optimal estimators for ambient air quality levels

7/25/2019 Optimal Estimators for Ambient Air Quality Levels

1/11

AtmosphericEnvironmentVol.26A,No. 1. pp. 113-123.1992. 0004-6981/92$3.00+0.00

Printed in Great Britain. 1991PergamonPress

p i

O P T I M L E S T IM T O R S F O R M B I E N T I R Q U L I T Y

L E V E L S

STUART A BATTERMAN

The School of Public Health, U niversity of Michigan, Ann A rbor, M I 48109-2029, U.S.A.

First received 1 September 1990 and in final form 14 March 1991)

Abstract--Procedures to estimate missing data, determine extrema, and deriv e uncertainties for data

collected in am bient air m onitoring networks are presented. The optimal linear estimators u sed obtain

unbiased, minimum variance results based on the tempo ral and spatial correlation o f the data and estimates

of sample uncertainty. The first estimator interpolates m issing data. The second estimator derives extrema,

e.g. minimum and maxim um concentrations, from the completed data set. Together the estimators can be

used to check the validity o f mon itored observations, identify outliers, and estimate regional and local

comp onents of pollutant levels. The estimators are evaluated usin g data collected in urban air quality

monitoring networks in Houston, Philadelphia and St Louis.

Key word index: Distribution, data imputation, optimal estimation, statistical models, uncertainty.

I. INTRODUCTION

A m b i e n t a i r q u a l i t y m o n i t o r i n g n e t w o r k s o p e r a t i n g

t h r oughou t t he w or l d ove r t he l a s t f e w de c a de s ha ve

c o l l e c te d a va s t a mo un t o f da ta . T he s e da t a po t e n t i a ll y

a r e u s e fu l f o r ma ny t ype s o f s tud i e s. H ow e ve r , s e ve ra l

i s sues should be addres sed before us ing h i s tor i ca l

da ta . T hese inc lude th e fo l lowing. (1) H ow represent -

a t i v e a r e t h e m e a s u r e m e n t s ? ( 2 ) H o w c a n s a m p l i n g

e r r o r s be e s t ima t e d? ( 3 ) C a n mi s s i ng o r i nva l i d da t a be

ident if i ed and es t im ated? (4) W ha t a re the ext rem a in

po l l u t a n t l e ve l s ? ( 5 ) C a n ' l oc a l ' a nd ' d i s t a n t ' ( o r

' ba c kg r oun d ' ) c omp one n t s be s e pa r a te d? T he s e is sues

m ay be c r i ti ca l in in te rpre t ing a i r qua l i ty da ta . D espi t e

t he i r i mpor t a nc e , f e w me t hods w h i c h a dd r e s s t he m

e x i s t , a nd none a r e i n c ommon us e .

T h i s pa pe r de ve l ops p r oc e du r e s t o de r i ve mor e

m e a n i n g fu l i n f o r m a t io n f r o m a m b i e n t n e t w o r k d a t a.

T he p r oc e du r e s u s e op t i ma l e s t i ma t i on t e c hn i que s

w h i c h e mpl oy t he s pa t i a l a nd t e mpor a l c o r r e l a t i on o f

a mbi e n t me a s u r e me n t s a nd r e l a t e d c ova r i a t e s , a nd

e s t ima t e s o f s a mp l i ng unc e r t a i n t y . T he p r oc e du r e s ,

w h i c h a r e qu i t e ge ne r a l , c a n p r ov i de a p r a c t i c a l w a y t o

enh ance the use fulness of h i s tor i ca l da ta .

Th e pap er i s orga nized as fo l lows . Sec t ion 2 reviews

a s pe c t s o f a mb i e n t a i r qua l i t y s a m pl i ng a n d s t at i st ic a l

p r oc e du r e s u s e d t o a na l y s e t he c o l l e c t e d da t a . S e c -

t i on 3 p r e s e n t s a c onc e p t ua l f r a me w or k f o r c ompon-

e n t s o f a mb i e n t po l l u t a n t l eve ls a nd t he n g i ve s t he

ma t he ma t i c a l de v e l opme n t o f t he e s ti ma t i on p r oc e d -

ures . Sec t ion 4 appl i es and ev a lua tes the proc edu res

us ing three urb an sca le case s tudies . Sec t ion 5 d i s-

c us s e s r e s u l t s a nd c onc l ude s t he pa pe r by s ugge s t i ng

f u r t he r a pp l i c a t i ons o f t he p r oc e du r e s .

2 B A C K G R O U N D

A m bi e n t a i r qua l i t y mon i t o r i n g ne t w or ks a r e e s t a b -

l is he d f o r pu r po s e s w h i c h i nc l ude ( 1) t he a s s e ss me n t o f

~ A ) m l . s

c onc e n t r a t i on l e ve l s a nd t he c ompl i a nc e s t a t u s w i t h

a i r qua l i t y s t a nda r ds ; ( 2) t he de t e r m i na t i on o f he a l th

a nd e nv i r onm e n t a l i mp a c t s ; a nd ( 3 ) t he s e l e ct i on a nd

mo n i t o r i ng o f e mi s s i on a ba t e me n t s tr a te g i es . M a n y

a i r qua l i t y mo n i t o r i ng ne t w or ks c ons i s t o f 5 - 20 s it es

ob t a i n i ng h ou r l y m e a s u r e me n t s o f c r i te r i a po l l u t a n t s

C O , 0 3 , N O x a n d S O 2 , a n d 2 4 - h m e a s u r e s o f t o t al

s u s pe nde d pa r t i c u l a t e s ( T S P ) . N e t w or ks w h i c h ha ve

be e n op e r a t i ng f o r t w o de c a de s m a y ha ve c o l l e c t e d 107

obs e r va t i ons . R e c e n t c onc e r ns a nd ne w a i r qua l i t y

s t a nda r ds ha ve i nc r e a s e d t he num be r o f po l l u t a n t s

mon i t o r e d . P a r t i c u l a t e ma t t e r l e s s t ha n 10 / ~ mdi a .

( P M - 10 ) , le a d , h y d r o c a r b o n s a n d o t h e r c o n t a m i n a n t s

ma y a l s o be r ou t i ne l y me a s u r e d .

A n a l y s i s o f a i r q u a l i ty d a t a

R e por t s ge ne r a t e d f r om c o l l e c t e d da t a i nc l ude

mo n t h l y a nd a nnu a l s umm a r i e s li s ti ng c onc e n t r a t i ons

a t va r i ous pe r c e n t i l e s a nd a ve r a g i ng t i me s . M a ny

more sophi s t i ca ted s t a t i s t i ca l ana lyses have been per -

f o r me d , a l t hough f e w p r oc e du r e s a r e u s e d r ou t i ne l y .

A pp l i c a t i ons o f a dva nc e d a na l y s e s ge ne ra l ly ha ve

been l imi ted to spec ia l s tudies , e .g . t rend ana lys i s ,

e xpos u r e s t ud i e s , r e c e p t o r mode l i ng , a nd d i s pe r s i on

mod e l va l i da t i on .

Ta ble 1 class i fies s tat is t ical ly-o riented ana lyses in

t he l it e r a tu r e by t w o f a c t o r s : t he num be r o f mo n i -

t o r i ng s it es , a nd t he n um be r o f var i ab l e s. A w i de r a nge

o f a na l y s e s ha ve be e n e mpl oye d , i nc lud i ng bo t h s t a nd -

a r d a n d i n n o v a t iv e m e t h o d s . T h e f o ll o w i ng s u m m a r y

gives a c ross -sec t ion of the l i t e ra ture . S ingle var i able

( i. e. s ingle p ol lu tan t ) - - s ing le s it e s tudies have in-

c luded c las s i ca l t ime se r ies Box and Jenkin s - type

mod e l s f o r s ho r t t e r m f o r e c a s ts ( e.g . M c C ol l i st e r a n d

Wi l son, 1975) , spec t ra l ana lyses indica t ing p er iod-

ici ti es o f pol lu tant da ta (e .g . Ha ya s et a l . , 1982),

r e g r e s s i on mode l s e s t i ma t i ng po l l u t a n t d i s t r i bu t i ons

( e. g. L a r se n , 1976 ) , P o i s s on r a n do m p r oc e s s mode l s

113


2/11

114

STUARTA. BATTERMAN

Table 1. Statistical methods applicable to network data

Single monitoring site M ultiple mon itoring sites

Single pollutant Tre nd analysis Upw ind/downw ind analysis

Analysis of distributions Kriging

Probabili ty of exceedance Spatial interpolation (1)

Extreme value statistics Op tim al estimation

Tim e se r ies (ARIM A)

Spectral analysis

Ma rk o v - ty p e m o d e l s

Correlation analysis

Fac tor ana lysis

Generalized linear models (2)

Receptor models

Cluster analysis

Time se r ies (ARIM AX)

M ultiple variables Ka lm an fil ter mod els

Co-kriglng (3)

Optimal estimation (4)

Notes: (1) Includes contouring, e .g. l inear (planer) and non-linear interpolation.

2 ) Includes l inear and non-linear regression.

(3) No studies identified using co-kriging.

(4) Could use procedure discussed with exogenous variables.

( e .g . Ba k e r

e t a l .

1 98 4) , p r o b a b i l i t y o f e x c e e d a n c e s a n d

r e t u r n p e r i o d m o d e l s ( e . g . D r u f u c a a n d G i u g l i a n o ,

1 9 7 7 ) , M a r k o v - t y p e m o d e l s b a s e d o n u p - a n d d o w n -

c r o s s i n g s o f a t h r e s h o l d c o n c e n t r a t i o n ( e. g. N o r t h e t

a l . 1 9 8 4 ) , a n d e x t r e m e v a l u e s t a t i s t i c s ( e . g . R o b e r t s ,

1 9 79 ; S h i v e l y , 1 9 9 0 ) . R e c e n t l y , t h e s i n g l e m o n i t o r i n g

s i t e - m u l t i p l e v a r i a b l e c a t e g o r y h a s r e c e i v e d t h e

g r e a t e s t a t te n t i o n d u e t o t h e a p p l i c a t i o n o f c h em i c a l

m a s s b a l a n c e r e g r e s s i o n - t y p e r e c e p t o r m o d e l s ( e . g .

H e n r y

e t a l .

1 9 8 4 ) . R e c e p t o r m o d e l s t u d i e s a l s o h a v e

u s e d p r i n c i p a l c o m p o n e n t a n d f a c t o r a n a l y s i s m e -

t h o d s ( e.g . L o w e n t h a l a n d R a h n , 1 9 8 7 ) . S i n gl e

s i t e - m u l t i p l e v a r i a b l e s t u d i e s h a v e e m p l o y e d t i m e

s e r i e s m o d e l s w i t h p o l l u t a n t a n d m e t e o r o l o g i c a l v a r -

i a b l e s ( e .g . F in z i

e t a l .

