 21, ' ""• . . IQ1 23 1, . . IRSCC1 24 2 . 2 5 , 0 . C 5 , 1 . 7 5 , 0 . 0 6 , . . C l H , C C i , P M U C l , S I G C l 25 2 , 0 . 0 5 0 , . . I P T C E . O C E 26 41 , . . ICE1 27 41 , . . I C C 28 1 , ICALC 29 1 ,1 ,1 , •.I PR I M ( 1 - 3 ) ( T C EE FOLLOWED BY DGLCTI END OF FILE 165 III-2 U D G L C T 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 2 0 21 22 2 3 24 . 2 5 26 27 28 29 3 0 31 3 2 33 3 4 35 36 3 7 38 3 9 4 0 41 4 2 43 4 4 4 5 4 6 47 4 8 4 5 50 51 52 53 54 55 1 . 0 , c , c . 9 , 9 , 9 , 9 , 5 . 5 , 5 . 5 . 5 . 5 . 5 . 5 , 4 . 4 , 4 , 4 , 1 2 , S S T A B I L I T Y 2 1 . L O G . S L C G . S T A B I L I T Y S L O G . S 0 . 0 * 1 , 0 , 9 , 9 , 9 , 5 , 5 . 5 , 5 . 5 , 5 . 5 . 5 . 5 , 4 , 4 , 4 , 4 , 1 9 . Li » I N C S P E E D ( M / S ) 2 2 , L C G . U L C G . k I N C S P E E D U C M / S I L O G U 1 . C C C 5 . 9 , 9 . 9 , 5 . 5 , 5 . 5 , 5 . 5 , 5 . 5 , 4 , 4 , 4 , 4 , 2 5 , 8 WINC C I R E C T I C N I D E G . N ) 2 6 , L O G . B L C G . k I N O D I R E C T I O N 8 ( C E G . N ) L O G . B 0 , 0 , 1 , 0 . 5 , 5 , 9 , 5 , 5 . 5 , 5 . 5 , 5 . 5 , 5 . 5 , 4 , 4 , 4 , 4 , 2 5 . 0 1 E M I I S I O N P A T E #1 ( G / S ) 2 8 , L C G . C I L O G . E M I S S I O N R A T E #1 0 1 ( G / S ) L O G . 0 1 C C , 0 , 0 , 5 , 9 , 9 , 9 . 5 . 5 , 5 . 5 , 5 - 5 , 5 . 5 , 4 , 4 , 4 , 4 , 2 5 . 0 2 E M I I S I C N P A T E *2 ( G / S ) 2 8 , L O G . 0 2 L O G . E M I S S I O N R A T E #2 0 2 ( G / S ) L C G . C 2 0 , 0 , 1 , 1 , 9 , 9 , 9 , 9 , 5 . 5 , 5 . 5 , 5 . 5 . 5 . 5 , 0 , 0 , 0 , 0 , 3 9 , E l C H A R A C T E R I S T I C F U N C T I O N #1 ( S * M * * - 3 ) 3 8 , L O G . E l L C G . C H A R A C T E R I S T I C F U N C T I O N #1 E l ( S * M * * - 3 ) L O G . E l 0 , 0 , 0 , 0 , 9 , 9 , 5 , 5 , 5 . 5 , 5 . 5 , 5 . 5 , 5 . 5 , 0 , 0 , C , C , 3 9 , E2 C H A R A C T E R I S T I C F U N C T I O N *2 < S * H * * - 3 ) 3 8 , L C G . E 2 L C G . C H A R A C T E R I S T I C F U N C T 1 C N #2 E2 ( S * M » * - 3 ) . . I P L 0 T S ( l - 4 ) o . X S I Z E S ( 1 - 4 ) . . Y S I Z E S I 1 - 4 I « . I S Y M B S ( 1 - 4 ) . . . N S I . L B L S 1 4 ) . . N S 2 . L B L S 2 . o L B L S 3 o e L 8 L S 4 « . I P L 0 T U ( 1 - 4 ) . . X S I Z E U I 1 - 4 ) « o Y S I Z E U ( 1 - 4 ) . . I S Y M B U I 1 - 4 ) . . N U l . L B L U l » . N U 2 , L B L U 2 c .LeLU3 • • L 8 L U 4 . . I P L O T B I 1 - 4 ) . . X S I Z E B I 1 - 4 ) - < » Y S I Z E B ( 1 - 4 ) . « I S Y M 8 B ( i - 4 ) - . N B l . L B L B I . . N 8 2 . L 8 L B 2 O . L B L B 3 . . L 8 L B 4 • . I P L T O K 1 - 4 ) - « X S I Z O l l 1 - 4 ) - - Y S I Z Q 1 ( 1 - 4 ) . . I S Y M O K 1 - 4 ) . . K C 1 1 . L B L C 1 1 . . N Q 1 2 . L B L 0 1 2 . . L B L Q 1 3 . . L 8 L Q 1 4 . . I P L T 0 2 ( 1 - 4 ) . . X S I Z 0 2 I 1 - 4 ) . . Y S I Z 0 2 1 1 - 4 ) . . I S Y M Q 2 U - 4 ) . . N 0 2 1 . L B L Q 2 1 . . N C 2 2 , L B L C 2 2 - . L B L 0 2 3 . . L B L C 2 4 . - I P L T E l ( 1 - 4 ) - - X S I Z E K 1 - 4 ) . . Y S I Z E K 1 - 4 ) . . I S Y M E H 1 - 4 ) . . K E l l . L B L E l l • . N E 1 2 , L B L E 1 2 . . L B L E 1 3 - . L B L E 1 4 . . I P L T E 2 I 1 - 4 ) . . X S I Z E 2 ( 1 - 4 ) • • Y S I Z E 2 I 1 - 4 ) . • I S Y M E 2 I 1 - 4 I . . N E 2 1 , L B L E 2 1 • . N E 2 2 , L B L E 2 2 S . L B L E 2 3 166 II1-3 56 LOG. E2 ..LBLE24 57 0,0.1.0, ..IPLCTCJ1-41 58 9,9,9,9, „.XSIZEC(1-4) 59 5.5,5.5,5.5,5.5, ..YSIZEC(1-41 60' 0,0,0,0, ..ISYMBCI1-4) 61 39,C GRCLNC LEVEL CCNCENTRATICS (C-*M**-3> ..NCl.LBLCl 62 38, LCG.C LCG. C-RCUNC LEVEL CONCENTRATION ..NC2.L3LC2 63 C (G*M**-3I ..LBLC3 64 LCG. C «oLBLC4 65 213, 156, 104, 66, 50.5, 34, ..(AY(I),1=1,6) 66 0.854, ..BY 67 440.8, 1C6.6, 61.C, 33.2, 22.8, 14.35, ..(AZ1(I I , I-1,6 68 1.941, 1.145, C.511, C.725, C.678. C.740. ..(BZ1 69 9.27, 3.3, Q.C, -1.7, -1.3, -0.35, ..(CZ1 70 459.7, ICE.2, 61.C, 44.5, 55.4, 62.6. •.(AZ2 71 2.094, 1 .09e, 0.911, 0.516, C.3C5, C.180, . . (BZ2 72 -9.6, 2.C, C C , -13.0,-34.0,-48.6, .. (CZ2I I ), 1= 1,6 73 11,PROBABILITY «.NP1,LBLPl 74 22,CUMULATIVE PRCBAEILITY ..NP2,LBLP2 E N O CF FILE 167 I I I-4 1L 03 1 2 7, s •.NTITLE 3 4 l.DPAc.ci m ? •? 5 QI,CI ==> e« 6 7 8 Q1.E* ==> CI 9 10 11 C, 2.3.1, 1.1, ..ITRACE .oICPTl, .. IREACA ICPT2,ICPT3 ,IPEAOB 12 13 14 19,33, O.C.l. 9,9,9, 0, 9, . . N A , N E .. IFLCTM1-4) ..XSIZEA 15 16 17 5.5,5. 5,5,5, 1, 5,5.5,5.5, 5, ..YSIZE* ..ISYMBA ..ISTATA 18 19 20 28,EMISSION RATE CI 27,LOG.EMISSION PATE 01 (G/S) (G/S) . .NA1,LELA1 LOG(Ql)..NA2,L8LA2 ..LB LA 3 21 22 23 ALOG10(C=) 2 0.3548123F+02 3 C.3981061E+02 ..LBLA4 0. 154S59SE+C1 0. 15595595+01 C.2960221E-C7 C. 1341C95E-05 0.2980221E-07 0.1370900F-05 24 25 26 4 5 6 C.4466826F+C2 0.5011867F+02 0.5623407r+C2 0. 165CCCCF+C1 0.17CCCCC5+C1 0.175CCC0E+C1 C.2288810S-04 C.2652330=-C3 C.2164147E-02 0.2425899E-04 0.2534919E-C3' 0.2457639F.-C2 27 28 29 7 8 9 C.6309557S+C2 C.7079440E.+ 02 C.7943265F+C2 0.ITeSSSSE+Cl 0.1849SSS5+C1 0. 19CCC005+01 C.1189412E-C1 C.4473002E-C1 0 .1151634=+00 0.1435176«=-C1 C.55C8177E-C1 0.1742451F+C0 30 31 32 10 11 12 0.8912491E+02 C.99995915+02 0.1122C145+C3 0. l S 5 C C 0 C e * 0 1 0.2CCCC005+01 0.2045555E+C1 C.2C30756E+CC C.24534185 + 00 C.2C30827E+0C C.3773245E+C0 0.6226667F+00 0.8257495F+00 33 34 35 13 14 15 0.1258522F+03 0.1412534E+C3 C.158489CP+C2 0.2055SS9E+Q1 0.215CCC0E+O1 0.22CCCCCS+CI C.1151667E+00 0.44 72198S-01 C. H 8 9 4 e C 5 - C l C.94C5161F + CC. 0.9856481E+C0 C.9975429E+C0 36 37 38 16 17 18 C.1778276E+C3 C.1995256F+C3 0.223'8714F + a 3 0.2250CC0E+C1 0.22SS5SSE+C1 0.23455555*01 C.2164206E-C2 C.2692628E-02 C.22E881C6-C4 C.999707CE+C0 0.9999763C+00 0.9999991E+00 39 40 43 19 20 0 ,C,1 , i C.2511381E,*02 O.2818379F+03 0, 0.24CCCCC5+01 C.1341099E-05 0.245CCCCE+C1 C.2980221E-07 . . I PL T 8 I.I ) ,1 = 1,4 0.10000005+01 0.1000000E+01 44 45 46 S,S,9, 5.5,5. 5,5,5, 9, 5,5.5,5.5, 5, . . X S I Z E S ..YSIZE5 . . . i S Y M a e 47 48 49 1, ••I STATE 40,GRCUNC LEVEL CONCENTRATION C1(G*M*— 31, LCG.CI •2) 50 51 52 CI (G*M**-3) LOG(Cl) 7 C. 1212435F-C3 -0.3916343E+01 0. 1758632E-1i 0.6139279E-C5 53 54 55 8 9 10 0.1360375E-02 C.1526366F-03 C.1712608^-03 -0.2866243E+C1 -0. 28162-,2E + 01 -0.3766343E+C1 C.87022995-10 C.1735622E-C8 C.2339995':-07 0.6139366E-C5 0.61411C1E-C5 0.6164501F-05 56 57 58 11 12 13 0.192157EE-03 C.2156C46E-C3 0.2419124^-03 -0.2716243E+Q1 -0.26663435+01 -0.36162435+C1 C . 2 2 6 6 3 6 1 5-C6 C. 1635984E-05 C.5112340S-C5 0.6391336F-C5 0.8027319E-05 0.1713965F-04 59 60 61 14 15 16 0.2714305F-03 C.3045495E-03 0.3417102E-C2 -0.2566242E+01 -0.35163433*01 >-0. 24663435+C1 C .4C89874F.-C4 0.1542154E-03 C.5C47766=-C3 C.56C3839E-C4 0.2122538F-C3 0.71703C4E-C3 62 63 64 17 18 19 C . 3 8 3 4 C 5 2 E - C 3 C.43C1S79P-C3 0.4826789E-03 -0.2416343E+01 -0.3366343E+01 -0.23162425+01 C .14635295-C2 C.3807270E-C2 C.9CC9550E-C2 0.218C56CE-C2 0.5987827E-02 0.14SS778E-C1 168 III-5 65 6 6 67 20 21 22 C.54157366-0.6076563E-0.6818019F-•C3 C3 •03 -0. -0. -0. 2266243E*C1 22162435+C1 2166243E+C1 C.1969362E-01 C.2976485E-C1 C .7156462=-01 0.3469140E-01 0.7445621F-C1 0. 14642C6F+-CC 6 6 69 70 23 24 25 0. 7649544E-C8583381F-C.9630695E-C3 C3 03 -0. -0. -0. 3116343<= + 0 i 3046342E+C1 3016243E+01 C. 1H6492E+0C C.1447091E*0C C.1577999E+CC 0.25807ClF-r00 0.4O27791F+00 C.560575CE+CO 71 72 73 26 27 28 C.1080582F-0.1212434F-C.1360274F-02 C2 C2 -0. -0. -0. 2966343E+C1 2916242E+C1 2aC6343e«-0l 0.1482992E+CC C.1211466E+CC C .8444065F-01 0.7088782 p*00 0.8200248E+CO 0.9 144655F + C0 74 75 76 29 30 31 0.1526365F-0.1712607E-C.1921577E-02 02 C2 -0. -0. -0. 2816242E*C1 2766243E+01 2716343E+01 C.450315BF-C1 C .22466645-01 C.9217646F-02 0.9634970E+C0 0.9869636F+C0 0.9961812E+C0 77 78 79 32 33 34 C2156C45F-C.2419123E-0.27143C4F-C2 02 C2 -0. -0. -0. 26«;6243F*C1 2616243E+01 2566342F+01 C2529376F.-C2 C.7225232E-C2 0.135654<5C.-G3 0.9991106E+00 0.9558421E+CC 0.9999827F»00 80 81 82 35 36 37 0.3045494F-C.3417101F-0.3834C51E-02 02 C2 -0. -0. -0. 2516243E+01 2466343E+01 2416343E+C1 C.1576779HI-C4 C.2032997E-05 C. 14564565-C6 C.1CCO0C2E+C1 0.1000004E+C1 C.1000004E+01 83 84 38 39 C.4301876F-0.4826788E-02 C2 -0. -0. 23*62436*01 2316342E+01 C.7164786E-C8 0.1385092F.-09 C.10C0CC4E+C1 0.1OO0004E+C1 85 q f C , l , 0, . .IFLCTCM-4) 86 87 88 9 5 5 ,S,9, .5,5. .5,5, 9, 5,5.5,5.5, 5, ..XSIZEC •.YSI2EC ..1SYMPC 90 40,CHARACTERISTIC FUNCTION E'<S*C**-3) 91 12, LCG (E'»  ..ISTATC ..NCl,LeLCl ..NC2.LBLC2 92 93 94 0, t(s*M**-3I LOG(E« ) CO, . .LBLC3 ..LBLC4 ..IRESCC.PELC,ITRIMC 95 96 97 1, 19, 0,C,1,0 . ..IREAOD ,.N0 ,.IPLOTO(1-41 98 5,5,9,9, 99 5.5,5.5,5.5,5.5, 100 5,5,5,5, ..XSIZEO ..YSIZED ..ISYMBO 101 102 103 1, 28 27 EMISSION RATE 02 (G/S) ,LOG. EMISSION RATE L0GU2) • ISTATD ..N01.LBLD1 ..KC2.LBLC2 104 105 106 -4 107 108 09 .10" 111 112 113 114 115 02 (G/S) ALCG1C<C2) 2 0.35481236*02 3 0.39"81061E*02" 4 C.4466826E*C2 5_ 0.5011S67F+02 ^ ~ "0 . 56 2340TFHJ2" 7 C.6309557F4-C2 8 C.7079440E+02 ..LBLD3 ..LBL04 C.1545555E*01 0.298022IE-0 7 9 C.7943 26 5FVC2 10 0.8912491F+C2 11 G.9999591E+02 0.1555955E+C1 C. 0.165CC00E+01 C. 0. 17CCCCOE + 01 C "Oil 7 5CCCGE401 C 0.17«5559E*01 0. 0.1845C55E + 01 C 1241099E-C5 C 2288810E-C4 C 2652330E-C3 0 2164147S-C2 0 1189412E-01 0 4473CC25-C1 C 0.19CCG00E+01 0. C1950CCCF + C1 C. 0. 2CCCCC0E+O1 C. 1151634E»CC 0 2C30758E-»0C 0 245341S6+CC 0 .2980221E-07 .137C90CE-C5 .24256S5E-C4 .2934915E-03 . 2 4 5 / 6 J S b - C 2 .1435176E-C1 .59C8177E-G1 .1742451E+CC .3773245E*00 .6226667F+00 116 117 118 119 120 121 122 123 124 128 129 130 12 0.1122014F+03 13 C1258922E+03 14_0.1412534F+03 15 C.1584890E+03 16 C.1778216E+C3 17 C.1995256F+03 0.2045555E+01 0. 0.20<5?59E+C1 C. 0.215CCCOE+01 C. 2030327E+0C 0, 1151667E*0C 0. 4473158E-C1 C. 18 C.2238714E+C2 19 0.2511881P+03 20 C.2818379F+C3 0.22CCCCOE + 01' 0.225CCC0E+C1 0.2255559E+01 C.1189480E-01 0, C.21642C6P-C2 0. C.2652628E-C3 0, 8257495E+C0 9409161F*00 98564eiErCC 0, 5, 6, 0,1,0 , 5,9,9, 6,6,6, 0 . 2 3 4 « c c C F * C 1 0 . 2 4 C C C C 0 E + 0 1 0 . 2 4 5 C C C C E + 0 1 C.2288810E-C4 0, C. I2410955-C5 C. C.29602215-07 0. 