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Estimation of Spatial Patterns and Inventories of Environmental Contaminants Using Kriging Jeanne C. Simpson Pacific Northwest Laboratory, Richland, WA 99352

Kriging is a relatively new statistical approach to spatial estimation. The kriging estimator is a weighted average which is the "best linear unbiased estimator." The derivation of the kriging weights takes into account the proximity of the observations to the point (or area) of interest, the "structure" of the observations (the relationship of the squared differences between pairs of observations and the intervening distance between them) and any systematic trend (or drift) in the observations. Additionally, kriging provides a variance estimate that can be used to construct a confidence interval for the kriging estimate. This paper will discuss the assumptions made in kriging and the derivation of the kriging estimator and variance. The application of kriging is demonstrated with lead measurements in soil cores from two sites near lead smelters and a third site in a control area. K r i g i n g ( g e o s t a t i s t i c s ) i s a r e l a t i v e l y new s t a t i s t i c a l approach t o s p a t i a l e s t i m a t i o n . Much o f t h e e a r l y t h e o r e t i c a l work was done by G. Matheron a t t h e P a r i s School o f Mines i n t h e 1960s. The development o f g e o s t a t i s t i c s was m o t i v a t e d by D. G. K r i g e and h i s e f f o r t s t o e v a l u a t e the ore i n South A f r i c a n g o l d mines. To t h i s day most o f t h e r e s e a r c h i n t o g e o s t a t i s t i c a l methods i s s t i l l aimed a t ore and o i l r e s e r v e e s t i m a t i o n . However, i n r e c e n t y e a r s i t s use has spread t o many o t h e r d i s c i p l i n e s i n c l u d i n g s e a - f l o o r mapping (_1), h y d r o l o g i e parameter e s t i m a t i o n (2), ground water s t u d i e s (3>) > a q u a t i c m o n i t o r i n g ( 4 J , gene frequency mapping ( 5 j and r a d i o n u c l i d e c o n t a m i n a t i o n from a t m o s p h e r i c n u c l e a r t e s t s (£-8). Assumptions Used i n K r i g i n g The e a r l y t h e o r e t i c a l work was done by Matheron (9-11) a t t h e P a r i s School o f Mines. J o u r n e l and H u i j b r e g t s ( 1 2 ) , Rendu (13) and David (14) have p u b l i s h e d books which d e s c r i b e t h e theoreticâT a s p e c t s o f k r i g i n g and t h e d e r i v a t i o n o f t h e k r i g i n g system o f l i n e a r e q u a t i o n s . 0097-6156/85/0292-0203$l 1.00/0 © 1985 American Chemical Society Breen and Robinson; Environmental Applications of Chemometrics ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

ENVIRONMENTAL APPLICATIONS OF CHEMOMETRICS

204

Pauncz (15) and B e l l and Reeves (16) have p u b l i s h e d b i b l i o g r a p h i e s o f the E n g l i s h language p u b l i c a t i o n s on k r i g i n g . T h i s s e c t i o n w i l l d e f i n e some o f t h e common t e r m i n o l o g y used i n k r i g i n g and g i v e an o v e r v i e w o f t h e assumptions made by k r i g i n g . Random F u n c t i o n s and R e g i o n a l i z e d V a r i a b l e s . In u n i v a r i a t e s t a t i s t i c s , an o b s e r v a t i o n y. i s d e f i n e d as a r e a l i z a t i o n o f a random v a r i a b l e Y, where Y has a p r o b a b i l i s t i c d i s t r i b u t i o n ( e . g . , n o r m a l , lognormal, exponential, e t c . ) . This d i s t r i b u t i o n i s generally c h a r a c t e r i z e d by c e r t a i n p a r a m e t e r s , such as t h e mean and v a r i a n c e , which a r e assumed t o e x i s t b u t a r e unknown. O f t e n t h e goal i s t o make i n f e r e n c e s about t h e s e unknown parameters. C o n s e q u e n t l y , a number o f r e a l i z a t i o n s , s a y {y^ y } , o f t h i s random v a r i a b l e 9

n

are o b t a i n e d and i n f e r e n c e s about t h e parameters o f t h e d i s t r i b u t i o n o f t h e random v a r i a b l e a r e made u s i n g t h e s e o b s e r v a t i o n s . For example, i f we assume t h a t Y has a normal d i s t r i b u t i o n then t h e mean p (μ) and v a r i a n c e (σ ) a r e e s t i m a t e d by