1 98 0) a n d o t h e r p r o c e d u r e s s u c h

a s c l u s t e r a n a l y s i s ( G e t h e r a n d S e i p , 1 9 7 9 ) . M o s t

m u l t i p l e s i t e s t u d i e s h a v e b e e n p e r f o r m e d f o r t w o

p u r p o se s . U p w i n d / d o w n w i n d a n a ly s e s h a v e b ee n

u s e d t o e s t im a t e c o n t r i b u t i o n s f r o m d i s t a n t a n d l o c a l

e m i s s i o n s o u r c e s ( e . g . B a t t e r m a n

e t a l .

1 9 8 7 ) . V a r io u s

c o n t o u r i n g r o u t i n e s h a v e b e e n u s e d t o d e r i v e p o l l u -

t a n t i s o p l e t h s o v e r a r e g io n t o e s t i m a t e e x p o s u r e s a n d

o t h e r i m p a c t s , i n c lu d i n g t h e u s e o f k r ig i n g ( L e f o h n e t

a l . 1 9 87 ; V e n k a t r a m , 1 9 88 ; H a a s , 1 9 9 0 ). A f e w p o t e n -

t i a l m u l t i p l e s i t e - m u l t i p l e v a r i a b l e t e c h n i q u e s a r e

i d e n t if i e d in th e t a b l e ; m o s t a p p l i c a t i o n s u s e a K a l m a n

f i l t e r i n g a p p r o a c h t o r e c o n c i l e m o d e l s a n d d a t a ( e . g .

M u l h o l l a n d , 1 9 8 9 ) .

B a c l ~ r o u n d e s t im a t e s

O f t e n i t i s i m p o r t a n t t o a p p o r t i o n p o l l u t a n t c o n t r i -

b u t i o n s a t t r i b u t a b l e t o l o c a l a n d d i s t a n t e m i s s i o n

s o u r c e s. L o n g - r a n g e t r a n s p o r t b y d i s t a n t s o u r c e s c a n

p r o v i d e a s i g n if i ca n t r e g i o n a l o r b a c k g r o u n d c o n t r i -

b u t i o n w h i c h r e s t r i c t s t h e c o n t r o l o p t i o n s a v a i l a b l e

t o l o c a l a u t h o r i t i e s . S u c h s i t u a t i o n s c a n o c c u r w i t h

P M - 1 0 , s u l f a t e , o z o n e a n d o t h e r p o l l u t a n t s .

A p p r o a c h e s f o r s e p a r a t i n g l o c a l a n d r e g i o n a l c o m -

p o n e n t s u s e e i t h e r d i s p e r s i o n m o d e l i n g o r a m b i e n t

m o n i t o r i n g . B o t h a p p r o a c h e s r e q u i r e t h a t m o n i t o r i n g

a n d m o d e l i n g e r r o r s a r e n e g li g ib l e o r k n o w n . T h e k e y

d i s a d v a n t a g e o f t h e d i sp e r s i o n m o d e l i n g a p p r o a c h i s

t h e u n c e r t a i n t y o f t h e p r e d i c t i o n s , w h i c h i s a b o u t a

f a c t o r o f t w o f o r s h o r t - t e r m a v e r a g e s ( A m e r i c a n

M e t e o r o l o g i c a l S o c i e t y , 1 9 81 ). A l s o , a s u i t a b l e m o d e l ,

a n a c c u r a t e s o u r c e i n v e n t o r y , a n d m e t e o r o l o g i c a l

o b s e r v a t i o n s f o r a r e p r e s e n t a t iv e p e r i o d a r e r e q u i re d .

T h u s , t h is a p p r o a c h i s n o t r e c o m m e n d e d ( E P A , 1 98 4) .

T h e s u g g e s t ed a p p r o a c h u s e s a m b i e n t m o n i t o r i n g a t

u p w i n d o r i s o l a te d r e g i o n a l s it es . U p w i n d o b s e r v a -

t i o n s s h o u l d e x c l ud e m e a s u r e m e n t s a f fe c te d b y l o c a l

s o u r c e s . R e g i o n a l s i t e s s h o u l d b e l o c a t e d a w a y f r o m

t h e a r e a o f i n t e r e s t a n d u n a f f e c t e d b y l o c a l s o u r c e s

( E P A , 1 9 8 4 ) . M o r e d e t a i l e d g u i d e l i n e s f o r s i te s t o

m o n i t o r r e g i o n a l a t m o s p h e r i c d e p o s i t i o n s p e c i fy a

m i n i m u m s e p a r a t i o n d i s t a n ce o f 1 0 k m f r o m i n d u s -

t r ia l a n d n a t u r a l s o u r c e s o f e m i s s i o n s e x c e e d in g

1 0 ,0 0 0 t y - 1 a n d p o p u l a t i o n c e n t e r s o f 1 0 ,0 0 0 o r m o r e

( A S T M , 1 9 8 9 ) . S e p a r a t i o n d i s t a n c es s h o u l d b e i n -

c r e a s e d d r a m a t i c a l l y i f t h e s a m p l e r li e s i n th e p r e -

v a i li n g d o w n w i n d d i r e c t io n o f e m i s s io n s o u r c e s .

B a c k g r o u n d e s t i m a t e s b a s e d o n m o n i t o r i n g m a y

h a v e s e v e r a l d e f i c i e n c i e s c a u s e d b y i n s u f f i c i e n t t e m -

p o r a l a n d s p a t i a l c o v e r a g e in t h e n e t w o r k . S i x e x -

a m p l e s a r e g i v e n . ( 1 ) B a c k g r o u n d c o n t r i b u t i o n s

d o m i n a t e s o m e p o l l u ta n t s ( e. g. P M - 1 0 a n d S O 4 ) , a n d

t h e r e m a y n o t b e e n o u g h m o n i t o r i n g s i t e s t o d e t e c t

r e l a t i v e l y s m a l l l o c a l i m p a c t s . ( 2 ) I t m a y b e d i f f ic u l t t o

d e s i g n a t e p a r t i c u l a r s it es a s u p w i n d o r r e g i o n a l s i te s

s i n ce s o m e p o l l u t a n t s ( e .g . P M - 1 0 , N O x , H C ) a r e

e m i t t e d b y m a n y w e l l - d i s p e r s e d s o u r c e s s u r r o u n d i n g

m o s t m o n i t o r s . ( 3 ) W i n d s h i f t s d u r i n g s a m p l i n g

p e r i o d s m a y i n v a l i d a t e u p w i n d d e s i g n a t i o n s , e s -

p e c i a l l y f o r p o l l u t a n t s c o l l e c t ed o v e r l o n g p e r i o d s , e . g.

2 4 - h p a r t i c u l a t e s a m p l e s . ( 4 ) P o l l u t a n t s s a m p l e d i n t e r -

m i t t e n t l y , e . g . T S P m e a s u r e d e v e r y s i x t h d a y , h a v e

t e m p o r a l r e s o l u t i o n t o o c o a r s e t o d e t e r m i n e b a c k -

g r o u n d . ( 5 ) T h e r e l at i v e a c c u r a c y o f m e a s u r e m e n t s


3/11

Optimal estimators for air quality mod els 115

d e c r e a s e s w i t h t h e l o w c o n c e n t r a t i o n s l i k e l y a t r e -

g iona l s i te s (e .g . Ev ans a nd Ry an, 1983) . (6)M iss ing

d a t a m a y b i a s r e s u l t s ( D a v i s o n a n d H e m p h i l l , 1 9 8 7 )

e s p e c ia l ly w h e n t h e r e a r e f ew m o n i t o r i n g s it es . A n y o f

these event s may cause se r ious e r rors .

A c c u r a c y a n d r e p r e s e n t a ti v e n e s s o f d a ta

A g e n e r a l g o a l o f s a m p l i n g i s t o o b t a i n r e p r e s e n t a t -

i v e m e a s u r e m e n t s , d e f i n e d b y G e i g e r ( 19 6 5) a s h a v i n g

a w i d e r a n g e o f v al id i ty . T h i s g o a l i s te m p e r e d b y t h e

n e e d t o o b t a i n a p p r o p r i a t e s p a t i a l a n d t e m p o r a l r e -

s o l u t i o n g i v e n ti m e a n d c o s t c o n s t ra i n t s , a n d t h e n e e d

t o a c c u r a t e l y m o n i t o r c o n c e n t r a t i o n s a t s p e c i fi e d p e r -

c e n t il e s a n d a v e r a g i n g t i m e s. F o r e x a m p l e , a i r q u a l i t y

r e g u l a t i o n s f o c u s o n s h o r t - t e r m p e a k c o n c e n t r a t i o n s

s u c h a s t h e s e c o n d h i g h e s t c o n c e n t r a t i o n i n a y e a r .

T h e s e p e a k c o n c e n t r a t i o n s o r e x t r e m a c a n b e d if fi cu lt

t o m e a s u r e a c c u r a t e l y .

I n m o s t a n a l y s e s , m o n i t o r i n g o b s e r v a t i o n s a r e a s -

s u m e d t o b e r e p r e s e n t a ti v e o f a m b i e n t l e ve ls a t t h e

m o n i t o r i n g s it e f o r th e a v e r a g i n g t im e o f t h e m e a s u r e -

m e n t . W i t h t h e e x c e p t io n o f s o m e r e c e p t o r m o d e l i n g

t e c h n i q u e s ( e .g . W a t s o n et al., 1984), obse rva t ion s a l so

a r e a s s u m e d t o b e e r r o r - f r e e . E r r o r s , h o w e v e r , c a n

a r i s e f r o m m a n y s o u r c e s i n c l u d i n g ( 1 ) a n a l y t i c t e c h -

n i q u e s ; ( 2 ) s a m p l e r b i a se s ; (3 ) l a c k o f s a m p l i n g r e p -

resenta t iveness ; and (4)misce l l aneous sources , e .g .

s a m p l e d e g r a d a t i o n , d a t a e n t r y m i s t a k e s , e t c . I n

t h e o r y , e r r o r s c a n b e p a r t i t i o n e d i n t o s y s t e m a t i c a n d

r a n d o m e l e m e n t s , a f f e c t i n g a c c u r a c y a n d p r e c i s i o n ,

r e sp e c ti v e ly . M o s t c o n c e n t r a t i o n s a r e b a s e d o n s e v e r a l

c o m p o n e n t s , e. g. s am p l e v o l u m e a n d p a r t ic l e m a s s f o r

p a r t i c u l a t e c o n c e n t r a t i o n s , e a c h o f w h i c h c o n t r i b u t e s

s y s te m a t ic a n d r a n d o m e r ro r s. A s c o m p o n e n t e r r o rs

m a y b e a d d i t i v e o r m u l t i p li c a ti v e , c o r r e l a t e d o r i n d e -

p e n d e n t , o r s i m p l y u n k n o w n , t h e t o t a l e r r o r i s o f t e n

u n c e r t a in . A g o o d m e a s u r e o f t h e t o t a l r a n d o m e r r o r i s

t h e s a m p l e v a r i a n c e o f re p l ic a t es ( D r a p e r a n d S m i t h ,

1 9 81 ), h o w e v e r , t r u e r e p e t it i o n s i n r o u t i n e m o n i t o r i n g

p r o g r a m s a r e r a r e . A s s u m p t i o n s o f r e p r e s en t a t iv e n e s s

a n d a c c u r a c y m a y b e p a r t i c u l a r l y p r o b l e m a t i c a l f o r

t h e e x t r e m a n e e d e d t o d e t e r m i n e r e g i o n a l a n d l o c a l

c o n t r i b u t i o n s . B o t h t h e l o w e s t a n d t h e h i g h e st c o n c e n -

t r a t i o n s m a y b e p r o n e t o m e a s u r e m e n t a n o m a l i e s .