9975425F*C0 995707CE*Q0 9555763E+C0 . .IfLCTE(1 ..XSIZEF ..YSIZEE 9599991 £-.00 10CCOCOE*01 10000006*C1 -4) 169 I I I-6 131 132 133 134 135 136 5,5,5,5, 1 . _3J^6R_CUND LEVEL CCNCENTRATICN C2tG*M*~ ? 1 12, LCG (C2) ~ ~ C2(G»M*-r-3) L0G(C2) . . ISYMeE . o I STATE .NE1.LELE1 ..NE2.LELE2 ..LELE3 ..LBLF4 137 138 139 0,0,0, 11,PROBABILITY _2_2^CUWULATIVE PROBABILITY > IRESCE,DELF,ITRI*£ 140 22.CUMULATIVE PROBABILITY* ENC OF FILE S L 04  * 7 » . . N T I T L E 2 **** 1.CPAC.C2 ?:m 4 ~ " 6 Ql.Ci ==> E• 7 0 2 . E ' ==> C2' 8 9 10 0, . . ITRACE 11 12 13 2.2,1, ' 1.1, 15,29, . . I 0 P T 1 , . . I R E A C A . .NA.NB ICFT3.ICFT3 .IREADB 14 15 16 C,C , l,Cf 9,9,9,9, 5.5,5.5,5.5,5.5. . . IF10TAU-4) . .XSI2EA . . Y S I Z F A 17 18 19 5,5,5,5, 1. 2e,EMISSION RATF CI .•IS Y M B A • . I S T A T - * (G/SI ..NA1.LELA1 -20 21 22 27,LOG.EMISSION PATE Ql C G/S» AL0G10IQU LOG(Ql...NA2,Lfil«*2 ..LBLA2 • • L EL A4 23 24 25 2 C.2511882E+02 3 0.2818379E*02 4 C.3162274?*C2 C.14CCCCCS+C1 Oo 145CCCQE+01 0. 15CCCCCE*01 C.2580218E C.2235163E C.8612329<= -07 -C5 -04 0.2980218E-07 C.2264965E-C5 0.8839325F-C4 26 27 - i f -30 31 5 0.3548122p+02 6 C.3981061F*02 7 0.4466826F + C2 8 G.5011fi67F*02 9 G.5623407?o02 10 0.6309557E+C2 C.15Aec59E>Cl 0.1559S99S+01 _0.L65CCCQE+QL 0.17C0CCOE+O1 0.1750CC06*0l 0. 17e?959 c *C l C. i6 602 77€-C2 C. 1684C02E-C1 0.87C36C1E-01 C.2328C58E*CC 0.3230803E+00 C.22281 565 + CC — • *^ ^ W b '.. w , 0.1768670E-C2 0.186C865F-C1 0.1056446F*00 0.3384544E+C0 O.6615347E+C0 0.8943503E+00 32 33 34 11 C.7079440F*C2~ 12 0.7943265E+C2 13 0.8912491E+02 0.1845555c4-01 0. 19CCCCCE*C1 0.195CCQ0?+01 C.e7C4C36c C.168413CE C. 16G0455F -01 -CI -C2 C.S8135C7EtCC 0.9982320E+00 0.9959124E+00 35 36 37 14 C.5999991F+C2 15 0.1122014F+C2 16 0.1258922F+03 o . 2 c c c c c c = » a i 0.204?=S9S*C1 0. 20C'jcc<3c,oi C 8612829c C.2235163E C.25802185 -04 -05 -07 0.9999985E+C0 c . ioaooooE+oi C. 1QCCCCCE+C1 40 41 42 O.C.1,0, S,e,9,5, 5.5,5.5,5.5,5.5, ..IPLTB (IJ ,1*1,4 ..XSIZES ..YSTZF« 43 5,5,5,5, .. ISYM3E 44 45 1, 4CGRCUNC LEV=L CONC* ..ISTATE NTRATICN CI (G*M» — 3 ) 46 47 48 31, LCG.CI CI (G*M**-3) LOGIC 1J 49 50 ^ 51 9 C.8583385P-C4 10 C.9630699F-04 11 C. 108C5e2<:-C3 -C. 4C662425*C1 -0.4016343E+01 -0.39663435*01 C.175e631E-C.13S7819E-C.57039675--11 -C9 -08 0.4351140E-C5 0.4251275E-C5 0.4356982F-C5 170 III-7 52 12 0.1212435E-C3 -0.29163425+C1 C. 1243024E-C6 0.4481284E-C5 53 13 C.1360375E-C3 -0.3866343E+01 ' C. 15331893-05 0.6014473E-C5 54 14 0.15263665-C3 -0.28163425+C1 C.1159978E-C4 0.1761425E-C4 55 15 C.1712608F-C3 -0.27663435+01 C.6C233045-04 0.7784725F-C4 56 16 0.192157EE-C3 -0.3716343E+01 0.2421115E-03 0.3199587E-03 57 17 C.2156C46E-03 -0.26e6243=+Cl C.E1552825-C3 0.1139887E-C2 56 18 C.2419124F-03 -0.26162425+01 C.24228415-C2 0.2562728E-C2 59 19 Oo2714305E-03 -0.35663425+01 0.63071515-C2 C.9869877F-C2 60 20 0.3045495F-C3 -0.35163425+01 C. 1464543E-C1 0.2451531E-C1. 61 21 C.34171025-C3 -0.24662435+01 C.3148326E-01 0.5599857E-01 62 22 0.3834C52E-C3 -0.24162435+C1 C.6435275E-01 0.1203512E+00 63 23 0.4301879*:-C3 -0.23662425+C1. C .1152942E + 0C 0.2356455E+CC 64 24 0.48267395-C3 -0.3316342E+0L C.1600949E+C0 O.39574C4E+C0 65 25 0.5415736E-C3 -0.2266343S+C1 C.1715631E+CC 0.5673036E+00 66 26 C.60765635-03 -0.22162435+01 C.15792245+0C 0.725226CE+CC 67 27 0.6818C19F-03 -0.31663435+01 0.1282C475+CC 0.8534307E+00 68 28 0.7649944E-05 -0.2116243E+C1 C.8208452E-C1 0.9365153E+C0 69 29 C.85833815-03 -0.3066242E+C1 C.4121940E-01 0.9777346E+00 70 30 C.9630655 c-C3 -0.30163435+C1 C.1620238E-C1 0.993937CF+00 71 31 0.1080582F-C2 -0.2966243E+01 0.48967675-02 0.9988337E+CC 72 32 0.1212434E-C2 -0.25163435+01 C. 1023465E-02 0.9998571E+00 73 33 C.1360274E-C2 -0.2866343E+C1 0. 1251155E-C3 C.5999922F+00 74 34 C.1526365F-02 -0.28162425+01 C.1C60920E-C4 C.1000C02E+C1 75 35 C.1712607E-C2 -0.27663435+01 0.4715180E-C6 C.1000002E+01 76 36 0.1921577F-02 -0.2716242E+01 C.1132CC5E-C7 0.1OOO0O2E+01 77 37 C.2156 045E-02 -0.2666343E+01 0.1385091E-09 0.1000002E+01 80 0,C,1, 0, ..IFLCTC(l-4) 81 5,5,9, 9, ..XSIZEC 82 5.5,5. 5,5.5,5.5, ..YSI2EC 83 5,5,5, 5, ..ISYMEC -84 1, ..ISTATC 85 40,CHARACTER! STIC FUNCTION E • (S *••***-3) ..NC1.LELC1 86 12, LCG (E«) ..NC2,LBLC2 87 EMS*M**-3I ..LELC3 88 LOG(E') ..L8LC4 89 0,C,0, ..IRESCC ,DELC,ITRIMC 90 1, ..IREAOO 91 15, • .NO 92 0 r C . l , 0 , ..IPL0TD(1-4I 9 3 ^5,5,9, 5 , . . X M i t U 94 6,6,6, 6, ..YSIZED 95 5,5,5, 5, ..ISYMBC 96 1, ..ISTATD 97 28,EHISSI0N RATE C2 (G/SI ..NCl.LBLCl 98 27,LCG . EMISSION RATE L0GJC2) ..NC2,LBLC2 99 02 (G/S) •.LBL03 100 AL0G10(C2) ..LBLC4 101 2 0.2511882F+C2 C.14CCCC0E+01 0.2580218E-07 0.2980218F-07 102 3 Q.2818379E+02 O.145CCC0E+01 C.22351635-C5 C.2264565E-C5 103 4 C.3162274E+C2 0.15CCGC06+01 C.66128295-04 0.88393255-C4 104 5 C.3548123F+C2 C . 1 5 4 5 C 5 S 5 + C 1 C.16802775-C2 0.1768670E-C2 105 6 C.3981C61E+02 0. 155S555E*01 C.1684C02E-C1 0.186C6655-C1 106 7 0.44668265+C2 0. 165CC00E + 01 C.8703601E-01 0.1O56446E+0O 107 8 C.50118671+02 0.17CCCCCS+C1 C.2228058E+0C 0.3384544E+C0 108 5 C.5623407F+02 O.175C0C0E+01 C.22308C35+CC C.6615347E+C0 109 10 C.6309557E+02 0.17=5«55E+01 0.2228156E+0C 0.8943503c+00 110 11 0.70794405+02 0.1945SS55+C1 C.E7C4C26E-C1 C.98139075+C0 111 12 C.7943265E+02 0.ISCCCCCc+Cl C . 16841305-C1 0.99823205 + 00 112 13 C.8912491E+02 0.155CCCC5+C1 C.1660455F-C2 0.9999124E+C0 113 14 C.9999991F»02 0.2CCCCC05+C1 C.E6123295-C4 0.5S99985E+CC 114 15 C.1122C14E+03 0.20455S95+01 0.2235163E-05 0.1000000E+CI 171 I I 1 - 8 115 16 0 . 1 2 5 8 9 2 2 F + C 3 1 1 8 0 , C , 1 , Q , 119 5 , 5 , 9 , 9 , 1 2 0 5 . 5 , 5 . 5 , 5 . 5 , 5 . 5 , 121 5 , 5 , 5 , 5 , 0 . 2 0 ' 5 5 5 5 5 + C l 1 2 2 1 , 1 2 3 3 7 , C2« 1 2 4 1 2 , L C G <P7»i 1 2 5 C 2 M G * M * * - 3 ) 1 2 6 L C G ( C 2 « ) -1? 7 c » c , o . C . 2 9 e 0 2 1 8 E - C 7 0 . 1 0 0 0 0 0 0 6 * 0 1 . . I P L C T E I 1 - 4 ) o . X S I Z E S . . Y S I Z E E . . I S Y M E E 128 I M P R O B A B I L I T Y 129 2 2 , C U M U L A T I V E P R C E « E I L I T Y 130 2 2 , C U M U L A T I V E P R C B A f l T l T T Y ENO GF F I L E ' " . . I S T A f E . . N E 1 . L B L E 1 . . N E 2 , L P L 5 2 . . L E L E 2 . . L 8 L E 4 • « IR E S C E . D E L E . I T R f fE J L 05 L 2 3 4_ 5 6 x 8 9 10 17 18 19 T O " 21 22 23 24 2 5 26 2 7 2 8 29 30 3 1 3 2 33 34 35 36 3 7 38 3 9 4 0 5 , 0 1 l . O S T P . E — G L C — > Q 1 , C 1 —PAD—> E ' 8 7 6 0 , 1 , It l t - 1 , 1 . 1 , - 1 . 1 . 11 9 . 0 , 5 . 5 , 4 , - 5 7 4 7 1 12 C O N C E N T R A T I O N E l , E l " _13 16 1 * 1 C . 3 0 4 5 5 0 4 6 - C 5 15 2 0 . 3 4 1 7 U 3 E - 0 5 AA.. 3 0 . 3 8 3 4 0 6 4 F - 0 5 _ M I R E A C I L I S T ( I P - L C T d . , 1 = 1 . 6 ) XS I Z E , YS I Z E , IS YMEA, I S Y M 8 8 , ISYI -BC 4 C . 4 3 0 1 8 9 5 5 - 0 5 5 C . 4 8 2 6 8 C 7 F - C 5 6 0 . 5 4 1 5 7 5 7 E - 0 5 ^ CTfcO;6581E-C5" 8 0 . 6 8 1 8 0 3 8 E - 0 5 9 C . 7 6 4 9 5 6 5 5 - 0 5 - 0 . 5 5 1 6 2 4 2 E + 0 1 C - 0 . 5 4 6 6 3 4 2 E + 0 1 C j ^ 0 _ . 5 4 1 6 2 4 2 F * 0 1 C 10 0 . 8 5 8 3 4 1 3 5 - 0 5 11 C . 9 6 3 0 7 2 2 c - 0 5 12 C . 1 0 g C 5 g 6 E - C 4 13 0 . 1 2 1 2 4 3 9 5 - C 4 14 C . 1 3 6 0 3 7 6 F - C 4 15 C . 1 5 2 6 3 6 9 E - C 4 - 0 . " 5 3T6"24 2 E * 0 1 C - 0 . 5 3 1 6 2 4 1 6 + C 1 C - 0 . 5 2 6 6 2 4 2 5 * 0 1 C ^ O V 5 2 1 6 2 4 2 E * C 1 C - 0 . 5 1 6 6 2 4 2 5 * 0 1 0 t_0. 5 1 1 6 2 4 2 E + 0 1 C - C . 5 C 6 6 2 4 1 5 + C 1 C - 0 . 5 0 1 6 2 4 2 5 * 0 1 C • 0 . 4 9 6 6 2 4 2 E * 0 1 _ 0 . C - 5 9 0 1 0 1 6 E - 0 4 • 2 6 4 5 6 2 2 E - C 3 _ . 1 0 1 2 5 4 8 5 - 0 2 . 2 3 1 1 2 7 5 E - C 2 • 5 2 4 2 C 5 1 5 - C 2 . 2 2 0 2 6 5 4 E " ^ C r r . 4 4 8 2 6 5 5 5 - 0 1 ^ 7 7 9 0 0 4 1 5 - 0 1 16 0 . 1 7 1 2 6 1 2 5 - 0 4 48 C . 3 0 4 5 4 8 3 6 - 0 5 49 . 0 . 3 4 1 7 C 9 0 E - 0 5 - 0 . 4 9 1 6 2 4 2 E * C 1 C - 0 . 4 8 6 6 2 4 2 F + 0 1 C • 0 . 4 8 1 6 3 4 1 5 * 0 1 0 . 2 6 2 0 5 3 8 5 + 0 C . 1 5 8 5 2 5 1 5 * 0 0 1 4 6 4 9 2 2 5 * 0 0 50 0 . 3 8 3 4 0 3 9 5 - 0 5 51 • C . 4 3 0 1 8 6 6 E - C 5 52 0 . 4 8 2 6 7 7 5 G - C 5 - 0 . 4 7 £ 6 2 4 2 6 * 0 1 C. - 0 . 5 5 1 6 3 4 5 E * 0 1 0 , - 0 . 5 4 6 6 2 4 5 5 * 0 1 - 0 , . I 5 2 5 C 1 4 5 + CC • 4 4 8 2 6 7 4 E - C 1 . 2 1 2 6 8 8 5 5 - C 1 0 . 0 0 . 5 9 0 1 0 1 6 E - 0 4 _ 0 . 3 2 3 5 7 2 2 5 - C 3 _ 0 . 1 3 3 6 5 2 0 5 - 0 2 0 . 4 6 4 7 7 9 5 E - C 2 0 . 1 3 8 8 5 S 5 5 - C 1 0 . 3 5 9 1 6 8 3 t - - c r 0 . 8 0 7 4 3 7 3 E - C 1 J L - J 5 8 6 4 4 1 6 * C C _ 0 . 4 2 0 7 3 8 0 5 * 0 0 0 . 5 7 5 2 6 3 1 E + C C 0 . 7 2 5 7 5 5 3 E * C Q 0 . 5 1 5 2 5 6 7 5 * 0 0 C . 9 6 4 0 8 2 4 E + C 0 0 . 9 9 5 3 5 2 2 E * 0 0 53 C . 5 4 1 5 7 3 2 6 - 0 5 54 0 . 6 0 7 6 5 5 2 F - C 5 55 0 . 6 8 1 8 0 0 5 5 - 0 5 C . 7 6 4 9 5 3 6 E - C 5 - 0 . 5 4 1 6 2 4 5 5 * 0 1 G . - 0 . 5 3 6 6 2 4 4 6 * 0 1 0 . - 0 . 5 3 1 6 2 4 4 5 • C l C ^ - 0 . 5 2 6 6 2 4 4 E + C 1 C . - 0 . 5 2 1 6 2 4 4 5 * 0 1 0 . • 0 . 5 1 6 6 3 4 4 5 * 0 1 0 . 4 6 4 7 6 1 6 E - C 2 C . 9 9 5 9 5 9 8 P * 0 0 2 1 2 6 2 2 7 6 - 0 3 0 . 4 3 5 4 8 5 3 5 - C 5 1 5 5 4 3 7 C 5 - C 3 - 0 . 1 5 1 0 4 2 2 F - C 3 4 8 1 2 6 2 7 5 - C 3 7 9 3 3 9 8 4 6 - 0 3 2 5 3 3 5 C 5 5 - C 2 5 C 1 7 5 6 4 ~ 5 - C 2 2 2 2 5 2 4 0 5 - 0 1 4 4 6 C 2 1 7 E - C 1 C . 3 3 C 2 2 C 2 F : - C 3 0 . 1 1 2 3 7 1 9 E - C 2 0 . 4 6 5 7 2 2 4 5 - G ? 0 . 1 3 6 7 4 756 -cT 0 . 3 5 9 2 7 1 9 E - C 1 0 . 8 0 5 2 5 3 3 E - C 1 41 4 2 43 56 57 58 0 . 8 5 8 3 2 7 2 E - C 5 0 . 9 6 3 0 7 0 4 5 - 0 5 4 4 45 ENO OF F I L E 59 C . 1 0 8 0 5 8 3 5 - C 4 -60 C . 1 2 1 2 4 3 5 E - C 4 -61 C . 1 3 6 0 3 7 5 F - C 4 -62 C . 1 5 2 6 3 6 6 E - C 4 -63 0 . 1 7 1 2 6 1 2 5 - 0 4 -0 . 5 1 1 6 2 4 3 E * 0 i C . 0 . 5 C 6 6 2 4 3 5 + C 1 C . 0 . 5 0 1 6 2 4 2 E + 01 C_. 6.4 5 6 6 3 4 3 5 * 0 I 0 . 0 . 4 9 1 6 3 4 3 6 + C 1 C . 0 . 4 8 6 6 3 4 3 E * 0 1 C . 7 8 1 2 2 0 8 5 - 0 1 2 6 1 8 7 4 7 E + 0 C 1 5 8 7 3 9 0 5 * 0 0 0 . 4 8 1 6 2 4 - 2 6 * 0 1 0 . 0 . 4 7 6 6 2 4 2 6 * 0 1 C . 1 4 6 2 3 C 5 E * 0 C 1 9 3 7 C 5 5 5 + C C 4 4 6 1 8 C 9 5 - C 1 0 . 1 5 8 6 5 1 4 E + C C 0 . 4 2 0 5 2 6 1 5 * 0 0 0 . 5 7 5 2 6 5 1 C * C C 2 1 4 7 7 2 8 5 - 0 1 4 4 4 C C 4 3 E - 0 2 0 . 7 2 5 5 4 5 6 6 * C 0 ~ 0 . 9 1 5 2 5 5 1 E * C 0 C 9 6 2 e 7 2 1 F * C C 0 . 9 9 5 3 5 0 4 6 * 0 0 C . 