σ

•

1

π

=1

Σ V i=l

1

1

i

χ

=

a

1

n

d

Σ V i=l 1

i

=

y

1

Λ 2 /\ The m i n i m i z a t i o n o f E [ ( Y - Y) ) ] w i t h t h e c o n s t r a i n t E[Y - Y] = 0 i s done u s i n g s+1 Lagrange m u l t i p l i e r s ( μ , ..., μ ) . 0

The r e s u l t i s t h e f o l l o w i n g

k r i g i n g system o f l i n e a r

m s Σ λ.γ(χ.;χ.) + I y f (x,) i=l J ~ J t=0 z 1 1 t

t

=

=

for

equations

j = 1, ..., m

1

Ί

^ V t M i )

_ Ύ(Χ,;Β)

$

(5) for

f (BÎ

t = 0

s

t

The k r i g i n g v a r i a n c e i s

°î

=

m ï λ,γ(χ.;Β) + l y . f . ( B ) - γ(Β;Β) i=l t=0τ τ 1

(6)

1

The s o l u t i o n o f t h i s system f o r t h e k r i g i n g w e i g h t s , λ., i s b e s t done using matrix algebra.

When Z(x} i s an IRF-1 d e f i n e

Ύ (2L2 '—2 ^

1

1

x

x

l

2

ç."

=

Fix^B),

μ

γ(χ ;Β), 2

x

1

x

1

x

1

0

0

0

x

0

0

0

0

0

0

"*"

y2

··· V

1

···

Y ( x^ 9^_2}

y

μ

μ

0* 1 ' 2

m

m

l 2

m

]

..., Ύ ( Χ ; Β ) , l , χ, y ] Π |

Breen and Robinson; Environmental Applications of Chemometrics ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

y

m


212

D'

=

l(xj

[Zixj), Z(x ),

9

2

0, 0, 0 ]

then t h e k r i g i n g system i s A Β

=

C

and t h e s o l u t i o n o f t h i s system f o r t h e k r i g i n g w e i g h t s and Lagrange multipliers i s

T h e r e f o r e , t h e k r i g i n g e s t i m a t o r and v a r i a n c e a r e Υ

=

o\

=

Β' D B*C - 7(B;B)

When t h e g e n e r a l i z e d c o v a r i a n c e i s used i n s t e a d o f t h e semi-variogram, t h e k r i g i n g system l o o k s almost t h e same as t h e one above because o f t h e r e l a t i o n s h i p between t h e g e n e r a l i z e d c o v a r i a n c e and semi-variogram shown i n E q u a t i o n 3. Thus, t h e semi-variogram γ(χ.;x.) i s r e p l a c e d by K(x.;x.) i n m a t r i x A and γ(χ,· ;B) i s r e p l a c e d * j • j * by K(_x..;B) i n v e c t o r V , and t h e k r i g i n g v a r i a n c e i s now Z