M i s s i n o d a t a

A n a d d i t i o n a l c o n c e r n i s t h e c o m p l e t e n e s s o f t h e

d a t a . M i s si n g ( o r i n v al id ) d a t a m a y r e s u lt f r o m i n s t r u -

m e n t f a i l u r e , c a l i b r a t i o n a n d m a i n t e n a n c e p r o b l e m s .

I n t h e c a s e s t u d ie s d e s c r i b e d l at e r, a b o u t o n e - q u a r t e r

o f th e d a t a w a s m i s s in g . A l a r g e r p e r c e n t a g e w a s

mis s ing a t spec if ic m on i tor in g s i te s , e spec ia l ly a t rura l

s i t e s which a re d i f f i cu l t t o s e rv ice . Many ne tworks

a c h i e v e c o m p a r a b l e r e c o r d s . M i s s in g d a t a i n c r e as e t h e

d i ff i cu l ty o f e s t a b li s h i n g t r e n d s a n d d e t e r m i n i n g c o m -

p l ia n c e w i th a m b i e n t s t a n d a r d s b a s e d o n t h e n u m b e r

o f e x ce e d a n c e s ( D a v i s o n a n d H e m p h i U , 1 9 8 7 ) . I n

m u l t i v a r i a t e a p p l i c a t i o n s s u c h a s r e c e p t o r m o d e l s , th e

o m i s s i o n o f a s i n g le e l e m e n t m a y n e c e s si ta t e t h e

re jec t ion of the en t i re obse r va t ion .

A n u m b e r o f m e t h o d s t o h a n d l e m i ss in g d a t a h a v e

b e e n d e v e l o p e d i n b i o s t a t i s t i c s w h e r e r e s p o n s e s t o

s u r v e y s , f o r e x a m p l e , o f t e n c o n t a i n l a r g e a m o u n t s o f

mis s ing or inc or rec t da ta (e .g . Gar f inke l , 1986; L i t t l e e t

al., 1 9 8 9 ) . G e o g r a p h e r s h a v e a l s o c o n f r o n t e d t h i s

p r o b l e m ( a s r e v i e w e d b y B e n n e t t et al., 1 9 8 4 ) . F e w

a p p l i c a t io n s o f t h e s e o r o t h e r m e t h o d s h a v e b e e n u s e d

f o r a i r q u a l i t y d a t a . O n e a p p r o a c h f o r e s t i m a t i n g o r

i n p u t i n g m i s si n g o z o n e d a t a u s e d o z o n e - t e m p e r -

a ture re l a t ionships (Davi son and Hemphi l l , 1987) .

M o r e g e n e r a l m e t h o d s , a s d e v e l o p e d i n t h e f o l l o w i n g

sec t ion , would be he lpfu l for o the r pol lu tan t s .

3. OPTIMALESTIMATORS

S t a t i s t i c a l f r a m e w o r k

A f r a m e w o r k f o r a m b i e n t a i r c o n c e n t r a t i o n s i s

d e v e l o p e d c o n s i d e r i n g a s i n g l e c o n s e r v a t i v e ( n o n -

r e a c t i v e ) p o l l u t a n t m e a s u r e d i n a n u r b a n s c a l e m o n i -

t o r i n g n e t w o r k . T h e c o n c e n t r a t i o n o b s e r v e d a t s i t e i

and t ime t , C~ ,z , cons i s t s of th ree com po nen t s :

Ci, t= Li . t Dt V/,t, (1)

w h e r e L i. t a n d D a r e lo c a l a n d r e g i o n a l c o m p o n e n t s ,

respec t ive ly , and Vii., i s measu rem ent e r ror . Lo ca l

c o n t r i b u t i o n s r e s u l t f r o m e m i s s i o n s o u r c e s s i t u a t e d

w i t h i n t h e u r b a n a r e a . T h e s e c o n c e n t r a t i o n s t y p i c a l l y

i n c r ea s e to w a r d s t h e s o u r c e . T h e r e g i o n a l c o m p o n e n t ,

p r o d u c e d b y l o n g - r a n g e t r a n s p o r t , h a s g r a d ie n t s t h a t

a re negl ig ib le on the loca l s ca le . Thus , D t i s t ime

v a r y i n g , b u t c o n s t a n t i n s p a c e a t t h e u r b a n s ca le .

T h e s p a t i a l a n d t e m p o r a l c o r r e l a t i o n p r e s e n t in t h e

d a t a i s u se d t o i m p r o v e t h e a c c u r a c y a n d r o b u s t n e s s

( insensi t iv i ty to out l i e r s ) of con cen t ra t io n es t ima tes . A

three -par t p rocedure i s used . F i r s t , an es t ima te of

measurement uncer t a in ty V~, i s de r ived . Next , an

o p t i m a l e s t i m a t i o n p r o c e d u r e e s t i m a t e s m i s s in g d a t a .

L a s t l y , t h e l o w e s t a n d h i g h e s t c o n c e n t r a t i o n s a r e

es t ima ted f rom the es t ima ted da ta s e t .

M e a s u r e m e n t u n c e r t a i n t y

S e v e r a l a p p r o a c h e s c a n b e u s e d t o e s t i m a t e e r r o r

V~, . Ra nd om er rors m ay be es t im a ted us ing rep l i ca te

obse rv a t ions , e .g . co lo ca ted sam ple rs , whi l e sys tem at i c

e r r o r s c a n b e e s t i m a t e d u s i n g r e f er e n c e o r c a l i b r a t e d

s a m p l e r s . A l t e r n a t i v e l y , e r r o r s m a y b e e s t i m a t e d b y

i s o l a ti n g u n c e r t ai n t i es i n t h e c o m p o n e n t m e a s u r e s a n d

then propaga t ing the i r e f fec t s , e .g . us ing Gauss ian

q u a d r a t u r e . L a s t l y , e m p i r i c a l m e a n s m a y b e u se d . T h e

f o l l o w i n g e x a m p l e s d e m o n s t r a t e t h e s e a p p r o a c h e s .

S ince 1981, federa l regula t ions have requi red s t a t e

a n d l o c a l a g e n ci e s t o a ss e ss t h e a c c u r a c y a n d p r e c i si o n

o f t h e ir a m b i e n t a i r q u a l i t y m e a s u r e m e n t s y s t e m s .

D a t a c o l l e c te d i n t h e P r e c i s io n a n d A c c u r a c y R e p o r t -

i n g S y s t e m ( P A R S ) a r e b a s e d o n b l i n d a u d i t s u s i n g

ca l ibra t ion gases for cont inuous ins t ruments (gases ) ,

a n d c o l o c a t e d s am p l e s f o r m a n u a l i n s t r u m e n t s ( T S P ,

P b , a n d o l d e r g a s m e a s u r i n g i n s t r u m e n t s ) . P A R S


4/11

116 STUART A BATTERMAN

resu l t s , expres sed as a 95% conf idence in te rva l , t yp i -

c a l ly s h o w a r e l a ti v e a c c u r a c y o f a b o u t 1 0 % f o r m o s t

o f th e c r i t e r ia p o l l u t a n ts . T h e p r e c i si o n o f t h e m e a s -

u r e m e n t s , o b t a i n e d b y r e p e a t e d m e a s u r e s , i s a b o u t

1 0 % f o r 0 3 , 1 2 % f o r T S P , 20 fo r S O 2 , a n d 4 6 % f o r

NO 2 (Rho des and Evan s , 1988). These s t a t is t i c s rep-

r e s e n t m a n y t h o u s a n d s o f a u d i ts .

O n e t h e o r e t ic a l s t u d y o f e r r o r s i n m a s s , f lo w r a t e

a n d t i m i n g m e a s u r e m e n t s s u g g e s ts e r r o r s a b o u t h a l f o f

t h a t o b t a i n e d i n f i e l d e v a l u a t i o n s ( E v a n s a n d R y a n ,

1 9 83 ). O t h e r e x a m p l e s o f c o m p o n e n t e r r o r s e s t i m a t e

f i l t e r m a s s m e a s u r e m e n t e r r o r s ( u s i n g b e t a g a u g e

a t t e n u a t i o n ) o f 3 g m - 3 f o r 1 2 - h s a m p l e s (J a k le v i c e t

al. 1 9 81 ), a n d b i a s es b e t w e e n g r a v i m e t r i c a n d b e t a

g a u g e m e a s u re m e n t s o f < 5 % ( C o u r t n e y et al. 1982).

W i t h a i r v o l u m e e r r o r s o f 5 - 1 0 % , t h e s e f ig u r e s y i e ld a

t o t a l e r r o r o f 1 0 - 2 0 % a t t y p i c a l p a r t i c l e c o n c e n t r a -

t ions .

A n e m p i r i c a l e s ti m a t e o f sa m p l i n g e r r o r s i s t h e

d i f f e r e n c e b e t w e e n t h e l o w e s t t w o c o n c e n t r a t i o n s i n

t h e n e t w o r k , a s s u m i n g t h a t t h e s e c o n c e n t r a t i o n s r e -

su l t ma in ly f rom regiona l sources . Whi le imper fec t ,

th i s e s t ima te may be use fu l in l a rge moni tor ing ne t -

w o r k s w h e r e t h e t w o l o w e s t c o n c e n t r a t i o n s c a n b e

cons id e red rep l ica tes . In the case s tudies (desc r ibed

later ), t h i s proce du re gav e re la t ive e r rors o f 15-20% . A

be t t e r , bu t ra re ly ava i l ab le measure , i s the va r i ance

b e tw e e n m e a s u r e m e n t s o b t a i n e d f r o m c o l o c a t e d s a m -

plers.

T h e t h r e e a p p r o a c h e s y i e l d r e l a t i v e e r r o r s i n t h e

r a n g e o f 1 0 - 4 6 % . I n m o s t c a s e s , e r r o r s t a t i s t i c s a r e

n o t a c c u r a t e l y k n o w n . A l s o , m e a s u r e m e n t s o b t a i n e d

u n d e r u n u s u a l c o n d i t i o n s m a y y i e l d m u c h l a r g e r

e r r o r s . F o r e x a m p l e , e r r o n e o u s p a r t i c u l a t e m e a s u r e -

m e n t s c a n b e c a u s e d b y h i g h l o a d i n g s w h i c h c l o g

f i l t e r s , unusua l s i ze d i s t r ibu t ions , and h igh wind

speeds which a f fec t in l e t pe r formance .