5 9 9 7 9 0 4 E + 0 0 11 L 1 1 1 - V SI 06 1 _2_ 3 4 5 1.0STP.C1 Q l — G L C — > <z Q1.C1 — P f l Q — > 5.5 6 7 8760, Q1,E1' — F A O — > C I ' COMPARE C I , C I ' M 8 9 10 I t 1, 1,-1,1. 1,-1.1. IREAC ILIST < t P L C T t l l , 1 - 1 , t ) 11 12 13 14 15 16 9.0,5.5,4,-5,4, CQNCENTRATICN Cl,Cl« XSIZE.YSIZE, ISYMBA,ISYMB8,ISYMBC N 7 0.1212435E-03 8 0.1360375F-C3 9 0.15263665-03 -0.39163435+01 C. -0.3866343E+01 0. -0 .3ei62425+01 C. 1758632E-11 0 8702299P-1C 0 17356225-C8 0 .6135275E-C5 •6139366E-05 61411C1E-C5 17 18 19 20 21 22 10 C.17126C8E-03 11 0.192157EC-C3 12 0.2156046F-03 13 C.2419124E-C3 14 0.2714305E-03 15 0.3045495F-C3 -0.37663435*01 C, -0.3716343E+01 C. -0.26663435*01 C, -0.3616343~E*01 0. -0.2566242S+C1 C. -0.35163435*01 C. 2239955E-C7 0 22683615-06 0 16359845-05 0 .6X64501S-C5 •6391336E-C5 •8027315E-C5 5112340E-G5 0 4C898745-C4 0 1542154E-C2 0 .17I356 5E-C4 .5803839F-C4 .2122S28F-C3 23 24 25 26 27 28 TT 30 31 32 33 34 16 0.3417102E-03 17 0.3334C52F-C3 18 C.4301879E-03 19 C.4826789E-03 20 0.5415736E-02 21 0.60765635-03 22 0.6818C15E-03"" 23 0.7649544E-03 2 4 C . 8 5 8 3 3 8 1 F - 0 3 -0.3466343S+01 C. -0.2416242E+C1 C. -0.33662435*01 C. 25 C.96306955-02 26 C.1080582E-C2 27 0.1212434E-C2 -0.2316242E*C1 C. -0.22663435*01 C. -0.22 1 6 3 4 3 5 * C 1 C . -0." 216 634~3E*"CT C". -0.3116343E*01 0. -0. 20662425+C1 -C. 5047766E-03 14635255-C2 3807270E-02 -0.301634 35+01 0 -0.29663435+01 C •0.25163435+01 C 5C09550E-02 1569362E-01 3576485E-01 715'6462E=XT~ 11164925+00 1447C515+0C .15779995+CC C .1482992E+00 0 ,12114665+CC 0 0.7170304F-03 0.2180560E-02 J3.5987827E-02 0.1499778E-01 0.3469140F-01 0.7445621E-01 ~0TT4'eT42C8F + CC 0.25807C1E+00 0.4027751E+00 560575CE+C0 7088782E+C0 83002485+CO 35 36 37 38 39 40 28 C.1360374F-C2 29 0.1526365F-02 30 0.17126075-02 31 C. 192X5775-02 32 0.2156C45F-C2 33 0.24191235-02 -0.2866243E+01 C -0.2816342E+01 C. •0.2766243E+01 C. -0.2716243E+01 C. -0.2666243E+C1 C. -0.26163435+01 C. 8444065E-01 0 4503158E-01 0 2346664E-G1 0 5217646E-02 0 2529376E-C2 0 7225232E-C2 0 ,91446555+CC .963497CF+00 •5865636E+C0 .99618125+00 .9951106E+00 .555e431E+CC 41 42 43 44 45 46 34 0.27143C45-C2 25 C.3045494E-02 36 C.34171015-C2 37 C.3834C5 lfc-CT 38 0.4301876C-02 39 0.4826788E-02 -0.25663426+01 0. -0.2516243E+C1 C. -0.24663435+01 C. -0.241634SE+Ci C. -0.2366243E+01 C. -0. 2316342E+C1 0. 13565495-03 15767795-C4 2C32957E-C5 1456456E-C6 "il647E66-C8 13850925-C5 0.9999827E+00 C.1000C02F*01 C.1000CC4E+C1 0.1000004C + C1" C.1000004F+C1 C.i000004E+01 47 48 49 50 51 52 31 0.1212419--C3 32 0.1360 356C—C3 33 0.1526345E-03 34 0.1712588F-C3 35 C.19215595-C3 36 C.2156C24F-C3 -0. 291624g>: + C l C. -0.23662485+01 - 0 . -0.3816348E+01 0. -0.2766T4 85+C1 C. -0.27162485+01 C. -0.36663485+01 0. 56C8825E-07 21167385-C7 28405345-07 2771856E-C7 0. 25056925-06 0, 16361585-C5 0. 0.8015259E-C6 0.78025855-C6 0.80876385-06 8464824E-06 1057C52E-C5 27332105-05 53 54 55 56 57 58 37 0.24190995-03 38 C.27142745-C3 39 0.304546 75-03 40 0.3417C74E-0 3 41 C.3834C2CE-C3 42 C.4301842^-03' -0.36163475+C1 C. -0.3566347E+01 C. -0.35162475+01 C. -0.3 4 66 2 475 +Ol" C. -0.3416347E+01 0. -0^23662465+CI C. 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PROB A EI L I TV 1 0.3162274E+02 0.15001CQF+01 0.0 0.0 2 0. 3b48123-+02 1. 15499<= 9<=+o 1 -1.2983 221 ?-37 0.2930 221"-) ' 3 0.3981061^+02 0.1599959=+01 0.13410995-05 0. 1 37C9005-05 4 G.4466826=+02 0.16500COF+01 0.228881OF-04 0.2' 25399<=-.)4 ~3 e.b01It?67* + H7 TjTTT^O TCrr^+TJI rT7Z&~57 31T-? -03 072934 919 =-T3~ 6 0.56234075+02 0.17500C05+01 0.21641475-02 0.24576355-02 7 C.*305P e7= + ' ; 7 0.1799959F+01 0.11894125-01 0.14351765-01 ~~B— 0. /«. /544fT + r2 ld4O5c<=r>01 0.4473" C. 5<50 Bl / / — J l 9 C.79432655+02 0.19C00C0E+01 0.1151634E + 30 0. 17424515+ CO 10 C.85124?l^+?2 O.igSOlOQg+Ol 0.2030^985+00__0.3773245i + 00_ 12 C.11220145+C3 0.20499995+01 0 . 203082 7E+00 0.82 57495? + 1C 13 0.1258022^+03 0.2099959^+01 0 . 11 51 66 7F+ Q Q 0.94J9161~ + ')0 T4" 0,1412544 ,-+03 U. 21 >'J KU r-+Ul 0. 44 / i ! ydfc-01 G.98564«lb + 'J'J 15 C.1584890 c + 03 0. 22030C0 t=*01 0 .118948nF-01 0.9975425* + JO 16 0.1778276 s + r3 0.22570.f C + 0 1 0 . 2 1 6 4 2 0 6 — 0 2 0 .«997070<= + :V) 1"7 CTT9*T2T&E:+P3 0TZT5T9T9"F+T!T 0T26 _926T8E^03 0.99597635 + O.0-18 C.223871*<=*03 0.2349999 c+01 0 . 2288310E-04 0.9°99?91=+0C 19 •0.251188l e' + "3 0. 2^">">?r,0~+nl O.1341O9°5-05 0«lO007C3"+01 TO" 0.28183 'afc + n? 0.24533C.3F+O1 0. 298G22lE"-fl7 C. lCCC3CTr + 01 21 C.31622735+03 0.25000COE+01 0.0 0.10C030J c+01 J E l (S*M**-3> LOG. E l PROBABILITY CUMULATIVE . PROBABILITY 1 0.30455045-C5 -0.55163425+01 0.0 0.0 2 C.3417113 r :-0S -0.5456342<=+0l 0.590 I'M 6E-04 0 . 5501316=-04 -3 C. 3834C64g-''5" -*.b416342=+Cl 3.264562 2^-33 3 2 3 5 /2 2 ^ - 3 3 4 C . 4 3 0 1 8 9 5 F - 0 5 - 0.5366342 = *01 0 . 1 0 1 2 9 4 8 5-02 0.133 652 05-02 5 C.4826807 :=-05 -0.531634 1 =+01 0 . 3311 2?5 =-02 o .46477955-02 E 0.541 — - C. b 27T63 42-+01 075T4"ZC9TE=02 0.T38 89 895-01™" 7 0.60765*15-05 - O . 5 21 6 342 c+01 C . 220 26Q45-01 0 . 3551683 C-01 8 Q,b818r>1S'z-r'5 -n.5166342^+01 0 • 4482ft95'=-'n 0 . 80 74 37 3 = -01 S 0. /64996b = - ' J b — - ?. b 116 U 2b+Jl 0. 7793C41fc-Jl 0.1586441F+3C 10 C.8583413~-C5 -C.5066341=+01 0 . 2 6 2 0 9 3 9 5+00 C.4207380~+C0 11 0.96307?2* :-' ,5 - 0. 50 1 63425+01 0 . 1 5 8 5 2 5 1 5 + 0 r 0 . 5792631 c + 00 TZ C.1080586E-C5—-0". *?6"0^4TFRJI 0TT4649?2F+• 0"H 0.7257 5535 + CO 13 C. 1212438'=-04 -0.49163425+01 0.19350145+00 0 .9 1925675+.00 14 P. 136C378~-r4 4866342=+01 0.44 82 6 745-01 0.9640834?1 + 00 T5 (J. lb26J6Ml--'}4 -0.4 f l 6 J 4 1 -+01 J . i 1 2 6 5 ^ - 0 1 3 . 'Vi b 3 52 I'- +30 16 C.1712612^-04 - O.47663425+01 0 .464761 6E-02 0 . 9999593^+ ?0 17 0.1921582 — 04 - 0.4716342 = + 01 0.0 0.9999993E + 00 177 I I 1 - 1 4 C (G*M**-3) LOG. C PROBABILITY CUMULATIVE "PROBABILITY"' 1 0.6176 I!67 ! :-W 1— U . 6 H i a U 2 2 « r - C 4 -0.421 6343 c+01 ""-0.4i66J4i^+01 0.61392785-05 "OTTJ • 0.61392765-05 ~U.6.3^2 /8E-J5 3 C 76459485-0*. 4 "0.8533385^-^4 "3 CT9~53 05~nc5~FrO"4" 6 0.108C582 c-03 7 C.1212435 e-"3 "S U . I 3 6 J J ^ e - P ' J -O.M16343 c+01 ^ r i . 4 0 6 6 3 4 2 e + 01 - 0. 401 6T4J5+0T -0. 39663^>3 c-r0l -0^391 6 34 3'-'•01 "~M.i«ftfeJ4J'-«-01-0 . 0 3.0 ~~ore — 0.0 0.17586325-11 J . a / 3 2 2 9 9 E - 1 0 0.17356225-^8 0.2339O95E-07 'J o 2268361E-05" 0.16359846-05 0.9112340E-C5 0.40 898 7 45-04 0 . 6 1 3 9 2 7 3 5 - 0 5 0 . 6 1 3 9 2 7 8 3 - 0 5 T ) T 6 T 3 9 2 7 8 F - 0 T ~ 0 . 6 1 3 9 2 7 3 5 - 0 5 0.61392795 - 0 5 5 C .15263665-"? 10 0.1712608^-03 ~n n . i v s i b / g g - T r r 12 0.2156046^-03 13 0.2419174^-0^ T4" L'.^/14J,Jb'?-C3 -0. 3 3 1 6 3 4 2 ^ + 0 1 -Oo 2 7 6 6 3 4 3 P + 0 1 ~0TT71"eT4T!TrTJT - 0 . 3 6 6 6 3 4 3 5 + 0 1 - 0 . 3 6 1 6 3 4 3 5 + 0 1 6 6 3 4 2 *":«• O T 0.6141101--05 0.61645C1E-05 ~0T6~79T73 £ 5 - J 5 ~ 0.8027319 5-75 0.1713965F-04 15 C.30454°5=-r3 16 0.34171Q2E-03 "T7 L'.3a34l!b24-r3 18 r.43018795-03 19 Q.4fl?67qqP-03 T O 'J . b41 b Ii 6'- —0 ? -0.3516343P+P1 -0.34663435+01 "'-0.3416T4"3T^+TJT~ -0.2366343 c+01 -0.3316347^+01 2<56634Jt- +01 0.15421545-03 3.50477665-03 TTTl4^y57~9F:=0~r 0.3807270E-02 C.50099505-02 i ) . H t u u A 2 =--111 O . b H 0 J 9 3 9 ' i — 3 4 0 . 2 1 2 2 5 3 8 5 - , r 3 0 . 7 1 7 0 3 0 4 5 - 0 3 OT2rST560T=T"2~ 0 . 5 9 8 7 8 2 7 5 - 0 2 0 . 1 4 9 9 7 7 8 — 3 1 0 . J 4 6 9 1 4 0 - - H 21 0.60765635-03 2 2 C . 6 8 1 8 0 1 9 5 - 0 3 "23 C 7 T O ? i ? s ' ^ m ^ -0.85833815-03 C.963C6 e l'5 c- r !3 (I. I t t i< !bt f?=-*2 ' 24 2 5 27 28 — 2 9 _ 30 31 -0.3216343F+O1 -0.3166343P+01 ~-T3ir63"43F+cr -0.20663425+01 -0.3O16343 c+01• '•'•o 296634 3- + 01 0.3976485E-01 0.71964625-01 0.14470915+00 0 . 15775995+00 0. 148^9921:+00 0.744562 15-31 _0. 1464203'=+00 0.2380701T+yn~ 0.4027791E+OC 0.56C579C5+30 J . ;UH8/82 r+30 0.1212434F-C2 C. 13603745-C2 0.17126075-^2 0^19215775-^2 ' U.<ilb604.S = -C? -0.2°16343P+01 -0.28663^3E+01 -"077817347^+01--0.2766343 c+01 - n.2716343^+01 ^666 344 F+01" 0.121146 65+00 0.844406 55-01 0. 490313 85"=rOT" 0.234666^5-01 0.°217646P-Q2 'J.^y24i/6b-02 0.8300248 c+00 0.91446555+00 ^3.96345705+OCT 0.98696365+00 ".99613125+30 0.999HC6E + 30 ' 33 0.241oi23'=-',2 34 0.27143045-^2 "35 0730434^^=07" 36 C3*171015-02 37 0.3834Q?l~- r2 3 E L . 4 3 0 1 B / 6 b - C 2 •r>.2616343P+01 -0.2566342c+01 ^ 7 7 3 1 6 3 4 3 F+OT -".2466343F+01 -o. 2416343 , : + 01 r'23663435+0r 0.7325232E-03 0.13 56549F-03 "gTT9TF779F-0* 0.203 29975-05 0. 14564565-06 0. /164 /86c-08 0.9998431<= + 0C 0.9999827^+00 rooooc25+or 0. lC r ,00C4 = + 01 0.1C000C4F+Q1 u. iL".)0JC4e+ai 39 G.4826788 c-C2 40 0o5415734=-02 "41 C76TJ7"S36J'T-CZ' -0.2316342 e+01 •0.22663435+01 -U. .^163435+01" 0. 1385092E-09 0.0 "07T7 0.10000045+01 0.10000G4E+01 C1000304F+0r 178 I I I - 1 5 A2 1 ! 1! It G.L.C. CATA GSN6RATICN PROGRAM 1 1 I 1 1 1 1 1 f 1 11 •VARYING HIND SPEED CU) 1 1 tl 1 1 - 1 1 1! PHISSICN RATE 10) 1 I 1 1 J 5 LG5. S PRCBA6IL I l"Y CUMULATIVE PROBABILITY 1 C.40CQ000": + C1 O.£02O6CDfc+9<: " 0 . 1 " ) i W n 0 D E + n l o . I O O T > O O E + O I J U IM/S) L U G U "PPnBAEIL!TY C U M U L A T I V -PPOB «BILITY 1 2 3 "0.15848O2=+01" 0. 1778278? ••-I 0.!995261=+0! 0.19 ' 9 9 C C 8 S V n C ' " 0. 249o<;cflc+.0C 0.29995585+0C 0.59010155-04 0.26456225-03 0.10123395-02 C.5901016C - 3 4 0.3235722* : :-)3 0.12364616-02 5 6 0.2238 Utir+OL c.25ii8-85-':+',i 0=281e380E+Cl 0.34 ^ c ^ + O U -C. 3959556~+CC 0.44959=35+00 J .33111^^;-'^ 0. 5242^6 2F.---2 0 . 2 2 O 2 6 7 C E - 0 1 / 6 1 e'--'J^ 0.13E89685-01 0 .3551647F-31 "T. 3074 215^-11"" 0.1536434 C+0C 0. 2742446= ••00 7 8 9 C.31622 /b1-*''! 0.3548129 r : + 'M 0.39fll06? c+01 0.4999958r+Ui: r*.5499958 c+r0 . 0 . 5 9 O 9 9 5 8 P + 0C 0 .44B26 l l r z ~ 0 1 0.7790C17 r-01 0.1156012 c+00 IC 11 1 2 C . 4 ^ 6 6 8 :U- + ul 0. 501!868'=+ 01 0.5623405~+01 0.6999958F+0C 0.7499953F+0C 0 . 1 4 6 4 9 ? 0. 1585251^+00 0 .14649246+00 '•'.4?!.i / 3 6 d ' - + ••).» 0.57926195+00 0.72575435+10 13 1 4 15 C.6305556'-: + !-r" 0.7079450 C+C1 0 79432755 + 01 0. 