o

k

=

K(B;B) - B'C

Model E s t i m a t i o n As seen i n E q u a t i o n 4, t h e k r i g i n g e s t i m a t o r o f t h e v a l u e o f Z(>0 a t a s p e c i f i c l o c a t i o n o r t h e average v a l u e o f Z(x.) over a s p e c i f i e d area i s a weighted average o f t h e d a t a . The k r i g i n g weights used i n the weighted average and t h e k r i g i n g v a r i a n c e a r e o b t a i n e d from s o l v i n g t h e k r i g i n g system o f l i n e a r e q u a t i o n s as shown i n Equation 5. When t h e model ( i . e . , d r i f t and semi-variogram o r g e n e r a l i z e d c o v a r i a n c e ) i s known, t h e k r i g i n g e s t i m a t o r i s a b e s t l i n e a r unbiased e s t i m a t o r (BLUE). However, t h e model i s g e n e r a l l y unknown and thus must be e s t i m a t e d u s i n g t h e o b s e r v a t i o n s . I f t h e model i s n o t i d e n t i f i e d c o r r e c t l y , t h e k r i g i n g e s t i m a t o r i s no l o n g e r BLUE. In t h i s s e c t i o n t h e e s t i m a t i o n o f t h e model i s d e s c r i b e d . Semi-variogram Models. The semi-variogram i s a f u n c t i o n o f d i s t a n c e ( h ) . That i s , t h e semi-variogram a t h i s one h a l f t h e expected squared d i f f e r e n c e between a p a i r o f o b s e r v a t i o n s Z(x) t h a t a r e s e p a r a t e d by a d i s t a n c e h ( s e e Equation 1 ) . T h i s f u n c t i o n ( o r model) must be c o n d i t i o n a l l y p o s i t i v e d e f i n i t e so t h a t t h e v a r i a n c e o f t h e l i n e a r f u n c t i o n a l o f Ζ(χ,) i s g r e a t e r than o r equal t o z e r o . F i v e o f the common semi-variogram models which s a t i s f y t h i s c o n d i t i o n a r e : 1.

Power F u n c t i o n ( F i g u r e l a ) y(h)

=

b|h|

p

with 0 < ρ < 2

(when ρ = 1, t h e semi-variogram

i s a l i n e a r model)


16.

2.

213

Estimation of Spatial Patterns

SIMPSON

S p h e r i c a l Model ( F i g u r e l b )

Y(h)

=

C[|M -

f o r |h| < a a

y(h)

=

C

f o r |h| > a

1

'a e q u a l s t h e range o f t h e s e m i - v a r i o g r a m and C e q u a l s t h e sill. The range can be thought o f as t h e "zone o f i n f l u e n c e . " I f t h e d i s t a n c e between two p o i n t s i s l e s s than t h e range, then the v a l u e a t one p o i n t i s c o r r e l a t e d w i t h t h e v a l u e a t t h e o t h e r p o i n t . I f t h e d i s t a n c e between two p o i n t s i s g r e a t e r than t h e range, then t h e p o i n t s a r e independent. The s i l l i s t h e bound on t h e semi-variogram and p r o v i d e s an e s t i m a t e o f t h e o v e r a l l variability. When a s e m i - v a r i o g r a m i s bounded then t h e random f u n c t i o n i s second o r d e r s t a t i o n a r y and C0V[Z(x + h ) , Z ( x ) ]

=

VAR[Z(x)] - y ( h )

When |hj > a C0V[Z(x + h ) , Z ( x ) ]

= 0

Y(h)

= C

and thus VAR[Z(x)] 3.

Cubic Model ( F i g u r e l c )

y[h) 4.

=

C

f o r |h| > a

E x p o n e n t i a l Model ( F i g u r e I d ) Y(h) (the

5.

= C

=

C[l - i

|

h

|

/

a

]

range o f t h e s e m i - v a r i o g r a m i s a p p r o x i m a t e l y 3a)

Gaussian Model ( F i g u r e l e )

y(h) (the

=

C[l - e "

( h / a )

]

range o f t h e s e m i - v a r i o g r a m i s a p p r o x i m a t e l y 2a)

When h i s s e t t o z e r o , γ(0) must a l s o be equal t o z e r o . However, i f t h e d i f f e r e n c e [ Z ( x j - Z(_x')] does n o t tend t o z e r o , f o r measurements t a k e n a t a r b i t r a r i l y c l o s e p o i n t s x. and _x , then t h e r e 1


F i g u r e 1. Common s e m i - v a r i o a r a m models: (a) power f u n c t i o n , (b) s p h e r i c a l , ( c ) c u b i c , ( d ; e x p o n e n t i a l , (e) Gaussian and ( f ) l i n e a r w i t h a nugget.


16.