E s t i m a t i n g m i s s i ng d a t a

T h i s s e c ti o n d e v e l o p s a n o p t i m a l l i n e a r e s t i m a t o r t o

e s t i m a t e m i s s i n g o b s e r v a t i o n s . M i s s i n g o b s e r v a t i o n s

a r e c o n s i d e r e d u n k n o w n r a n d o m v a r i a b l e s . T h e s t a t -

i st ic s of these va r i ab les a re b ased on ava i l ab le da ta ,

a n d a r e s e le c t ed t o p r e s e r v e t h e o b s e r v e d s p a t i a l a n d

t e m p o r a l c o r r e l a t i o n .

C o l u m n v e c to r Z i s a r r a n g e d t o c o n t a i n l e a d i n g ,

l a g g i n g a n d s i m u l t a n e o u s o b s e r v a t i o n s a t a l l m o n i -

tor ing s i t e s in the ne twork:

z , = [ c , , , _ , . . c . . , - , I I c x , , c . . , I

. . I C ~ , , + . . . c . , , . ] , 2 )

where C~ ,, i s the c on cen t ra t io n a t s i te i and t ime t , n i s

t h e n u m b e r o f m o n i t o r i n g s i te s, m is t h e n u m b e r o f

l e a d i n g a n d l a g g i n g t i m e p e r i o d s , a n d t h e q u o t e

d e n o t e s t r a n s p o s e . T h e l e a d i n g a n d l a g g i n g e l e m e n t s

p e r m i t i n t e r p o l a t i o n s i n t im e , w h i le t h e s i m u l t a n e o u s

o b s e r v a t i o n s a l l o w s p a t i a l a v e r a g i n g . A s s h o w n l a te r ,

one l ag a nd l ead pe r io d i s genera l ly suf fi ci en t , so m = 1

and Zt inc ludes 3n e l ement s ( l agging , s imul t aneous ,

and l eading concent ra t ions a t n s i t e s ) . As desc r ibed

e a r li e r i n E q u a t i o n (1 ) , o b s e r v a t i o n Z , i n c lu d e s t h e

t rue pol lu tan t l eve l Xt p lus e r ror V , :

Z~ = X, + V,. (3)

I f s o m e d a t a a r e m i ss in g , t h e c o r r e s p o n d i n g e l e m e n ts

in vec to rs Z , and V, have m is s ing values , bu t these can

b e e s ti m a t e d a s t h e c o r r e s p o n d i n g e l e m e n t s o f v e c t o r

Xt and mat r ix S t , a s desc r ibed be low.

E r r o r c o v a r i a n c e m a t r i x R , is d e fi n e d a s :

R, = E l 'Vt V ;]. (4)

E r r o r V a n d c o v a r i a n c e R t m u s t b e e s t im a t e d . I f

e r rors a re uncor re la t ed , R f i s a d iagona l ma t r ix . Dia -

g o n a l e l em e n t s o f R t a r e s e t t o t h e m e a s u r e m e n t

var i ance . As d i scussed in s ec t ion 2 , measurement e r r -

ors can be es t ima ted in s evera l ways . Here , e r rors a re

a s s u m e d t o b e t i m e i n v a r i a n t , u s i n g a r e la t iv e e r r o r o f

3 0 % a n d t h e m e a n c o n c e n t r a t i o n . T h e d i a g o n a l e l e -

m e n t s c o r r e s p o n d i n g t o m i s s i n g o b s e r v a t i o n s a r e s e t

to a much l a rge r va lue , e .g . 1000 t imes the measure -

m e n t v a r i a n c e , t o r e p r e s e n t t h e l a r g e ( p ri o r) v a r i a n c e

of the mis s ing da ta .

F i r s t a n d s e c o n d m o m e n t s t a t i s t i c s , n a m e l y , m e a n

v e c t o r M a n d c o v a r i a n c e m a t r i x P , a r e s a m p l e e s t i -

m a t e s f r o m a v a i l a b l e d a t a :

M = T -

i y., X, (5)

P = r - '~ X J - ( X t - M ) ( X , - M ) ' ] , (6 )

w h e r e T is th e n u m b e r o f o b s e r v a t i o n s u s e d t o e s ti -

m a t e M a n d P . M a t r i x P c o n t a i n s i n f o r m a t i o n r e g a r -

d i n g t h e s p a t ia l a n d t e m p o r a l c o r r e l a t i o n o f t h e d a t a .

A s s u m i n g u n b i a se d n e s s ( E [ V , ] = 0 ) a n d u n c o r r e la t e d

e r r o r s ( E [ ' X , V ; ] = 0 ) , t h e b e s t l i n e a r , u n b i a s e d a n d

m i n i m u m v a r i a n c e e s t im a t e X o f t h e m i s s in g o b s e r v a -

t ions is :

'.Kt= M + P ( P + R , ) - 1 Z t - M ) . ( 7)

T h i s B a y e s i a n e s t i m a t o r w e i g h t s th e i n f o r m a t i o n p r o -

v ided by the obse rva t ions ( the so-ca l l ed inf luence

v e c t o r Z t - M ) t o y i e ld t h e e s t im a t e X v R e s u l t s w il l b e

i d e n t ic a l t o m e a n M i f t h e r e i s z e r o c o r r e l a t i o n b e -

tween obse rva t ions , i . e . P = 0 . The (pos te r ior ) e r ror of

es t ima t ion mat r ix S i s :

S , = E [ ( X t - X t ) ( X , - X t ) ] = P - P ( P + R t ) - 1p . ( 8 )

T h e e s t i m a t o r i n E q u a t i o n ( 7) m i n i m i z e s th e d i a g o n a l

t e r m s S , f o r t h e s t a t e d a s s u m p t i o n s , a s s h o w n b y

S c h w e p p e ( 1 9 73 ). I n E q u a t i o n ( 7) , m i s s i n g d a t a i n Z t

a r e s e t t o z e r o , h o w e v e r , s o l u t i o n s ( f o r m i s s in g d a t a )

a re insens i t ive to the v a lue spec if ied s ince the va r i ance

t e r m s i n R t f o r c o r r e s p o n d i n g e l e m e n t s a r e s o l a r ge .

A s a m b i e n t d a t a a r e g e n e r a l l y h i g h l y c o rr e l a te d , t h e

es t ima te of the i th m is s ing obse rv a t ion ) (~ . t o f t en i s

ve ry d i f fe ren t f rom the mean, wi th a va r i ance S~ , l . ,

w h i c h i s g r e a t l y r e d u c e d f r o m t h e a s s u m e d p r i o r

v a r i an c e . C o n v e r s e l y , i f t h e m e a s u r e m e n t v a r i a n c e i s

z e r o f o r t h e i t h ( k n o w n ) o b s e r v a t i o n , t h e n e s t im a t e X t . t

i s unc han ged f rom obse rva t io n Z~ .t and S~. ~ . ,= 0 .


5/11

Optim al est imators

E s t i m a t e X t is th e m e a n o f t h e c o n d i t i o n a l d i s t r i b u -

t i on g iven Z f (Schweppe , 1973) , t ha t i s , i t i s t he

e x p e c te d v a l u e o r a v e r a g e o f t h e m i s s i n g d a t a b a s e d o n

m a n y i n s t a n c e s i n w h i c h t h e s a m e p o l l u t a n t c o n d i -

t i o n s p r e v a i l e d . T h e a c t u a l p o l l u t a n t l e v el s f o r a n y

s i n g l e i n s t a n c e ( i f a v a i l a b l e ) w o u l d d i f f e r f r o m t h i s

m e a n a n d s h o w a b r o a d e r d i s t r i b u t i o n t h a n o b t a i n e d

f r o m t h e e s t i m a t o r . C o n d i t i o n a l s i m u l a t i o n i s u s e d t o

o b t a i n t h e o r i g i n a l d i s t r i b u t io n b y a d d i n g a r a n d o m

t e r m t o t h e m e a n , e . g . p o s t - w h i t e n i n g :

t ^

X i = E [ X i . , l Z t ] + ~ ti. ,, (9)

w h e r e , i s a z e r o m e a n r a n d o m v a r i a b l e w i t h

v a r i a n c e S ~,~ ., o b t a i n e d i n E q u a t i o n ( 8 ). B e c a u se p o l -

l u t a n t o b s e r v a t i o n s a r e r o u g h l y l o g - n o r m a l l y d is t r i-

b u t e d w i t h p r i m a r i l y m u l t i p l i c a t i v e e r r o r s , t h e l o g -

a r i t h m s o f o b s e r v a t i o n s a r e u s e d i n E q u a t i o n s ( 1 ) -( 9 ).

T h e f in a l e st i m a t e e m p l o y s c o n d i t i o n a l s i m u l a t io n a n d

e x p o n e n t i a t i o n :

~tn t

X , . ,= ex p X , ,t + S~/~2tw ),

(10)

w h e r e w i s a n o r m a l l y d i s t r i b u t e d , u n i t v a r i a n c e

r a n d o m v a r i a b l e .

E s t i m a t i n g e x t r e m a

E x t r e m a a r e f o u n d u s i n g a n o p t i m a l e s t i m a t i o n

p r o c e d u r e s i m i l a r t o E q u a t i o n s ( 2 ) - ( 1 0 ) . T h e p r o c e d -

u r e i s d e v e l o p e d f o r t h e h i g h e s t t h r e e c o n c e n t r a t i o n s .

L e t Ht , t , H2 . t and Ha .e , r e spec t i ve ly , r ep re sen t t he

h i g h e s t, s e c o n d a n d t h i r d h i g h e s t c o n c e n t r a t i o n s i n t h e

n e t w o r k a t t i m e t . R e d e f i n i n g t h e s y m b o l s u s e d e a r l ie r ,

Z , c o n t a i n s t h e t o p t h r e e c o n c e n t r a t i o n s i n t h e n e t -

w o r k f o r t h e c u r r e n t, a n d m l e a d i n g a n d l a g g i n g

p e r i o d s :

Z , = [ H I . , _ , , , H 2 . , _ m H 3 a - m l . . . [ H L , H 2 . t H 3 , t l

11)

T h e o b s e r v a t i o n Z , o f t h e s e c o n c e n t r a t i o n s d i f f e rs

f r o m t h e t r u e c o n c e n t r a t i o n X t b y s a m p l i n g e r r o r V t

( E q u a t i o n ( 3 )) . C o v a r i a n c e m a t r i x P i n E q u a t i o n ( 6)

d o e s n o t i n c l u d e s a m p l i n g e r r o r s . L e t R r e p r e s e n t t h e

e r r o r c o v a r i a n c e m a t r i x o f s a m p l i n g e r r o r s . I f R i s

k n o w n a n d u n c o r r e l a t e d w i th t h e m e a s u r em e n t s , t h e n

P c a n b e e s t i m a t e d a s

P ,,~ T - ~ Z r ( Z t - M ) ( Z , - M ) ' - R , (12)

w h e r e M i s t h e s a m p l e m e a n . O b s e r v a t i o n s ( n o t

e s t i m a t e d v a l u e s ) a r e u s e d t o c o m p u t e M a n d P .