79 S^SB'r+CTC 0.84999585+0C 0 . 89°«959P+nC U . 11 i6<H b S + G O 0.77°0n<v ic-01 0 ,A4e2695P-01 • ) .8411557"+ 30 C.91925626+00 0.964C331' : + 00 I t 17 18 c y s i ^ y y . - * " ' 0.9°99996 C+C1 0 .1122 n15=+02 C. ,T4y l^ t.B ' -^• r , L , " n.99OQ953E+0C P.1049C<=96+01 H . 2 ^ l * 2 f c 9 4 - c - ' i l 0.92420915-02 0 .33112755-02 - . 9 b 6 11C0!-+'.V' 0.955252C5+00 0.9986632^02 ~0i99967627+00 0.9999407^+00 0. 99999975+00 15 20 21 P. 12bH923r:+C2 0.1412535P+ n 2 0.1584891 r+C2 0 . 1059555!: + 01 0 .11500C05+01 O 1 2 C 0 0 C 0 c + 0 l T . n l 294 5>-rjz 0.2645622E-03 3.59010165-04 q (CFG.NJ LOG.P P R O B A B I L I T Y C U M U L A T I V E HKL'dftiL 1 I Y C.18000005 + 03 0.2255272E«-C1 _0_.1000000S+01 0 . 10000C05+01 179 111-16 J 01 ( G / S ) L O G . 01 P R O B A B I L I T Y C U M U L A T I V E — ! P P O B A B T L T T Y 1 0 . 2 2 3 8 7 1 5 g » 0 2 1 3 4 9 9 5 0 6 + 0 1 O . G C u O —2 C . 2 5 1 1 S P 2 E + 0? 0 . 1 4 0 ^ 0C O E + O l 0 . 2^58021 S5"-07 0 . 29S!"2l S n - u ? " 3 0 . 2 8 1 8 3 7 9 ^ + 0 2 0 . 1 4 5 3 0 0 0 ? + 0 1 0 . 2 2 3 5 1 6 3 5 - 0 5 0 . 2 2 6 4 9 6 5 ^ - 0 5 4 C . 3 1 6 2 2 7 4 ~ + 0 2 0 . 1 5 0 0 0 C 0 c + 01 0 . 8 6 1 2 3 2 9 5 - 0 4 0.0 8 3 5 3 2 5 5 - 0 4 —5 C T 3 5 ¥ B 1 2 3 h + 0 2 0TJ3"49 9 9 ^ + 0 1 0~7T53l>"277^u2 0 7 1 T 6 8 S 7 0 T - D " 2 ~ ' 6 C . 3 9 8 1 0 6 1 c + P 2 n . 1 5 5 9 9 5 9 5 + 0 1 0.1684.10 2 5 - 0 1 0 . 1 8 6 0 3 6 9 = - ) 1 7 0 . 4 4 6 6 8 2 6 5 + 0 2 0 . 1 6 5 0 0 C 0 6 + 0 1 0 . 8 7 0 3 6 0 1 5 - 0 1 0 . 1 0 5 6 4 4 6 F + CO B C . b u l l a * r-±"l O.l'MCn CO c + r n 3 . 2 3 2 8 0 9 e.P+00 0 . 3254=4: ; - - + '30 -9 n.56234^7 C+C2 0.17500CO6+C1 0.323OOO35+O0 0 . 6 6 1 5 3 4 7 5 + 0 0 1 0 C . 6 3 C 9 5 5 7 5 + 0 2 0 . 1 7 5 9 9 5 9 5 + C 1 0 . 2 3 2 8 1 5 6 5 + 0 0 0 . 8 9 4 3 5 0 3 = + 0 0 T I G . 7 C 7 9 4 ^ 0 ' : f r 2 O T r 3 4 9 9 T 9 c + 0 1 O T B T O ^ O l " * - ^ ! 0 7 9 8 1 3 5 0 7 C + 0 0 -12 0 . 7 9 4 3 2 6 5 5 + 0 2 0 . 1 9 0 0 0 C 0 5 + 0 1 0 . 1 6 8 4 1 3 0 5 - 0 1 0 . 9 9 8 2 3 2 0^+00 13 0 . 3 9 1 2 4 9 1 5 + 02 0 . 19 500 C05+01 0 . 1 6 8 0 4 5 5 5 - 0 2 0 . 9 9 5 5 1 2 4*=+00 T4" Q . 59°5°Q1': + 'J'J 0 . 2 0?00f 0"+tl J . E612329 5--J4 0 . 9 9 9 9 9 8 57- + J " ' 1 5 0 . 1 1 2 2 0 1 ^ + 03 0.20499c<je < . c i 0.223 51635-05 C.1CC00C06+31 16 0 . 1 2 5 8 9 2 2 5 + 0 3 0 . 2 0 5 9 9 5 9 5 + 0 1 0 . 2 9 8 0 2 1 8 5 - 0 7 0 . 1 C C O O C 0 5 + 3 1 T 7 T ! T T 4 T 2 5 T 4 - - + r 3 0 . 21b'>T* ,0*r+01 D T O O T T O 0 0 0 0 3 * + T T ~ 1 8 0 . 1 5 8 4 8 9 0 5 + 0 3 C . 2 2 0 0 0 C 0 F + 0 1 0.0 0 . 1 C C 0 0 C 3 5 + 0 1 19 0 . 1 7 7 8 2 7 6 - + 0 3 0 . 2 2 * 3 0 0 0 6 + 0 1 0 . 0 0 . 1 C 0 0 0 C 0 = + 0 1 J 51 ( S * M * * - 3 ) : L O G . 61 P P C 3 A B I L T T Y C UMUL A T I V 6 ' 1—PROBAtJIL ITY 1 0 . 3 0 4 5 5 0 4 = - C 5 - 0 . 5 5 1 6 3 425+01 0 . 0 0 . 0 ~2 Orj4T7113--0^—=7T75^"66"34 2^+0T 3T5S7yr0T6T-O* 0 - . 5 9 3 i a r * = - 3 4 — 3 0 . 3 8 3 4 0 6 4 C - - < 5 - 0 . 5 4 1 6 3 4 2 c + 0 1 2 6 4 * 6 2 2 c - 0 3 0 . 3 2 3 5 72 2 5 - ^ 3 4 0 . 4 3 O 1 8 9 5 s - ' ' B ; - 0 . 5 3 6 6 3 4 2 5 + 0 1 3 . 1 0 1 2 9 4 F 5 - 0 2 0 . 1 3 3 6 5 2 3 e - 0 2 C . 4 B 2 f c P 0 7 r . i ^ — 5 3 1 6 5 4 1 ^ + 0 1 A . 3 3 1 1 2 7 5 r - 0 2 0 . 4 6 4 7 7 9 5 e - 0 2 6 0 . 541 5 7 5 7 ' I - ' " 5 - 0 . 5 2 6 6 3 4 2 ^ + 0 1 0 . 9 2 4 2 " 9 1 ==-02 0 . 13 8 8 9 3 9 c - 3 1 7 C . 6 0 7 6 5 8 1 5 - " 5 - 0 . 5 2 1 6 3 4 2 6 + 0 ! C . 2 2 0 2 * 9 4 F - 0 1 0 . 2 5 5 1 6 8 2 C - 3 1 ~ 8 C . 6 8 1 8 0 3 ^ r = " 0 = — - < : . 5 1 6 6 34 2r+rTI T7"4 i826 '5 5 ;-"01 0 . 3 0 7 4 37 3 r - T I 9 C . 7 6 4 9 ° 6 5 ' * - 0 5 - 0 . 5 1 1 6 3 4 2 c + 01 0 . 7 7 c 0 0 4 1 <=-01 0 . 1 5 8 6 4 4 1 5 + 00 10 0 . 8 5 8 3 4 1 3 = - C 5 - 0 . 5 0 6 6 2 4 16+01 0 . 2 6 2 0 5 3 8 5 + 0 0 C . 4 2 Q 7 3 3 0 ' = » C 3 TT C . 9 6 3 C 7 2 2 " i - 0 5 — - 0 . 16 342=+<.vl 3 . 1 5 S 5 25 1F+00 ~~rrrrTF&Tr+T'r~ 12 0 . 1 0 8 0 5 8 6 ^ - 0 4 - 0 . 4 9 6 6 3 4 2 5 + C 1 0 . 1 4 6 4 9 2 2 5 + 0 0 0 . 7 2 5 7 5 5 3 6 + 0 0 13 0 . 1 2 1 2 4 3 8 c - 0 4 - 0 . 4 9 1 6 3 4 2 5 + 0 1 0 . 1 9 3 5 3 1 4 5 + 0 0 3 . 9 1 5 2 5 6 7 5 + 0 0 T 4 C 7 1 3 5 0 3 7 8 = ^ 4 — = 7 r T 4 S 6 " 6 T 4 ' Z F + i r i 0~4"4"82"674r^Jl 0 . 5 6 4 0 8 3 4 c + y > " 15 0 . 1 5 2 6 3 6 9 6 - 0 4 - 0 . 4 8 1 6 3 4 1F+01 0 . 2 1 ? 6 3 3 5 F - 0 1 0 . 9 9 5 2 5 2 2 5 + OC 16 0 . 1 7 1 2 6 1 2 5 - 0 4 . - Q . ^ 7 6 6 3 4 2 c + 0 1 0 . 4 6 4 7 f 1 6 6 - 0 2 0 . 5 ° 9 9 9 9 8 - : + 00 T 7 n . 1 9 2 1 3 0 2 b - — - 0 . 4 / 1 6 3 4 2 - + 0 1 0 . ^ 9 9 9 5 9 8 ^ + 0 0 180 111-17 C (G*M**-3) LCG. C PROBABILITY CUMULATIVE PROBABILITY " I 073417104F=7T4"" 2 0.38340.54 C-C4 3 0.43018<U e-C4 ~4" L 1.48^6 7 9 ^ - 0 4 -O . * 4 T 6 3"43"5T0T - 0 . 4 4 1 6 3 4 3 F + 0 1 - 0 . 4 3 6 6 3 4 3 c » 0 1 4 3 1 6 3 4 2 5 + ^ 1 0 . 4 3 5 H 3 9 E - 0 5 0 . 0 O.C TTo" "T74"35rr3*3 e-05~ 0.43511395-05 0.43511395-05 0 . 4 J b l i 3 9 E - 3 5 0.43511395-05 0.43511395-Q5 5 C.54157405-OA 6 6.60765*75-04 —7 C.fc8ia02T !=^TT 8 0.76499485-04 9 0.85a3335 c-C* T75 <J.9&31699--"4 11 0.10805825-03 12 C. 1212435'=-03 T I 0.1360375= - 0 3 14 0.15263665-03 15 0.171 2608=-"3 0.1921578"-03 16 17 18 I T 20 21 -0.42663435*01 -0.4216343^+01 C. 21560465-03 0.2419124^-03 U.2/I4305E-C3 0.30454t>5'=-03 0.3417102--O3 C.33340S2E-O3 23 0.43018795-C3 24 0.4826789 C-C3 T 5 CT54"r5T3-6T=7T3-26 C.60765635-03 27 C.6813O19=-03 T S 0.7649*44^-03 29 ' 0.8583381P-' ,3 30 C.96306955-03 ~ ^ r T 4 T 6 T 3 ^ 5 ~ + i n " -0.4116343E+01 -0.40663*25*01 -' ,,4P16343 C+01 -0.3966343E*01 -0.3916343E+01 -0. ?8~6"63??B~+0T~ -0.33163425*01 -0.37663435*01 -0.37163435*01 -0.36663435+01 -O.36163435+01 -0.2566^42f r + 0 T " -0.35163435+01 -0.2466343' : + 01 O.C 0 . 0 "~0"70" O.C 0.1753*315-11 •V. 13 e-7819'":-n)9 0.57039675-08 0.12430245-06 0 . i533nre==o~r 0.11599785-04 0.6023304E-O4 3.242111 55-03" 0.81992325-03 0.24228415-C2 0.14645435-01 0.31433265-01 0.643b27*6-01 0.42511395-05 0.43511405-35 0.43512795-05 0.43569835-05 0.44812845-05 _0T'6"0r447 3^-05" 0. 1761425P-04 0.77847295-04 0.31995875-33 0.1139837=-02 0.3562723 c-02 15V9 86"9*77E-aT 0.24515315-01 0.5599357^-Cl C. 12023l3'":+C0 0.23564555+00 _0.39574C4=+0C 0 r5"6'73 0 3 6" +"00"~ 0.72522605+0O 0.35343075+OQ C.93651535+0C 0.97773465+00 0.99353705+00 _0.'9989337 = + 0Q" 0.99985715+00 0.99999225+00 0.1 COO 1025+31 0.1C000025+O1 C.1C000C25+01 -OTICOOOO25+01-0.10000025+01 O.1COO0C25+O1 0.10007C25+01 0.1000002E+Q1 31 32 33 "34" -0.i416343E+m -0.33663435+01 -0.3316342=+01 r0T3263T43"5+'C -0.32163435+01 •0.31663435+01 T7TJTH75TJ7« r=7T' C. 12124345-02 0.136037*=-02 'U. lt>!63'-b c-02 0.17126075-02 0.1921577E-02 21b6045 c-C2~ 0.2419123 c-02 C.2714304=-02 -0.;1162435+G1 -0.30663425+01 -0.30163435+01 : rOT7T6^6T4rjF+0T -0.29163435+01 -0.28663435+01 •'J.28l6342'=+Cl 0.11529425+00 C.16009495+00 rr553"pr*D-r 0 . 15792245+00 0. 128204~E+C0 0.83084525-01 0.41219405-01 0. 162C23BE-01 D'.43 957 6~7 5^0 2"" 0. 10234655-n? 0.13511555-03 d . 1060923=-04 35 36 1 7 " 38 39 ~4~T" (J. J ( ' ; 4 b 4 y 4 - - r 2 0 . 3 4 1 7 1 O 1 E - 0 2 -0.27663435+01 -0.2716343 c+01 =OV26663"2f 3 E*ZTT -0. 2616343"= + 01 -0.25663*2=+01 -'J. ^b!6343 = + 0 i 0.47151805-06 0 .11320055-07 ~37T3E50~9T^0"9~" 0.0 0.0 41 -0.24663435+01 O.C 181 111-18 A3 1%X X 1.0PA0.C1 C l 0 1 ,C1 t e * = = > >=• = 0 C1 • • *=*=*S* 3* = *= *<=*=*=*=* = * = * = J 0 1 (G/SJ AL0G1C(G=) PROBABILITY CUVULATIV C i c .354ei23 c+' ,2 ~Z CTFiin9blr-.*-<ll 3 0.44668265+02 4 C.5-T1 l«67= + r2 - 3 C.b62343 't + 0? 6 c.63C95576+02 7 0.7P7944«6+C2 8 077943Z65 6+02 5 C.89124915+02 10 C.9999991=+02 15499555+01 0.16500GO?+01 0. 1 7 0 T 0 O 6 + C1 0.17bu*C0fc+01 0. 1799999<=+01 0. 18495C9^ + G1 0.I903OT0E+C1 0.19500006 + 01 0.2OC00CO5+O1 0.^499=9^+01" 0. 2059559=+"l 0.21500C06+01 "0T2 rrrorjTjo'* r*-c «.2250' ,CCE + C1 0. Z2 q99595 + 01 0.299 A2215-^7 0,1341'09191"=0'5~ 0.22893105-04 0.269233*5-03 0.216414 75-02 0.1189412S-01 0.4473002E-01 0.11br6"3"4-5+W-0 .2030 79 36+00 0.2453418^+00 U . 2 U 3 U H 2 7 r + 0 0 0.115166 75+tK; 0.44731986-01 3'54305T-0'1"' 0.216^2n6P-i2 0.26926235-03 >i . 2 2 « H 3 n - - 0 4 0.13410955-05 0.2980 221E-07 0.29802216-07 ~Tr.T37C900E-T5-0.24258595-04 ".2934915 c-33 0.245763^5-02 0. 14351765-01 C.59C81775-01 "CTI74245IF+CC" 0.37732495+10 0.62266676+00 "fl'.'BZ: /455=+-.3C 0.94091616+00 0.98564815+00 ' C. e975425-f+ 00 0.99970705+00 G.9cce7625+00 0.55999515+00 C l CCOOCOP + 01 C.1CC00C36+01 T T 0 11220I46  i2 -125«»922c  313 0.1412534^+03 T 4 C T T ^ P ^ a o ^ ^ 15 C.17782766+03 16 0.19952566+03 T 7 C.22337145+C3" 18 0.25118316+03 19 0.28183795+03 0.23499P«, = +C1" 0.24030C05+C1 O.245G0C06+O1 STATISTICAL PA"AMETER S PGR DISTRIBUTION OF Cl <G/S» SK'WN^'Sb C05PP.KU3. r e s t s O.57285096+0O 0.35 874892+01 MF AN ' 0.10176726+03 S i D . n t V . — ' — 0.1921445F+C2 M6AN STD.DEV. 3DTTTJ'N~OT SKEKNESS COEFF.KURTOSTS 0.1959993'i+01 0.812906^-01 0.36/bl24&-J4 C. 299381H+01 C l ( G * * * * - 3 J L0G(C1) PROBABILITY CU^ULATTV6 PK'IB Ab IL I 1 Y 1 0.121243?=-"3 3 0.1526366=-03 4 0.17126'«8'=-03 -0.3916343=+C1 -0;3566'2 43t+01 -0.28163425+01 37663436+01 -u. i (-16343c+01 -0.3666343 C+P1 -0.36163435+01 ~0T35cTc7T42'*:+Cl~ - P . 3 5 1 6 3 4 3 c+ri •0.34663436+C1 0. 17536326-11 "TrrH70229 9F^-TCr 0.17356225-08 0.23399955-07 'J. 22683615-06 0.16359346-05 0.91123405-05 -7J7ZT0T9 8T«P^a4-0.15421546-03 0.5C47766E -03 0.61392796-05 "0.6T253666-05' 0.61411C15-05 0.6164501 c-05 U.6 •i91,J3 6'--05 0.8f273196-05 C. 17139656-04 "0.58033395^C4-0.2122538 ,=-02 0.71703C45-03 ~5 C. 1921b '85-03 6 C.2156046=-C3 7 C.2419124f=-03 8 0 . 2 7 r 4 3 0 5 5-03 9 0.3045495^-03 10 0.34171026-03 182 ~ i l 1 C.38340*2= -C3 3416343*"+01 0.146352°5-02 O.218056OC-32 12 0. 4301879': -03 -0.23663436+0] 0.38072706-02 0.55878276-32 13 C.48267395 -03 -0. 3316342=+01 - C.5C099506-02 0.14557785-31 T4" C.5415736? -03 - 0 . T26"6T4Tr>fir 3.19693625-01 0.3469140 6-01 15 C.60765636 -03 -0. 32163435+01 0.39764856-01 0.74456216-01 -16 0.6818019= -r>3 -0. 31*6343=+01 0.7196462 c-01 0. 1464203= + 3C 17 0./64 S944= -"3 -n. 2116343r" + r i 3. l l l 6 4 9 2 - + '>3 r ,.?58O70l c+Ol 18 0.85833815 -03 - o . 