215


SIMPSON

i s a d i s c o n t i n u i t y o f the semi-variogram a t the o r i g i n . This d i s c o n t i n u i t y i s c a l l e d t h e nugget e f f e c t . I f t h e r e i s a nugget e f f e c t , t h e s e m i - v a r i o g r a m model i s a d j u s t e d t o t a k e i t i n t o a c c o u n t . For example i f t h e model i s l i n e a r w i t h a nugget o f s i z e Κ ( F i g u r e I f ) then y(h)

=

b|h| + Κ

for

|h| > 0

y(h)

=

0

for

|h| = 0

E s t i m a t i o n o f t h e Semi-variogram When t h e D r i f t i s C o n s t a n t . In p r a c t i c e , t h e s e m i - v a r i o g r a m a t each d i s t a n c e h i s e s t i m a t e d as fol1ows:

Y ( h )

=

1 2NlhT

N

i

h

)

[

Z

2 (

^ i

"

Z

(

* i

)

]

(

7

)

where N(h) i s t h e number o f p a i r s o f p o i n t s , a d i s t a n c e h a p a r t , a c t u a l l y taken i n t o t h e sum. Then one o f t h e above common models i s f i t t e d t o t h e e s t i m a t e s o f t h e s e m i - v a r i o g r a m a t each h. U s u a l l y y ( h ) f o r each h i s based on a range o f d i s t a n c e s , s i n c e i n s u f f i c i e n t d a t a e x i s t s f o r a s p e c i f i c d i s t a n c e h. Once y ( h ) has been e s t i m a t e d , t h e c o r r e c t s e m i - v a r i o g r a m model, u s u a l l y one o f t h e f i v e models d i s c u s s e d above, has t o be s e l e c t e d and t h e parameters o f t h e model need t o be e s t i m a t e d . Most model f i t t i n g i s done by " t r i a l and e r r o r . " G e n e r a l l y t h e a p p r o p r i a t e model can be chosen v i s u a l l y . F o r example, i f t h e v a r i o g r a m has a s i g m o i d shape, then e i t h e r t h e c u b i c o r G a u s s i a n model i s a p p r o p r i a t e . To d i s t i n g u i s h between t h e s e two models, note t h a t t h e r e l a t i o n s h i p between t h e s i l l and range a r e d i f f e r e n t : f o r the cubic model, y ( l / 3 a ) = 0.47C, w h i l e f o r t h e G a u s s i a n model, y ( l / 3 a ) = 0.36C. The e s t i m a t i o n o f t h e parameters ( t h e s i l l and range) f o r a g i v e n s e m i - v a r i o g r a m model i s a g a i n governed by t h e r e l a t i o n s h i p between t h e s e parameters. Once t h e model i s chosen and t h e s i l l i s e s t i m a t e d , then t h e range i s s e t . The e s t i m a t e o f t h e s i l l and range can be a d j u s t e d t o some e x t e n t t o improve t h e " f i t " o f t h e model. However, i t s h o u l d be noted t h a t " s m a l l " changes i n t h e parameters o f the s e m i - v a r i o g r a m model do n o t make a s i g n i f i c a n t d i f f e r e n c e i n t h e k r i g i n g w e i g h t s and v a r i a n c e which a r e c a l c u l a t e d by s o l v i n g t h e k r i g i n g system o f l i n e a r e q u a t i o n s . Thus, t h i s p r o c e d u r e seems t o be no worse than any o t h e r t e c h n i q u e . E s t i m a t i o n o f t h e Semi-variogram When t h e D r i f t i s Not C o n s t a n t . When a n o n - c o n s t a n t d r i f t i s p r e s e n t , t h e e s t i m a t i o n o f t h e s e m i - v a r i o g r a m model i s confounded w i t h t h e e s t i m a t i o n o f t h e d r i f t . That i s , t o f i n d t h e o p t i m a l e s t i m a t o r o f t h e s e m i - v a r i o g r a m , i t i s n e c e s s a r y t o know t h e d r i f t f u n c t i o n , b u t i t i s unknown. D a v i d (14) recommended an e s t i m a t o r o f t h e d r i f t , m*(iO> d e r i v e d from l e a s t - s q u a r e methods o f t r e n d s u r f a c e a n a l y s i s ( 1 8 ) . Then a t e v e r y d a t a p o i n t a r e s i d u a l i s g i v e n by Y*(x)

=

Z ( x ) - m*(x)