M a t r i x R i s e s t i m a t e d a s :

R = k l , ( 1 3 )

w h e r e I i s a n i d e n t i t y m a t r i x a n d k is t h e s a m p l i n g

e r r o r w h i c h i s a s s u m e d c o n s t a n t a t a l l s i te s . P a r a m e t e r

k m a y b e s e l e c t e d b a s e d o n t h e e x p e c t e d r e l a t iv e e r r o r

f o r e x t r e m a i n t h e n e t w o r k , e .g . if t h e r e l a t i v e e r r o r w a s

3 0 , k i s t h e s q u a r e o f t h e p r o d u c t o f 0 .3 a n d t h e m e a n

c o n c e n t r a t i o n . R e s t r i c t i o n s m u s t b e p l a c e d o n R t o

ensure t ha t P i s pos i t i ve sem i -de f in i t e , and t e s t s fo r

p o s i t i v e d e f i n i t e n e s s m a y b e m a d e . A g a i n , E q u a -

f o r a i r qu ality models 117

t i o n ( 7) p r o v i d e s a n u n b i a s e d m i n i m u m v a r i a n c e e s -

t i m a t e o f X .

T h e l o w e s t c o n c e n t r a t i o n s i n t h e n e t w o r k a r e a n

e s t i m a t e o f t h e r e g i o n a l c o m p o n e n t D e a s d e s c r i b e d

e a r l i e r in E q u a t i o n ( 1) . T h e s e c o n c e n t r a t i o n s a r e d e -

t e r m i n e d i n t h e s a m e m a n n e r a s t h e p e a k c o n c e n t r a -

t i o n s e x c e p t t h a t t h e l o w e s t t h r e e c o n c e n t r a t i o n s

r e p l a c e t h e t h r e e p e a k v a l u e s . H i g h a n d l o w e x t r e m a

c a n b e s i m u l t a n e o u s l y e s t i m a t e d b y i n c l u d i n g t h e

h i g h e s t a n d t h e l o w e s t c o n c e n t r a t i o n s i n t h e o b s e r v a -

t i o n v e c t o r . L o c a l i m p a c t s L i, t c a n t h e n b e e s t i m a t e d

a s t h e d i f fe r e n c e b e t w e e n e s t i m a t e d p e a k s X t a n d t h e

e s t i m a t e d b a c k g r o u n d l e v e l D r .

4. APPLICATION

I m p l e m e n t a t i o n a n d e v a l u a t i o n

T h e e s t i m a t o r s d e s c r i b e d i n s e c t i o n 3 w e r e c o d e d i n

F O R T R A N a n d r u n o n a n 8 0 3 8 6 -b a se d c o m p u t er . A

L U i n v e r s io n a l g o r i t h m ( P r e s s et al., 1987) was used.

B o t h s i n g l e a n d d o u b l e p r e c i s i o n p r o g r a m s w e r e

w r i t t e n . A ja c k - k n i f e p r o c e d u r e w a s u s e d t o e v a l u a t e

t h e e s t i m a t o r s ' p e r f o r m a n c e . A p o r t i o n o f t h e d a t a i n

t h e c a s e s t u d i e s , s e l e c te d r a n d o m l y , w a s i n t e n t i o n a l l y

d e l e t e d . D e l e t e d d a t a w e r e t h e n p r e d i c t e d u s i n g t h e

f i rs t e s ti m a t o r . E s t i m a t e s w e r e m a d e i f h a l f o r m o r e o f

t h e e l e m e n t s i n v e c t o r Z t w e r e a v a i l a b l e . T h i s c r i t e r i o n

p r o v i d e s a c o m p r o m i s e b e t w e e n t h e r e l i a b i l i ty o f t h e

e s t i m a t o r a n d d e m a n d s p l a c e d o n t h e p r o c e d u r e .

P r e d i c t i o n s w e re c o m p a r e d t o t h e a c t u a l o b s e r v a t i o n s

u s i n g l i n e a r c o r r e l a t i o n c o e f f ic i en t s , m e a n b i a s , a n d

s c a t t e r p l o t s . B e ca u s e th e a c t u a l e r r o r s w e r e u n k n o w n ,

t h e e s t i m a t o r s w e r e t e s te d w i t h e r r o r s r a n g i n g f r o m 0

t o 6 0 .

C as e s tud ie s

T h r e e c a s e s t u d i e s w e r e u s e d t o e v a l u a t e t h e e s t i -

m a t o r s . T h e f i r s t e m p l o y e d p a r t i c u l a t e d a t a c o l l e c t e d

i n S t L o u i s , I L f r o m M a y t o S e p t e m b e r 1 9 76 a s p a r t o f

t h e R e g i o n a l A i r P o l l u t i o n S t u d y ( S t r o t h m a n n a n d

S c h i e r m e i e r, 1 9 7 9 ) . I n t h i s s t u d y , d i c h o t o m o u s s a m -

p le r s a t 10 s i t e s co l l e c t ed 12-h sam ple s i n f i ne and

c o a r s e s i ze f ra c t i o n s . B e c a u s e o f l o n g g a p s o f m i s s i n g

da t a , f ew er s i te s a re used he re (T ab l e 2 ). Mo s t s i t e s

w e r e u r b a n ; c o v e r a g e e x te n d e d t o a b o u t 4 5 k m f r o m

t h e c i t y c e n t e r . T h e s e c o n d s t u d y i s t h e P h i l a d e l p h i a

A r e a F i e l d S t u d y ( T o o t h m a n , 1 98 4) i n w h i c h a m b i e n t

d a t a w e r e c o l l e c t e d f r o m 1 4 J u l y t o 1 3 A u g u s t 1 98 2 a t

s i x s i te s . A s i n S t L o u i s , d i c h o t o m o u s s a m p l e r s c o l l e c -

t e d 1 2 - h p a r t i c u l a t e s a m p l e s , a l s o i n t w o s i ze f r a c ti o n s .

T h i s u r b a n a r e a w a s c o n s i d e r a b l y l a r g e r th a n S t

L o u i s , y e t t h e m o n i t o r i n g n e t w o r k w a s s m a l l e r . M o s t

s i te s w e r e u r b a n a n d i n d u s t r i a l . T h e s t u d y i n c l u d e d a

' s p e c i a l s t u d i e s ' s it e w i t h i m p a c t s f r o m a n e a r b y o i l

t a n k f a r m , l o c a l t r u c k t r a f f i c a n d o n g o i n g c o n s t r u c -

t i o n , a n d a r u r a l s i t e i n N e w J e r s e y . T h e t h i r d c a s e

s t u d y u s e d O 3 o b s e r v a t i o n s t a k e n a t 1 1 s it e s in

H o u s t o n , T X f r o m A p r i l t o S e p t e m b e r 1 9 8 7 . I n t h i s

s t u d y , m o n i t o r i n g s i t e s r a n g e d o v e r a d i s t a n c e o f


6/11

118

STUART A . BATTERMAN

Table 2. Num ber o f available and deleted d ata in case s tudies . Range is shown in parentheses

Num ber Num ber of observations per s i te

mo nitor ing Maximum Average Average

sites possible available deletions

St Louis

Fin e particles 9

Co arse particles 7

Philadelphia

Fin e particles 6

Co arse particles 6

Houston

Oz one average 11

Ozone pea k 11

Average percentage o f total

92 62 (58-79) 23 (16-32)

92 62 (55-82) 22 (17-27)

62 54 (38-60) 7 (1-13)

62 49 (30-95) 5 (4-6)

152 120 (97-14 7) 7 (6--8)

152 120 (9 147 ) 17 (9-22)

100 78 14

6 0 k m ; m o s t w e r e l o c a t e d w it h i n u r b a n H a r r i s

C o u n t y . D a i l y a v e r a g es a n d d a i l y h o u r l y p e a k s w e re

c a l c u l a t e d a t e a c h s i t e f r o m h o u r l y o b s e r v a t i o n s . I f

f e w e r t h a n 1 2 - h w e r e a v a i l a b l e a t a s i t e i n a g i v e n d a y ,

t h e o b s e r v a t i o n w a s c o n s i d e r e d t o b e m i s s in g .

T h e d a t a c a p t u r e r a t e s o f t h e t h r e e n e t w o r k s a r e

s u m m a r i z e d in T a b l e 2 . F o r t h e m o n i t o r i n g n e t w o r k s

a n d t i m e p e r i o d s s e le c t ed , n e t w o r k d a t a c a p t u r e a v e r -

a g e d 7 8 . F o r s i n g le s i te s , 4 8 - 9 7 o f t h e d a t a w e r e

a v a i l a b l e .

s t imates o f miss ing data

T a b l e 2 a l s o s h o w s t h e n u m b e r o f o b s e r v a t i o n s

d e l e t e d f r o m e a c h d a t a s e t i n o r d e r t o e v a l u a t e t h e

e s t i m a t o r . A n a v e r a g e o f 1 4 o f t h e a v a i l a b l e o b s e r -

v a t i o n s w a s d e l e t e d . T h i s p o s e d a s e v e r e t e s t o f

p e r f o r m a n c e s i n ce a n a v e r a g e o f 2 2 o f t h e d a t a w a s

a l r e a d y m i ss i n g , t h u s a n a v e r a g e o f o n l y 6 4 o f t h e

d a t a w a s u t i l i z e d t o p r e d i c t m i s s in g o b s e r v a t i o n s .

L i n e a r c o r r e l a t i o n c o e f f ic i e nt s a n d b i a s e s b e t w e e n

e s t i m a t e d a n d a c t u a l d a t a a r e s h o w n i n T a b l e 3 . T h e s e

c a l c u l a t i o n s w e r e o b t a i n e d f o r e a c h d a t a s e t u s i n g 0 , 1

a n d 2 l e a d / l a g s . A r e l a t iv e e r r o r o f 3 0 w a s u s e d i n

e a c h c a s e . C o r r e l a t i o n s w e r e h i g h , b e t w e e n 0 . 6 6 a n d

0 .9 0 , a n d b i a s e s w e r e s m a l l . D e s p i t e t h e i m p r o v e m e n t

e x p e c te d , t h e n u m b e r o f l e a d s / l a g s d i d n o t h a v e

d r a m a t i c e f fe ct s. I n f a c t , p e r f o r m a n c e s o m e t i m e s d e -

g r a d e d a s t h e n u m b e r o f l e a d / l a g p e r i o d s i n c re a s e d a s

i l l u s t r a t e d b y r e s u l t s f o r S t L o u i s . T h i s r e s u l t e d a s

r o u n d - o f f e r r o r s i n c r e a s e d w i t h a d d i t i o n a l l a g s a n d

n e g a t e d t h e s m a l l ( a n d d i m i n i s h i n g ) i n f o r m a t i o n p r o -

v i d e d w i t h l o n g e r l e a d a n d l a g p e r i o d s . F o r e x a m p l e ,

w i t h 2 l e a d s / l a g s a n d n = 1 0 ( 10 m o n i t o r i n g s i t es ), a n

i l l - c o n d i t i o n e d m a t r i x o f r a n k 5 0 m u s t b e i n v e r te d .

