on *n*=2 measurements *Z*(*x*1) and *Z*(*x*2), where the

equivalent to j *w*i = for any , which holds

three locations (*x*0, *x*1, and *x*2) are distinct. Using

Equations 2-23 and 2-28, note that the correlation

if and only if j *w*i = 1 . Therefore, all linear

function is

unbiased estimators need to have weights that sum

1&

exp & 3 , *h *> 0

to one. There are many sets of weights that satisfy

D (*h*) =

(2-37)

this condition, including the set with all the weights

equal to 1/*n*, as in the sample mean, Equation

1,

2-34. However, the unique set of weights that

minimize the prediction variance (Equation 2-33)

can be shown to satisfy the following set of *n*+1

For illustrative purposes, suppose that

and Srivastava (1989)):

= *p*, 0 # *p *# 1

(2-38)

j *w*j Dij %

8

(2-35a)

= Di0, *i*=1, 2 ..., *n*,

where *p *is a fixed proportion. The quantity *p *is

sometimes referred to as a **relative nugget**.

j *w*j = 1

(2-35b)

(b) The ordinary kriging Equations 2-35a and

2-35b are given by

where Dij = D(*h*ij) is the correlation between obser-

8

= D10

vations *i *and *j*, *h*ij is the distance between locations

(2-39a)

constrained optimization. Furthermore, the

resulting **ordinary kriging variance **is

8

= D20

(2-39b)

2

^

2

Fk (*x*)0 = *+ Z *(*x *) & *Z *(*x *)

0

0

(2-36)

(2-39c)

= *s *1 & j *w*jDj0

&8

These three equations have three unknowns: *w*1,

(c) The system of Equations 2-35a and 2-35b

can easily be solved for the wi's and 8, after which

1 D10 & D20

1

the kriging variance can be obtained from Equa-

%

(2-40a)

2 1 & D12

tion 2-36. Note that the ordinary kriging variance

2

changes depending on the prediction location *x*0,

even though the variance of *Z*(*x*0) itself (Equa-

1 D10 & D20

tion 2-6) is constant for all *x*0.

1

&

(2-40b)

2 1 & D12

2

(2) Example 1.

and

(a) Let the mean of *Z(x) *satisfy Equation 2-30,

and suppose that the residual *Z*(x) *(Equa-

8=

(D10 % D20 & D12 & 1)

tion 2-16) has an isotropic exponential variogram

(2-41)

(Equation 2-23). Consider predicting *Z*(*x*0) based

2

2-11

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