216

An e x p e r i m e n t a l v a r i o g r a m o f e s t i m a t e d r e s i d u a l s y * ( h ) can then be c a l c u l a t e d . However, t h i s v a r i o g r a m d i f f e r s from t h e u n d e r l y i n g v a r i o g r a m o f t h e t r u e r e s i d u a l s , y ( h ) , and t h e b i a s i s a f u n c t i o n o f the form o f t h e e s t i m a t o r m*(_x). In o r d e r t o f i n d y ( h ) from y * ( h ) , Y*(h) i s g r a p h i c a l l y compared w i t h a s e t o f y g ( h ) d e f i n e d from v a r i o u s t y p e s o f variograms "ÏQW

and t h e same type o f e s t i m a t o r

m*(_x). I f t h e f i t i s " r e a s o n a b l e " ( t h e r e i s no t e s t f o r t h e goodness of f i t ) , t h e model y ( h ) i s assumed c o r r e c t . I f the f i t i s not " r e a s o n a b l e " t h e p r o c e s s s t a r t s a g a i n w i t h a new e s t i m a t o r o f t h e drift. Neuman and Jacobson (19) have developed a s t e p - w i s e i t e r a t i v e r e g r e s s i o n p r o c e s s f o r s i m u l t a n e o u s l y e s t i m a t i n g t h e g l o b a l d r i f t and r e s i d u a l semi-variogram. E s t i m a t e s o f t h e f u n c t i o n a r e o b t a i n e d by s o l v i n g a modified s e t o f simple k r i g i n g equations w r i t t e n f o r the r e s i d u a l s . The m o d i f i c a t i o n s c o n s i s t o f r e p l a c i n g t h e t r u e semi-variogram i n t h e k r i g i n g e q u a t i o n s by t h e semi-variogram o f t h e r e s i d u a l e s t i m a t e s as o b t a i n e d from t h e i t e r a t i v e r e g r e s s i o n p r o c e s s . G e n e r a l i z e d C o v a r i a n c e Models. When Z(_x) i s an i n t r i n s i c random f u n c t i o n o f o r d e r k, an a l t e r n a t i v e t o t h e semi-variogram i s t h e g e n e r a l i z e d c o v a r i a n c e (GC) f u n c t i o n o f o r d e r k. L i k e t h e semi-variogram model, t h e GC model must be a c o n d i t i o n a l l y p o s i t i v e d e f i n i t e f u n c t i o n so t h a t t h e v a r i a n c e o f t h e l i n e a r f u n c t i o n a l of Z(x_) i s g r e a t e r than o r equal t o z e r o . The f a m i l y o f p o l y n o m i a l GC f u n c t i o n s s a t i s f y t h i s requirement. The p o l y n o m i a l GC o f o r d e r k i s

K(h)

Co -

=

ι (-i)V|h| 1-0

2 i + 1

1

where C i s t h e nugget e f f e c t which was d e s c r i b e d e a r l i e r and c 0

_ "

1 0

i f i f

h = 0 h t 0

When k _< 2 and x. i s t w o - d i m e n s i o n a l , t h e c o e f f i c i e n t s ou have t h e following constraints: a

l

-

CXQ >_ 0, α

2

^_ 0 and

^T^tft!

The o r d e r o f t h e p o l y n o m i a l GC model i s t h e same as t h e o r d e r o f the d r i f t . Thus t h e a v a i l a b l e models can be summarized as f o l l o w s : DRIFT

k

POLYNOMIAL GC MODEL

Constant

0

C6 - a | h |

Linear

1

Co - a | h | + a | h |

Quadratic

2

C6 - a | h | + a ^ h j

Q

Q

Q

3

1

3

- a |h|

5

2


16.

SIM PSON

217


As can be seen above, when t h e d r i f t i s c o n s t a n t , t h e GC models a r e q u i t e l i m i t e d ( i . e . , K(h) = Co, K(h) = - a | h | o r K(h) = Co - a | h | ) . Q

Q

Thus, when t h e r e i s a c o n s t a n t d r i f t , t h e semi-variogram s h o u l d be used i n s t e a d o f GC models.