3066342E+01 0.14470916 + 00 0.40277515+3C 19 0.963C6955 -03 -0. 3016343=+01 3 .15779996+03 0.56C575C6+00 20 -C2 -0.29rT6341 cTcT 0TTTE2^92^+Oir" 0rr0887325 + 00~ 21 0.12124345 -02 -0. 2916343F+01 0.1211466E+03 0.8200243^+00 22 0. 1360374= -02 -0. 2866343=+01 0.844406 55-01 0.9144655=+0C 23 IU 1 5Z6J * 5 5 ' -02 -0.28163425+C1 C.49031595-01 U.5634573E+00 24 C.17126076 -02 -0. 27663436+0! 0 .23466645-01 0.98696365+03 25 01921577= -02 -0. 2716343=001 0.92176465-02 0.9961812=+0O 26 0.21560455 -02 - 0 . 26 66343"P+0T iTT2^2-93T6F=02 0;(5"9511C65 + 0'3~ 27 C.2415123= -02 -0. 26163436+01 0.7225232E-03 0.9998431r+30 28 0.27143*4= -0. 25663425+01 0.1356549^-^3 r>.99993275 + 30 29 0. 304t>494E -02 -0. 2516343F+C1 3. 197677cp_04 C. 1COOOC2C + 01 30 0.3417101= -02 -0. 2466343^+01 0.2032997F-05 0.1C000046+01 31 0.3834P?1': -C2 -0. 2416343=+01 0.1496«56=-06 C.1C003046+01 32 C.4301876E -02 -0. 23663435+01 0.71647865-08 O lC0C0C4 c + 0l 32 0.48267885 -02 -0. 2316342P+C1 0.13850926-39 0.1 COOTIE+01 *=* b 1 A 1 Ib 1 I CAL PARAMETERS FDR Dial "BUT ION OF Cl (G*M**-3J « MEAN STO.OFV. SK = V«NESS C05 CF.KU°TCSIS TJ71"0T5546T=02 0T296 2"C5"3E=TJ3 OTT88T3T4 cT00 0.40178325 + 01"" »«» STATISTICAL P A P A M 5 T 5 P S FOP DISTRIBUTION C = LCG(Cl) < MEAN STO.OEV. SK5WN6SS CO =66. KL'RTC 3 IS -0.3011333 c+01 n .1257464E+CC -o .95800945-31 0.30562315+31 * =*=* =*=* ' = *=* = *=* = *«*• =*=*=* J E'(S*M**-3) LOG(E') PROBABILITY CUMULATIVF PROBABILITY 1 0.13603415 -07 -0. 78663545+01 -0. 1179810E-02 -0.1175310F- 02 2 0.1526348= -07 -n. 7816354=+01 * o =5745225-03 -0.22235757- 03 3 C l 71 25 88-: - f l -0. "? 766354c + 0] - J . /641460F-33 -0.5e65Q39 e- 33 4 0.19215525 -07 -0. 77163535+01 0. 63245955-33 -0.38404^4=- 03 5 0.21560135 -07 -0. 76663535+01 -0. 47<-.4171E-03 -0.35446155-03 ' 6 0.2415032= -C7 -0. ('6163,:ii«- + 01 u . 35390226-33 -0.4905593 6-"33""" 7 0.2714251= -07 -0. 75663536+01 -0. 27936286-03 -0.76552205-C3 8 0.3O45434!: _ o 7 -0. 75163535+ni 0. 21451596-03 -n.55540615- 33 9 C.34170505 -07 -0. /466 3525+?! -0. 167 l^^it-yTi -C. / 2 i l b O i 5 - 33 10 C.3834020= -07 -0. 74163525+01 0. 13 807786-03 -0.5850724F- 33 11 0.4301834= -rn 7366352^+01 -0. 12510516-03 -0.71017756- 03 T2 " 0.4B26725E -07 • -0.7316352.E+0! 0. 12797546-03 -0. 5 82 2 3T9E-3 3 ~ 13 0.5415669= -07 -0. 7266352^+01 -0. 14496916-03 -0.72717085-03 14 0.60764H = -07 -0.72163525+01 0. 17365536-03 -0.55351556- 03 J. 3 "C.681 /'902E -n t - y . /166Jil!-+Cl - ' ) . 2112i30r:-33 -0.764 /43 25-'J3 16 0.7646°135 -07 -C. 71163516+01 0. 25516685-03 -0.50959155-03 17 0.8583328= -r.7 -0.7C663515+C1 -0. 30366216-03 -0.81324375-03 70163bl=+01' '•J m 333T2555"-03 -=^.-55811715--33—-19 0.1080573= -06 -o. 69663516-+C1 -0. 40714216-03 -0.86525925-33 20 3.12124146 -06 - n . 69163506+01 0. 45700796-03 -0.40825126- 03 183 111-20 T I 7 j . Y 3 6 C 3 ^ 2 c - r 6 - " . 6 8 6 6 3 505 + 01 - O . 5 0 2 4 5 2 7 5 - 0 3 - 0 . 9 10 7 0 4 0 5 - 3 3 22 0 . 1 5 2 C 3 4 2 5 - 0 6 - 0 . 6 e i 6 3 5 0 E + 0 1 0 . 5 4 3 4 « 1 4 5 - 0 3 - 0 . 3 6 7 2 1 2 6 E - 0 2 23 0 . 1 7 1 2 5 8 4 = - 0 6 - 0 . 6 76 63 505+01 - 0 . 5 8 0 1 5 4 * 5 - 0 3 - 0 . 9 4 7 3 6 6 9 P - 0 3 -24" ? T 7 T 9 7 T 5 5 1 ' - - ' " S — - A 7 6 7 r n 3 ! W P ) l 3 X 6 X 2 3 4 3 3 = = 1 3 - 0 . 3 3 5 0 1 9 6 5 - 0 3 — 25 0 . 2 1 5 6 0 1 6 5 - C 6 - 0 . 6 6 6 6 3 4 9 5 + 0 1 - 0 . 6 3 G 5 0 7 2 5 - 0 3 - C . 9 7 2 5 2 5 5 5 - C 3 2 6 C . 2 4 1 9 0 9 C ~ - O f - 0 . 6 6 1 6 3 4 < 5 ' : + 0 1 3 . 6 5 8 7 4 3 1 5 - 0 3 - 0 . 3 1 4 7 8 2 8 F - 0 3 " T 7 ? . 2 i l 4 ? 6 a - — 7 6 — - 0 . 6 5 6 6 3 4 ^ + 0 1 — - U . 6 / 4 7C9 ^ - 1 3 - 0 . 9 S 9 4 52 5 ^ - 3 3 — 2 8 0 . 3 0 4 5 4 5 9 5 - 0 6 - 0 . 6 5 1 6 3 4 9 F + 0 ! 0 . 6 9 9 2 3 3 3 5 - 0 3 - 0 . 3 0 C 2 5 4 2 C - C 3 "29 C . 3 4 1 7 " 6 1 5 - 0 6 - 0 . 6 4 6 6 3 4 9 F + 0 1 - 0 . 7 0 3 0 8 1 5 5 - 0 3 - 0 . 1 0 0 3 3 3 6 5 - 0 2 " T O O T T H l ^ O T r T ^ T J E — = T J 7 F 4 T £ 3 T 8 r + r n D T ^ T T C ' 4 ~ B ! ~ 0 3 — = 0 7 2 8 5 4 5 0 9 5 - 0 3 31 C . 4 3 0 1 8 ? 8 ' : - 0 6 - 0 . 6 3 6 6 3 4 8 ^ + 0 1 - 0 . 7 1 7 5 6 6 8 5 - 0 3 - 0 . 1 0 3 7 3 1 8 5 - 3 2 3 2 0 . 4 8 2 6 7 3 6 C - C 6 - " . 6 3 1 6 3 4 8 5 + 01 0 . 7 1 1 0 1 7 8 5 - 0 ? — 3 . 2 9 5 9 9 9 9 C - 3 3 " 3 3 C . S 4 1 b i B 6 - P . £ 2 6 6 3 4 8 5 + 0 1 - ' J . 6 ^ 4 / 8 8 l f c - 0 3 - C . 9 9 0 / 8 8 ' J t - O i 3 4 C . 6 0 7 6 5 0 3 = - " 6 - " . 6 2 1 6 3 4 8 ^ + 0 1 0 . 6 7 3 2 1 5 8 E - 0 3 - 0 . 3 1 7 5 7 2 2 c - 0 3 3 5 C . 6 8 1 7 9 5 C = - , : ' 5 - 0 . 6 1 6 6 3 4 8 r + r i - 0 . 6 5 3 4 3 0 6 5 - 0 3 - 0 . 9 7 1 0 0 3 0 c - 0 3 ~ 3 6 0 . 7 6 4 9 8 - A 7 h - 0 6 -176" IT63"4Tr+TJ I 0 7 T ^ T " 6 T 7 5 F = 0 3 — = 0 7 3 2 5 3 9 5 5 5 - Zl 3 7 C . 8 5 8 3 3 0 2 5 - 0 6 - 0 . 6 0 6 6 3 4 7 5 + 0 1 - 0 . 6 4 0 3 4 4 8 5 - 0 3 - 0 . 9 6 9 7 3 0 3 5 - 0 3 3 8 0 . 5 6 2 C 6 ? l c - ' - 6 - " . 6 0 1 6 3 4 7 5 + 01 0 . 6 4 7 9 6 3 5 5 - 0 3 - 0 . 3 2 1 7 6 6 8 5 - 0 3 ^ - 3 9 ( J . 1 0 V f ) b / 4 b - U b — - 0 . 5 < * 6 6 J 4 . ' f + 0 1 — - 0 . 6 6 C C 8 / 9 i r - Q 3 — - 3 . 9 6 1 9 5 4 / ^ - O i 4 0 0 . 1 2 1 2 4 2 5 5 - C ^ - 0 . 5 c 1 6 3 4 7 " + n 1 0 . 6 7 2 3 9 3 0 5 - 0 3 - 0 . 3 D 9 4 6 1 7 c - 0 3 41 0 . 1 3 6 0 3 6 4 ' - - 0 S - 0 . 5 8 6 6 3 4 6 ^ + 01 - 0 . 6 8 2 4 1 5 9 5 - 0 3 - 0 . 9 9 1 8 7 7 6 E - 1 3 ~A~2 0 . 1 5 2 ' 6 3 5 4 = - r 5 — = T J 7 r B X 6 ~ 3 " 4 T r + 0"I 0 X 6 ? O 6 CT£TO 5 =03 = 0 . 3 0 1 2 7 1 7 5 - 3 3 4 3 0 .1712598'=-05 - 0 . 5 7 6 6 3 4 6 ^ + 0 1 - 0 . 6 9 8 3 3 6 8 5 - 0 3 - 0 . 1 0 0 0 1 5 8 5 - 0 2 4 4 0 . 1 9 2 1 ^ 6 7 5 - 0 5 - 0 . 5 7 1 6 3 4 6 5 + 0 1 0 . 7 C 6 9 2 4 1 5 - 0 3 - 0 . 2 5 3 2 3 4 4 E - 3 3 " 5 3 t.aibfcJj*.--'"- - 0 . b 6 6 i ^ ) 4 6 ! - + J I — - J . ')/M9<J / = - J 3 - J . 10 0 l l 3 4 * - j v 4 6 0 . 2 4 1 5 1 1 2 — C 5 - " . 5 6 1 6 3 4 5 F + 0 1 0 . 6 3 7 7 4 6 8 5 - 0 3 - 0 . 3 1 3 3 3 7 3 5 - 0 3 4 7 C . 2 7 1 4 2 ° 0 5 - O 5 - 0 . 5 5 6 6 3 4 5 F + 0 1 - 0 . 6 2 7 5 0 5 1 5 - 0 3 - 0 . ^ 4 C 8 9 " 2 4 5 - 0 3 - 3 3 C 7 ? 0 4 3 4 3 3 7 ^ 3 — = ^ 7 5 3 T " £ 3 * 5 5 + O T 373^772'9I = = 3 3 — = 0 . 4 3 2 1 5 3 3 5 - 3 3 4 5 0 . 3 4 1 7 0 ° 0 = - 0 5 < - 0 . 5 4 6 6 3 4 5 F + 0 1 - 0 . 2 5 1 6 6 2 O 0 3 - 0 . 6 8 4 8 2 5 2 5 - 0 3 5 0 Q .38340395 - 0 * - 0 . 5 4 1 6 3 4 * ^ + 0 1 0 . 2 9 6 6 4 3 e 5 - 0 3 - O . 3 9 8 1 31 4 F - 1 3 ~5T C . 4 3 5 1 9 6 6 " — — - 1 7 5366344r-+ci ^ . 1 3 3 2 2 3 1 5 - 0 2 0 . 9 8 4 0 9 9 2 c - 3 3 5 2 0 . 4 8 2 6 7 7 5 5 - 0 5 - 0 . 5 3 1 6 3 4 4 5 + 0 1 0 . 2 4 4 2 3 1 5 5 - 0 2 0 . 2 4 2 6 4 1 4 5 - 0 2 5 3 0 . 5 4 1 5 7 3 2 c - O 5 - 0 . 5 2 6 6 3 4 * e + 0 1 Q . 1 0 7 1 1 5 5 5 - 0 1 0 . 1 4 1 3 7 5 6 E - 3 1 "5~4 0 . 6 0 7 6 5 5 2 " - 0 5 — = Trr5ZY6J*~5T7+VZ Trr= 3 8 5 ' 8 ' v F = 0 1 — o . 3 4 0 2 3 7 7 c - 3 1 5 5 0 . 6 8 1 8 0 0 5 5 - 0 5 - 0 . 5 1 6 6 3 4 4 5 + 0 1 0 . 4 7 6 5 0 3 1 5 - 0 1 C . 8 1 6 7 4 0 4 E - 0 1 5 6 0 . 7 6 4 9 9 3 S - - " * - 0 . 5 1 U 3 4 3 C + 0 1 0 . 7 4 * 7 0 2 2 5 - 0 1 0 . 1 5 6 1 4 * 3 ^ + 3 0 " 5 7 0 . P 5 8 3 3 72"- f b—•~Q .5< 6 6 3 4 3 ' + <"l j . 2 6 5 9 6 9 4 ^ + 0 0 0 . 4 2 2 1 l3')R.*!K 5 8 C . 9 6 3 0 7 0 4 5 - 0 5 - " . 5 0 1 6 3 43 5+01 0 . 1 5 4 3 c 8 9E+ 0 0 0 . 5 76 512 5"=+3 0 55 C . 1 0 8 C 5 8 3 5 - O - 0 . 4 9 6 6 3 4 3 - + 0 1 0 . I 5 O 6 8 3 35+O0 0 . 7 2 7 1 9 575 + 10 - 5 C 0 . 1 2 1 2 4 3 5 " - 0 ~ 4 — = T 7 4 ~ 5 T S 3 4 3 F + T 1 1 7 T 3 5 3 9 9 1 3 + 0 C 0 . 9 1 6 5 9 4 7 5 * 3 0 61 0 . 1 3 6 0 3 7 5 5 - 0 4 - 0 . 4 8 6 6 3 4 3 r : + C l 1 . 4 9 7562 7 5 - 0 1 0 . 9 6 5 3 4 1 0 5 + 0 3 6 2 0 . 1 5 2 6 3 6 6 r - 0 4 - " . 4 e i f 3 4 2 r + 0 1 0.2 7 5 7 9 7 2 - — 0 1 0 . 9 9 2 9 2 0 7 5 + 0 0 S3 0 « 1 7 1 V ! 6 I 2 ' - ; - 0 4 — - 0 . 4 / 6 6 3 4 2 E + 01 C . 8 0 6 6 7 3 6 : - 0 2 0 . ! 3 C C 9 8 7 b + 3 1 * = * S T A T I S T I C A L P A R A M E T 5 B S F G F D I S T R I B U T I O N OF 5 M S * * * * - 3 ) * M ? A N S T D . D E V . <?K5W*J5SS C O E F F . K L R T O S I S 1 . 1 0 0 0 8 8 3 5 - 0 4 O . 2 1 8 9 6 3 C 5 - 0 5 0 . 5 2 2 2 5 1 3 5 + 0 0 0 . 2 9 8 5 7 8 5 5 + 0 1 *=* STATISTICAL ° A R A M g T £RS FOR DISTRIBUTION OF LOG(E') * "FffN STD.DEV. SKFWNFS3 CUFF TV K QR T CSXS~ - 0 . 5 0 1 3 9 7 8 E + 0 1 0 . 6 3 0 2 0 2 9 E - O 1 0 . 6 1 2 4 1 5 2 E + 0 2 - 0 . 2 7 9 1 2 7 9 5 + 0 4 1 Ol -IG/SJ A L O G 1 0 IC11 P T B A B I L T T Y . C T J M U U 5 T T V E ~ P R O B A B I L I T Y 184 111-21 -~I 0 . 3 5 4 e i 2 3 c + r 2 0 .15499CCF + 01 0.2980221E-37 0.2980221=-0 7 2 C.3981061F+02 0 .1599959F+ 0 1 0.13410995-05 C.13705005-05 3 C . 4 4 6 6 8 ? 6 c + 0 2 0 . 16 50000 = +0 1 3.22389105-04 0.2*253595-3* OTOTTBST* Tr 2 * . T 7 ' 0 W f « P * C l 07269'2330n=-=?T3 0.2 934915 ' - 0 3~~ 5 0.56234075+02 0 .17500C0E+01 0.21641475-02 0.24576395-32 6 , C.6305557F+0? 0 . 1 7 9 9 9 5 ° = + 0 1 0.1199412 5-01 0.14351765-01 ~~7 0 . 7 0 7 9 4 4 0 ^ + 0 2 0. 1849c>5 = f: + 01 0.44730025-01 C. 59031775-31 8 0.79*32655+02 0.19000005+01 0.11516345+00 0 . 174 24515 + CO 9 C. 991 2*915 + 02 0 . 1 9 5 0 0 f 05-t-C 1 0 . 20307995+OO 0.37722495+00 TO CT999991977r+rJ2 0T?0OCr0TTJF+ 01 0T2'4534T8T+TJ7J 0V6 2 2 666 7 E + 30~~ 11 O.U22014"+03 0.20*99595+01 0 . 2030927E+0O 0 .8257495E + CC 12 0.12599225+"? 0.2O599C9F+01 0.11516675+nO 0 . 9*091615+00 T3 C.1412534&+03 C.2l5"C0CC c+Ol 0.44?3! 99P-01 C.9S56431E+30 . 14 C.15848O0'= + 03 0.22000""P+01 0.118 9*805-31 0.9975425=+33 15 C. 1778276^ + 03 0 .2250OC0P + C1 0.2164 20 65-02 0 .995737C5 + 00 T5 0TT9-95756E+O3 0T2TS 9 93^5+01 0T26<3 262 95 - 03 o; 9 9 59 7 6 3 =+00 17 C.22387145+03 0 . 2349959F+01 0.2 2 888105 - 0 * 0 . 9999991 18 0.251 1881' + C3 0 .24000C0~ + C1 0 . 134109°5-05 0 .IC00300F+01 T5 0.2818379E + C3 0.24b00C15+01 <3 . 295022 l r - 0 7 C. 1CO030O5 + 0 1 — *=* STATISTICAL PAPAMETEPS FCR DISTRIBUTION OF 01 (G/SJ * MEAN STO.DFV. SKGWN5SS COEFF.KU'TOSIS 0.10176725+03 0.19216455+C2 0.57285C9E+03 - 0.358748«=+01 * = « S T A T I S T I C A L P A R A M E T E R S FOP C I S T R T B U T T Q N OF A L O G 1 0 ( 0 4 > MEAT! STDTDTV7 5KEWN=SS C O E F F; kUP.TCSTS" 0.19999985+01 0.81290605-01 0.86751245-34 C.29993112+01 -3 ClT7J*M**-3 J — : CTJGTCn PTrCTATTTTTY CQlMTjCffTTVe PROBABILITY -1 a , 3H33*J!?A=-^S—-o.S4163i4"+Cl—-0. 