W h i l e t h e d o u b l e p r e c is i o n v e rs i o n o f th e p r o g r a m

r e d u c e d t h e s e e r r o r s , m o r e a c c u r a t e i n v e r s i o n r o u t i n e s

m i g h t b e a d v a n t a g e o u s .

M e a n s a n d s t a n d a r d d e v i a t io n s o f t h e o b s e r v a t io n s

a n d e s t im a t e s m a t c h e d c lo s el y . O v e r 6 0 - 8 0 o f t h e

p r e d i c t i o n s w e r e w i t h i n 2 5 o f t h e o b s e r v a t i o n , a n d

8 0 - 9 0 w e r e w i t h i n 5 0 . S c a t t e r p l o t s s h o w v e r y

g o o d a g r e e m e n t w i t h t h e T e x a s d a t a s e t (F i g s le a n d

I f) . H o w e v e r , t h e v a r i a b i l i t y o f th e p a r t i c u l a t e d a t a i s

n o t f u l l y r e p r o d u c e d , e . g . p e a k v a l u e s a r e u n d e r -

e s t i m a t e d a n d l o w v a l u e s a r e o v e r e s t i m a t e d ( e s p e c i al l y

F i g s l b a n d l c ). T h i s a n a l y s i s a s s u m e s t h a t m o n i t o r i n g

o b s e r v a t i o n s a r e e r r o r- f re e . B e t t e r a g r e e m e n t c o u l d b e

o b t a i n e d b y ( 1 ) c h a n g i n g t h e r e l a t iv e e r r o r ; ( 2) c h a n g -

i n g t h e n u m b e r o f l e a d / l a g s ; ( 3 ) a l t e r i n g t h e l o g -

n o r m a l a s s u m p t i o n ; ( 4 ) d e c r e a s i n g t h e f r a c t i o n o f

m i s s i n g d a t a ; a n d ( 5 ) i n c r e a s i n g p o s t - w h i t e n i n g . F o r

e x a m p l e , e x c e l l e n t a g r e e m e n t c o u l d b e o b t a i n e d

( r i > 0 .9 5 ) f o r t h e P h i l a d e l p h i a f i n e f r a c t i o n p a r t i c u l a t e

d a t a u s i n g 1 l e a d / l a g , a r e l a t i v e e r r o r o f 0.1 0, a n d

a s s u m i n g n o r m a l , r a t h e r t h a n l o g - n o r m a l d i s t ri b u -

t i o n s . I n p r a c t i c e , s u c h p a r a m e t e r s c o u l d b e c a l i b r a t e d

Table 3. Correla t ion coef fic ient ( r) and bias (b) of the es t im ator

Philadelp hia St Louis Hou ston

F ine Coar se F ine Coar se Peak A verage

Lags =0

r 0.84 0.84 0.73 0.66 0.90 0.90

b 0.88 2.11 4.10 4.74 0.36 0.05

Lags = 1

r 0.84 0.82 0.68 0.67 0.86 0.89

b 0.07 1.00 3.25 5.65 0.39 0.04

Lags = 2

r 0.81 0.82 0.33 0.76 0.88 0.82

b 0.6 1 1.23 3.50 3.68 0.06 0.24


7/11

O p t i m a l e s t i m a t o r s f o r a i r q u a l i t y m o d e l s 1 19

ESTIMATED (uglm 3)

80 I COARSE PARTICLES

PHILADELPHIA STUDY

s o a )

4 o / ~ ,,

2 1 u

0 1

0 20 60 80

0

ORIGINAL (ug/m 3)

ESTIMATED (uglm3 )

80 i FINE PART CLES J

PHILADELPHIA STUDY

60

(b)

o i .

O

0 20 40 60 80

ORIGINAL (ug/m 3)

8 0

60

4 0

20

0

0

ESTIMATED (ug/m 3)

COARSE PARTICLES / / ~

ST. LO UIS STUDY

c) , ~

20 40 60 80

ORIGINAL (ug/m 3)

ESTIMATED (ug/m 3)

100

FINE PARTICLES /

ST. LO UIS STUDY

80

(d) ~ , /

.o

20 S ~ g ~ 1 u

O

0 20 40 60

80 100

ORIGINAL (ug/m 3)

ESTIMATED (pphm )

10

12 Hr. AVERAGE OZONE ~ / ~

8 HOUSTONSTUDY

(e) ~ u

6 s ~ B

4 B ~ sa mz

f

I A l I

0

O 2 4 6 8

10

ORIGINAL (pphm)

ESTIMATED (pphrn )

20~

/

HOURLY PEA K OZONE

/

HOUSTON STUDY

158

f)

_ ~

101 ~ ~ ~

u = i r l l B . ~

s

0 I I ;

0 5 10 15 20

ORIGINAL (pphrn)

F i g . 1 . S c a t t e r p l o t s o f p r e d i c t e d v s e s t i m a t e d d a t a f o r t h e t h r e e c a s e s t u d i es .

using s i te-specif ic resul ts . Even without such cal ib-

r a t io n s , t h e p r e d i c t io n s p r e s e r v e t h e a c t u a l s p a t i a l a n d

t e m p o r a l t r e n d s u s i n g s o l e l y th e d a t a s c o r r e l a t i o n

s t ruc ture .

T h e l a r g e s t r e l a t i v e e r r o r s r e s u l t f r o m o v e r p r e d i c -

t i o n o f v e r y lo w o b s e r v a t i o n s , e .g . p a r t ic u l a t e c o n c e n -

t r a t i o n s a v e r a g i n g 6 a n d

9 g m -

3 i n c o a r s e a n d f in e

f rac t ions , re spec t ive ly . These obse rva t ions were s ev-

e ra l t imes smal l e r than those a t o the r s i t e s , and a l so

s m a l le r t h a n p r e c e d i n g a n d f o l l o w i n g c o n c e n t r a t i o n s .

T h e s e m e a s u r e m e n t s a r e a n o m a l i e s a n d p o s s i b l y e r r o -

n e o u s . T h e l a r g e s t e r r o r s a t h i g h c o n c e n t r a t i o n s o c c u r


8/11

120 STUARTA. BATTERMAN

w i t h t h e c o a r s e f r a c t i o n p a r t i c u la t e d a t a s et s, p r o b a b l y

a resu l t o f loca l in f luences which were un cor re la t ed

w i t h o b s e r v a t i o n s a t o t h e r s it es . S u c h c a s e s c a n n o t b e

p r e d i c t e d w i t h o u t h i g h l y d e t a i le d i n f o r m a t i o n . I n

g e n e r a l, t h e t e c h n i q u e o f d e le t i n g a n d t h e n p r e d i c t in g

e a c h o b s e r v a t i o n c a n p r o v i d e a g o o d c h e c k o n d a t a

val idi ty.

xtrema estimates

E x t r e m a e s t i m a t e s w e r e c a l c u l a t e d f o r e a c h d a t a s et ,

u s i n g v a r i o u s l e a d / l a g s a n d r e l a t iv e e r r o rs . F i g s 2 a n d

3 show typica l re su l t s for h ighly cor re la t ed ( f ine f rac -

t i o n ) a n d p o o r l y c o r r e l a t e d ( c o a r s e f r a c t i o n ) d a t a

u s i n g t h e P h i l a d e l p h i a d a t a f o r t h e p e r i o d o f 1 6 - 3 1

July 1982. Al l ava i l ab le da ta were used , and no

d e l i b e r a t e d e l e t i o n s w e r e m a d e ( a s i n t h e p r e v i o u s

a p p l i c a t i o n ) . O n e l e a d / l a g p e r i o d i s u s e d . P o l l u t a n t

o b s e r v a t i o n s a n d e s t i m a t e s a r e s h o w n a s p o i n t s , w h i l e

e x t r e m a e s t i m a t e s a r e d r a w n a s l i n e s . T h r e e e x t r e m a

e s t i m a t e s a r e s h o w n . ( T h e s e c o n d a n d t h i r d h i g h e s t

a n d l o w e s t e x t r e m a e s t i m a t e s h a v e b e e n o m i t t e d f r o m

t h e g r a p h f o r c l ar i ty . ) E x t r e m a w i th 0 r e la t iv e e r r o r

p r o v i d e a n e x a c t m a t c h b e t w e e n a c t u a l o r e x p e c t e d

e x t r e m a , e . g . t h e e s t i m a t e i s u n c h a n g e d f r o m t h e

o b s e r v a t i o n , a n d t h e t w o l in es s i m p l y c o n n e c t s e q u e n -

t ia l m a x i m a o r m i n i m a . A s t h e r e la t iv e e r r o r i n c re a s es

to 20 and the n 60 , the va r i a t ion in pol lu tan t l eve ls i s

d a m p e n e d , n e i t h e r c o m p l e t e l y ri s in g t o t h e p e a k c o n -

cent ra t ions n or fa l l ing to the low es t values . I n genera l ,

t h e u p p e r a n d l o w e r e n v e l o p e s c o n v e r g e t o w a r d s t h e

C o n c e n t r a t i o n ( u g / m 3 )

1 0 0

Obse rv ed

stimated

8 0 ~ E x t r e m e : 0

- - - - E x t r e m e : 2 0

. .. .. .. E x t r e m e : 6 0

6 0

4 0

2 0

0 I J J J I

0 5 1 0 1 5 2 0 2 5 3 0

O b s e r v a t i o n N u m b e r (1 2 h o u r p e r i o d )

Fig. 2. Observations, estimates and extrema en velopes for fine fraction p articulate

data in Philadelphia.

1 4 0

1 2 0

1 0 0

8 0

6 0

C o n c e n t r a t io n ( u g /m 3 )

O b s e r v e d

E s t i m a t e d / I

- - E x t r e m e =

0 /

/

- - E x t r e m e : 2 0 [

4 0

0 5 1 0 1 ~ 2 0

O b s e r v a t i o n

N u m b e r ( 1 2 h o u r p e r i o d )

F i g . 3 . O b s e r v a t i o n s , e s ti m a te s a n d e x t r c m a e n v e l o p e s f o r c o a r se f r a c t io n p a r t i c -

u l a t e data in Philadelphia.

2 5 3 0


9/11

Optimal estimator s for air quality models 121

m e a n a s t h e r e l a t i v e e r r o r i n c r e a s e s , t h u s m a x i m u m

c o n c e n t r a t i o n s a r e u n d e r p r e d ic t e d a n d m i n i m u m c o n -

c e n t r a t i o n s a r e o v e r p r e d i c t e d . L o c a l i m p a c t s L ~ ,t a r e

r e p r e s e n t e d a s t h e w i d t h o f t h e e n v e lo p e . E s t i m a t e s o f

t h e l o w e s t c o n c e n t r a t i o n a p p e a r s m o o t h e d o r d a m -

p e n e d i n t i m e , p o s s i b l y r e f l e c t i n g t h e r e g i o n a l c o m -

p o n e n t w h i c h u s u a l l y c h a n g e s s l o w l y w i t h r e s p e c t t o

t h e s a m p l i n g f r e q u e n c y .