models

E s t i m a t i o n o f t h e G e n e r a l i z e d C o v a r i a n c e Model. An a l g o r i t h m f o r t h e e s t i m a t i o n o f t h e o r d e r o f t h e d r i f t and t h e c o e f f i c i e n t o f t h e p o l y n o m i a l GC f u n c t i o n has been developed by D e l f i n e r ( 1 7 ) . T h i s a l g o r i t h m , termed "Automatic S t r u c t u r e I d e n t i f i c a t i o n (7\SI)" i s used i n BLUEPACK 3D (a p r o p r i e t a r y computer package s o l d by t h e P a r i s School o f Mines) and i s o n l y b r i e f l y d e s c r i b e d i n t h e l i t e r a t u r e ( 1 7 ) . A s i m i l a r a l g o r i t h m has been developed by Hughes and L e t t e n m e i e r ( 2 0 ) , who i n c l u d e d t h e computer programs i n t h e i r publication. The ASI a l g o r i t h m i s broken down i n t o t h r e e s t e p s . F i r s t t h e o r d e r o f t h e d r i f t ( k ) i s e s t i m a t e d . Then a l l t h e p o s s i b l e p o l y n o m i a l GC models a r e e s t i m a t e d . Thus i f k = 0, t h e r e a r e 3 models e s t i m a t e d ; k = 1, t h e r e a r e 7 models e s t i m a t e d and i f k = 2, t h e r e a r e 15 models e s t i m a t e d . The i n a d m i s s i b l e models ( t h o s e models whose parameter e s t i m a t e s do n o t meet t h e c o n s t r a i n t s o f t h e p o l y n o m i a l GC model) a r e d i s c a r d e d and t h e t h r e e b e s t models a r e chosen. The t h i r d s t e p compares t h e r e m a i n i n g models and makes t h e f i n a l choice. The ASI method has t h e advantage o f b e i n g automated. However, t h i s method has i t s problems: t h e o r d e r o f t h e d r i f t tends t o be u n d e r e s t i m a t e d when samples a r e from a symmetric g r i d (symmetric neighborhoods tend t o f i l t e r p o l y n o m i a l s by i t s e l f ) ; t h e f i n a l c h o i c e o f t h e model depends on an ad-hoc d e c i s i o n procedure ( t h e r e a r e a g a i n no goodness o f f i t t e s t s ) ; and when t h e d r i f t i s c o n s t a n t , t h e o n l y model i s l i n e a r w i t h a nugget which i s n o t a l a r g e enough c l a s s o f models, thus t h e u s e r needs t o go back t o t h e v a r i o g r a m a n a l y s i s described e a r l i e r . P r o b a b l y i t s b i g g e s t weakness i s i t s l a c k o f r o b u s t n e s s a g a i n s t v a r i a b l e s t h a t do n o t w e l l s a t i s f y t h e i n t r i n s i c hypothesis. K r i g i n g A n a l y s i s o f Lead Measurements i n S o i l

Cores

The l e a d measurement (Z(x.) = ppm l e a d i n a s o i l c o r e ) a r e from t h r e e s i t e s ; RSR and DMC a r e c e n t e r e d around l e a d s m e l t e r s w h i l e REF i s a reference or control s i t e . RSR has 208 measurements, DMC has 206 measurements and REF has 100 measurements. F i g u r e s 2 through 4 d i s p l a y t h e s p a t i a l d i s t r i b u t i o n o f t h e measurements a t RSR, DMC and REF, r e s p e c t i v e l y . The d i s t r i b u t i o n o f t h e data f o r a l l t h r e e s i t e s a r e skewed t o t h e r i g h t . T h e r e f o r e , t h e n a t u r a l l o g a r i t h m (LN) o f the data i s used i n t h e k r i g i n g a n a l y s i s . I t i s assumed t h a t t h e i n t r i n s i c h y p o t h e s i s h o l d s w i t h i n t h e l i m i t e d neighborhood t h a t i s used i n c a l c u l a t i n g t h e k r i g i n g e s t i m a t e s . That i s , e i g h t measurements w i l l be used by t h e k r i g i n g e s t i m a t o r and w i t h i n t h e l i m i t e d neighborhood ( w i t h a r a d i u s o f a p p r o x i m a t e l y 1000 f e e t ) t h a t these measurements o c c u r i t i s assumed t h a t t h e r e i s no d r i f t o r s y s t e m a t i c t r e n d . Semi-variogram E s t i m a t i o n . The f i r s t s t e p i n t h e k r i g i n g a n a l y s i s i s t o e s t i m a t e t h e semi-variogram f o r each s i t e . The sample