69 3491 9*-v"8- -0.69 8491 55-09 2 0.4301773 c-05 -o.53663545+C1 0.2840534E-07 0.21420425-37 3 0.48266695-05 -0.5316354 =+01 0 . 3720173--07 C. 586221 55-)7 —c, C7T*T56"T'3^05—=0732 66354^ + 01 07271231 65-07 0.8 57452 85-37 — 5 C.6076*19 c-05 -0.5216353^+01 0.1357512=-07 C.99323405-37 6 0.6817855' :-r5 .5166353=+01 n . 74" 3287 = -08 0 . 1C 67 23 75-06 ~1 0.7649769^-0=—-0. 5116353F+01 3.2?1 f i575E-07 0. 1299194E-C6 8 C.85831*35-05 -0.50663535+01 0 .1705484^-07 0.I*65743=-C6 9 C.96304925-^5 - 0. 5016353F + 01 0.43626645-07 0.19060095-06 TO CTT7rBTJ55"95=-04^—-0.49663525+01 37T6T2T3 IF=07 0T2 27126 25-36 11 0.1212409--O* -0.4516352 c+01 3.61467295-07 0.2885935=-06 12 Co 13602455-C4 - r . A866352 c + 01 0.2654269-5-07 0. 31 513625-06 T3 C.152633 3 ^ - 0 4 — - C . 42163525+01 3.5960464^-07 0 . 3 74 74 C9E-&6 14 C.171257*5-0* -0.*766352F+01 0.67055235-C7 0.44179615-06 15 C.19215*05-0* -0.47163525+01 0.6705 52 35-07 C.5G885135-36 T6 CT2T3SC 06^04"—=TJ74'6~6~6~3T1'F*+0'I 0T56 9575 F T-0 7 0.6C5 7 0 8 95 - C 6""" 17 C.2415078=-"* -0.46163515+01 0.5960*645-07 0.66 5 313 55-06 18 0.27142525-C* -C.45663515+01 0.5560*645-Q7 0.7 2451825-36 T5 C. 30454*15-04 - 0 .4M6331 +01 3.14501165-^ / 0. /39y193T-3= 20 n.34170435-0* -0.446635l r+01 0.223 51745-07 0.76217115-06 21 0.38339995-04 -0.A41635C5+01 0.218860 85-07 0.''8405715-36 2? C7^3TJT8075-oz:—-0.43F63T0TT+TJ1—=OT2T4?042F=07 0.76263675-0 6 22 0.48267095-C4 - 0.4316350r + 01 0.6984919F-08 0.769621 6=-06 24 0.5415657E-C4 -0.42663 50 c+01 -0.258-5592=-07 0.733 7657^-06 185 111-22 -—75 r.fcn - , 6 4 6 n J--f 4 - o . 4 21 63 5ni=+T7.—-0. 1 ">r 69935-07" 0.72369b3 c-06 26 0.68179176-04 -0.41663496+01 -0.7828°306-08 0.715 8 6696-06 27 C.76458115 - 0 4 -0.^116349^4.01 0.190758 *5-07 0.734542 76-06 — 2 8 C «T?5 S I Z T ^ - * * — - A . - 4 0 6 6 3 4 9 P + O T = 0 T T 3 I 7 2 2 ? ^ 0 7 Q ."7 21 7 2 0 5 c - 0 6" 29 0.9630571F-04 -0.40!6349F+01 0 . 31160265-07 0.752fi8C76-06 3 0 C.10805695 - 0 3 -Q.39663* Q C+01 -0.7443^2 05-03 0.7454377 r-06 —31 0 . m 2 4 i n c - r n — - U . 3916348^+01 0. 56CHS2 bfc-0 / 0.801 b2b9f:--6 3 2 C.136C3565-03 -0.3866343F+01 -0.2116738E-07 C.78035855-36 ' 3 3 0.1526345=-n3 -0.38163486*01 0.2840 5346-07 0.80376335-36 —ZA C'^r25Kt3T=-*T3—- 0T^6"6"3THF* '01 0~7Z7TT/956"5^0T—0 .54643 246-CE " 35 0.1921568 , ;-C3 -0.37163^8^+01 0.25056925-06 0 .109705 26-C5 3 6 0.2156024^-03 - 0 . 2 6 6 6 3 4 e c + 0 1 0.163615 85-05 C.2733210 c-35 ~ T 7 O 2 M 9 0 a ° 5 - n ? -0.361634 /C+Cl 0.51613*4*!-05 0. 11894b /'r-04 3 8 C.2714274C-03 - 0.35663^76+01 3.4390866E-04 0.5230323=-34 3 9 0 . 304546 7<=-03 3516 3 4 7 F + C i 0. 1 5 4 2 7 9 1 F-03 0. 207032 3 c-33 —4"0 a.34170 74 = -03 =TT"."2"45'63"4~7F+C1 0 . 5C47 89'4"C=03 0.71187156-03 41 C.38340205-03 - 0 .2416347^+r1 0.14635776-02 0.21754495-02' 4 2 C.4301842 c-03 -C.2366346E+01 0.33072325-02 0.5982727E-02 —4-3 0.4826747^-03—-0.3316346 «•• +01 0.9310^0 26-32 0. 1455272F - U I 44 t*. 5 4 157<U * - 0 3 -C. 2 2 6 6 3 1 6=+ni 0 . 1 9 6 9 3 5 55—01 0 . 3 4 6 3627^-31 4 5 0.60765246-03 - 0.32163466+01 0.3976478E-01 0.74451025-01 — 4 5 0;'6"81797 3*--0 3 — - 0 . 31663^6~fn :01 0T71"9F42"0F^3I 0 . 14£4T52F+aTJ 47 0 . 7 6 4 e f l 5 3 c - 0 3 - 0 . 211 6345F + 0 1 0. 111 649 6E+00 0.25 9063 56 + 00 4 8 0.8583323E-O3 -0.30663455+01 0.14470836+00 0.40277215+0C —179 ( ] . M 6 J C 6 b c " - - ' , , 3 — J 0 1 6 3 ^ b h + U l J . 15 Ofc+^O 0 . 560 b n 1'.-+ 10 50 O . !08057a*=-02 -0.29663456 + 01 0.14829846+00 0.7C836956+0C 51 0.12124?9c-02 -0.29163^55+01 0.12114585+00 C83001545+0C —52 C7TT6"C3"68^-0Z—-0T2H563"44r + 01 0T8444C23T-01 0 .91445565 + 00 53 0.152635 8^-02- -0.28163446+01 0.4903127E-01 0.9634969E+0G 54 C.1712604<=-02 -0.27663^46+01 0.23^66625-01 C . 9 869 5345+30 — 5 3 O . l ^ l b M 1 : - ^ - 0.2/16344^+01 0. 521 759O<;-02 0.9961 71 ^  + 33 5 6 0.21560419-C2 -0.2666344F+01 0.29294215-02 0.99510C46+30 57 O.2419119 c-02 -0.26163^3 C+01 0.73 2510 85-03 0 . 9 9 9 3 22 55+00 ~5"B 0'.77I«TZ5T6T=^r ,2—=TTZ 56"6~3 4"3^+Tl 0TT3 56 B'SU^Ql "0 .-9 5 5 572.5 r + 3C 59 C.30454946-02 -0.25163436+01 0.19742826-04 0.599992 26+30 60 0.34171115 - 0 2 -0.24663*3 C+P1 0.205342?=- r5 0 . ° 9 9 5 9 4 3 c + 00 - 5 1 0. 38340'-1'--02—-0. 2 4 1 f c i 4 i ' . + 0 i — 0. 160014b"-'J6 0 . 9 9 9 S 9 4 4 t + oO 62 C . 4 3 0 1 8 7 6 r - C 2 - 0 .2366343F+01 0.623 275 06 - 0 7 0.99999455+CC 6 3 C.482 679 8=-02 - 0.23163426+01 0.262 835 46-07 C.99999455+30 *=* S~ATISTTCAL P A R A M E T E R S FOR DISTRIBUTION 0= C1'(G*M**-3J * TETN SID.Ut V. L^WN^SS Cnfcf-P.Ku'J I5> 0.1015536E-02 0.29620785-03 0.7881763E+30 0.40155525+Cl "•^B STATISTICAL PAPAHtrfcRS F-UK DISTR'. BUTTON Cf6 LTGTC i ' l M6AN STO.OEV. SK5WN6SS C0 6 C F . K U 9 TQSIS - ' ) . 30 11.4 24l-+?U !).12b/b /bk + E ' l —-0.9 55156^-3 1 0. 30^32334+01 186 111-23 A 4 -l.0PAn.C2 CI,CI = = > b ' 02,5« ==> C2« 01 (G/S) ALOG 1C(01) P RCBA B I L I T Y C U M I J L A T T V P P R O B A B I L I I i 1 0.25118925+C2 ~Z 0. ^ a r E 3 7 9 c + <"2 ' 3 0.31622745+02 4 0.35481235+02 0.14000005+01 -PTr43Tj^ro=+7vT~ 0.15000CO C+01 0.1549959=+01 0 .29 80 2195-07 0.22351535-0? 0.86123295-04 0.16802775-02 0. 16340025-01 0.87036015-01 0.232RO985+00 0.323TJ803~C+Tj1~ 0.23231565+0J O.87O40365- -*! U. 1684130^-01 0. 298021 85-07 "TJ722 6496 55-03" 0 . 9 e 3 5 2 2 5 5 - 0 4 0 .17686705-02 0. 1860369b-'Jl" 0. 1056446S+C3 0 .33 845445+00 "0.£6r534TE+TC 0.39425025+00 n.98139'>75+-v) "3 C.398 1061^ + 02 6 C.44668365+02 7 C.50118S7=+"2 8 0,562340 /= +1'^  9 0.63095=75 + 02 10 0.7079440= + <*2 "0. 15999<r9t- + 01 0.16 5OOC0r+Oi 0. 1 7 n 3 C O F » 0 1 'J. 1 /5<J0C3F- + U1 0.17999595+01 0. 18499«=9 C + 01 •j. l'j'juijcob+m. 0.19500CJ5+01 0 . 2 0 C T C 0 5 + O1 0. 59 8 2 Jilb+'JC 3.95591245+0C C.59959855+0C ' C.ICCOOCOF + OI" 0. 10011005+01 "TI 'J. i ,94326bl- + '.'^ ' 12 C.89124Ql c+02 13 0.55999915 + '*? T 4 O.I123014b+03 15 C.12589225+C3 0 . 16904555-02 0.66123295-04 0.2235T63==15~ 0.20999595+01 0 .298021 85-0 7 01 IG/bJ -C055F.KJRT0SIS ' ~ "0.33 27 27 93+OT -ALCGIO(OI) * S T A T I M 1L-AL P A H A M b l b'-'.b H I p U1S ! K1HU ' l ' J N Ur-M5AN STO.OEV. SK5UN=SS TJT^305534F+00 — 0.5690470E+02 0.8112 3 83F*TJT *=* STATISTICAL PA3AM5TERS ^OP DISTRIBUTION C= M5 AN STD.DEV. SK5WN5SS C0E5F.KU 1? TOSIS 0.17499985 +01 ".61710985-01 0.11566125 - 1 3 0.29964005+01 *=* =*=* *=* = * = *=•* J CI (G*M** -3) L O G I C l ) PP0BA3ILITY CUMULATIV5 4 PROBABILITY 1 0. 85933355- 04 -0 . 40663425+01 0.1758631E- 11 0.42511405-05 2 0. 563C6 5°' :-0* -0.4016343 c+01 0.13578195- 09 0.4351279=-05 3 0. l-jaCb8 2" ' - 03 -'i„ :^66i4i*-+Ul ' U. b /UiV6 lr.- i6 U . t J 5 b S « i : - ' J 5 4 0. 12124355- 03 -0 . 3916343E+01 0.12430245- 06 0.44812345-05 5 w * 1360375=- C3 — n 38663435+01 0.15331895- 05 0. 60144735-05 n mirTZ^—"V/ 6 u. 152 63"5'65"-U3 3816i42'- + ' . i i 0 . 1 1 S 7 8b-fj'+ U.I !cl4Z3C-U4 7 0. 17126095- 0 3 - 0 . 37663435+01 0.60233045- 04 0.7784725=-04 8 0. 19215785- 03 27163435+01 0.24211155-03 C. 31595375-03 ( J . ^IbfcU4fcb-TT3 J6fc6443b*Ul Di u. i L - vtjg /*:-u<! 10 o. 24191245- 03 -0. 3616343=+01 0.2422841E- 02 0.35627235-02 11 0. 27143"55- C3 — n • . 3566342F+C1 0.63071515-•02 C.9869977E-02 «—*i r"c T"ii 'j ^  ^ — n T 12 — 0 . 30454755-••03 - 0 . 3bl6343b+01 0.14645435- •Ui U . 4 5 I - 3 15 01 13 0. 3*171"2»=-•03 - 0 . 34663435+-01 0.31433265-•01 0. 55558572-01 14 0. 3834*525- "3 -0 . 3416343=+01 0.6435275=- •01 C.1203513=+00 187 15 16 17 "T8~ 19 20 T T C . 4 3 0 1 8 79i=-<;3 0 . 4 8 2 6 7 8 9 - - ' , 3 0 . 5 4 1 5 7 ? 6 5 - 0 3 ~0T5TJT55'6"3-«r=TJT 0.6818119=-*? 0.7649Q44F-03 <! .Pbt Jij yi«-c-5 - 0 . 3 3 6 6 3 4 3 c + 0 1 - 0 . 3 3 1 6 3 4 2 5 + 0 1 - 0 . 3 2 6 6 3 4 3 E + 0 1 ^ T 3 2 r 6 ~ 3 4 3 F + 0 -o. 3 1 6 6 3 4 3 5 + 0 1 • 0 . 3 1 1 6 3 4 3 = + 0 1 2 0 6 6 3 4 2 r + C' l J . 11329426+n0 C. 16009495+00 0.17156316+00 f7ir79'2"24~E+P~9~ 0. 128204-/5 + 00 0.8308452=-01 '(!.4121940 S-OT O . 2 2 5 6 * 5 5=+'0C 0 . 3 9 5 7 4 " » 4 6 + 0 0 _ 0 . 5 6 7 3 0 3 6 6 + 0 0 0 . 7 2 5 226 C = +00" 0 . 8 5 3 4 3 0 7 5 + O C 0 . 9 3 6 5 1 5 3 6 + 0 0 0 . ' ) / / M 4 6 = +1^ 22 0.963O695=-O3 23 0.10 805S2=-0? "74 07T7T24T4F=TT^ 25 0 . 1 3 6 0 3 7 4 E - C 2 26 Q.1526365=-02 "27" M.'l / i y^n /r-»2 28 0.1921577P -0? 29 0.21560455-C2 • 0 . 3 0 1 6 3 4 3 6 + 0 1 • 0 . 2 9 6 6 3 4 3 6 + 0 1 Z 9 T 6 3 4 T e " + 0 T - 0 . 2 8 6 6 3 4 3 5 + 0 1 • 0 . 2 8 1 6 3 4 2 = + " ! 2 f"66343:- + 01 - 0 . 2 7 1 6 3 4 3 6 + 0 1 - 0 . 2 6 6 6 3 4 3 5 + 0 1 0 . 1 6 2 0 2 3 8 6 - 0 1 0 . 4 8 9 6 7 6 7 5 - 0 2 0".l-T2T563r=S"2~ 0 . 1 3 5 1 1 5 5 6 - 0 3 0 . 1 0 609 2 0 6 - 0 4 0 . 4 / 1 5 H O = - - » 6 O . 1 1 3 2 0 O 5 E - 0 7 0 . 1 3 8 5 0 9 1 E - C 9 0 . 9 9 3 9 3 7 0 6 + 0 0 0 . 9 9 8 8 3 3 7 ^ + 0 0 " 0 . 9 9 9 3 5 7 1 c+'0'0"" 0 . 9 9 9 9 9 2 2 6 + 00 O . l O O O O C Z C + O l 0 . 1 0 0 0 0 C 2 = + 0 1 C . 1 C 0 0 3 C 2 S + 0 1 0 . 1 C 0 C 0 0 2 5 + 0 1 STATISTICAL PARAMETERS FOR DISTRIBUTION OF C l ( G * M * * - 3 ) Mb A N — !  0.56 6 8 5 9 ? E - 0 3 i I U . D F V . i « c » « N = S S I ' L ' I r r - r - . K U R T C ' S I S 0 . 1 4 8 9 9 1 9 5 - 0 3 0 .6605535E+00 0.36437615+01 S ' A r i S l l L A L P A g A P f c t f c V S h j n DISTRIBTJ7TT>N~DF' TCGTCTT H5AN J^tlJ33f:+01 ST0.D5V. SK5WN5SS C066F.KURTOSIS ' . ' . l i 4 o a i f t + C C -0.1282482E+O0 -0.30330 35 =+01 E i { S * M * * - 3 ) LCG(F') PRC8ABILTTY C U MUL ATIV6 " ; ' ! ! ! PR0BA6ILITY 1 0 . 1 3 6 0 3 4 1 = -07 -0 . 7 8 6 6 3 5 4 6+01 - 0 . 2 O 5 8 2 3 5 E-03 - 0 . 2 0 5 8 2 3 5 6 -•03 z " 0 . I 5 Z £ 3 4 - F = -07 " " - 0 . / 8 1 O T ' 4 F+0T~ •7T"r732T==7J3 - 0 . ; 6 0 9 7 2 5 2 5 - r C 7 ~ 3 0 . 1 7 1 2 5 8 8 5 - C 7 -1 . 7 7 6 6 3 5 4 6+C1 2057521=-03 - o . 2 0 5 8 1 3 1 5 -•03 4 0 . 1 9 2 1 5 5 2 5 - C 7 - 0 . 7 7 1 6 3 5 3 6 + 0 1 0 . 2 0 5 8 3 3 6 6-03 0. 7 C 5 1 8 5 85-•C7 3 C.2136913 C - C .' -0.76 i43 53f+01 ' - J . 20b . ' 5 7 5 E - 0 3 — n • 2057274?-"33 6 0. 2419082 '= ~rj -1 . 