P e a k c o a r s e f r a c t i o n c o n c e n t r a t i o n s ( F i g . 3 ) o c c u r

a t t h e F i r e b o a t s i t e , d u e t o l o c a l i m p a c t s w h i c h a r e

l a r g e l y u n c o r r e l a t e d t o c o n c e n t r a t i o n s a t o t h e r s i t e s .

T h i s i s c l e a r l y s h o w n b y t h e e x t r e m a e s t i m a t e d w i t h

/ > 2 0 e r r o r w h i c h d o n o t a p p r o a c h t h e s e p e a k s .

C o n v e r s e l y , t h e f i n e f ra c t i o n d a t a i s h i g h l y c o r r e l a te d

a n d t h e e n v e l o p e m a i n t a i n s a n e a r l y c o n s t a n t w i d t h .

I n b o t h c a s e s , u p p e r e x t r e m a a r e s e n s i t i v e t o t h e

r e l a t i v e e r r o r u s e d , w h i l e l o w e r e x t r e m a a r e n o t . I n

par t , t h i s re su l ts f rom the use of a logar i thm ic t rans -

f o r m a t i o n .

A s d is c u s se d , t h e l o w e s t a n d h i g h e s t c o n c e n t r a t i o n s

a r e p r o n e t o m e a s u r e m e n t a n o m a l i e s , u n u s u a l

m e t e o r o l o g i c a l f a c t o r s a n d s o u r c e c o n d i t i o n s . L a r g e

di f fe rences be tween obse rved and es t ima ted ex t rema

m ay ind ica te such event s ; def in i te ly , these cases need

fur the r inves t iga t ion . Thi s s imple s t ra t egy can i so la t e

a t y p i c a l e x t r e m a w h i c h m a y a ff ec t e s t im a t e s o f l o c a l

a n d r e g i o n a l i m p a c t s .

C o m p u t a t i o n a l d e m a n d s

T h e c o m p u t a t i o n a l d e m a n d s o f e st im a t i o n d e p e n d

o n m , th e n u m b e r o f l e a d s/ la g s , a n d n , t h e n u m b e r o f

s it es . C o v a r i a n c e m a t r i x P a n d m e a n v e c t o r M a r e

d e t e r m i n e d o n c e f o r e a c h d a t a s e t . T o i n t e r p o l a t e

m i s s i n g d a t a f o r e a c h s a m p l i n g p e r i o d r e q u ir e s i n v e r -

s io n o f a r a n k n ( 2 m + l ) m a t r i x a n d s e ve ra l m a t r i x

m u l t i p li c a ti o n s . A b o u t 2 s o f c o m p u t e r t i m e w e r e

r e q u i r e d f o r e a c h s a m p l i n g p e r i o d u s i n g a f a s t

(3 3 M H z ) 8 0 3 8 6 / 7 m i c r o c o m p u t e r a n d a h i g h a c c u r -

a c y L U d e c o m p o s i t io n i n v e r si o n a l go r i th m . E x t r e m a

es t imates requi re a s ingle invers ion for the en t i re da ta

s e t . O n l y s i m p l e m a t r i x o p e r a t i o n s a r e n e e d e d f o r

e a c h o b s e r v a t i o n , t h u s t h e s e e s t i m a t e s a r e q u i c k l y

ca lcu la ted .

5 DISCUSSION AND CONCLUSION

T h i s p a p e r h a s d e v e l o p e d l i n e a r e st i m a t o r s t o e s t i m -

a t e d a t a a n d e x t r e m a w i t h t h e p u r p o s e s o f h a n d l i n g

m i s s i n g d a t a a n d a c c o u n t i n g f o r e r r o rs . E x t r e m a a r e

es t ima te d f ro m the ful l (e s t ima ted) d a ta s e t. Es t im a tors

o f t h e l o w e s t c o n c e n t r a t i o n i n t h e n e t w o r k m a y r e p r e s -

ent reg ion a l leve ls i f t he m on i tor in g sys tem inc ludes

s i t e s which a re l a rge ly unaf fec ted by loca l sources .

P e a k e s t im a t e s , p r o v i d e d b y t h e s a m e e s t i m a t o r , i n d. i-

c a t e t h e c o n t r i b u t i o n o f lo c a l s o u r c e s. B o t h e s t i m a t e s

s h o u l d b e m o r e r o b u s t t h a n o b s e r v a t io n s f r o m s in g le

s t a t i o n s s i n c e s p a t i a l a n d t e m p o r a l i n f o r m a t i o n f r o m

all sites is utilized.

I n a p p l i c a t i o n t o t h r e e d i v e r se d a t a s e ts i n S t L o u i s ,

P h i l a d e l p h i a a n d H o u s t o n , t h e e s t i m a t o r s p r o v i d e d

re l iab le resu l ts . T he h igh spa t i a l and t e mp ora l co r re la -

t i o n p r e s e n t i n a m b i e n t p o l l u t a n t l e v e l s a t t h e u r b a n

sca le makes such es t ima tors prac t i cable . The es t i -

m a t o r s m a y b e le ss a p p r o p r i a t e f o r o b s e r v a t i o n s w i t h

l i t t l e cor re la t ion , e .g . some coarse f rac t ion pa r t i c -

u l a t e s , n e t w o r k s w i t h i n t e r m i t t e n t s a m p l i n g , o r n e t -

wo rks cove r ing ve ry la rge spa t i a l s ca les. T he B ayes ian

e s t i m a t o r i n E q u a t i o n ( 7) , w h i c h a l s o c a n b e v i e w e d a s

a l i n e a r c o n t r a c t i o n o p e r a t o r , t e n d e d t o r e d u c e t h e

sca t t e r in the or ig ina l da ta . In genera l , t h i s i s an

u n d e s i r a b l e p r o p e r t y . H o w e v e r , t h e o r i g i n a l d i s p e r -

s i o n o f t h e d a t a c a n b e r e s t o r e d b y c h a n g i n g t h e d e g r e e

of pos t -w hi t en ing , a l t e r ing the re l a t ive e r ror , o r b y

u s i n g a d i f f e r e n t d a t a t r a n s f o r m a t i o n . S u c h n e t w o r k -

spec i f i c ca l ibra t ions may fur the r inc rease the accuracy

of the es t ima tors .

T h e e s t i m a t o r s v i ew h i s to r i c a l o b s e r v a t i o n s a s i m -

p e r f ec t (e r r o r - c o n t a in i n g ) r a n d o m v a r ia b l es , a f u n d a -

m e n t a l l y d i f fe r e n t p e r s p e c ti v e t h a n t h e u s u a l a s s u m p -

t i o n s t h a t t h e o b s e r v a t i o n s a r e r e p r e s e n t a t i v e a n d

e r r o r- f re e . S p a t ia l a n d t e m p o r a l i n f o r m a t i o n h a s b e e n

used in the oppos i t e manner to s e lec t s i t e s in the

o p t i m a l d e s ig n o f a ir m o n i t o r i n g n e t w o r k s ( e . g .

S h i n d o et al. 1990) . Resul t s ob ta ined in the case

s t u d i e s i m p l y t h a t m o n i t o r i n g o b s e r v a t i o n s , t o v a r y -

ing degrees , a re redundant in provid ing s i t e - spec i f i c

i n f o r m a t i o n s i n c e o b s e r v a t i o n s a t s o m e s i t e s c a n b e

used to predic t concent ra t ions a t o the r s i t e s .

T h e b e h a v i o r o f t h e e s ti m a t o r d e p e n d s o n t h e

r e la t iv e s t r e n g t h s o f t h e t e m p o r a l a n d s p a t ia l c o r r e l a -

t io n . I f t e m p o r a l c o r r e l a t i o n i s d o m i n a n t , a m i s si n g

o b s e r v a t i o n b o t h p r e c e d e d a n d f o l l o w e d b y v a l i d

obse rva t ions a t the s ame s i t e i s e s t ima ted us ing pr im-

a r il y a w e i g h t e d s u m o f l e a d i n g a n d l a g g i n g o b s e r v a -

t ions a t tha t s i te . I f l ead ing an d l agging ob se rva t ions

a re mis s ing , the es tima te i s de r ived f rom s imu l taneou s

obse rva t io ns a t o th e r s i te s . I f spa t i a l cor re la t ion i s

d o m i n a n t , r e s u l t s d e p e n d o n s i m u l t a n e o u s m e a s u r e -

ment s t aken a t o the r s i t e s and to a l e s se r ex ten t on

l e a d i n g a n d l a g g i n g o b s e r v a t io n s . I f m a n y s i m u l t a n -

e o u s m e a s u r e m e n t s a r e m i s s i n g , l e a d i n g a n d l a g g i n g

o b s e r v a t i o n s a n d t h e c o n s t a n t ( m e a n ) a r e e m p h a s i z e d .

I n e a c h c a s e , w e i g h t s g i v e n t o l e a d i n g a n d l a g g i n g

observa t ions can be s igni f i cant , and the coef f i c i en t s

a re s i t e - spec i f i c and depend on the da ta ava i l ab le . In

c o m p a r i s o n w i t h t h e e s t i m a t o r s f o r m i s s i n g v a l u e s ,

e x t r e m a e s t i m a t e s p r i m a r i l y d e p e n d o n s i m u l t a n e o u s

o b s e r v a t i o n s . T h u s , t h e s e e s t i m a t o r s m i g h t b e s im p l i -

f i ed to use obse rva t ions a t on ly the cur ren t t ime . The

e s t i m a t o r a u t o m a t i c a l l y d e te r m i n e s t h e w e i g h t i n g s s o

a s t o m i n i m i z e t h e v a r i a n c e o f t h e e s ti m a t e . T h e

es t imat ion procedure i s f l ex ib le and appl i cable to

o t h e r t y p e s o f d a t a .

S e v e r a l r e f i n e m e n t s t o t h e e s t i m a t i o n p r o c e d u r e s

a r e p o s si b le . A l t h o u g h n o t a t t e m p t e d h e r e , a d d i t i o n a l

v a r i a b l e s c o u l d b e u s e d t o a u g m e n t t h e p o l l u t a n t

v a r i a b l e s a n d i m p r o v e p e r f o r m a n c e , F o r e x a m p l e ,

a m b i e n t t e m p e r a t u r e c o u l d b e u se d t o h e lp p r e d i c t 0 3

c o n c e n t r a t i o n s . M o r e a c c u r a t e e s t i m a t o r s m i g h t d i s -

a g g r e g a t e b y s e a so n , w i n d d i r e c ti o n , o r o t h e r f e a t u r e s


10/11

122 STUARTA. BATTERMAN

- - i f t h e d a t a a r e s u f fi c ie n t t o e s t i m a t e c o v a r i a n c e

m a t r i x P . I n t h e c a s e s t u di e s , h o w e v e r , e s t i m a t e s b a s e d

o n s e a s o n a l d a t a , f i r s t , s e c o n d a n d t h i r d o r d e r l a g s

w e r e s i m i l a r . I n t h e c a s e s t u d i e s , e s t i m a t e s w e r e o b -

t a i n e d w h e n u p t o 5 0 o f t h e d a t a w e r e m i s s i n g . A

m o r e r e s t r i c t i v e p a r a m e t e r w o u l d i m p r o v e p e r -

f o r m a n c e . A s t e p w i s e p r o c e d u r e m i g h t b e u s e d t o

d e t e r m i n e h o w m a n y s i te s a n d h o w m a n y l e a d / l a g

p e r i o d s a r e n e c e s s a r y f o r e s t i m a t i o n . W h i l e a d d i t i o n a l

d a t a w o u l d b e e x p e c t e d t o i m p r o v e r e s u l t s , i t a l s o

i n t r o d u c e s g r e a t e r n u m e r i c a l e r r o r s i n m a t r i x i n v e r -

s i o n . A n a u t o m a t e d p r o c e d u r e c o u l d b e u s e d t o te s t

v a r i o u s o r a l l p o s s i b l e s u b s e t s o f s i te s a n d v a r i a b l e s,

a n d h a v e t h e a d d e d b e n e f i t o f s h o w i n g t h e s e n s i t iv i t y

o f r e s u l t s t o t h e s e f a c t o r s .