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SIMPSON

semi-variogram i s v e r y s e n s i t i v e t o " o u t l i e r s , " thus t o o b t a i n a good e s t i m a t e o f the u n d e r l y i n g s e m i - v a r i o g r a m , the data needs t o be s c r u t i n i z e d and any " o u t l i e r s " removed. I t can be seen i n F i g u r e s 2 through 4 t h a t some o f the " o u t l i e r s " a r e easy t o s p o t and g e n e r a l l y the j u s t i f i c a t i o n f o r removing them i s e v i d e n t . F o r example, t h e measurements o v e r 10,000 ppm a t DMC was taken near a junk y a r d . I t can be argued t h a t the l e a d i n t h i s sample was not p r i m a r i l y from the DMC s o u r c e , but a r e s u l t o f a source w i t h i n the junk y a r d . A d d i t i o n a l l y , t h r e e measurements a t REF are b a s i c a l l y o u t s i d e the range o f the sampling p a t t e r n . Since these three points are a l l w i t h i n approximately a f o o t o f each o t h e r and would be c o n s i d e r e d as o n l y one measurement i n the k r i g i n g a n a l y s i s , they would have l i t t l e impact on t h e k r i g i n g e s t i m a t e s . In the end, 7 " o u t l i e r s " a r e removed from RSR, 5 " o u t l i e r s " a r e removed from DMC and 8 " o u t l i e r s " a r e removed from REF. F i g u r e s 5 through 7 show the sample semi-variograms f o r each s i t e a f t e r t h e " o u t l i e r s " a r e removed. When l o o k i n g a t the sample semi-variograms f o r RSR and DMC, they appear t o be d i f f e r e n t r e a l i z a t i o n s o f the same u n d e r l y i n g phenomena. Both s i t e s a r e i n the same g e n e r a l a r e a ( c i t y ) and the v a r i a b l e ( l e a d c o n c e n t r a t i o n ) i s d i s p e r s e d by the same p r o c e s s (a s m e l t e r ) . T h e r e f o r e , t o g e t a b e t t e r e s t i m a t e o f the s e m i - v a r i o g r a m , t h e semi-variograms f o r DMC and RSR are combined (see F i g u r e 8 ) . Among the common semi-variogram models, t h e e x p o n e n t i a l model b e s t f i t s the sample semi-variograms. T h e r e f o r e , f o r REF y(h) i s used.

=

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i s used. K r i g i n g E s t i m a t e s and S t a n d a r d D e v i a t i o n s . The k r i g i n g a n a l y s i s i s performed on the n a t u r a l l o g a r i t h m s o f the measurements, the k r i g i n g e s t i m a t o r Y.., i s i n l o g s c a l e .

EXP[Y..] i s not an unbiased

estimate

o f t h e mean c o n c e n t r a t i o n i n the b l o c k , i t i s an e s t i m a t e o f t h e median b l o c k v a l u e . Rendu (21) shows t h a t the unbiased k r i g i n g e s t i m a t o r o f the mean c o n c e n t r a t i o n i n the o r i g i n a l s c a l e i s Y*

=

EXP[Y

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L

2 where λ., a r e the k r i g i n g w e i g h t s and

i s the l o g a r i t h m i c k r i g i n g

variance. The k r i g i n g e s t i m a t e s o f the mean c o n c e n t r a t i o n (ppm l e a d ) o v e r a 250 f o o t by 250 f o o t b l o c k and the k r i g i n g s t a n d a r d d e v i a t i o n f o r each b l o c k a r e shown i n F i g u r e s 9 through 14. A t RSR and DMC the e s t i m a t e d b l o c k means a r e shown f o r b l o c k s whose m u l t i p l i c a t i v e k r i g i n g s t a n d a r d d e v i a t i o n was l e s s than 2. ( S i n c e the measurements a r e t r a n s f o r m e d u s i n g the n a t u r a l l o g a r i t h m , the s t a n d a r d d e v i a t i o n s


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Estimation of Spatial Patterns and Inventories of ... - ACS Publications

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