7616353=+01 0 . 2 0 5 4 4 3 1 6-03 - 0 . 2 8 4 2 5 7 1 ? - •06 7 C.2 7 1 4 2 5 1 6 -07 -0 . 7 5 6 6 3 5 3 5+01 - o . 20499645-03 - 0 . 2 C 5 1 7 C 7 5 -•03 8 —0T3"0"4 543A= - 0 7 -C.7"5I 63 53 = + C1~ 0 T 2 0 - 4 3 7 4 5 T-03" ""-""779 5 7 2 7 8 = -"36"" 9 0 . 3 4 1 7 0 8 0 r - 0 7 - ^ . 7 4 6 6 3 5 2 5 + 01 -0 . 2 0 3 9 7 7 5 6-03 - 0 . 2 0 4 7 7 3 2 E - 03 1 0 0 . 3 8 3 4 0 2 0 5 - 0 7 - 0 . 7 4 1 6 3 5 2 = + C 1 0 . 2 0 3 8 1 1 9 5-03 - C . 9 6 1 2 5 7 7 = - . j f.-11 U.4 3'.7 1934= - r i -0. / 3 6 6 3 5 2 F + 0 1 -0. 2J3b /4b = -03 -0. 2 0 4 5 3 5 9 * : - 33 12 0 . 4 8 2 6 7 2 9 5 - 0 7 - 0 . 7 3 1 6 3 5 2 6 + 0 1 0 . 2 0 3 2 7 6 2 6-03 - 0 . 1 1 5 9 7 C 0 5 - 05 13 0 . 5 4 1 5 6 6 9 5 - 0 7 -0 . 7 2 6 6 3 5 2=*01 - 0 . 2 0 3 1 6 5 1 6-03 - 0 . 2042243F- 03 14 0 . 6 0 / 6 4 l\\ - C 7 - r . /216352r + 01 0 . • 2 0 3 X 7 4 ^ c-T3 - 0 7 T T 5 0 4 I -05~ 15 0 . 6 8 1 7 9 0 2 5 - 0 7 -0 . 7 1 6 6 3 5 1 6+01 - 0 . 2 0 2 9 2 9 2 6-03 - 0 . 2 0 4 0 7 9 6 5 - 33 16 0 . 7 6 4 5 9 1 3 = - 0 7 - 0 . 7 1 1 6 3 5 1=+01 0 . 202487=5-03 -0. 1 5 9 1 7 1 3 = - 3 5 l / 18 0. B33aj-;S'T' -(,' i - ! ) . / 0 f c 6 i i l ? - + C l - 0 . '2JId6i-55-03 - c . 2 C i 4 b 3 6 t - C3 0 . Q 6 3 0 6 3 9 5 - 0 7 - 0 . 7 0 1 6 3 5 1 5 + 0 1 0 . 2 0 1 4 9 3 4 5-03 - 0 . 1 9 6 2 1 8 0 5 - 35 15 0 . 1 0 8 C 5 7 3 5 -06 - C . 6 9 6 6 3 5 1 F + C 1 - 0 . 2 0 1 3 3 196 - 0 3 - 0 . 2^-33440= - 0 2 za fJ. l<i 1241 4^ =TJ6 - 0'. 6 9 T 5 3 T 0 " F + 0T~~ 0 . 2 0 T 5 9 7 8 F=u3~ — C " . T 7 4 6 1 4 3 5 - 35 21 0 . 1 3 6 C 3 5 2 = -C6 -i. 6 8 6 6 3 5 0 6+01 - 0 . 2 0 1 7 5 3 5E-03 - 0 . 2 0 3 5 0 5 1 5 - r 1 22 0.1 5 2 6 3 4 2 5 ' - 06 - 0 . 6 8 1 6 3 5 0 6 + 01 0. 2 0 1 9 7 6 4 6-03 - o . 1 5 2 8 6 7 9 = - 0 5 £ i U. 1 11 2534*- - C 6 -U.67663b0r+0i - 0 . ki)l^Zb65-03 - 0 . 2 G 2 4 5 4 2 E - 03 24 0 . 1 9 2 1 5 5 1 5 ' -C6 - 0 . 6 7 1 6 3 5 0 6 + 0 1 0 . 2 0 1 7 4 6 8 5-03 - 0 . 1 7 0 7 4 9 3 E - 05 25 0 . 2 1 5 6 r t i 6 = --G6 -0. 6 6 6 6 3 4 9 6 + 01 - o . 2 0 1 2 6 2 5 5-03 - 0 . 2 C 2 5 7 C 0 5 - 03 ""75 0 . 2 4 l 9 r T 9 0 F - - ' O F - 70"015"6&5C^03 - o v 2 2 0 2 5 3 9 5 - "05"' 27 C.2 7 1 4 2 6 8 5 - - 0 6 - 0 . 6 5 6 6 3 4 9^+01 - 0 . 2 0 0 1 4 4 9 5-03 - 0 . 2C 2 4 4 8 5 5 -02 28 0 . 3 0 4 5 4 5 ° = - -06 -0 . 6 5 1 6 3 4 9 6+01 0 . 2 0 014495-03 - o . 2 3 0 3 5 3 9 6 - 05 188 29 0.3417061=-"6 -0.6466349=+01 -0.20058455-03 -0.20288815-0 2 30 0.39340075-*6 -0.6416348=+C1 0.2012 62 56-03 -0.162553 66-05 31 0.43019?8=-G6 - 0 . 6 3663 4 86+01 -0.20 211195-03 -0.20272745-02 "72 U7A~ZTtrn.-**r=T:e—- 075716"348F+-0 1 0T273 23047-07—-0.507 01735 - 0 6" 33 0.541 5687=-06 -0.62663486 + C1 - 0 . 20437646-03 - 0.20488345-03 34 0.6O765O3C-O6 -0.6216348=+01 0 .20547075-03 0.5868496=-06 -315 C.6B1 / o y - f f c — - n . fcl66i4(J? + c!l -<) .2062424!.- J J -0 . ZOSibbt 1— J J 3 36 0.7649867 — C6 -0.61163476 + 01 0.20675006-03 0. 10944796-05 37 C . 8 5 8 3 3 O 2 c - 0 6 -0.6066347F+01 -0.20692406-03 -O.2O58255P-02 — 3 7 079-67X1571=^6—-OT£0T5347=+0I 0T2069736E=03 ~ C . T144072^-03" 39 0.10805745-C5 -0.59663475+01 -0.20716176-03 - 0.20601775-03 40 0. 121242 c= g- r5 -0.5916347c <-Ql 0.20746195-03 0. 14441905-35 41 — y . 1360364'— *b—=B66J46r+«M—-0.2075~60=-03 -0.2C6461 d - - 'J 3 " 42 C.15263546-05 -0.58163465+01 0.20826725-03 0. 1 80521 15-05 43 0. 1 7 1 2 5 « 3 c - o 5 -0.5 766 346 5+01 -3.20948795-03 -0.20668265-33 -4"4 0, 192156 7 r-0b =TJT5Tr6346T+01 377036 35?5=03— 0.200324 65-33" 45 0.2156034=-05 -0.56663466+C1 -0.20899316-03 -0.20693585-33 &6 0.241511 2 r-O5 - C . 56163*5=+P1 0.20582926-03 o.2839 31 l=-05 — 4 7 0«<!/14^m.l — - C.bb66 34b!:+Ul—-3 . 21106 /1 h-Q'i - 0 .2032 2 / yb-UJ 48 0.30454835-C5 -o. 5516345 5+01 0 .21262276-03 0.4394953=-G5 49 0 . 3 4 1 7 0 0 0 5 - 0 5 - 0 . 54663*5 r+01-0.15543706-03 -0.1510422 = -33 50 0.3B343J9b-O5—-0. i-41 5745E+7JI 0 . 48 1 26776=03 0.32022035=33" 51 C.4301866=-05 -0.53663445*0! 0.7933934=-03 0.112 37195-02 52 0.48267755-O5 -0. 5316344<"+01 0. 35335095-02 0. 46 57J2V--02 — 5 7 0 . b 4 1 b r 3 2 t—0b -0. b 2663446+01' 0.531 7b64«—02 0.136 / 4 /"»'• -31 54 0.6076552--05 -0.5216344 C+C1 0.22252405-01 0.35 9 27196-31 55 0.6818005^-05 -0.51663445+01 0.44602176-01 0.9052933 5-01 — 5 6 07764^0-3 05 -^07371X347?""* 01 OV79T2208F-OI 0 . 158631 4= +07~ 57 0.8583372 c-"5 -0.50663436 + 01 0.2613747=+00 0 .420 5 261 6 + 00 58 0.963C7O46-C5 -0.501634 36+0! 3.158739C5+00 0.57926515+00 — 5 3 C . 1 0 a C b ° - 3 b-('4—-0.4966343P+C1— 0 . 1462S05fc+o<) C .725545 5 = + 33""' 60 0.1212435=^ r4 -0.49163436+01 0.19370955+0O 0.9192551 T + 00 61 C.1360375=-04 -0.4866343 5+0! 3.44619095-01 C.96287315+30 — 5 2 C.l52 6 3 3 6 u - 0 4 — - 0.4 816347=WI 0731477283-01 0.95535045+00" 63 C.17126126-04 -0.47663426+01 0.44400435-02 0.99979C45+00 S ' A ! 1 b T 1 L AL "A^AMH F-.Wb> l-JK U l S i V l B L H i U N u> £ ' I b *M * *- i 1 MEAN STD.OEV. SKEWMFSS C0666.KURTdSIS 0.9577196P-05 077X755293=03 0747 5°474=+C 0 0.70 2 2 5243 + 01 *=* STATISTICAL "APAMETEPS FOR 0 1 S T 9 T 9 U T T ^ 06 LCO(E') HEAN STO.OEV. SKEWNSSS CC66F.KURT03 IS -0.50100236+01 0.9135801E-C1 0.29220205+01 -0.97770315+02 J 02 (G/S) AL0G10(C2) PPCBAETL TTY CUMULATIVE PRQBAEILITY 1 C.25118926+02 0.14030006+01 0.29302186-07 0.29802135-07 2 0.?9183?9=+r2 0 . 14500CO= + 0 ] 0.22351635-05 0.226496 5^-05 3 U.ilbll tUh-Hi; O.lbOUHHb+Ul T735773757=TJZf 0733337733^34-4 G.35481235+02 0.154996 Q5+0I 0.16802776-02 0.17686735-02 5 C.39810616+02 0.15999695+01 0.16 8400 26-01 C.1860969=-01 "6 0.44668 26FTP3 TTT6300C05+01 ~0737'C76"0T3-01 0.105 c446" + C0 7 O.5011867=+02 0.17C00C0E+O1 0.23280935+00 0.33845445+30 8 C.56234075*0? 0.17500C05+01 0.32303036+00 0.6615347 c+00 189 •• 9 • 10 11 ~T.biC<i*5 fr+CZ H.707544OF+02 0.79432656+02 0.1 /-qqoceiK+oi <">. 1 849959-- + 01 1.19COOC05+0! "> .2'32BI'i6f:+00 0.87040365-01 • 0. 16841305-01 O.89425O3S+00 0.98139C75+00 0.998222C5+0C T 2 13 14 T T B 9X2497^+*" 2 0.9°999<U6+P2 O.1122014=+03 "' 0.19 510C3F+0I 0. 20C10C0F+01 0.2049 Q59F+01 0.16 80455=02 0.£6128295-04 0.2235163F-Q5 —0~.999912 45+7C" 0.9999985E+0C 0. 10C00C05 + 01 IS C. 1259922';+"3 0.29802185-0 t O.lQCOOUJfe+Ql • *=* " STATISTICAL PAPAP^ETEP.S FOP DISTRIBUTION OF 02 (G/S) MEAN 0.56804706+02 STD.OFV. 0.8112398E+01 SKEhN5SS 0.4305534F+00 CQ5FF.KUPT0SIS 0.33272793+01 STATISTICAL PARAMETERS FOR DISTRIBUTION CF ALCG10(02) M4AN 0.17499985+01 a T D.DEV . fi.6171098E-01 SK5WN555 0.11566125-03 T O F F F . K L ' P TOSIS. 0.29564 C05.+01 =*=*=*=*=*=»=»=*= * = • = * = » = * J C2' IU*M**-J J L U G ! C 2 ' J " PRL'BABIL I TY CUMULA r i v:-PROBABILITY 1 u . inzbbi'— -D . b /663bbr: + CJ U.2 i'J4 / y v i - - 0 6 U. <UU4 iti^- •J5' 2 C.1921529=- 05 -0.571635*5+01 -0.1969747F -C6 0.1350413=- 07 3 0.21555935- 05 -0.56663545+01 0.2486059= -06 0.26211415- 06 4 C.24"T9064'£-Ub -0.'5616354F+C1"~ -0.1956652F -06 ' C.56 44353T-TJ7" 5 C.27142355- 0= -0.5566354F+01 0.2384H475 -06 0 .3048536=-06 6 0.30454225- 05 ' -0.5516354-^ + 01 -3.13053365 -06 0.1243201E- 06 / " 0.3417021=-Ob " -C .b466353. l-+01 .3. i i ifiSlv-. -06 U. i 6 2 U l ' 3 4 r - Lb 8 0.3833«655- 05 -0.54163535+01 -0.18760776 -36 0.1744077=- 06 9 C.43017815- 05 -0.5366353F+01 0.22458335 -06 0.39899105- 06 "10 . C.4B266785- Cb -0.b316353'F+0I -0 .19296935 -06 0.2060217^-067 11 0. 5415623=-05 - C . 5 2 6 6 3 5 3 5 + 01 0.22124735 -06 0.42726905- 06 12 0.60764365- "5 -0.5216352F+01 - 0 . 15768465 -06 0.26953455- 06 13 0.681 VH fb-- 'Jb - n . b 1663T2'-.-+y 1 J . Z 3 Z 36 -V.b 'J . SU 1 5 4 9 4 r - 06 14 C.7645784=- 05 -0.5116352E+C1 -0.1967302= -06 0.31521935- 06 15 0.85832005- 05 -0.50663525+01 0.25331 o 7= -06 0.56853535- 06 16 C.96305115- n>s -0. 5016352=+rT~ -0 . 1 J 8 87 3 95 -0"6" ~0-.38972515-17 C.10805625- 04 -0.4966352F+01 3.2533197E -06 0.64304485- 06 18 0.1212411=- 04 -0.4916351F+C1 -0.19371515 -06 0.44922575- C6 "19 '0.I36" '34E'=- C4 4 8 6 6 3 iir-l+ni •' . 2 6 8 2 2 0 9 c — 0 6 C.7 1 755C6--06 20 0.15263365- 04 -C.4816351E+01 -0.15646225 -06 0.5610884E- 06 21 C.1712577=- 0* -0.4766351P+C1 0 .26077335-06 0.8218587=- 06 i i " u . i y 2 i b 4 « 3 c -04 -r'.4 /I6351F + 01 - 0 . I 71363*5-06 •0.65045545-•0"6~ 23 0.21560105- 04 -0.46663505+01 0.250O601E -06 0.90055555- C6 24 C.2415094=- 0* -0.4616350F+01 -0.17648565 -06 0.7240699=- 06 25 "'C . ^ 7 1 4 2 b 75-U4 - i J . 4 b 6 6 3 b O > + ' j i 0.2o!96J65 -06 C.9860J32E— 06 26 0.3045447E- 0* -0.45163505+01 -0.17177085 -06 0.81426245- 06 27 C . 34170=2=-04 -P.*466350=+01 0 . 2 5 9 5 1 7 r 5 -06 0. 10737795-0 5 28 0.3833*396='= 04 -0.*416349F+TJT~ -=OT20TT172E : =05" 0i866C67 26-06 25 0.4301815=- 04 -0.4366349F+01 0.2560 2905 -06 0.11223955- 05 30 0.492671 7=-04 -0.4316349P+01 -0.2185578E -06 0.9025376=- 06 i i U.b41b6 735- (J4 - 0 . 4 ^ 6 6 3 S V t + 'JJ. 'J. 2 0 8 4 4 2 bfc -U6 U. I l l l'-jSdfi- Gb 32 C.6076*856- C4 -0.42163495+01 -0.21891785 -36 0.89 20623=-06 33 3 4 -C 6 8 1 7 9 3 0 5 -• 04 -0.41663485+01 0.2053566E -06 0.10584196- 05 C.T6498465- (14 '-0.41I6343F+01 -0.20 67536E -06' " 0.8916650=- •CF"" 35 C.85832705- c * -0.40663485+01 0.222766 2 5-^6 0.11144315- 05 36 0.9630599=- 04 -0 . 40163435+01 -0.23098306 -06 0.88344816- 06 190 111-27 3 1 0.10805 715- C3 29663^86+Pi 0 .231809 /fr-06 0. 111525a t:-05 38 0.1212421=- 03 -0. 3916348^+01 -0.3340465E-07 0.1081853=-05 39 C.13603596- 03 - 0 . 38663475+01 0.17613326-05 0.28431855-05 40 C.152634 _E-- 03 -0. 3'ffr6-?57F+n-l 0TTT4TT92f=-04 0."14 261136-0 4~ 41 C.17125Q3P- 03 -0. 37663476+01 0.60464926-04 0.74726026-34 , 42 0.1921561=- "3 -rt • - • 3716347=+01 0 .2419392E-03 0.21666506-03 4 - n J -'.I. i66634 /i:+01 0.52^2111E-D3 i l . l l 368 76 E-')2 ; 44 0.2419103=- 03 -0. 36163466+01 0.24226326-02 0.25555C86-02 : 45 G.. 2714279=- 03 - o . 35663*66+01 0.63073675-02 0.96668755-02 46 U. J!)4b4 /6'— Vi -P. 3516346^+M 0 .1464516^-01 0T"24512n3H-"01~ 47 0.34170816- C3 - 0 . 3466346E+01 0.2148347E-01 0.55555516-C1 48 0.3834027=- 03 -0. 34163466+01 0.6435227E-01 0.12034775+C3 4V 0 . 4 i l l i a b 1=- r 3 1 - t . ii6634br-+01 0. l l i Z ' H O r + C O 0.2ib641 /t- + 30 50 C.48267615- 03 -0. 