T h e e s t i m a t o r s h a v e s e v e r a l a p p l i c a t i o n s . T h e y c a n

b e u s e d t o e s t i m a t e m i s s i n g d a t a , t h u s p r o v i d i n g a

m o r e c o m p l e t e d a t a s e t , A l t h o u g h t h e e s t i m a t o r s

p e r f o r m e d w e l l , t h e u s e o f e s t i m a t e d d a t a m u s t b e

c a r e f u l l y c o n s i d e r e d i n i n t e r p r e t i n g r e s u l t s , e s p e c i a l l y

i f f ew d a t a a r e a v a i l a b l e . S e c o n d , t h e y m a y b e u s e d t o

c h e c k t h e v a l i d i t y o f o b s e r v a t i o n s . S u s p i c i o u s l y lo w o r

h i g h o b s e r v a t i o n s m a y b e e a s i l y i d e n t i f i e d . A n a u t o -

m a t e d b o o t s t r a p p i n g p r o c e d u r e i s s u g g e s te d t o a c -

c o m p l i s h t h is t as k . T h i r d , t h e e s t i m a t o r s m a y p r o v i d e

m o r e r o b u s t e s t i m a t e s o f e x t r e m a f r o m w h i c h l o c a l

a n d r e g i o n al c o n t r i b u t i o n s m a y b e d e t e rm i n e d . S u c h

e s t i m a t e s c a n b e u s e d i n t r e n d a n a l y s i s a n d t o d e t e r -

m i n e c o m p l i a n c e w i t h a i r q u a l i t y s t a n d a r d s .

Ac k now l e dge m e n t s - - The

author s incere ly thanks Dr Hap

Hemphi l l fo r p rov id ing the Ho us ton da ta s et , and M r A da rs h

Kulshre tha for h is ass is tance in performing the computa-

t ional work. The two ano nym ous reviewers provided a

nu m be r of helpful suggestions.

R E F E R E N E S

Am erican Meteorological Socie ty (1 981 ) Air q ual i ty

mode l ing and the c l ean a i r a c t: r e comm enda t ions to EPA

on dispers ion model ing for regula tory appl ica t ions .

Boston, MA.

ASTM (1988) Proposed s tandard guide for choos ing loca-

t ions and s ampl ing me thods to mon i to r a tmos phe r i c

depos i t ion. American Socie ty of Tes t ing and Materia ls ,

Bos ton, MA.

Baker M . B. , Eyland er M. and Harr iso n H. (1984) The

statistics of chemical trace co nce ntra tion s in the steady

state. Atmospheric Environment lg, 969-975.

Benn et t R. J. , Hain ing R. P. an d G riff ith D . A. (1984) The

problem of miss ing da ta o n spat ia l surfaces .

Ann. Ass. Am.

Geograph.

74, 138-156.

Bat term an S. , Fay J . and Go lom b D. (1987) Signif icance of

reg iona l sou rce co n t r ibu t ions to u rban PM -10 concen t ra -

tions. J. Air Pollut . Control Ass. 37, 1286-1292.

Courtney W. J . , Shaw R. W. and Dzubay T. G. (1982)

Prec is ion and accuracy of a be ta gauge for aerosol mass

determinat ions .

Envir. Sci. Technol.

16, 236-239.

Dav ison A. C. and Hem phil l M. W. (1987) On the s ta t ist ica l

ana lysi s o f amb ien t ozone da ta when m eas u remen t s a re

missing.

Atmospheric Environment

21, 629-639.

Draper N. R. and Smith H. (1981) Applied Regression Ana-

lysis. John Wiley, New York.

Dru fuca G . and Giug l i ano M . (1977) The du ra t ion o f h igh

SO2 con centra t ion s in an urb an a tmosphere. Atmospheric

Environment 11, 729-735.

EPA (1984) Proposed rule for 40 CFR, Part 50, Federa l

Register, pp. 10,407-10,462, 20 M arch.

Evan s J . S. and Rya n P. B. (1983) Sta tist ica l uncer ta inty in

ae ros o l mas s concen t ra t ions meas u red by v i r tua l mas s

impactors .

Aerosol Sci. Technol.

2, 531-536.

F inz i G . , F ron za G . and Sp i r ito A . (1980) M ul t iva r i a te

s tochast ic models of sulphur dioxide pol lut ion in an urb an

area.

J. Air Pollut . Control Ass. 30

1212-1215.

Ga r f inke i R ., Ku nn a thu r S . and Liep ins A . S . (1986) Op t ima l

im puta t ion of e rroneo us da ta : ca tegorical da ta , genera l

edits. J. Oper. Res. 34 744-751.

Geiger R. (1965) The Climate near the Ground. Harva rd

Unive rs i ty Press , Cam bridge , MA.

Geth er J . and Seip H. M. (1979) Analys is of a i r pol lu t ion da ta

by the combined use of in terac t ive graphic presenta t ion

and a clustering technique.


13,

87-96.

Haas T . C . (1990) Kr ig ing and au toma ted va r iog ram mode l -

ing within a m oving widow.

Atmospheric Environment 24A

1759-1769.

Hayas A . , Gonza lez C . F . , Pa rdo G . and M ar t inez M . C .

(1982) App l ica t ion of spectra l analys is to a tm ospheric d us t

pol lut ion. Atmospheric Environment 16, 1919-1922.

Hen ry R. C. , Lewis C. W. and Ho pke P. K. (1984) Review of

receptor model fundamenta ls . Atmospheric Environment

18, 1507-1515.

Jaklevic J . M., Gat t i R . C. , Gould ing F. S., Leo B. W. and

Th om pson A. C. (1981) Aerosol Analys is for the Regional

Air P ol lut ion Study. P B 81-157 141, Env ironm enta l Sc ien-

ces Research Labo ra tory, Off ice of Research and Develop-

men t , U.S. Environ me nta l Protec t ion Agency, Research

Tr iang le Pa rk , NC .

Larsen R. I . (1976) A mathematica l model for re la t ing a i r

qua l i ty meas u remen t s to a i r qua l i ty s tanda rds . R epor t AP-

89, U.S. Env ironm enta l Protec t ion A gency, Research

Tr iang le Pa rk , NC .

Lefohn A. S. et al. (1987) An evaluat io n of the kr iging metho d

to predic t 7-h seasonal mean ozone con centra t ion s for

es t imat ing crop losses .

J. Air Pollut . Control Ass.

37,

5 9 5 - - 6 0 2 .

L it tl e R . J., R u b i n D . B . a n d S t r a w d e r m a n W . E . 1 9 8 9 )

Statistical Analysis with Missing Data. John Wiley, New

York.

Lowen tha l D . H . an d R ahn K . A . 0987) App l i ca t ion o f the

factor-analysis receptor model to s imula ted urban - and

regional-scale data sets.


21,

2005-2013.

McCollister G. M . an d W ilson K. R. (1975) Lin ear stochastic

mod els for forecas ting da i ly maxim a and h ourly concen-

t ra t ions of a i r pol lu tants .

Atmospheric Environment 9

417-423.

M ulhol la nd M . (1989) An autoregress ive a tmospheric d isper-

s ion model for f i t ting com bined source and receptor d a ta

sets. Atmospheric Environment 23, 1443-1458.

Nor th M . , He rnandez E . and Ga rc ia R . (1984) Frequency

analys is of h igh CO concen tra t ions in M adrid by s tochas-

tic process modeling.


18, 2049-

2054.

Press W. H.

et al.

(1987)

Numerical Recipes The Ar t of

Scientific Computing.

Cam bridge U nivers i ty Press , New

York.

Rhodes R. C. and Evan s E. G. (1988) Prec is ion and accuracy

assessments for s ta te an d loca l a i r mo ni tor in g ne tworks :

1986 . EPA/600 /S4-88-007 . Env i ronmen ta l M o n i to r ing

Sys tems Labo ra tory, U.S. Env ironm enta l Protec t ion

Agency, Research Triangle Park, NC.

Rob erts E. M . (1979) Review of statistics o f extrem e values

with appl ica t ions to a i r qual i ty da ta : P art I . Review. J . Air

Pollut. Control Ass. 29, 632-637.


11/11

Op t imal es t imators for a i r qual i ty models 123

Schwepp e F. C . 1973) Uncertain Dynamic Systems. Prent ice-

Hal l , Eng iewoo d Cl i ff s , NJ .

Sh i ndo J ., O t K . a nd Ma t s um ot o Y . 1990) C ons i de r a t ions on

ai r pol lu t ion moni tor ing ne twork des ign in the l ight of

spat io- tempora l var ia t ions of da ta . Atmospheric Environ

ment 24B, 335-342.

Shively T. S. 1990) An analys is of the long- term t rend in

ozone data f rom two Houston, Texas moni tor ing s i t es .

Atmospheric Environment 24B, 293-301.

S t r o t hm a nn J . A . a nd Sc h i e rm e i e r F . A . 1979) D oc u-

men ta t ion o f the regional a i r qual i ty s tudy and re la ted

inves t iga t ions in the St . Lo uis a i r qua l i ty cont rol region.

EPA-600/4-79-076, Envi ronmenta l Sciences Research

Lab ora tory , Office of Research and Develo pme nt , U.S.

Envi ronmenta l Protec t ion Agency, Research Tr iangle

Pa r k , N C .

Too t hm a n D . 1984) D e ve l opm e n t o f a n e m i s s ion i nve n t o ry

for urban par t i c le model va l ida t ion in the Phi ladelphia

AQCR. Envi ronmenta l Sciences Research Labora tory ,

U.S. Env i ronm enta l Protec t ion Agency, Research Tr iangle

Pa r k , N C .

Ven kat ram A. 1988) On the use of kr ig ing in the spat ia l

analysis of acid precip i tat ion data. Atmospheric Environ

ment 22, 1963-1975.

W atson J . G. , Coo per J . A . and Hun tz icker J . J . 1984) The

effect ive variance weighting for least squares calculat ions

appl ied to the mass ba lance receptor model . Atmospheric

Environment 18, 1347-1355.

optimal estimators for ambient air quality levels

Documents