33163*55+01 0.1600938E+00 0.39573555+CC . 51 0.5415715=- 03 -0. 3266345F+01 0.17156235+00 0.56729785+00 3<i C . f c T C6b3b !— UJ -' J . 32I634b» +01 U. 15 /92126+00 0. /2521905+00" 53 0.6817987=- 03 -0. 21663455+01 0.1282C41E+00 0.85342315+30 54 0.764=907=- 03 -0. 3116344=+P1 O.83O83925-01 0.936537 15+00 is 5 (J. Sb 03348'=-Ci - o . 30b6344F- + C l y.4121939E-01 U.977726 45+00 56 0.96306765- 03 - 0 . 30163*4=+01 0.1620218E-01 0.99252865+0C 57 0. 1 0 8 0 5 9 O C -^2 -0. 29663445+C1 0.48969465-02 0.9988255 r+00 bti '0.12124315- C'2 - 0 . 29163*46+01 0.1023 244F=OT~ "~0T995 a 4 87 6+00" 59 0.1360370=- C2 -0 . 28663436+01 0.13531085-03 0.999994C5+00 60 0.1526362=- "2 - o . 2816343= + r i n.10448156-04 0.99959455+00 61 U. l /1263/E-•M - n . 2/66i4iF+CI ' '0.6/8/0/95-06 "•U.99999bl5 + C0" 62 C.19215775- C2 -0. 2716343=+01 -0 .21459075-06 0 .99999455 + 30 63 C.21560455- 02 - o . 26663*3=+01 0.2276777E-C6 0.99999516+0O *=* STATISTICAL< PARAMETERS FOP DISTRIBUTION OF C2« ( G * M * * - 3 ) MEAN COEFF.KURTOSIS 0.56685415 -03 0.1489937E-03 0.6607035G+00. 0.36457735+01 *=* S I A I I S I I C A L PARAMh' E1?? FUF DISTP ItfUTION OF LGG'(C2'T H5AM STD.OFV. $K = V«N5SS C0E6F. KUR TCS IS - U . i 2 6 i i Z 4 b + ' J l U.1140869E+00—- 0«1ii13£ H f c + 00 U. i 166 b345+01 191 111-28 A5 l . n s r p . 1 2 T 4 5 "TT 01 — G L C — > E o i , c i—= = P r a o - - > =' — — COMPARE C 1 ( T ) , C 1 M T ) P(5XP5C T5D) P(CALCULATED) CP ( E X » E C T E D ) CP(CALCULATED) 0.0 ' 0.59010166-04 "C7TT6-4-5-6T2c-P'J 0. l n i 29485-02 0.33!1275F-C2 '-1.^4 2*: "->!'--12 0.2202694=-01 0.44826 C'5=-C1 0 . 77VOOyT^~=T'T~ 0.26205395+0C 0. 15852 51 5+^'•, 0.2126227P-03 -0.1554370E-03 nT7*30"6Z7P=r3T-0.7933584 C-C3 0.35335C°E-02 ''!'. /564=-02 0.22252406-01 0.*4602I7=-01 " n . 78122r35 - (TT-0.2618747E+OC 0.1587390 C+OC lio Ltit>ZQ'C5r*OQ 0. 1 9 3 7 0 c 5 c + P r i 0.4461809=^01 0.3147728^-01 1.23841865-06 0.5924857E-04 "TrT37T8T0"6l--03 0.13367566-02 0.46480345-0? 1.1389~7 25-01 0.35917075-01 O.8074397E-01 1 .1 bB6"4'4^F*"0Tr 0.42072826+CO 0.57«26335+0 r> o. rib("sbbt + cn 0.9192569^+00 1.96408365+00 0. 99535T*5 + O0r 0.21399365 0. 58.55662 = "0-53991915 0.1333317F 0.48668245 U.1J88439" -03 04 -IT -02 -02 7 8 — 9 10 11 T Z 0. 1464SV2 t- + '"' 13 0.1935"145 + r>o 14 0 . 4 4 8 2 6 7 4 5 - 0 1 T 5 r j . 3 1 2 6 8 9 5 = - 0 T 0.36126796 0.80 7 3 8905 ~C"." 15 88611^ C.4207357= 0.57947465 U. lib lbz>l-0.91946465 ".96409275 T J . 9 95550!: 6 0.10003C05 -01 -01 +1u • CO + 00 TOTJ + 00 +00 +-0TJ +01 16 0 . 4 6 4 7 6 1 6 5 - 0 2 0 . 4 4 4 0 0 4 3 E - 0 2 0 . 1 0 C 0 0 O O E + 0 1 ~ » * r r A T T S T T O L -•PABAM'STcpj • (IN TERMS nc j , 'HE CLASS *) MPAN STD.DEV. t x P c C T 5 U 11.0999870 CALCULATED 11.0983C47 1.9210615 ? EP90R 0.015169! -C.12048* ** DIFFPF=NTIAL P CCRA8ILI7TFS * * X% LEAST S0UAR5 LIN5AP REGRESSION 3! 5? v(CALCULA 1fcu) = S l O F 5 * P(EXPFCTEC) SL0P6 = 0.9999 1MT6° C 5 P T = 0.0000 + IN TE"nLb , , J 1 tUKKfclA 1 I ON CUEf-P ICI fcM 1 = 0. %% L H i - i u A U b G ' juuNbiJ* nt- h i ! TEST T0T4L FREQUENCY = 8760 n LL "^bG^'PEC'LLAbi'IbS _ CHI-S0AR60 = ii 0.0812112 ** CUMULA^IV= PROBABILITIES ** Xt LEAST SCUAR? LINFAP REGRESSION 55? CPrCALCULATFD)=SLCPC *CP(EXPCCTED) • INTERCEPT sicpc = 1.0000 TVTzPTnn = -O.nOCl CORRELATION COEFFICIENT = 0.5995956 %X KOLMOGOROV-SMIRNOV GCODNFSS OF FIT T?$T %l # LI- LL AySfr-S (N> : =-= T 6 ~ MAX.DEVIATION T M CUMULATIVE PPC3AEILITY 0.0C02157 DTCUPPD AT CLASS * 7 ;  — CRITICAL K-S VALUES F H P, N>40 4 O C 5 ZTJ"? T7T. ' 0.26749S9 10? 1.2n49S«:8 5* 0.3399 c5<5 n 0.4074958 193 111-30 A6 1.DSTP.L1 Ql — G L C — > E 0 1 , C l — P A D — > E« O l t S l 1 — PAD—> C l * COMPARE C l t C l * I P(EXPECTFO) P(CALCULATED) CP(EXPECTED) CP(CALCULATED) 1 0.1758632E-•11 C. 5608825E-07 C. 149C117E-C5 0.6374180E-05 2 0.87022996- 10 -0. 2116733E-C7 C. 149G204E-05 C. 6353012E-05 3 0.1735622E-•08 0. 2840534E-07 0. 14919405-C5 C. 6381418E-05 4 0.23399956- 07 0. 37718565^07 0. 151534OE-05 0. 64191365-05 5 0.22683616- •06 0. 2505692=-06 0. 1742176f:-C5 C. 66697C6E-C5 6 0.1635584=- 05 0. 16361585-05 c . 33731605-05 0. 83058646-05 7 0.91123406- 05 0. 9161364 c-05 c . 12490506-C4 0. 1746722E-04 8 0.40898745- •04 0. 4090866E-04 0. 53389235-C4 0. 5837588E-04 9 0. 1542154E-•03 C. 1542751E-03 c . 20760466-03 0.21265506-03 10 0 .5047766=-03 0. 50478545-03 c . 71238125-C3 0. 71744435-03 11 0.14635295- 02 0. 14635775-02 c. 21759105-02 C. 2181C21E-02 12 0.38072705- 02 0. 38072826-02 0. 5983178S-C2 c- 5988200E-02 13 0.9009950=- 02 0. 90100025-02 0. 14953125-01 C. 1499830E-01 14 0.19653625- 01 0. 19693555-01 c . 34686756-01 0.3469135E-01 15 0.39764356- 01 0. 39764786-01 c . 74451575-C1 C. 74456635-01 16 0.71964625- 01 0. 71964206-01 0. 14641626+OC 0. 14642086+00 17 0.11164525+OC 0. U-16486E + 00 0. 25306546+C0 0. 25806956+00 18 0.14470515+00 0. 14470835+00 c . 40277456+CC C. 40277775+00 19 0.15775596 + 00 0. 157799C5+00 0. 5605744E+0C 0. 56057676+00 20 0.14829525+0C c. 14829845+00 0. 706S7365+CC 0. 70887516+00 21 0 .12114666+00 0. 12114586+00 c . 830C202E+CC C. 83002056+CO 22 0.8444C65E- 01 0. 8444C235-01 C. 91446095+CC 0. 91446115+00 23 0.49031585- 01 0. 4903127E-01 c . S6345256+CC 0. 96345246+00 24 0.23466645- 01 0. 2346662E-01 0.93695905+OC 0. 98695905+00 25 0.9217646E- 02 0. 52175505-02 0. 996 17675+00 0. 99617656+00 26 0.29293766- 02 0. 29294215-02 C. 5551060E+CC 0. 9951 C55E+.CO 27 0.7225222E- C3 . 0. 73251085-03 C. 5598385E+CC 0. 99932845+00 28 0.13565496- 03 0. 1356680=-03 0. '5597316+CC c . 99997815+00 29 0.19767796- 04 0. 1974282E-04 0. 55559795+CC C. 9999=786+00 3C 0.2C32997E- 05 0. 20534235-05 0. 55599995+00 0. 99995986+00 31 0.14964565- 06 0. 16001455-06 c . 1CGCCCC5+C1 C. 9959599E+00 32 0.71647865- 08 0. 62327505-07 C. 1C00000E+C1 C. 1000CC05+01 33 0.1385092E- 05 0. 2628354E-07 0. 1CG0000E+01 0. 10000005+01 ** STATISTICAL PARAMETERS ** U N TERMS OF It THE CLASS H) MEAN STC.OEV. EXPECTED CALCULATED ? ERROR 19.0999146 19.0999298 -0.00008? 2.5149307 2.514945C -C.CCC57? ** DIFFERENTIAL PROBABILITIES ** 194 111-31 X X LEAST SQUARE LINEAR REGRESSION XX P(CALCULATED ) = SLOPE * P(EXPECTED) • INTERCEPT SLOPE = 1.0000 INTERCEPT = O.nooo CORRELATICN COEFFICIENT - 0.9999992 XX CHI-SQARE GOCDNESS OF FIT TEST J? """ TOTAL FREQUENCY 8760 # OF REGROUPED CLASSES = CHI-SQARED 18 O.CCCCOCO * * CUMULATIVF PROBABILITIES ** XX LEAST SQUARE LINEAR REGRESSION %% CPICALCULATEDJ=SLOPE *CP<EXPECTED) • INTERCEPT SLOPE = 1.0000 INTERCEPT = 0.0000 CORRELATICN COEFFICIENT = l.OOCCCOO XX KOLPOGCROV-SMIRNOV GCOCNESS OF FIT TEST X? # OF CLASSES (N1 MAX.DEVIATION IN CUMULATIVE 33 PROBABILITY 0.CCC0052 OCCURED AT CLASS * 13 — CRITICAL K-S VALUES FCR N>40 ARE: 20X . . . . 0.1862630 10X 0.2123746 5? 0.2367455 1? 0.2827464 195 A7 Ql —r,i_c—> *£ ! 0T7TTI ~VTU--^~Tr — •. Q 2 t E l ' — P a n — > C2' CCHPAPe C2.C2' 1 PTFxTE7r^ F D T P ( C ALCUL AT6Q J — f e X P S C T E D i : D > (CALCULATED) 1 -0.17586315 -11 0.22276626-06 0.13113035-05 0.60640216-05 3 4 u . i j w a i v ' - : 0.57039675 0.1243^2*^ — rq -08 -06 -r. Z3G98305^-'06 0.23180576-06 -0.33*0*655-07 0.13114436-05 ' 0.13171486-05 0.14414505-05 0.5 833 '03B''^T5 — 0.60648475-35 C.60 214435- "35 3 6 7 <J. i 53 31 3 9 -0 . 1 1599786 C.60233r>*= -•."5 -04 -04 l / 6 l 3 32"-C 5 0.1141752F-04 0.60464526-04 U.29 746396-0 5 0.14574426-04 0.74807456 -04 " 0. / 792 7 7 5T:- 35 0. 1921068=-04 0.79675615-04 8 9 1C C.Z421115r 0.81992825 0.24228* 15 -03 -03 -C2 0. 2419"3'52F^0_3— 0.82021115-03 0.2422632=-02 ~~TT3i6'919 86-03 0.11368475-02 0.35596885-^2 C.32l6143=-33 """ 0.11413265-02 0.35644535-32 11 12 13 U.630fl51= 0.14645435 0.31483265 - 0 2 -01 - n i 0.630736 76-02 0.146*5165-01 0.3148347=-01 0.93668376 -02 " 0.24512275-01 0.55595535 - 0 1 0.987132 2E -02 0.2*51693=-01 0.56noo*5=-31 1 14 15 16 U.64352755 0.1152942'= C.16009495 -01 + 00 0.64352276-01 " 0.11529*0G+OQ C. 16005385 + 0 f 0.12034'93E'+00 0.2356*255+00 0.39573745+00 0. 1 2335276+30 0.2356467=+33 0.35574045+00 1 7 18 19 t i . l ^lb621E C.15792245 0.128204 7= + O r > • OC 0.17156236+00 '" 0. 1579212E+0C 0.128204I=+00 0-.56730.05F + 00" 0.72522295+00 0.85342765+00 0 .56730285+00 0.725 22405+00 0.8534281E+00 20 21 22 0.337J3432"6 0.41219405-0. 1620238=' -or~ -0.1 0.8 3083937^0"! n.41216396-01 0.16202186-01 C.53651215+00 0.97773156+00 0.99393386+00 0 . 9 2 6 5 1 2 ^ + 33 O.9777313 r + 0 0 0.99393356+0C 23 24 25 0.4896767 = -0.10234655-0.13511555-- 0 2 -"2 -03 0 .48969466-02 ' 0. 10232445-02 0.13531086-03 0.99883065+00 0.69985*05+00 0.995°8905+00 0.99883C45+3C 0.9998537^+30 0.O99989 CE + 00 26 27 28 .0. 10609205-0.4715190 s-0.11320056--04 -^6 -07 0.104481J=-04^— 0 .67870796-Q6 -0.2145907F-06 ' 0.999999T6+00' -0.10000005+01 1.1000OO05+01 0.999999 *5+33 0.1CO0OC35+31 0 . 9 9 9 0 9 9 1 5 = 29 Q.13850915--C9 0.22767775-G6 "' 0.13000006+11 O.l'.'3-J JOJ5 + 01 •* STATISTICAL PACAMCTPQS ** ( I N TERMS OF I t T HE CLASS *) MEAN STD.DEV. E X P E C T E D 17.09996^3 2.2814255 CALCULATED 3J ERROR 17.099929« 0.00018? 2.2914436 -0.00079? ** aihF6R6NT I«L P°.OBABIL!T1£S' 5-5 %%' L" F A S T~ 5 00* R T~L IN £ A"P—PTFG'RE S S l O N t~i P(CALCULATEDJ = SLOF= * »(2XPECTEC> • T?-e<JC5P T SLOPE = 1.0000 I-NTERCEPT = 0.0000 CQRRFLAT7.CUJ LJtr-h 1L1 !-N i = U. iggi j t j t n XX CHI-SOARC GOODNESS OF FIT TEST S35 TOTAL FREQUENCY = " 8 7 6 0 " « OF REGROUPEC CLASSES - • - • 1 6 T^=5OTRE1T— = 0T757JCT3W CUMULA'TVF OROBABILITIES ** 3535 1FAST SQUARE LINEAR REGRESSION %35 C P ( C A L C U L A T C - D » = SL^Ps ^CPlEypECTEO. • INTERCEPT SLOPF. = 1.0000 INTERCEPT = 0.0000 "TORRELATT1?N~CT,"EFP ICI FNT * 0.9999995 [ %% K C L M U G C » O V - S M ! » N L : V GUOONESS Of- P I T TEST %t # OF CLASSES (N) == 29 H A'XTD F VI A T f O^T"! N CUMUL AT I vE P»OftAfiTL!TY 0.0000050 OC CURED AT r L 4 S S * 11 — CRITICAL K-S VALUES FQR N>40 A R E : 20 3; . . . . 0.1986939 " ~ 1"0~I 77. .. o~.?26"5T37. 5 * 0.2525455 1 ? 0.3026822 196 H I . 3 3 

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